Modelling Biomedical Signals

Ibmedical

Editors

Giuseppe Nardulli Sebastiano Stramaglia

1

J

Mod^tm^omedicaJ •t^-f

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pmedica

igna s Bari, Italy

19-21 September 2001

Editors

Giuseppe Nardulli Sebastiano Stramaglia Center of Innovative Technologies for Signal Detection and Processing University of Bari, Italy

V | f e World Scientific wb

New Jersey • London • Sim Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

MODELLING BIOMEDICAL SIGNALS Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4843-1

Printed in Singapore by Mainland Press

V

Preface In the last few years, concepts and methodologies initially developed in theoretical physics have found high applicability in a number of very different areas. This book, a result of cross-disciplinary interaction among physicists, biologists and physicians, covers several topics where methods and approaches rooted in physics are successfully applied to analyze and to model biomedical data. The volume contains the papers presented at the International Workshop Modelling Bio-medical Signals held at the Physics Department of the University of Bari, Italy, on September 19-21th 2001. The workshop was gathered under the auspices of the Center of Innovative Technologies for Signal Detection and Processing of the University of Bari (TIRES Centre); the Organizing Committee of the Workshop comprised L. Angelini, R. Bellotti, A. Federici, R. Giuliani. G. Gonnella, G.Nardulli and S. Stramaglia. The workshop opened on September 19th 2001 with two colloquia given by profs. N. Accornero (University of Rome, la Sapienza), on Neural Networks and Neurosciences, and E. Marinari (University of Rome, la Sapienza) on Physics and Biology. Around 70 scientists attended the workshop, coming from different fields and disciplines. The large spectrum of competences gathered in the workshop favored an intense and fruitful exchange of scientific information and ideas. The topics discussed in the workshop include: decision support systems in medical science; several analyses of physiological rhythms and synchronization phenomena; biological neural networks; theoretical aspects of artificial neural networks and their role in neural sciences and in the analysis of EEG and Magnetic Resonance Imaging; gene expression patterns; the immune system; protein folding and protein crystallography. For the organization of the workshop and the publication of the present volume we acknowledge financial support from the Italian Ministry of University and Scientific Research (MURST) under the project (PRIN) "Theoretical Physics pf Fundamental Interactions", from the TIRES Centre, the Physics Department of the University of Bari and from the Section of Bari of the Istituto Nazionale di Fisica Nucleare (INFN). We also thank the Secretary of the Workshop, Mrs. Fausta Cannillo and Mrs. Rosa Bitetti for their help in organizing the event.

Giuseppe Nardulli Sebastiano Stramaglia University of Bari

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VII

CONTENTS

Preface

v

ANALYSIS AND MODELS OF BIOMEDICAL DATA BY THEORETICAL PHYSICS METHODS The Cluster Variation Method for Approximate Reasoning in Medical Diagnosis H. J. Kappen*

3

Analysis of EEG in Epilepsy K. Lehnertz, R. G. Andrzejak, T. Kreuz, F. Mormann, C. Rieke, P. David and C. E. Elger

17

Stochastic Approaches to Modeling of Physiological Rhythms Platnen Ch. Ivanov and Chung-Chuan Lo

28

Chaotic Parameters in Time Series of ECG, Respiratory Movements and Arterial Pressure E. Conte and A. Federici

51

Computer Analysis of Acoustic Respiratory Signals A. Vena, G. M. Insolera, R. Giuliani, T. Fiore and G. Perchiazzi

60

The Immune System: B Cell Binding to Multivalent Antigen Gyan Bhanot

67

Stochastic Models of Immune System Aging L. Mariani, G. Turchetti and F. Luciani

80

NEURAL NETWORKS AND NEUROSCIENCES Artificial Neural Networks in Neuroscience N. Accornero and M. Capozza

Italicized name indicates the author who presented the paper.

93

VIII

Biological Neural Networks: Modeling and Measurements R. Stoop and S. Lecchini Selectivity Property of a Class of Energy Based Learning Rules in Presence of Noisy Signals A. Bazzani, D. Remondini, N. Intrator and G. Castellani

107

123

Pathophysiology of Schizophrenia: fMRI and Working Memory G. Blasi and A. Bertolino

132

ANN for Electrophysiological Analysis of Neurological Disease R. Bellotti, F. de Carlo, M. de Tommaso, O. Difruscolo, R. Masssafra, V. Sciruicchio and S. Stramaglia

144

Detection of Multiple Sclerosis Lesions in Mri's with Neural Networks P. Blonda, G. Satalino, A. D'addabbo, G. Pasquariello, A. Baraldi and R. de Blasi

157

Monitoring Respiratory Mechanics Using Artificial Neural Networks G. Perchiazzi, G. Hedenstierna, A. Vena, L. Ruggiero, R. Giuliani and T. Fiore

165

GENOMICS AND MOLECULAR BIOLOGY Cluster Analysis of DNA-Chip Data E. Domany

175

Clustering mtDNA Sequences for Human Evolution Studies C. Marangi, L. Angelini, M. Mannarelli, M. Pellicoro, S. Stramaglia, M. Attimonelli, M. de Robertis, L. Nitti, G. Pesole, C. Saccone and M. Tommaseo

196

Finding Regulatory Sites from Statistical Analysis of Nucleotide Frequencies in the Upstream Region of Eukaryotic Genes M. Caselle, P. Provero, F. di Cunto and M. Pellegrino

209

Regulation of Early Growth Response-l Gene Expression and Signaling Mechanisms in Neuronal Cells: Physiological Stimulation and Stress G. Cibelli

221

Geometrical Aspects of Protein Folding C. Micheletti

234

ix

The Physics of Motor Proteins G. Lattanzi and A. Maritan

251

Phasing Proteins: Experimental Loss of Information and its Recovery C. Giacovazzo, F. Capitelli, C. Giannini, C. Cuocci and M. lanigro

264

List of Participants

279

Author Index

281

ANALYSIS AND MODELS OF BIOMEDICAL DATA BY THEORETICAL PHYSICS METHODS

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3

T H E C L U S T E R VARIATION M E T H O D FOR A P P R O X I M A T E R E A S O N I N G IN M E D I C A L DIAGNOSIS H.J. KAPPEN Laboratory

of Biophysics, bertSmbfys.

University kun. nl

of

Nijmegen,

In this paper, we discuss the rule based and probabilistic approaches to computer aided medical diagnosis. We conclude that the probabilistic approach is superior to the rule based approach, but due to its intractability, it requires approximations for large scale applications. Subsequently, we review the Cluster Variation Method and derive a message passing scheme that is efficient for large directed and undirected graphical models. When the method converges, it gives close to optimal results.

1

Introduction

Medical diagnosis is the a process, by which a doctor searches for the cause (disease) that best explains the symptoms of a patient. The search process is sequential, in the sense that patient symptoms suggest some initial tests to be performed. Based on the outcome of these tests, a tentative hypothesis is formulated about the possible cause(s). Based on this hypothesis, subsequent tests are ordered to confirm or reject this hypothesis. The process may proceed in several iterations until the patient is finally diagnosed with sufficient certainty and the cause of the symptoms is established. A significant part of the diagnostic process is standardized in the form of protocols. These are sets of rules that prescribe which tests to perform and in which order, based on the patient symptoms and previous test results. These rules form a decision tree, whose nodes are intermediate stages in the diagnostic process and whose branches point to additional testing, depending on the current test results. The protocols are defined in each country by a committee of medical experts. The use of computer programs to aid in the diagnostic process has been a long term goal of research in artificial intelligence. Arguably, it is the most typical application of artificial intelligence. The different systems that have been developed so-far use a variety of modeling approaches which can be roughly divided into two categories: rulebased approaches with or without uncertainty and probabilistic methods. The rule-based systems can be viewed as computer implementations of the protocols, as described above. They consist of a large data base of rules of the form: A -) B, meaning that "if condition A is true, then perform action B"

4

or "if condition A is true, then condition B is also true". The rules may be deterministic, in which case they are always true, or 'fuzzy' in which case they are true to a (numerically specified) degree. Examples of such programs are Meditel 1 , Quick Medical Reference (QMR) 2 , DXplain 3 , and Iliad 4 . In Berner et al. 5 a detailed study was reported that assesses the performance of these systems. A panel of medical experts collected 110 patient cases, and concensus was reached on the correct diagnosis for each of these patients. For each disease, there typically exists a highly specific test that will unambiguously identify the disease. Therefore, based on such complete data, diagnosis is easy. A more challenging task was defined by removing this defining test from each of the patient cases. The patient cases were presented to the above 4 systems. Each system generated its own ordered list of most likely diseases. In only 10-20 % of the cases, the correct diagnosis appeared on the top of these lists and in approximately 50 % of the cases the correct diagnosis appeared in the top 20 list. Many diagnoses that appeared in the top 20 list were considered irrelevant by the experts. It was concluded that these systems are not suitable for use in clinical practice. There are two reasons for the poor performance of the rule based systems. One is that the rules that need to be implemented are very complex in the sense that the precondition A above is a conjunction of many factors. If each of these factors can be true or false, there is a combinatoric explosion of conditions that need to be described. It is difficult, if not impossible, to correctly describe all these conditions. The second reason is that evidence is often not deterministic (true or false) but rather probabilistic (likely or unlikely). The above systems provide no principled approach for the combination of such uncertain sources of information. A very different approach is to use probability theory. In this case, one does not model the decision tree directly, but instead models the relations between diseases and symptoms in one large probability model. As a (too) simplified example, consider a medical domain with a number of diseases d = ( d i , . . . ,d„) and a number of symptoms or findings / = (/i, • • • , / r o ) One estimates the probability of each of the diseases p(di) as well as the probability of each of the findings given a disease, p(fj\di). If diseases are independent, and if findings are conditionally independent given the disease, the joint probability model is given by: P(d,f)=P(d)p(f\d)=Upwiipifjidi) i

(i)

j

It is now possible to compute the probability of a disease dj, given some

5

findings by using Bayes' rule:

mft) =

-7W

(2)

where ft is the list of findings that has been measured up to diagnostic iteration t. Computing this for different di gives the list of most probable diseases given the current findings ft and provides the tentative diagnosis of the patient. Furthermore, one can compute which additional test is expected to be most informative about any one of the diagnoses, say di, by computing

for each test j that has not been measured so-far. The test j that minimizes Iij is the most informative test, since averaged over its possible outcomes, it gives the distribution over di with the lowest entropy. Thus, one sees that whereas the rule based systems model the diagnostic process directly, the probabilistic approach models the relations between diseases and findings. The diagnostic decisions (which test to measure next) is then computed from this model. The advantage of this latter approach is that the model is much more transparent about the medical knowledge, which facilitates maintenance (changing probability tables, adding diseases or findings), as well as evaluation by external experts. One of the main drawbacks of the probabilistic approach is that it is intractable for large systems. The computation of marginal probabilities requires summation over all other variables. For instance, in Eq. 2 p(/») = !>,/,*>( a ( * ) logp(:r) = £ 5 i x

(6)

0

where /3 runs over all subsets of variables d. The cluster variation method approximates the total entropy by restricting this sum to only clusters in U and re-expressing Sp in terms of Sa, using the Moebius formula and the definition Eq. 5.

s

~ ]C sl= Yl H aaSl= 12aaSa 0<EU

0€U aZ>0

(7)

a£U

Since Sa is a function of pa (Eq. 5) we have expressed the entropy in terms of cluster probabilities pa. The quality of this approximation is illustrated in Fig. 2. Note, that the both the Bethe and Kikuchi approximation strongly deteriorate around J = 1, which is where the spin-glass phase starts. For J < 1, the Kikuchi approximation is superior to the Bethe approximation. Note, however, that this figure only illustrates the quality of the truncations in Eq. 7 assuming that the exact marginals are known. It does not say anything about the accuracy of the approximate marginals using the approximate free energy. Substituting Eqs. 4 and 7 into the free energy Eq. 3 we obtain the approximate free energy of the Cluster Variation method. This free energy must be minimized subject to normalization constraints 5Zx pa(xa) = 1 and consistency constraints Pa(xp) = p0(xp),

/3eM,a£B,l3ca.

(8)

Note, that we have excluded constraints between clusters in M. This is sufficient because when /?,/?' £ M, f3 C (3' and ft' C a 6 B: pa{xp') — Pp'{x0') c

T h i s decomposition is similar to writing a correlation in terms of means and covariance. For instance when a = (i), S^ = SjL is the usual mean field entropy and S^jf = SjL + ST.. + S,.., defines two node correction. 0) (y) O n n variables this sum contains 2 n terms.

d

10

Figure 2. Exact and approximate entropies for the fully connected Boltzmann-Gibbs distribution on n = 10 variables with random couplings (SK model) as a function of mean coupling strength. Couplings viij are chosen from a normal Gaussian distribution with mean zero and standard deviation J'/'s/n. External fields di are chosen from a normal Gaussian distribution with mean zero and standard deviation 0.1. The exact entropy is computed from Eq. 6. The Bethe and Kikuchi approximations are computed using the approximate entropy expression Eq. 7 with exact marginals and by choosing B as the set of all pairs and all triplets, respectively.

and Pa(xp) = pp{xp) implies pp>{xp) = pp{xp). In the following, a and /? will be from B and M respectively, unless otherwise stated e . Adding Lagrange multipliers for the constraints we obtain the Cluster Variation free energy: Fcvm{{pa(xa)},

{Xa}, {Xapixp)})

~ H A° ( X ^ * ^ )

-1

= ^2

a

a ^Pafca)

{Ha(xa)

+

\ogpa(xa))

) ~ X ^2^2XMx0)(Pa(xp) -P0(x0)) (9)

3

Iterating Lagrange multipliers

Since the Moebius numbers can have arbitrary sign, Eq. 9 consists of a sum of convex and concave terms, and therefore is a non-convex optimization problem. One can separate F c v m in a convex and concave term and derive an e

In fact, additional constraints can be removed, when clusters in M contain subclusters in M. See Kappen and Wiegerinck 1 6 .

11

iteration procedure in pa and the Lagrange multipliers that is guaranteed to converge 17 . The resulting algorithm is a 'double loop' iteration procedure. Alternatively, by setting QF7™\,U £ U equal to zero, one can express the cluster probabilities in terms of the Lagrange multipliers:

Pa{xa) = — exp I -Ha(xa)

exp P0 (xp) = — Z

?

-H0{xp)

\

+ ^2 ^ap{xp)

(10)

V Xa0{x0)

a

^p

(11)

J

The remaining task is to solve for the Lagrange multipliers such that all constraints (Eq. 8) are satisfied. There are two ways to do this. One is to define an auxiliary cost function that is zero when all constraints are satisfied and positive otherwise and minimize this cost function with respect to the Lagrange multipliers. This method is discussed in Kappen and Wiegerinck 16 . Alternatively, one can substitute Eqs. 10-11 into the constraint Eqs. 8 and obtain a system of coupled non-linear equations. In Yedidia et al. 12 a message passing algorithm was proposed to find a solution to this problem. Here, we will present an alternative method, that solves directly in terms of the Lagrange multipliers. Consider the constraints Eq. 8 for some fixed cluster /? and all clusters a D (3 and define Bp = {a E B\a D /?}. We wish to solve for all constraints a D /?, with a € B0 by adjusting Xap,oi € Bp. This is a sub-problem with |2?/3||:r3| equations and an equal number of unknowns, where \Bp\ is the number of elements of B0 and \x0\ is the number of values that x0 can take. The probability distribution p0 (Eq. 11) depends only on these Lagrange multipliers, up to normalization. pa (Eq. 10) depends also on other Lagrange multipliers. However, we consider only its dependence on \ap,ct G Bp, and consider all other Lagrange multipliers as fixed. Thus, Pa(xa) = exp{Xap(xp))pa(xa),a

6 Bp

(12)

with pa independent of \ap, a € Bp. Substituting, Eqs. 11 and 12 into Eq. 8, we obtain a set of linear equations for \ap(xp) which we can solve in closed form: Xap{xp) =

/.

n

,Hp(x0) -Y]Aaa,

logpa>{xp)

12 with a/3 + \B0\ We update the probabilities with the new values of the Lagrange multipliers using Eqs. 11 and 12. We repeat the above procedure for all /? G M until convergence. 4

Numerical results

We show the performance of the Lagrange multiplier iteration method (LMI) on several 'real world' directed graphical models. For undirected models, see Kappen and Wiegerinck 16 . First, we consider the well-known chest clinic problem, introduced by Lauritzen and Spiegelhalter 15 . The graphical model is given in figure 3a. The model describes the relations between three diagnoses (Tuberculosis(T), Lung Cancer(L) and Bronchitis(B), middle layer), clinical observations and symptoms (Positive X-ray(X) and Dyspnoea(D)(=shortness of breath), lower layer) and prior conditions (recent visit to Asia(A) and whether the patient smokes(S)). In figure 3b, we plot the exact single node marginals against the approximate marginals for this problem. For LMI, the clusters in B are defined according to the conditional probability tables, i.e. when a node has k parents, a cluster of size k + 1 on this node and its parents is included in the set B. Convergence was reached in 6 iterations. Maximal error on the marginals is 0.0033. For comparison, we computed the mean field and TAP approximations, as previously introduced by Kappen and Wiegerinck 10 . Although TAP is significantly better than MF, it is far worse than the CVM method. This is not surprising, since both the MF and TAP approximation are based on single node approximation, whereas the CVM method uses potentials up to size 3. Secondly, we consider a graphical model that was developed in a project together with the department of internal medicine of the Utrecht Academic hospital. In this project, called Promedas, we aim to model a large part of internal medicine 18 . The network that we consider was one of the first modules that we built and models in detail some specific anemias and consists of 91 variables. The network was developed using our graphical tool BayesBuilder 19 which is shown with part of the network in figure 4. The clusters in B are defined according to the conditional probability tables. Convergence was reached in 5 iterations. Maximal absolute error on the marginals is 0.0008. The mean field and TAP methods perform very poorly on this problem. Finally, we tested the cluster variation method on randomly generated

13 — •

x MF O TAP + CVM

o

.•• o

r

:

x x

0.1

s® t-m^

• 0.1

(a) Chest clinic graphical model

,••-

0.2 0 .3 0.4 Exact marglinals

(b) Approximate inference

Figure 3. a) The Chest Clinic model describes the relations between diagnoses, findings and prior conditions for a small medical domain. An arrow a —> b indicates that the probability of b depends on the values of a. b) Inference of single node marginals using MF, TAP and LMI method, comparing the results with exact.

directed graphical models. Each node is randomly connected to k parents. The entries of the probability tables are randomly generated between zero and one. Due to the large number of loops in the graph, the exact method requires exponential time in the so-called tree width, which can be seen from Table 1 to scale approximately linear with the network size. Therefore exact computation is only feasible for small graphs (up to size n = 40 in this case). For the CVM, clusters in B are denned according to the conditional probability tables. Therefore, maximal cluster size is k + 1. On these more challenging cases, LMI does not converge. The results shown are obtained with the auxiliary cost function as that was briefly mentioned in section 3 and fully described in Kappen and Wiegerinck 16 . Minimization was done using conjugate gradient descent. The results are shown in Table 1. 5

Conclusion

In this paper, we have described two approaches to computer aided medical diagnosis. The rule based approach directly models the diagnostic decision tree. We have shown that this approach fails to pass the test of clinical

14

Figure 4. BayesBuilder graphical software environment, showing part of the Anemia network. The network consists of 91 variables and models some specific Anemias.

n 10 20 30 40 50

Iter 16 30 44 48 51

\c\ 8 12 16 21 26

Potential error 0.068 0.068 0.079 0.073 -

Margin error 0.068 0.216 0.222 0.218 -

Constraint error 5.8e-3 6.2e-3 4.5e-3 4.2e-3 3.2e-3

Table 1. Comparison of CVM method for large directed graphical models. Each node is connected to k = 5 parents. \C\ is the tree width of the triangulated graph required for the exact computation. Iter is the number of conjugate gradient descent iterations of the CVM method. Potential error and margin error are the maximum absolute error in any of the cluster probabilities and single variable marginals computed with CVM, respectively. Constraint error is the maximum absolute error in any of the constraints Eq. 8 after termination of CVM.

relevance and we have given several reasons t h a t could account for this failure. T h e alternative approach uses a probabilistic model to describe the relations between diagnoses and findings. This approach has the great advantage t h a t it provides a principled approach for the combination of different sources

15 of uncertainty. T h e price t h a t we have t o pay for this luxury is t h a t probabilistic inference is intractable for large systems. As a generic approximation m e t h o d , we have introduced the Cluster Variation m e t h o d and presented a novel iteration scheme, called Lagrange Multiplier Iteration. W h e n it converges, it provides very good results and is very fast. However, it is not guaranteed t o converge in general. In those more complex cases one must resort to more expensive methods, such as C C C P 1 7 or using an auxiliary cost function 1 6 . Acknowledgments T h i s research was supported in p a r t by t h e Dutch Technology Foundation ( S T W ) . I would like to t h a n k Taylan Cemgil for providing his M a t l a b graphical models toolkit, and Wim Wiegerinck and Sebino Stramaglia (Bari, Italy) for useful discussions. References 1. 2. 3. 4. 5.

6.

7.

8.

9. 10.

Meditel, Devon, Pa. Meditel: Computer assisted diagnosis, 1991. CAMDAT, Pittsburgh. QMR (Quick Medical Reference), 1992. Massachusetts General Hospital, Boston. DXPLAIN, 1992. Applied Informatics, Salt Lake City. ILIAD, 1992. E.S. Berner, G.D. Webster, A.A. Shugerman, J.R. Jackson, J. Algina, A.L. Baker, E.V. Ball, C.G. Cobbs, V.W. Dennis, E.P. Frenkel, L.D. Hudson, E.L. Mancall, C.E. Racley, and O.D. Taunton. Performance of four computer-based diagnostic systems. N-Engl-J-Med., 330(25):1792-6, 1994. D.E. Heckerman, E.J. Horvitz, and B.N. Nathwani. Towards normative expert systems: part I, the Pathfinder project. Methods of Information in medicine, 31:90-105, 1992. D.E. Heckerman and B.N. Nathwani. Towards normative expert systems: part II, probability-based representations for efficient knowledge acquisition and inference. Methods of Information in medicine, 31:106-116, 1992. M.A Shwe, B. Middleton, D.E. Heckerman, M. Henrion, Horvitz E.J., H.P. Lehman, and G.F. Cooper. Probabilistic Diagnosis Using a Reformulation of the Internist-1/ QMR Knowledge Base. Methods of Information in Medicine, 30:241-55, 1991. H.J. Kappen and F.B. Rodriguez. Efficient learning in Boltzmann Machines using linear response theory. Neural Computation, 10:1137-1156, 1998. H.J. Kappen and W.A.J.J. Wiegerinck. Second order approximations for probability models. In Todd Leen, Tom Dietterich, Rich Caruana, and Virginia de Sa, editors, Advances in Neural Information Processing Systems 13, pages 238244. MIT Press, 2001.

16 11. R. Kikuchi. Physical Review, 81:988, 1951. 12. J.S. Yedidia, W.T. Freeman, and Y. Weiss. Generalized belief propagation. In T.K. Leen, T.G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13 (Proceedings of the 2000 Conference), 2001. In press. 13. J. Pearl. Probabilistic reasoning in intelligent systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco, California, 1988. 14. Kevin P. Murphy, Yair Weiss, and Michael I. Jordan. Loopy belief propagation for approximate inference: An empirical study. In Proceedings of Uncertainty in AI, pages 467-475, 1999. 15. S.L. Lauritzen and D.J. Spiegelhalter. Local computations with probabilties on graphical structures and their application to expert systems. J. Royal Statistical society B, 50:154-227, 1988. 16. H.J. Kappen and W. Wiegerinck. A novel iteration scheme for the cluster variation method. In T.G. Dieterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems, volume 14, 2002. In press. 17. A.L. Yuille and A. Rangarajan. The convex-concave principle. In T.G. Dieterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems, volume 14, 2002. In press. 18. W. Wiegerinck, H.J. Kappen, E.W.M.T ter Braak, W.J.P.P ter Burg, M.J. Nijman, Y.L. O, and J.P. Neijt. Approximate inference for medical diagnosis. Pattern Recognition Letters, 20:1231-1239, 1999. 19. B. Kappen, W. Wiegerinck, and M. Nijman. Bayesbuilder. In W. Buntine, B. Fischer, and J. Schumann, editors, Software Support for Bayesian Analysis. RIACS, NASA Ames Research Center, 2000.

17 ANALYSIS OF EEG I N E P I L E P S Y K. LEHNERTZ 1 -*, R. G. ANDRZEJAK 1 - 2 , T. KREUZ 1 - 2 , F. MORMANN 1 ' 3 , C. RIEKE 1 ' 3 , P. DAVID 3 , C. E. ELGER 1 1 Department of Epileptology, University of Bonn John von Neumann Institue for Computing, Forschungszentrum Jiilich Institute for Radiation and Nuclear Physics, University of Bonn Germany 'E-mail:

[email protected]

We present potential applications of nonlinear time series analysis techniques to electroencephalographic recordings (EEG) derived from epilepsy patients. Apart from diagnostically oriented topics including localization of epileptic foci in different anatomical locations during the seizure-free interval we discuss possibilities for seizure anticipation which is one of the most challenging aspects in epileptology.

1

Introduction

The disease epilepsy is characterized by a recurrent and sudden malfunction of the brain that is termed seizure. Epileptic seizures reflect the clinical signs of an excessive and hypersynchronous activity of neurons in the cerebral cortex. Depending on the extent of involvement of other brain areas during the course of the seizure, epilepsies can be divided into two main classes. Generalized seizures involve almost the entire brain while focal (or partial) seizures originate from a circumscribed region of the brain (epileptic focus) and remain restricted to this region. Epileptic seizures may be accompanied by an impairment or loss of consciousness, psychic, autonomic or sensory symptoms or motor phenomena. Knowledge about basic mechanisms leading to seizures is mainly derived from animal experiments. Although there is a considerable bulk of literature on the topic, the underlying electrophysiological and neurobiochemical mechanisms are not yet fully explored. Moreover, it remains to be proven whether findings from animal experiments are fully transformable to human epilepsies. Recordings of the membrane potential of neurons under epileptic conditions indicate an enormous change, which by far exceeds physiological changes occurring with neuronal excitation. This phenomenon is termed paroxysmal depolarization shift (PDS 1 ' 2 ' 3 ) and represents a shift of the resting membrane potential that is accompanied by an increase of intracellular calcium and a massive burst of action potentials (500 - 800 per second). PDS originating from a larger cortical region are associated with steep field potentials (known

18

as spikes) recorded in the scalp E E C Focal seizures are assumed to be initiated by abnormally discharging neurons (so-called bursters 4>5>6) that recruit and entrain neighboring neurons into a "critical mass". This build-up might be mediated by an increasing synchronization of neuronal activity that is accompanied by a loss of inhibition, or by facilitating processes that permit seizure emergence by lowering a threshold. The fact that seizures appear to be unpredictable is one of the most disabling aspects of epilepsy. If it was possible to anticipate seizures, this would dramatically change therapeutic possibilities 7 . Approximately 0.6 - 0.8% of the world population suffer from epilepsy. In about half of these patients, focal seizures originate from functional and/or morphological lesions of the brain. Antiepileptic drugs insufficiently control or even fail to manage epilepsy in 30 - 50% of the cases. It can be assumed that 10 - 15% of these cases would profit from epilepsy surgery. Successful surgical treatment of focal epilepsies requires exact localization of the epileptic focus and its delineation from functionally relevant areas. For this purpose, different presurgical evaluation methodologies are currently in use 8 . Neurological and neuropsychological examinations are complemented by neuroimaging techniques that try to identify potential morphological correlates. Currently, the gold standard for an exact localization of the epileptic focus, however, is to record the patient's spontaneous habitual seizure using electroencephalography. Depending on the individual occurrence of seizures, this task requires long-lasting and continuous recording of the E E C In case of ambiguous scalp EEG findings, invasive recordings of the electrocorticogram (ECoG) or the stereo-EEG (SEEG) via implanted depth electrodes are indicated. This procedure, however, comprises a certain risk for the patient and is time-consuming and expensive. Thus, reliable EEG analysis techniques are required to localize and to demarcate the epileptic focus even during the seizure-free interval 9 . 2

EEG analysis

In recent years, technical advantages such as digital video-EEG monitoring systems as well as an increased computational power led to a highly sophisticated clinical epilepsy monitoring allowing to process huge amounts of data in real-time. In addition, chronically implanted intracranial electrodes allow continuous recording of brain electrical activity from the surface of the brain and/or within specific brain structures at a high signal-to-noise ratio and at a high spatial resolution. Due to its high temporal resolution and its close relationship to physiological and pathological functions of the brain, electroencephalography is regarded indispensible for clinical practice despite the rapid

19

development of imaging techniques like magnetic resonance tomography or positron emission tomography. Usually EEG analysis methods are applied to long-lasting multi-channel recordings in a moving-window fashion. The time length of a window is chosen in such a way that it represents a reasonable tradeoff between approximate stationarity and sufficient number of data points. Depending on the complexity of the analysis technique applied, computation times vary between a few milliseconds up to some tenths of seconds. Thus, most applications can be performed in real-time using standard personal computers. However, analyses cannot be applied in a strict mathematical sense because the necessary theoretical conditions cannot be met in practice - a common problem that applies to any analysis of short (and noisy) data segments or nonstationary data. Linear EEG analysis methods 10 can be divided into two main conceptbased categories. Nonpar am etric methods comprise analysis techniques such as evaluation of amplitude, interval or period distributions, estimation of autoand crosscorrelation functions as well as analyses in the frequency domain like power spectral estimation and cross-spectral functions. Parametric methods include, among others, AR (autoregressive) and ARMA (autoregressive moving average) models 11 , inverse AR-filtering and segmentation analysis. These main branches are accompanied by pattern recognition methods involving either a mixture of techniques mentioned before or, more recently, the wavelet transform 12,13 ' 14 . Despite the limitations mentioned above, classical EEG analysis has significantly contributed to and still advances understanding of physiological and pathophysiological mechanisms of the brain. Nonlinear time series analysis techniques 15 ' 16 ' 17 have been developed to analyze and characterize apparently irregular behavior - a distinctive feature of the EEG. Techniques mainly involve estimates of an effective correlation dimension, entropy related measures, Lyapunov exponents, measures for determinism, similarity, interdependencies, recurrence quantification as well as tests for nonlinearity. During the last decade a variety of these analysis techniques have been repeatedly applied to EEG recordings during physiological and pathological conditions and were shown to offer new information about complex brain dynamics 18 ' 19 ' 20 ' 21 . Today it is commonly accepted that the existence of a deterministic or even chaotic structure underlying neuronal dynamics is difficult if not impossible to prove. Nevertheless, nonlinear approaches to the analysis of the system brain have generated new clinical measures as well as new ways of interpreting brain electrical function, particularly with regard to epileptic brain states. Indeed, recent results provide converging evidence that nonlinear EEG analysis allows to reliably characterize different

20

states of brain function and dysfunction, provided that limitations of the respective analysis techniques are taken into consideration and, thus, results are interpreted with care (e.g., only relative measures with respect to recording time and recording site are assumed reliable). In the following, we will concentrate on nonlinear EEG analysis techniques and illustrate potential applications of these techniques in the field of epileptology. 3

Nonlinear EEG analysis in epilepsy

In early publications 22 ' 23 evidence for low-dimensional chaos in EEG recordings of epileptic seizures was claimed. However, accumulating knowledge about influencing factors as well as improvement of analysis techniques rendered these findings questionable 24>25>26'27. There is, however, now converging evidence that relative estimates of nonlinear measures improve understanding of the complex spatio-temporal dynamics of the epileptogenic process in different brain regions and promise to be of high relevance for diagnostics 28 ' 29 ' 30 ' 31 . 3.1

Outline of analysis techniques

In the course of our work attempting to characterize the epileptogenic process we investigated the applicability of already established measures and developed new ones. Results presented below were obtained from extracting these measures from long-lasting ECoG and SEEG recordings from subgroups of a collective of about 300 patients with epileptogenic foci located in different anatomical regions of the brain. Apart from linear measures such as statistical moments, power spectral estimates, or auto- and cross-correlation functions several univariate and bivariate nonlinear measures are currently in use. Since details of different analysis techniques have been published elsewhere, we here only provide a short description of the measures along with the respective references. Univariate measures: Based on the well known fact that neurons involved in the epileptic process exhibit high frequency discharges that are scarcely modulated by physiological brain activity 5 , we hypothesized that this neuronal behavior should be accompanied by an intermittent loss of complexity or an increase of nonlinear deterministic structure in the corresponding electrographic signal even during the seizure-free interval 33 . To characterize complexity, we use an estimate of an effective correlation dimension 32 £>|ff and the derived measure neuronal complexity loss L*33'34. These measures are accompanied by estimates of the largest Lyapunov exponent Ai 35 ' 36 ' 37 , by entropy

21

measures 38 , by the nonlinear prediction error 39 ' 40 , and by different complexity measures 41 derived from the theory of symbolic dynamics 42 . Detection and characterization of nonlinear deterministic structures in the EEG is achieved by combining tests for determinisms 43 and for nonlinearity 44 resulting in a measure we have termed fraction of nonlinear determinism £ 45 . 46 . 47 . Bivariate measures: As already mentioned, pathological neuronal synchronization is considered to play a crucial role in epileptogenesis. Therefore, univariate measures are supplemented by bivariate measures that aim to detect and characterize synchronization in time series of brain electrical activity. The nonlinear interdependence 5 4 8 , 4 9 characterizes statistical relationships between two time series. In contrast to commonly used measures like cross-correlation, coherence and mutual information, S is non-symmetric and provides information about the direction of interdependence. It is closely related to other attempts to detect generalized synchronization 50 . Following the approach of understanding phase synchronization 51 in a statistical sense 52 we developed a straight-forward measure for phase synchronization employing the circular variance 53 of a phase distribution. We have termed this measure mean phase coherence i? 5 4 ' 5 5 . 3.2

Localizing the epileptic focus

Several lines of evidence originating from studies of human epileptic brain tissue as well as from animal models of chronic seizure disorders indicate that the epileptic brain is different from the normal, even between seizures - during the so called interictal state. In order to evaluate the efficacy of different analysis techniques to characterize the spatio-temporal dynamics of the epileptogenic process and thus to localize the epileptic focus during the interictal state, we applied them to long-lasting interictal ECoG/SEEG recordings covering different states of normal behavior and vigilance as well as different extents of epileptiform activity. We retrospectively analyzed data of patients with mesial temporal lobe epilepsy (MTLE) and/or neocortical lesional epilepsy (NLE) undergoing presurgical evaluation. We included data of patients for whom surgery led to complete post-operative seizure control as well as of patients who did not benefit from surgery. Nonlinear EEG analysis techniques allow to reliably localize epileptic foci in different cerebral regions in more than 80 % of the cases. This holds true regardless whether or not obvious epileptiform activity is present in the recordings. Results obtained from our univariate measures indicate that the dynamics of the epileptic focus during the seizure-free interval can indeed be characterized by an intermittent loss of complexity or an increased nonlinear determin-

22

istic structure in an otherwise stochastic environment. Bivariate measures indicate that the epileptic focus is characterized by a pathologically increased level of interdependence or synchronization. Both univariate and bivariate measures thus share the ability to detect dynamical changes related to the epileptic process. It can be concluded that our EEG analysis techniques approach the problem of characterizing the epileptogenic process from different points of view, and they indicate the potential relevance of nonlinear EEG analysis to improve understanding of intermittent dysfunctioning of the dynamical system brain between seizures. Moreover, our results also stress the relevance of nonlinear EEG analyses in clinical practice since they provide potentially useful diagnostic information and thus may contribute to an improvement of the presurgical evaluation 9 ' 56 ' 57 ' 29 . 3.3

Anticipating seizures

In EEG analysis the search for the hidden information predictive of an impending seizure has a long history. As early as 1975, researchers considered analysis techniques such as pattern recognition analytic procedures of spectral data 58 or autoregressive modeling of EEG data 5 9 for predicting epileptic seizures. Findings indicated that EEG changes characteristic for pre-seizure states may be detectable, at most, a few seconds before the actual seizure onset. None of these techniques have been implemented clinically. Apart from applying signal analysis techniques the relevance of steep, high amplitude epileptiform potentials (spikes, the hallmark of the epileptic brain) were investigated in a number of clinical studies 60 ' 61 ' 62 . While some authors reported a decrease or even total cessation of spikes before seizures, reexamination did not confirm this phenomenon in a larger sample. Although there are numerous studies exploring basic neuronal mechanisms that are likely to be associated with seizures, to date, no definite information is available as to the generation of seizures in humans. In this context, the term "critical mass" might be misleading in the sense that it could merely imply an increasing number of neurons that are entrained into an abnormal discharging process. This mass phenomenon would have been easily accessible for conventional EEG analyses which, however, failed to detect it. Recent research in seizure anticipation has shown that evident markers in the EEG representing the transition from asynchronous to synchronous states of the epileptic brain {pre-seizure state) can be detected on time scales ranging from several minutes up to hours. These studies indicate that the seizure-initiating process should be regarded as an unfolding of an increasing number of critical, possibly nonlinear dynamical interferences between

23

neurons within the focal area as well as with neurons surrounding this area. Indeed, there is converging evidence from different laboratories that nonlinear analysis is capable to characterize this collective behavior of neurons from the gross electrical activity and hence allows to define a critical transition state, at least in a high percentage of cases 63,64,65,67,34,68,69,70,55,71,28 4

Future perspectives

Results obtained so far are promising and emphasize the high value of nonlinear EEG analysis techniques both for clinical practice and basic science. Up to now, however, findings have been mainly obtained from retrospective studies in well-elaborated cases and using invasive recording techniques. Thus, on the one hand, evaluation of more complicated cases as well as prospective studies on a larger population of patients are necessary. The possibility of defining a critical transition state can be regarded as the most prominent contribution of nonlinear EEG analysis to advance knowledge about seizure generation in humans. This possibility has recently been expanded by studies indicating accessibility of critical pre-seizure changes from non-invasive EEG recordings 65 ' 71 . Nonetheless, in order to achieve an unequivocal definition of a pre-seizure state from either invasive or non-invasive recordings, a variety of influencing factors have to be evaluated. Most studies carried out so far have concentrated on EEG recordings just prior to seizures. Other studies 33 ' 48 ' 66 ' 55,47,28 , however, have shown that there are phases of dynamical changes even during the seizure-free interval pointing to abnormalities that are not followed by a seizure. Moreover, pathologically or physiologically induced dynamical interactions within the brain are not yet fully understood. Among others, these include different sleep stages, different cognitive states, as well as daily activities that clearly vary from patient to patient. In order to evaluate specificity of possible seizure anticipation techniques, analyses of long-lasting multi-channel EEG recordings covering different pathological and physiological states are therefore mandatory 67,34,28 . Along with these studies, EEG analysis techniques have to be further improved. New techniques are needed that allow a better characterization of non-stationarity and high-dimensionality in brain dynamics, techniques disentangling even subtle dynamical interactions between pathological disturbances and surrounding brain tissue as well as refined artifact detection and elimination. Since the methods currently available allow a distinguished characterization of the epileptogenic process, the combined use of these techniques along with appropriate classification schemes 72 ' 73 ' 74 can be regarded a promising venture.

24

Once given an improved sensitivity and specificity of EEG analysis techniques for both focus localization and seizure anticipation, broader clinical applications on a larger population of patients, either at home or in a clinical setting, can be envisaged. As a future perspective, one might also take into consideration implantable seizure anticipation and prevention devices similar to those already in use with Parkinsonian patients 75 ' 76 . Although optimization of algorithms underlying the computation of specific nonlinear measures 77,69 already allows to continuously track the temporal behavior of nonlinear measures in real time, these applications still require the use of powerful computer systems, depending on the number of recording channels necessary to allow unequivocal characterization of the epileptogenic process. Thus, further optimization and development of a miniaturized analyzing system are definitely necessary. Taking into account the technologies currently available, realization of such systems can be expected within the next years.

Acknowledgments We gratefully acknowledge discussions with and contributions by Jochen Arnhold, Wieland Burr, Guillen Fernandez, Peter Grassberger, Thomas Grunwald, Peter Hanggi, Christoph Helmstaedter, Martin Kurthen, Hans-Rudi Moser, Thomas Schreiber, Bruno Weber, Jochen Wegner, Guido Widman and Heinz-Gregor Wieser. This work was supported by the Deutsche Forschungsgemeinschaft. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

E. S. Goldensohn and D. P. Purpura, Science 139, 840 (1963). H. Matsumoto and C. Ajmone-Marsan, Exp. Neurol. 9, 286 (1964). H. Matsumoto and C. Ajmone-Marsan, Exp. Neurol. 9, 305 (1964). R. D. Traub and R. K. Wong, Science 216, 745 (1982). A. R. Wyler and A. A. Ward, in Epilepsy, a window to brain mechanisms, eds. J. S. Lockard and A. A. Ward (Raven Press, New York, 1992). E. R. G. Sanabria, H. Su and Y. Yaari, J. Physiol. 532, 205 (2001). C. E. Elger, Curr. Opin. Neurol. 14, 185 (2001). J. Engel Jr. and T. A. Pedley, Epilepsy: a comprehensive text-book (Philadelphia, Lippincott-Raven, 1997). C. E. Elger, K. Lehnertz and G. Widman in Epilepsy: Problem solving in clinical practice, eds. D. Schmidt and S. C. Schacter (Martin Dunitz Publishers, London, 1999).

25

10. F. H. Lopes da Silva in Electroencephalography, eds. E. Niedermeyer and F. H. Lopes da Silva (Williams & Wilkins, Baltimore, 1993). 11. P J. Franaszczuk and G. K. Bergey, Biol. Cybern. 8 1 , 3 (1999). 12. S. J. SchiSetal, Electroencephalogr. clin. Neurophysiol. 91,442(1994). 13. R. R. Coifman and M. V. Wickerhauser, Electroencephalogr. clin. Neurophysiol. (Suppl.) 45, 57 (1996). 14. A. Effern et al, Physica D 140, 257 (2000). 15. H. G. Schuster, Deterministic chaos: an introduction (VCH Verlag, Basel, Cambridge, New York, 1989). 16. E. Ott, Chaos in dynamical systems (Cambridge University Press, Cambridge, UK, 1993). 17. H. Kantz and T. Schreiber, Nonlinear time series analysis (Cambridge University Press, Cambridge, UK, 1997). 18. E. Ba§ar, Chaos in Brain Function (Springer, Berlin, 1990). 19. D. Duke and W. Pritchard, Measuring chaos in the human brain (World Scientific, Singapore, 1991). 20. B. H. Jansen and M. E. Brandt, Nonlinear dynamical analysis of the EEG (World Scientific, Singapore, 1993). 21. K. Lehnertz , J. Arnhold, P. Grassberger and C. E. Elger, Chaos in brain? (World Scientific, Singapore, 2000). 22. A. Babloyantz and A. Destexhe, Proc. Natl. Acad. Sci. USA 83, 3513 (1986). 23. G. W. Frank et al, Physica D 46, 427 (1990). 24. J. Theiler, Phys. Lett. A 196, 335 (1995). 25. J. Theiler and P. E. Rapp, Electroencephalogr. clin. Neurophysiol. 98, 213 (1996). 26. T. Schreiber, in Chaos in brain?, eds. K. Lehnertz , J. Arnhold, P. Grassberger and C. E. Elger, (World Scientific, Singapore, 2000). 27. F. H. Lopes da Silva et al., in Chaos in brain?, eds. K. Lehnertz , J. Arnhold, P. Grassberger and C. E. Elger, (World Scientific, Singapore, 2000). 28. B. Litt et al, Neuron 30, 51 (2001). 29. K. Lehnertz et al, J. Clin. Neurophysiol. 18, 209 (2001). 30. M. Le Van Quyen et al, J. Clin. Neurophysiol. 18, 191 (2001). 31. R. Savit et al, J. Clin. Neurophysiol. 18, 246 (2001). 32. P. Grassberger, T. Schreiber and C. Schaffrath, Int. J. Bifurcation Chaos 1, 521 (1991). 33. K. Lehnertz and C. E. Elger, Electroencephalogr. clin. Neurophysiol. 95, 108 (1995). 34. K. Lehnertz and C. E. Elger, Phys. Rev. Lett. 80, 5019 (1998).

26

35. M. T. Rosenstein, J. J. Collins and C. J. de Luca Physica D 65, 117 (1994). 36. H. Kantz, Phys Lett A 185, 77 (1994). 37. J. Wegner, Diploma thesis, University of Bonn (1998). 38. R. Quian Quiroga et al., Phys. Rev. E 62, 8380 (2000). 39. A. S. Weigend and N. A. Gershenfeld, Time Series Prediction: Forecasting the Future and Understanding the Past (Addison-Wesley, Reading, 1993). 40. R. G. Andrzejak et al., Phys. Rev. E 64, 061907 (2001). 41. T. Kreuz, Diploma thesis, University of Bonn (2000). 42. B. L. Hao, Elementary Symbolic Dynamics and Chaos in Dissipative Systems (World Scientific, Singapore, 1989). 43. D. T. Kaplan and L. Glass, Phys. Rev. Lett. 68, 427 (1992). 44. T. Schreiber and A. Schmitz, Phys. Rev. Lett. 77, 635 (1996). 45. R. G. Andrzejak, Diploma thesis, University of Bonn (1997). 46. R. G. Andrzejak et al, in Chaos in brain?, eds. K. Lehnertz , J. Arnhold, P. Grassberger and C. E. Elger, (World Scientific, Singapore, 2000). 47. R. G. Andrzejak et al, Epilepsy Res. 44, 129 (2001). 48. J. Arnhold et al., Physica D 134, 419 (1999). 49. J. Arnhold, Publication Series of the John von Neumann Institute for Computing, Forschungszentrum Jiilich, Vol. 4 (2000). 50. N.F. Rulkov et al., Phys. Rev. E 51, 980 (1995). 51. M. G. Rosenblum et al., Phys. Rev. Lett. 76, 1804 (1996) 52. P. Tass et al., Phys. Rev. Lett. 81, 3291 (1998). 53. K. V. Mardia Probability and mathematical statistics: Statistics of directional data. (Academy Press, London, 1972). 54. F. Mormann, Diploma thesis, University of Bonn (1998). 55. F. Mormann et al, Physica D 144, 358 (2000). 56. C. E. Elger et al., in Neocortical epilepsies, eds P. D. Williamson, A. M. Siegel, D. W. Roberts, V. M. Thadani and M. S. Gazzaniga (Lippincott, Williams & Wilkins: Philadelphia, 2000). 57. C. E. Elger et al., Epilepsia 41 (Suppl. 3), S34 (2000). 58. S. S. Viglione and G. O. Walsh, Electroencephalogr. clin. Neurophysiol. 39, 435 (1975). 59. Z. Rogowski, I. Gath and E. Bental, Biol. Cybern. 42, 9 (1981). 60. J. Gotman et al, Epilepsia 23, 432 (1982). 61. H. H. Lange et al., Electroencephalogr. clin. Neurophysiol. 56, 543 (1983). 62. A. Katz et al., Electroencephalogr. clin. Neurophysiol. 79, 153 (1991). 63. L. D. Iasemidis et al., Brain Topogr. 2, 187 (1990).

27

64. C. E. Elger and K. Lehnertz, in Epileptic Seizures and Syndromes, ed. P. Wolf (J. Libbey & Co, London, 1994). 65. L. D. lasemidis et al. in Spatiotemporal Models in Biological and Artificial Systems, eds. F. H. Lopes da Silva, J. C. Principe and L. B. Almeida (IOS Press, Amsterdam, 1997). 66. M. Le Van Quyen et al, Physica D 127, 250 (1999). 67. C. E. Elger and K. Lehnertz, Eur. J. Neurosci. 10, 786 (1998). 68. J. Martinerie et al, Nat. Med. 4, 1173 (1998). 69. M. Le Van Quyen et al, Neuroreport 10, 2149 (1999). 70. H. R. Moser et al, Physica D 130, 291 (1999). 71. M. Le Van Quyen et al, Lancet 357, 183 (2001). 72. Y. Salant, I. Gath and O. Henriksen, Med. Biol. Eng. Comput. 36, 549 (1998). 73. R. Tetzlaff et al, IEEE Proc. Eur. Conf. Circuit Theory Design , 573 (1999). 74. A. Petrosian et al, Neurocomputing 30, 201 (2000). 75. A. L. Benabid et al, Lancet 337, 403 (1991). 76. P. Tass, Biol. Cybern. 85, 343 (2001). 77. G. Widman et al, Physica D 121, 65 (1998).

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S T O C H A S T I C A P P R O A C H E S TO MODELING OF PHYSIOLOGICAL R H Y T H M S P L A M E N CH. IVANOV Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 Cardiovascular Division, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA 02215, USA E-mail: [email protected] CHUNG-CHUAN LO Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA E-mail: [email protected] The scientific question we address is how physiological rhythms spontaneously selfregulate. It is fairly widely believed, nowadays, deterministic mechanisms, including perhaps chaos, offer a promising avenue to pursue in answering this question. Complementary to these deterministic foundations, we propose an approach which treats physiologiocal rhythms as fundamentally governed by several random processes, each of which biases the rhythm in different ways. We call this approach stochastic feedback, since it leads naturally to feedback mechanisms that are based on randomness. To illustrate our approach, we treat in some detail the regulation of heart rhythms and sleep-wake transitions during sleep — two classic "unsolved" problems in physiology. We present coherent, physiologically based models and show that a generic process based on the concepts of biased random walk and stochastic feedback can account for a combination of independent scaling characteristics observed in data.

1 1.1

M o d e l i n g scaling features in heartbeat

dynamics

Introduction

The fundamental principle of homeostasis asserts that physiological systems seek to maintain a constant output after perturbation 1 ' 2 - 3 ' 4 . Recent evidence, however, indicates that healthy systems even at rest display highly irregular dynamics 5-6>7>8>9,io_ Here, we address the question of how to reconcile homeostatic control and complex variability. We propose a general approach based on the concept of "stochastic feedback" and illustrate this approach by considering the neuroautonomic regulation of the heart rate. Our results suggest that in healthy systems the control mechanisms operate to drive the system away from extreme values while not allowing it to settle down to a constant (homeostatic) output. The model generates complex dynamics and

29 successfully accounts for key characteristics of the cardiac variability not fully explained by traditional models: (i) 1 / / power spectrum, (ii) stable scaling form for the distribution of the variations in the beat-to-beat intervals and (hi) Fourier phase correlations Ii>i2,i3,i4,i5,i6,i7_ Furthermore, the reported scaling properties arise over a broad zone of parameter values rather than at a sharply-defined "critical" point. 1.2

Random walks and feedback mechanisms

The concept of dynamic equilibrium or "homeostasis" x ' 2 ' 3 led to the proposal that physiological variables, such as the cardiac interbeat interval r(n), where n is the beat number, maintain an approximately constant value in spite of continual perturbations. Thus one can write in general r{n) =T0+n,

(1)

where TQ is the "preferred level" for the interbeat interval and 77 is a white noise with strength a, defined as the standard deviation of rj. We first re-state this problem in the language of random walks. The time evolution of an uncorrelated and unbiased random walk is expressed by the equation r ( n + l ) —r(n) = 77. At every step the walker has equal probability to move "up" or "down." The deviation from the initial level increases as n 1 / 2 18 , so an uncorrelated and unbiased random walk does not preserve homeostasis (Fig. la). To maintain a constant level, there must be a bias in the random walk 19 , Tin + 1) - Tin) = I(n),

(2)

T(„\ - j W tt + V) , if Tin) < T0, W - \ - « , (1+77) , if r ( n ) > r 0 .

,,,

with J

{6)

The weight w is the strength of the feedback input biasing the walker to return to its preferred level To- When away from the attraction level To, walker has an higher probability of moving towards the attraction level. This behavior represents Cannon's idea of homeostasis (dynamical equilibrium), where a system maintains constancy even when perturbed by external stimuli. Note that Eqs. (2 & 3 ) generate dynamics similar to Eq. (1) but through a nonlinear feedback mechanism. The dynamics generated by these rules correspond to a system with time-independent feedback. As expected in this case, for short time scales (high frequencies), the power spectrum scales as l / / 2 (Brownian noise) with a crossover to white noise at longer time scales due to the attraction to level TQ (Fig. lb). Note the shift

30

n

—». n

A

lnf

'

». Inf

'

>A

Figure 1. (a) Schematic representation of the dynamics of the model, (a) Evolution of a random walk starting from initial position TO. The deviation of the walk from level TO increases as n 1 ' 2 , where n is the number of steps. The power spectrum of the random walk scales as l / / 2 (Brownian noise). The distribution P(A) of the amplitudes A of the variations in the interbeat intervals follows a Rayleigh distribution. Here the amplitudes are obtained by: (i) wavelet transform of the random walk which filters out trends and extracts the variations at a time scale a; (ii) calculation of the amplitudes of the variations via Hilbert transform, (b) Random walk with a bias toward TO. (C) Random walk with two stochastic feedback controls. In contrast to (b), the levels of attraction TO and n change values in time. Each level persists for a time interval X; drawn from a distribution with an average value Ti oc i(. Each time the level changes, its new value is drawn from a uniform distribution. Perturbed by changing external stimuli, the system nevertheless remains within the bounds defined by A T even after many steps. We find that such dynamical mechanism based on a single characteristic time scale T\oc^ generates a 1 / / power spectrum over several decades. Moreover, P(A) decays exponentially, which we attribute to nonlinear Fourier phase interactions in the walk.

of the crossover to longer time scales (lower frequencies) when stronger noise is present. For weak noise the walker never leaves the close vicinity of the attraction level, while for stronger noise, larger drifts can occur leading to

31

longer trends and longer time scales. However, in both cases, P(A) follows the Rayleigh distribution because the wavelet transform filters out the drifts and trends in the random walk (Fig. lb). For intermediate values of the noise there is a deviation from the Rayleigh distribution and the appearance of an exponential tail. We find that Eqs. (2 & 3) do not reproduce the statistical properties of the empirical data (Fig. lb). We therefore generalize them to include several inputs Ik {k = 0,1, • • • , m), with different preferred levels Tk, which compete in biasing the walker: m

r(n + l)-T(n) = X)/*(«).

(4)

ife=0

where lk[

>~ \-wk

(1+v),

iir(n)>rk.

{b

>

From a biological or physiological point of view, it is clear t h a t t h e preferred levels Tk of t h e inputs Ik cannot remain constant in time, for otherwise the system would not be able t o respond t o varying external stimuli. We assume t h a t each preferred interval T>. is a random function of time, with values correlated over a time scale T]* ck . We next coarse grain t h e system and choose T^(TI) t o be a random step-like function constrained t o have values within a certain interval and with t h e length of t h e steps drawn from a distribution with an average value Xlock (Fig. l c ) . This model yields several interesting features, including 1 / / power spectrum, scaling of the distribution of variations, and correlations in t h e Fourier phases. 1.3

Neuroautonomic regulation of heartbeat dynamics

To illustrate the approach for the specific example of neuroautonomic control of cardiac dynamics, we first note that the healthy heart rate is determined by three major inputs: (i) the sinoatrial (SA) node; (ii) the parasympathetic (PS); and (hi) the sympathetic (SS) branches of the autonomic nervous system. (i) The SA node or pacemaker is responsible for the initiation of each heart beat 20 ; in the absence of other external stimuli, it is able to maintain a constant interbeat interval 2 . Experiments in which PS and SS inputs are blocked reveal that the interbeat intervals are very regular and average only 0.6s 20 . The input from the SA node, Is A , thus biases the interbeat interval r toward its intrinsic level TSA (see Fig. lb).

32

beat 10 r •

beat 10 3 .

,

10""

10"3

,10"'

/[beat]

10"1

10°

10s

10"4

10"3

,10"2

/[beat 1 ]

10~1

Figure 2. Stochastic feedback regulation of the cardiac rhythm. We compare the predictions of the model with the healthy heart rate. Sequences of interbeat intervals r from (a) healthy individual and (b) from simulation exhibit an apparent visual similarity, (c) Power spectra of the interbeat intervals r(n) from the data and the model. To first approximation, these power spectra can be described by the relation S(f) ~ l / / 1 1 . The presence of patches in both heart and model signals lead to observable crossovers embedded on this 1 / / behavior at different time scales. We calculated the local exponent B from the power spectrum of 24h records ( « 10 5 beats) for 20 healthy subjects and found that the local value of P shows a persistent drift, so no true scaling exists. (This is not surprising, due to the nonstationarity of the signals), (d) Power spectra of the increments in r(n). The model and the data both scale as power laws with exponents close to one. Since the non-stationarity is reduced, crossovers are no longer present. We also calculated the local exponent for the power spectrum of the increments for the same group of 20 healthy subjects as in the top curve, and found that the exponent Bj fluctuates around an average value close to one, so true scaling does exist.

(ii) The PS fibers conduct impulses that slow the heart rate. Suppression of SS stimuli, while under PS regulation, can result in the increase of the interbeat interval to as much as 1.5s 20>21. The activity of the PS system changes with external stimuli. We model these features of the PS input, Ips, by the following conditions: (1) a preferred interval, Tps(n), randomly chosen from an uniform distribution with an average value larger than TSA , and (2) a correlation time, Tps, during which Tps does not change, where Tps is drawn

33

from a distribution with an average value T]0Ck. (iii) The SS fibers conduct impulses that speed up the heart beat. Abolition of parasympathetic influences when the sympathetic system remains active can decrease the interbeat intervals to less than 0.3s 2 0 . There are several centers of sympathetic activity highly sensitive to environmental influences 2 1 . We represent each of the N sympathetic inputs by I3SS (j = 1, • • •, N). We attribute to I3SS the following characteristics: (1) a preferred interbeat interval TgS(n) randomly chosen from a uniform distribution with an average value smaller than TSA, and (2) a correlation time Tj in which r3ss(n) does not change; Tj is drawn from a distribution with an average value Xiock which is the same for all N inputs (and the same as for the PS system), so Ti ock is the characteristic time scale of both the PS and SS inputs. The characteristics for the PS and SS inputs correspond to a random walk with stochastic feedback control (Fig. lc). Thus, for the present example of cardiac neuroautonomic control, we have N + 2 inputs and Eq. (4) becomes: N

r{n + 1) - r{n) = ISA(n)

+ IPS (n,TPS(n))

+ ^2lJss

[n,TJss(n)j

,

(6)

where the structure of each input is identical to the one in Eq. (5). Equation (6) cannot fully reflect the complexity of the human cardiac system. However, it provides a general framework that can easily be extended to include other physiological systems (such as breathing, baroreflex control, different locking times for the inputs of the SS and PS systems 5 ' 2 2 , etc.). We find that Eq. (6) captures the essential ingredients responsible for a number of important statistical and scaling properties of the healthy heart rate. Next we generate a realization of the model with parameters N = 7 and WSA = wss — W P S / 3 = O.Olsec (Fig. 2b). We choose Tj randomly from an exponential distribution with average T)OCk = 1000 beats. (We find that a different form of the distribution for Tj does not change the results.) The noise r\ is drawn from a symmetrical exponential distribution with zero average and standard deviation a = 0.5. We define the preferred values of the interbeat intervals for the different inputs according to the following rules: (1) TSA = 0.6sec, (2) Tps are randomly selected from an uniform distribution in the interval [0.9,1.5]sec, and (3) the r | s ' s are randomly selected from an uniform distribution in the interval [0.2,1.0]sec. The actual value of the preferred interbeat intervals of the different inputs and the ratio between their weights are physiologically justified and are of no significance for the dynamics — they just set the range for the fluctuations of r , chosen to correspond to the empirical data.

34

/[beat"]

lnS(f)

~ \/ TB

\/ \

\ // ^\ 2lock

A

TAl

TB1

Tc D \C Ax

B \ A

TC In/ Figure 3. (top) Effect of the correlation time T[oc^ on the scaling of the power spectrum of T(TI) for a signal comprising 10 6 beats, (b) Schematic diagram illustrating the origin of the different scaling regimes in the power spectrum of r ( n ) .

1.4

Experimantal findings and results of simulations

To qualitatively test the model, we first compare the time series generated by the stochastic feedback model and the healthy heart 23 and find that both signals display complex variability and patchiness (Fig. 2a,b). To quantitatively test the model, we compare the statistical properties of heart data with the predictions of the model: (a) We first test for long-range power-law correlations in the interbeat intervals, which exist for healthy heart dynamics 24 . These correlations can be uncovered by calculating power spectra, and we see (Fig. 2) that the model simulations correctly reproduce the power-law correlations observed in data over several decades. In particular, we note that the non-stationarity of both the data and model signals leads to the existence of several distinct scaling

35

regimes in the power spectrum of r(n) (Figs. 2c and 3). We find that with increasing Tiock, the power spectrum does not follow a single power law but actually crosses over from a behavior of the type l / / 2 at very small time scales (or high frequencies), to a behavior of the type 1//° for intermediate time scales, followed by a new regime with l / / 2 for larger time scales (Fig. 3). At very large time scales, another regime appears with flat power spectrum. In the language of random walkers, T is determined by the competition of different neuroautonomic inputs. For very short time scales, the noise will dominate, leading to a simple random walk behavior and l / / 2 scaling (regime A in Fig. 3(bottom). For time scales longer than T^, the deterministic attraction towards the "average preferred level" of all inputs will dominate, leading to a flat power spectrum (regime B in Fig. 3(bottom), see also Fig. lb). However, after a time TB (of the order of T Iock /iV), the preferred level of one of the inputs will have changed, leading to the random drift of the average preferred level and the consequent drift of the walker towards it. So, at these time scales, the system can again be described as a simple random walker and we expect a power spectrum of the type l / / 2 (regime C in Fig. 3(bottom)). Finally, for time scales larger than Tc, the walker will start to feel the presence of the bounds on the fluctuations of the preferred levels of the inputs. Thus, the power spectrum will again become fiat (regime D). Since the crossovers are not sharp in the data or in the numerical simulations, they can easily be misinterpreted as a single power law scaling with an exponent /3 « 1. By reducing the strength of the noise, we decrease the size of regime A and extend regime B into higher frequencies. In the limit a — 0, the power spectrum of r(n), which would coincide with the power spectrum of the "average preferred level", would have only regimes B, C and D. The stochastic feedback mechanism thus enables us to explain the formation of regions (patches) in the time series with different characteristics. (b) By studying the power spectrum of the increments we are able to circumvent the effects of the non-stationarity. Our results show that true scaling behavior is indeed observed for the power spectrum of the increments, both for the data and for the model (Fig. 2). (c) We calculate the probability density P(A) of the amplitudes A of the variations of interbeat intervals through the wavelet transform. It has been shown that the analysis of sequences of interbeat intervals with the wavelet transform 25 can reveal important scaling properties 26 for the distributions of the variations in complex nonstationary signals. In agreement with the results of Ref. 2 7 , we find that the distribution P(A) of the amplitudes A of interbeat interval variations for the model decays exponentially—as is observed for healthy heart dynamics (Fig. 4). We hypothesize that this decay arises

36

10°

:

0"'

;

a. oe

i Data • Model

AP Figure 4. Analysis of the amplitudes A of variations in r{n). We apply to the signal generated by the model the wavelet transform with fixed scale a, then use the Hilbert transform to calculate the amplitude A. The top left panel shows the normalized histogram P(A) for the data (6h daytime) and for the model (with the same parameter values as in Fig. 2), and for wavelet scale a — 8 beats, i.e., ft) 40s. (Derivatives of the Gaussian are used as a wevelet function). We test the generated signal for nonlinearity and Fourier phase correlations, creating a surrogate signal by randomizing the Fourier phases of the generated signal but preserving the power spectrum (thus, leaving the results of Fig. 2 unchanged). The histogram of the amplitudes of variations for the surrogate signal follows the Rayleigh distribution, as expected theoretically (see inset). Thus the observed distribution which is universal for healthy cardiac dynamics, and reproduced by the model, reflects the Fourier phase interactions. The top right panel shows a similar plot for d a t a collected during sleep and for the model with N < wps/wssWe note that the distribution is broader for the amplitudes of heartbeat interval variations during sleep compared wake activity indicating counterintuitively a higher probability for large variations with large values deviating from the exponential tail 2 8 . Our model reproduces this behavior when the number of sympathetic imputs is reduced in accordance with the physiological observations of decreased sympathetic tone during sleep 2 0 . The bottom panel tests the stability of the analysis for the model at different time scales a. The distribution is stable over a wide range of time scales, identical to the range observed for heart data 2 7 . T h e stability of the distributions indicates statistical self-similarity in the variations at different time scales.

37

from nonlinear Fourier phase interactions and is related to the underlying nonlinear dynamics. To test this hypothesis, we perform a parallel analysis on a surrogate time series obtained by preserving the power spectrum but randomizing the Fourier phases of a signal generated by the model (Fig. 4); P{A) now follows the Rayleigh distribution P(A) ~ Ae~A , since there are no Fourier phase correlations 2 9 . (d) For the distribution displayed in Fig. 4, we test the stability of the scaling form at different time scales; we find that P(A) for the model displays a scaling form stable over a range of time scales identical to the range for the data (Fig. 4) 2 7 . Such time scale invariance indicates statistical selfsimilarity 3 0 . A notable feature of the present model is that in addition to the power spectra, it accounts for the form and scaling properties of P(A), which are independent of the power spectra 31 . No similar tests for nonlinear dynamics have been reported for other models 12>13>14. Further work is needed to account for the recently reported long-range correlations in the magnitude of interbeat interval increments 32 , the multifractal spectrum of heartrate fluctuations 33 and the power-law distribution of segments in heart rate recordings with different local mean values 34 . The model has a number of parameters, whose values may vary from one individual to another, so we next study the sensitivity of our results to variations in these parameters. We find that the model is robust to parameter changes. The value of Xiock and the strength of the noise a are crucial to generate dynamics with scaling properties similar to those found for empirical data. We find that the model reproduces key features of the healthy heart dynamics for a wide range of time scales (500 < TiOCk < 2000) and noise strengths (0.4 < a < 0.6). The model is consistent with the existence of an extended "zone" in parameter space where scaling behavior holds, and our picture is supported by the variability in the parameters for healthy individuals for which similar scaling properties are observed. 1.5

Conclusions

Scaling behavior for physical systems is generally obtained for fixed values of the parameters, corresponding to a critical point or phase transition 3 5 . Such fixed values seem unlikely in biological systems exhibiting power law scaling. Moreover, such critical point behavior would imply perfect identity among individuals; our results are more consistent with the robust nature of healthy systems which appear to be able to maintain their complex dynamics over a wide range of parameter values, accounting for the adaptability of healthy

38

systems. The model we review here, and the data which it fits, support a revised view of homeostasis that takes into account the fact that healthy systems under basal conditions, while being continuously driven away from extreme values, do not settle down to a constant output. Rather, a more realistic picture may involve nonlinear stochastic feedback mechanisms driving the system.

2 2.1

Modeling dynamics of sleep-wake transitions Introduction

In this Section we investigate the dynamics of the awakening during the night for healthy subjects and find that the wake and the sleep periods exhibit completely different behavior: the durations of wake periods are characterized by a scale-free power-law distribution, while the durations of sleep periods have an exponential distribution with a characteristic time scale. We find that the characteristic time scale of sleep periods changes throughout the night. In contrast, there is no measurable variation in the power-law behavior for the durations of wake periods. We develop a stochastic model, based on biased random walk approach, which agrees with the data and suggests that the difference in the dynamics of sleep and wake states arises from the constraints on the number of microstates in the sleep-wake system. In clinical sleep centers, the "total sleep time" and the "total wake time" during the night are used to evaluate sleep efficacy and to diagnose sleep disorders. However, the total wake time during a longer period of nocturnal sleep is actually comprised of many short wake intervals (Fig. 5). This fact suggests that the "total wake time" during sleep is not sufficient to characterize the complex sleep-wake transitions and that it is important to ask how periods of the wake state distribute during the course of the night. Although recent studies have focused on sleep control at the neuronal level 36>37,38,39^ v e r y little is known about the dynamical mechanisms responsible for the time structure or even the statistics of the abrupt sleep-wake transitions during the night. Furthermore, different scaling behavior between sleep and wake activity and between different sleep stages has been observed 40>41. Hence, investigating the statistical properties of the wake and sleep states throughout the night may provide not only a more informative measure but also insight into the mechanisms of the sleep-wake transition.

39

120 Wake REM

180 240 Time (min)

L

cc 2

g3 z 4

-^W

Wake Sleep 50

70

90 110 Time (min)

130

150

Figure 5. The textbook picture 4 3 of sleep-stage transitions describes a quasi-cyclical process, with a period of « 90 min, where the wake stage is followed by light sleep and then by deep sleep, with transition back to light sleep, and then to rapid-eye-movement (REM) sleep—or perhaps to wake stage. Sleep-wake transitions during nocturnal sleep, (a) Representative example of sleep-stage transitions from a healthy subject. Data were recorded in a sleep laboratory according to the Rechtschaffen and Kales criteria 5 2 : two channels of electroencephalography (EEG), two channels of electrooculography (EOG) and one channel of submental electromyography (EMG) were recorded. Signals were digitized at 100 Hz and 12 bit resolution, and visually scored by sleep experts in segments of 30 seconds for sleep stages: wakefulness, rapid-eye-movement (REM) sleep and non-REM sleep stages 1, 2, 3 and 4. (b) Magnification of the shaded region in (a), (c) In order to study sleep-wake transitions, we reduce five stages into a single sleep state by grouping rapid-eye-movement (REM) sleep and sleep stages 1 to 4 into a single sleep state.

2.2

Empirical analysis

We analyze 39 full-night sleep records collected from 20 healthy subjects (11 females and 9 males, ages 23-57, with average sleep duration 7.0 hours). We first study the distribution of durations of the sleep and of the wake states during the night (Fig. 5). We calculate the cumulative distribution of

40

durations, defined as p(r)dr,

(7)

where p(r) is the probability density function of durations between r and r + dr. We analyze P(t) of the wake state, and we find that the data follow a power-law distribution, P(t) ~ t~a .

(8)

We calculate the exponent a for each of the 20 subjects, and find an average exponent a = 1.3 with a standard deviation a = 0.4. It is important to verify that the data from individual records correspond to the same probability distribution. To this end, we apply the KolmogorovSmirnov test to the data from individual records. We find that we cannot reject the null hypothesis that p(t) of the wake state of each subject is drawn from the same distribution, suggesting that one can pool all data together to improve statistics without changing the distribution (Fig. 6a). Pooling the data from all 39 records, we find that P(t) of the wake state is consistent with a power-law distribution with an exponent a — 1.3 ± 0.1 (Fig. 7a). In order to verify that the distribution of durations of wake state is better described by a power law rather than an exponential or a stretched exponential functional forms, we fit these curves to the distributions from pooled data. Using Levenberg-Marquardt method, we find that both exponential and stretched exponential form lead to worse fit. The x 2 error of power-law fit, exponential fit and stretched exponential fit are 3 x 10~ 5 , 1.6 x 10~ 3 and 3.5 x 10~ 3 , respectively. We also check the results by plotting (i) log P(t) versus t and (ii) log(| logP(t)\) versus logt a and find in both cases that the data are clearly more curved than when we plot log P(t) versus logt, indicating that power law provides the best description of the data b. We perform a similar analysis for the sleep state and find, in contrast to the result for the wake state, that the data in large time region (t > 5 min) exhibit exponential behavior P(t) ~ e-*/ r .

(9)

"For the stretched exponential y = aexp(—bxc), where a, b and c are constants, the log(| log2/|) versus logx plot is not a straight line unless a = 1. Since we don't know what the corresponding value of a is in our data, we can not rescale y so that a — I. The solution is to shift x for a certain value to make y = 1 when x — 0, in which case a = 1. In our data, P(t) = 1 when t = 0.5, so we shift t by —0.5 before plotting log(| logP(t)|) versus logt. b According Eq. 7, if P(t) is a power-law function, so is p(t). We also separately check the functional form of p(t) for the data with same procedure and find that the power law provides the best description of the data.

41

Figure 6. Cumulative probability distribution P(t) of sleep and wake durations of individual and pooled data. Double-logarithmic plot of P(t) of wake durations (a) and semilogarithmic plot of P(t) of sleep durations (b) for pooled data and for data from one typical subject. P(t) for three typical subjects is shown in the insets. Note that due to limited number of sleep-wake periods for each subject, it is difficult to determine the functional form for individual subjects. We perform K-S test and compare the probability density p(t) for all individual data sets and pooled data for both wake and sleep periods. For both sleep and wake, less than 10% of the individual d a t a fall below the 0.05 significant level of disproof of the null hypothesis, that p(t) for each individual subject is very likely drawn from the same distribution. The K-S statistics significantly improves if we use recordings only from the second night. Therefore, pooling all data improves the statistics by preserving the form of p(t).

We calculate the time constants r for the 20 subjects, and find an average r = 20 min with a = 5. Using the Kolmogorov-Smirnov test, we find that we cannot reject the null hypothesis that p(t) of the sleep state of each subject of our 39 data sets is drawn from the same distribution (Fig. 6b). We further find that P(t) of the sleep state for the pooled data is consistent with an exponential distribution with a characteristic time r = 22 ± 1 min (Fig. 7b). In order to verify that P(t) of sleep state is better described by an exponential functional form rather than by a stretched exponential functional form, we fit these curves to the P(t) from pooled data. Using Levenberg-Marquardt method, we find that the stretched exponential form lead to worse fit. The X2 error of exponential fit and stretched exponential fit are 8 x 10~5 and 2.7 x 10~ 2 , respectively. We also check the results by plotting log(| logP(t)\) versus logt (1) and find that the data are clearly more curved than when we plot logP(i) versus logt, indicating that an exponential form provides the best description of the data. Sleep is not a "homogeneous process" throughout the course of the night

42

Figure 7. Cumulative distribution of durations P(t) of sleep and wake states from data, (a) Double-logarithmic plot of P(t) from the pooled data. For the wake state, the distribution closely follows a straight line with a slope a — 1.3 ± 0.1, indicating power-law behavior of the form, cf. Eq. (8). (b) Semi-logarithmic plot of P(t). For the sleep state, the distribution follows a straight line with a slope 1 / T where T = 22 ± 1, indicating an exponential behavior of the form, cf. Eq. (9). It has been reported that the individual sleep stages have exponential distributions of durations 53 > 54 > 55 . Hence we expect an exponential distribution of durations for the sleep state. 42 43

' , so we ask if there is any change of a and r during the night. We study sleep and wake durations for the first two hours, middle two hours, and the last two hours of nocturnal sleep using the pooled data from all 39 records (Fig. 8). Our results suggest that a does not change for these three portions of the night, while r decreases from 27 ± 1 min in the first two hours to 22 ± 1 min in the middle two hours, and then to 18 ± 1 min in the last two hours. The decrease in r implies that the number of wake periods increases as the night proceeds, and we indeed find that the average number of wake periods for the last two hours is 1.4 times larger than for the first two hours.

2.3

Model

We next investigate mechanisms that may be able to generate the different behavior observed for sleep and wake. Although several quantitative models, such as the two-process model 44 and the thermoregulatory model 45 , have been developed to describe human sleep regulation, detailed modeling of frequent short awakening during nocturnal sleep has not been addressed 46 . To model the sleep-wake transitions, we make three assumptions (Fig. 9) 4 r :

43 10

o

a=1.3 O

-

a. Wak e 3

10"

,

1

A».

°

, 10

time (min)

.X^ 100

o

• Middle 2 hr A Last 2 hr

£\o ,

o

>,

o First 2 hr

Cumulativ e probabilit

\g,

o

Cumulativ e probabili

>,

50

time (min)

Figure 8. P(t) of sleep and wake states in the first two hours, middle two hours and last two hours of sleep, (a) P(t) of wake states; the power-law exponent a does not change in a measurable way. (b) P(t) of sleep states; the characteristic time r decreases in the course of the night.

Assumption 1 defines the key variable x(t) for sleep-wake dynamics. Although we consider a two-state system, the brain as a neural system is unlikely to have only two discrete states. Hence, we assume that both wake and sleep "macro" states comprise large number of "microstates" which we map onto a continuous variable x(t) defined in such a way that positive values correspond to the wake state while negative values correspond to the sleep state. We further assume that there is a finite region — A < x < 0 for the sleep state. Assumption 2 concerns the dynamics of the variable x(t). Recent studies 37 ' 39 suggest that a small population of sleep-active neurons in a localized region of the brain distributes inhibitory inputs to wake-promoting neuronal populations, which in turn interact through a feedback on the sleep-active neurons. Because of these complex interactions, the global state of the system may present a "noisy" behavior. Accordingly, we assume that x(t) evolves by a random-walk type of dynamics due to the competition between the sleepactive and wake-promoting neurons. Assumption 3 concerns a bias towards sleep. We assume that if x(t) moves into the wake state, then there will be a "restoring force" pulling it towards the sleep state. This assumption corresponds to the common experience that in wake periods during nocturnal sleep, one usually has a strong tendency to quickly fall asleep again. Moreover, the longer one stays awake, the more difficult it may be to fall back asleep, so we assume that the restoring force becomes weaker as one moves away from the transition point x = 0. We

44

model these observations by assuming that the random walker moves in a logarithmic potential V(x) = b\uxJ yielding a force f(x) = -dV(x)/dx = —b/x, where the bias b quantifies the strength of the force. Assumptions 1-3 can be written compactly as: c /,x ,. -.N /.x f e W> if-A51, Hence, for large times, the cumulative distribution of return times is also a power law, Eq. (8), and the exponent is predicted to be a=\+b.

(11)

From Eq. (11) it follows that the cumulative distribution of return times for a random walk without bias (b = 0) decreases as a power law with an exponent a = 1/2. Note that introducing a restoring force of the form f(x) = —b/x1 with 7 ^ 1 , yields stretched exponential distributions 51 , so 7 = 1 is the only case yielding a power-law distribution. Similarly, the distribution of durations of the sleep state is identical to the distribution of return times of a random walk in a space with a reflecting boundary. Hence P(t) has an exponential distribution, Eq. (9), in the large time region, with the characteristic time r predicted to be r ~ A2 .

(12)

Equations (11) and (12) indicate that the values of a and r in the data can be reproduced in our model by "tuning" the threshold A and the bias b (Fig. 10). The decrease of the characteristic duration of the sleep state as the night proceeds is consistent with the possibility that A decreases (Fig. 9. Our calculations suggest that A decreases from 7.9 ± 0.2 in the first hours of sleep, to 6.6 ± 0.2 in the middle hours, and then to 5.5 ± 0.2 for the final hours of sleep. Accordingly, the number of wake periods of the model increases by a factor of 1.3 from the first two hours to the last two hours, consistent with the data. However, the apparent consistency of the power-law exponent for the wake state suggests that the bias b may remain approximately constant during the night. Our best estimate is b = 0.8 ± 0.1.

45

C. Wake Sleep Wake

Data

1 II 11 1II 1

Moclei

11

Sleep I

100

.

If

1 I

.

200

I

.

300

400

Time (min)

Figure 9. Schematic representation of the dynamics of the model. T h e model can be viewed as a random walk in a potential well illustrated in (a), where he bottom flat region between —A < x < 0 corresponds to the area without field, and the region x > 0 corresponds to the area with logrithmic potential, (b) The state x(t) of the sleep-wake system evolves as a random walk with the convention that x > 0 corresponds to wake state and —A < x < 0 corresponds to the sleep state, where A gradually changes with time to account for the decrease of the characteristic duration of the sleep state with progression of the night. In the wake state there is a "restoring force," f(x) = —b/x, "pulling" the system towards the sleep state. The lower panel in (b) illustrates sleep-wake transitions from the model, (c) Comparison of typical data and of a typical output of the model. The visual similarity between the two records is confirmed by quantitative analysis (Fig. 10).

To further test the validity of our assumptions, we examine the correlation between the durations of consecutive states. Consider the sequence of sleep and wake durations { 5i W\ Si Wi....Sn Wn }, where Sn indicates the duration of n-th sleep period and Wn indicates the duration of n-th wake period (Fig. 9b). Our model predicts that there are no autocorrelations in the series Sn and Wn, as well as no cross-correlations between series Sn and Wn, the reason being that the uncorrelated random walk carries no information about previous steps. The experimental data confirms these predictions,

46

Duration (min)

Duration (min)

Figure 10. Comparison of P(t) for data and model (two runs with same parameters), (a) P(t) of the wake state, (b) P(t) of the sleep state. Note that the choice of A depends on the choice of the time unit of the step in the model. We choose the time unit to be 30 seconds, which corresponds to the time resolution of the data. To avoid big jumps in x(t) due to the singularity of the force when x(t) approaches x = 0, we introduce a small constant A in the definition of the restoring force f(x) — —b/(x + A). We find that the value of A does not change a or T.

within statistical uncertainties. 2.4

Conclusions

Our findings of a power-law distribution for wake periods and an exponential distribution for sleep periods are intriguing because the same sleep-control mechanisms give rise to two completely different types of dynamics—one without a characteristic scale and the other with. Our model suggests that the difference in the dynamics of the sleep and wake states (e.g. power law versus exponential) arises from the distinct number of microstates that can be explored by the sleep-wake system for these two states. During the sleep state, the system is confined in the region —A < x < 0. The parameter A imposes a scale which causes an exponential distribution of durations. In contrast, for the wake state the system can explore the entire half-plane x > 0. The lack of constraints leads to a scale-free power-law distribution of durations. In addition, the l/x restoring force in the wake state does not change the functional form of the distribution, but its magnitude determines the power-law exponent of the distribution (see Eq. (11)). Although in our model the sleep-wake system can explore the entire halfplane x > 0 during wake periods, the "real" biological system is unlikely to generate very large value (i.e., extreme long wake durations). There must be

47

a constraint or boundary in the wake state at a certain value of a:. If such a constraint or boundary exists, we will find a cut-off with exponential tail in the distribution of durations of the wake state. More data are needed to test this hypothesis. Our additional finding of a stable power-law behavior for wake periods for all portions of the night implies that the mechanism generating the restoring force in the wake state is not affected in a measurable way by the mechanism controlling the changes in the durations of the sleep state. We hypothesize that even though the power-law behavior does not change in the course of the night for healthy individuals, it may change under pharmacological influences or under different conditions, such as stress or depression. Thus, our results may also be useful for testing these effects on the statistical properties of the wake state and the sleep state. 3

Summary

We show that a stochastic approach based on general phenomenological considerations can successfully account for a variety of scaling and statistical features in complex physiological processes where interaction between many elements is typical. We propose a "common framework" to describe diverse physiological mechanims such as heart rate control and sleep-wake regulation. In particular, in the context of cardiac dynamics we find that the generic process of a random walk biased by attracting fields, which are often functions of time, can generate the long-range power-law correlations, and the form and stability of the probability distribution observed in heartbeat data. A process based on the same concept, in the context of sleep-wake dynamics, generates complex behavior which accounts both for the scale-free power-law distribution of the wake periods, and for the scale-dependent exponential distribution of the sleep periods. Further studies are needed to establish the extent to which such approaches can be used to elucidate mechanisms of physiologic control. Acknowledgments We are greatful to many individuals, including L.A.N. Amaral, A.L. Goldberger, S. Havlin, T. Penzel, J.-H. Peter, H.E. Stanley for major contributions to the results reviewed here, which represent a collaborative research effort. We also thank A. Arneodo, Y. Ashkenazy, A. Bunde, I. Grosse, H. Herzel, J.W. Kantelhardt, J. Kurths, C.-K. Peng, M.G. Rosenblum, and B.J. West for valuable discussions. This work was supported by NIH/National Center

48

for Research Resources (P41 RR13622), NSF, NASA, and The G. Harold and Leila Y. Mathers Charitable Foundation. References 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18.

19.

20.

C. Bernard, Les Phenomenes de la Vie (Paris, 1878) B. van der Pol and J. van der Mark, Phil. Mag 6, 763 (1928). W. B. Cannon, Physiol. Rev. 9, 399 (1929). B. W. Hyndman, Kybernetik 15, 227 (1974). S. Akselrod et al, Science 213, 220 (1981). M. Kobayashi and T. Musha, IEEE Trans, of BME 29, 456 (1982). M. F. Shlesinger, Ann. NY Acad. Sci. 504, 214 (1987); M. F. Shlesinger and B. J. West, in Random Fluctuations and Pattern Growth: Experiments and Models (Kluwer Academic Publishers, Boston, 1988). M. Malik and A. J. Camm, Eds. Heart Rate Variability (Futura, Armonk NY, 1995). J. Kurths et al, Chaos 5, 88 (1995). G. Sugihara et al., Proc. Natl. Acad. Sci. USA 93, 2608 (1996). R. deBoer et al, Am. J. Physiol. 253, H680 (1987). M. Mackey and L. Glass, Science 197, 287 (1977); L. Glass and M. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton Univ. Press, Princeton, 1981); L. Glass et al, Math. Biosci. 90, 111 (1988); L. Glass and C. P. Malta, J. Theor. Biol. 145, 217 (1990); L. Glass, P. Hunter, A. McCulloch, Eds. Theory of Heart (Springer Verlag, New York, 1991). M. G. Rosenblum and J. Kurths, Physica A 215, 439 (1995). H. Seidel and H. Herzel, in Modelling the Dynamics of Biological Systems E. Mosekilde and O. G. Mouritsen, Eds. (Springer-Verlag, Berlin, 1995). J. P. Zbilut et al, Biological Cybernetics 75, 277 (1996). J. K. Kanters et al, J. Cardiovasc. Electrophysiology 5, 591 (1994). G. LePape et al, J. Theor. Biol. 184, 123 (1997). E. W. Montroll and M. F. Shlesinger, in Nonequilibrium Phenomena II: From Stochastics to Hydrodynamics, L. J. Lebowitz and E. W. Montroll Eds. (North-Holland, Amsterdam, 1984), pp. 1-121. N. Wax, Ed. Selected Papers on Noise and Stochastic Processes (Dover Publications Inc., New-York, 1954); G. H. Weiss, Aspects and Applications of the Random Walk (Elsevier Science B.V., North-Holland, NewYork, 1994) R. M. Berne and M. N. Levy, Cardiovascular Physiology 6th ed. (C.V. Mosby Company, St. Louis, 1996).

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21. M. N. Levy, Circ. Res. 29, 437 (1971). 22. G. Jokkel et al, J. Auton. Nerv. Syst. 5 1 , 85 (1995). 23. MIT-BIH Polysomnographic Database CD-ROM, second edition (MITBIH Database Distribution, Cambridge, 1992) 24. C. -K. Peng et al, Phys. Rev. Lett. 70, 1343 (1993); J. M. Hausdorff and C. -K. Peng, Phys. Rev. E 54, 2154 (1996). 25. A. Grossmann and J. Morlet, Mathematics and Physics: Lectures on Recent Results (World Scientific, Singapore, 1985); I. Daubechies, Comm. Pure and Appl. Math. 4 1 , 909 (1988). 26. J. F. Muzy et al., Int. J. Bifurc. Chaos 4, 245 (1994); A. Arneodo et al., Physica D 96, 291 (1996). 27. P. Ch. Ivanov et al., Nature 383, 323 (1996). 28. P. Ch. Ivanov et al, Physica A 249, 587 (1998). 29. R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1981). 30. J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford Univ. Press, New York, 1994). 31. P. Ch. Ivanov et al., Europhys. Lett. 43, 363 (1998). 32. Y. Ashkenazy et al, Phys. Rev. Lett 86, 1900 (2001). 33. P. Ch. Ivanov et al, Nature 399, 461 (1999). 34. P. Bernaola-Galvan et al., Phys. Rev. Lett 87, 168105(4) (2001). 35. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, London, 1971). 36. M. Chicurel, Nature 407, 554 (2000). 37. D. Mcginty and R. Szymusiak, Nature Med. 6, 510 (2000). 38. J. H. Benington, Sleep 23, 959 (2000). 39. T. Gallopin et al, Nature 404, 992 (2000). 40. P. Ch. Ivanov et al, Europhys. Lett. 48, 594 (1999). 41. A. Bunde et al, Phys. Rev. Lett. 85, 3736 (2000). 42. J. Born et al, Nature 397, 29 (1999). 43. M. A. Carskadon and W. C. Dement, Principles and Practice of Sleep Medicine (WB Saunders Co, Philadelphia) 2000, pp. 15-25. 44. A. A. Borbely and P. Achermann, J. Bio. Rhythm. 14, 557 (1999). 45. M. Nakao et al, J. Biol. Rhythm. 14, 547 (1999). 46. D. -J. Dijk and R. E. Kronauer, J. Bio. Rhythm. 14, 569 (1999). 47. C.-C. Lo et al, pre-print: cond-mat/0112280; Europhys. Lett. , (2002) in press. 48. S. Zapperi et al, Phys. Rev. B 58, 6353 (1998). 49. S. Havlin et al, J. Phys. A 18, 1043 (1985). 50. D. Ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and

50

Disordered Systems (Cambridge Univ. Press, Cambridge) 2000. 51. J. A. Bray, Phys. Rev. E 62, 103 (2000). 52. A. Rechtschaffen and A. Kales, A Manual of Standardized Terminology, Techniques, and Scoring System for Sleep Stages of Human Subjects (Calif: BIS/BRI, Univ. of California, Los Angeles) 1968. 53. R. Williams et al, Electroen. Clin. Neuro. 17, 376 (1964). 54. V. Brezinova, Electroen. Clin. Neuro. 39, 273 (1975). 55. B. Kemp and H. A. C. Kamphuisen, J. Biol. Rhythm. 9, 405 (1986).

51

CHAOTIC PARAMETERS IN TIME SERIES OF ECG, RESPIRATORY MOVEMENTS AND ARTERIAL PRESSURE E. CONTE, A. F E D E R I C I 0 Department

of Pharmacology

and Human Physiology-University

70100 Bari-Italy. '^Center ofInnovative

ofBari, P.zza G. Cesare,

Technologies for Signal Detection and

Processing,

Bari, Italy. E-mail: fisio2@fisiol. uniba. it Correlation Dimension, Lyapunov exponents, Kolmogorov entropy were calculated by analysis of ECG, respiratory movements and arterial pressure of normal subjects in spontaneous and forced conditions of respiration. We considered the cardiovascular system arranged by a model of five oscillators having variable coupling strengths, and we found that such system exhibits chaotic activity as well as its components. In particular, we obtained that respiration resolves itself in a non linear input into heart dynamics, thus explaining that it is a source of chaotic non linearity in heart rate variability.

1. Introduction A recent, relevant paradigm is that, due to the complexity of biological matter, chaos theory should represent a reasonable formulation of living systems. Chaotic behaviour should be dominant and non chaotic states should correspond more to pathological than normal states. Fundamental results and theoretical reasons sustain the relevant role of chaos theory in explaining the mechanisms of living matter. This is so because many physiological systems may be represented by the action of coupled biological oscillators. It has been evidenced 4 that, under suitable conditions, such stimulated and coupled oscillators generate chaotic activity. We retain that, in different physiological conditions, a stronger or lower coupling among such oscillators takes place, determining a modification in the control parameters of the system, with enhancement or reduction of the chaotic behaviour of an oscillator respect to the other mutually coupled. Such dynamical modification will be resolved and observed by a corresponding modification of the values of the chaotic parameters (i.e. Lyapunov exponents) usually employed in the analysis of experimental time series. Recent studies' of the cardiovascular system emphasize the oscillatory nature of the processes happening within this system. Circulatory system is represented by the heart, and the systemic and pulmonary vessels. To regulate vessels resistance, myogenic activity operates to contract the vessels in response to a variation of intra-

52

vascular pressure. This generates a rhythmic activity related to periodicity in signals of blood pressure5 and of blood flow7'9. Neural system also realizes the activity of the autonomic nervous system and it is over-imposed to the rhythmic activity of pacemaker cells. Rhythmic regulation of vessels resistance is also realized by the activity of metabolic substances in the blood. In conclusion, the dynamics of blood flow in its passage through the cardiovascular system, is governed by five oscillators, the heart, the lungs, myogenic-, neural-, metabolic-activities. We may consider this system to be a spatially distributed physical system constituted by five oscillators. Each oscillator exhibits autonomous oscillations but positive and negative feedback loops take place so that the continuous regulation of blood circulation is realized through the coherent activity of such mutually coupled oscillators. This is the model that we employ in the present study. We have all the elements to expect such system to be a non linear and complex system. So, we arrive to the central scope of the present work. We intend to ascertain the following points: using the methods of non linear analysis, we intend to establish if cardiovascular system exhibits chaotic activity as well as its components; we aim to ascertain also if the model of five oscillators is or not supported by our analysis, and, in particular, if we may arrive to the final conclusion that respiration resolves itself as a non linear input from respiration into heart dynamics of the cardiovascular oscillator. It is well known the importance to give a definitive and rigorous answer to this last problem. Let us specify in more detail the nature of our objective. Analyzing data regarding ECG time series, several authors2'6 obtained results indicating that the normal sinus rhythm in ECG must be ascribed to actual low-dimensional chaos. By the same kind of analysis, it was also obtained evidence for inherent non linear dynamics and chaotic determinism in time series of consecutive R-R intervals. The physiological origins for such chaotic non linearity are unknown. The purpose of our study was to establish whether a non linear input from spontaneous respiration to heart exists and if it may be considered one of the sources for the chaotic non linearity in heart rate variability. 2. Methods We measured signals of ECG, respiratory movements, arterial pressure in six normal nonsmoking subjects in normal (NR) and forced (FR) conditions of respiration, respectively. The condition FR was obtained by asking the subjects to perform inspiratory acts with a 5 s periodicity, at a given signal. The signal for expiration was given 2 s after every inspiration. The measured ECG signals were

53

sampled at 500 Hz for 300 s. Signals vs time tracings for respiration, ECG, Doppler, and R-R intervals are given in Fig.l for the subject #13-07. Peak to peak values were considered for time series. Programs for noise reduction were utilized in order to use noise reduced time series data only. In order to follow variability in time of the collected data, the obtained time series were re-sampled in five intervals (subseries), each interval containing 30.000 points. All the data were currently analyzed by the methods of non linear prediction and of surrogate data. Correlation dimension, Lyapunov spectrum, Kolmogorov entropy were estimated after determination of time delay T by auto-correlation and mutual information. Embedding dimension in phase space was established by the method of False Nearest Neighbors (FNN) (for chaotic analysis, see, as example, ref. 3, 8). 3. The Results The main results of chaotic analysis are reported in Table 1. For cardiac oscillations time delays resulted ranging from 14 to 60 msec both in the two cases of subjects in NR and FR. Embedding dimension in phase space resulted to be d = 4, thus establishing that we need four degrees of freedom to correctly describe heart dynamics. Correlation dimension, D2, established with saturation in a D 2 -d plot, resulted to be a very stable value during the selected intervals of experimentation. It assumed values ranging from 3,609 ± 0,257 to 3,714 + 0,246 in the case of five intervals for normal subjects in NR, and D2-values ranging from 3,735 ± 0,228 to 3,761 ± 0,232 in the case of subjects in FR. On the basis of such results, we concluded that normal cardiac oscillations as well as cardiac oscillations of subjects under FR follow deterministic dynamics of chaotic nature. Soon after we estimated Lyapunov exponents: ^-i and X2 resulted to be positive; X3 and X4 assumed negative value; the sum of all the calculated exponents resulted to be negative as required for dissipative systems. We concluded that cardiac oscillations of normal subjects under NR and FR, represent an hyper-chaotic dynamics. X] and X2 positive exponents in Table 1 represent the rates of divergence of the attractor in the directions of maximum expansion. These are the direction in which the cardiac oscillating system realizes chaoticity. ^,3 and ^.4 negative values in Table 1 represent the rates of convergence of the attractor in the contracting directions. The emerging picture is that cardiac oscillations, as measured by ECG in normal subjects, are representative of a large ability of heart to continuously cope with rapid changes corresponding to the high values of its chaoticity. Looking in Table 1 at the calculated values of X{ and X2 along the five different time intervals that we have analysed, we deduce that such values remained substantially stable in different intervals. Thus, we may conclude that, due to the constant action of the oscillators defined in our model, in

54

the NR and FR conditions, heart chaotic dynamics remains substantially stable in time. The same tendency was confirmed examining the results obtained for Kolmogorov entropy, K, (see Table 1) and characterizing the overall chaoticity of the system. Thus, we may arrive to establish the first conclusion of the present paper: heart dynamics exhibits chaoticity, and this remains substantially stable in time for normal subjects in NR and FR. However, we have to respond here also to the question to ascertain if respiration resolves itself in a non linear input given from the respiratory system into the heart dynamics of cardiac oscillator. To this regard we must remember that, as previously explained in the introduction, the estimation of Lyapunov exponents, and, in particular, of positive Lyapunov exponents must be considered, in chaotic analysis of physiological systems, a sign to confirm the presence of a physiological mechanism of control acting through a modification of the control parameters of the considered system via a stronger or lower coupling between the oscillators assumed to act in the system. According to this thesis, an existing non linear input from respiration into the heart dynamics of the cardiovascular system should be actually realised through a modification of the control parameters of the system via a modification of the coupling strength between two considered oscillators, and it would be evidenced through evident modifications of positive and negative Lyapunov exponents in the two cases of NR and FR. In fact, for A.! values we obtained, in five normal subjects, an increase, in FR with respect to NR, varying from 6% to about 36%. For X2, the same increase was more relevant, varying from about 12% to about 61%. Also the corresponding negative values, A,3 and X4, were increased in FR with respect to NR. Only in one subject a decrease was obtained in the values of Lyapunov exponents after passing from NR to FR. Also in this case, appreciable percent differences were observed. In conclusion, the substantial differences in Lyapunov exponent values in NR with respect to FR, is a result of the present work. Increasing values of positive Lyapunov exponent reveal an increasing degree of chaoticity. Decreasing values reveal, instead, decreasing chaoticity. Increased and (in only one case) decreased values of positive Lyapunov exponents that we found in FR with respect to NR, indicate that in the first condition we had an increasing (in only one case decreasing) chaoticity, and this establishes that respiration acts as a non linear input of the respiration on the cardiovascular oscillator. According to our model, such non linear input from respiration to cardiovascular oscillator, resolves itself in a greater (or lower) coupling strength between such two considered oscillators. Obviously, in order to confirm this conclusion, we need to emphasize that also respiration is characterised by chaotic dynamics. We so performed chaos analysis of respiration movements time series data obtained from the previously considered normal

55

subjects, and the analysis was executed following the same previous methodological criteria. Time delays resulted to vary from 4 to 76 msec, embedding dimension in phase space resulted to be d = 3. As previously said, such dimension reflects the number of degrees of freedom necessary for description of respiratory system. We deduced from d = 3 that it is necessary to consider the action of three possible oscillators determining the behaviour of such system. Mean value of correlation dimension, D2, resulted to be 2,740 ± 0,390 in the case of NR in the first interval of investigation. A rather stable mean value, ranging from D 2 = 2,579 + 0,340 to D2 = 2,665 ± 0,346, was also obtained in the case of the four remaining intervals. We concluded that respiratory system of the examined normal subjects exhibits chaotic determinism. As expected, during FR, we obtained a reduction of the mean values of correlation dimension respect to NR. The mean value resulted to be D2 = 2,414 ± 0,417 in the first interval and varied between D2 = 2,339 ± 0,314 and D2 = 2,389 ± 0,383 in the remaining four intervals whit a decreasing percent about 10-12 % respect to NR. Then, we had reduction of chaotic dynamics of respiration during FR respect to NR physiological condition. A clear discrimination of such two conditions was also obtained by calculation of the dominant Lyapunov exponent. We obtained a mean value of such parameter, XD, XD = 0,028 ± 0,023 in the case of NR and A.D = 0,009 ± 0,004 in the case of FR in the first interval of experimentation whit a percent decreasing in the case of FR about 68%. Evident discrimination was also obtained in the other four intervals, in the second interval it resulted A,D = 0,029 ± 0,020 for NR and XD = 0,012 ± 0,004 for FR (decreasing percent about 59%), in the third interval it resulted ^ D = 0,030 ± 0,022 for NR against XD = 0,008 ± 0,003 for FR (decreasing percent about 73%), in the fourth interval we had ^,D = 0,026 ± 0,022 for NR against XD = 0,009 ± 0,004 for FR (decreasing percent about 65%), and in the fifth interval XD = 0,022 + 0,020 for NR and A.D = 0,011 ± 0,008 for FR (decreasing percent about 50%). In conclusion, we had a great stability of the dominant Lyapunov exponents calculated along the intervals of experimentation and both in the two cases of NR and FR, while instead we determined a decreasing percent of values in the case of FR respect to NR. These results indicated that respiratory systems exhibit chaotic dynamics. Without any doubt, chaoticity resulted strongly reduced during FR respect to NR. This results clearly supports our thesis, based on the model of five oscillators: during forced respiration, we have a reduction of chaoticity of respiratory system respect to spontaneous respiration, to such reduction it corresponds instead an increasing of chaoticity of cardiac oscillations in consequence of a greater non linear input from the respiratory system to heart dynamics. In other terms, a stronger coupling between the two oscillators is realized

56

and it is resolved in an enhancement of cardiac chaoticity correlated to a simultaneous reduction of chaoticity of the respiratory oscillatory system. The final aim was to test for the possible chaotic dynamics of blood pressure. We analyzed arterial pressure time series data following the same previous methodology. Time delay resulted to be about 2 msec. Embedding dimension in phase space resulted to be d = 5. We retain that this is the best our result to confirm the correctness of the model of the cardiovascular system based on five oscillators. Blood pressure signal reflects the action of the five oscillators that we considered, and, in fact, our calculated embedding dimension resulted to be just d = 5. Calculation of correlation dimension, D2, again gave rather stable values along the five intervals of experimentation and resulted to vary between 3,661 and 3,924 in the case of NR, and between 3,433 and 3,910 in the case of FR. Thus, we may conclude that blood pressure signal indicates to have the behaviour of a very chaotic deterministic system, as confirmed also from Kolmogorov entropy values. The calculation of Lyapunov exponents are given in Table 1, and they confirm that blood pressure is a deterministic hyper-chaotic dissipative system. The values of the exponents Xu X2, X4, a n d A.5 resulted to be very stable along the five intervals of experimentation with an evident their similarity also in the two conditions of experimentation. Considering the model of five oscillators, we may say that we have constant non linear inputs acting from the oscillators that determine such constant level of chaoticity. X3 showed instead a very great variability along the five considered intervals as well as in comparison of NR respect to FR. The variability of A.3 happened with three characteristic times that resulted to be respectively about 3-4 seconds, about 10 seconds and finally about 20-30 seconds corresponding to 0,3-0,4 Hz, to 0,1-0,2 Hz and to 0,04-0,06 Hz. We concluded that the first frequency should be due to the action of the respiratory oscillator, and the two other remaining frequencies should correspond to the action of the myogenic and neural (baroreceptors) oscillators.

57 Table 1 CHAOS

ANALYSIS

LYAPUNOV

OF

R-R

OSCILLATIONS.

SPECTRUM

Kolmogorov entropy

h

h

X2

^2

^3

^3

/>4

K

K

K

NR

FR

NR

FR

NR

FR

NR

FR

NR

FR

Five intervals of data m.v.

0,271 0,343 0,089 0,119 -0,135 -0,133 -0,548 -0,639 0,360 0,462

s.d.

0,063 0,075 0,043 0,036

m.v.

0,278 0.349 0,092 0,138 -0,124 -0,119 -0,537 -0,606 0,370 0,488

s.d.

0,086 0,127 0,059 0,071

m.v.

0,262 0,346 0,091 0,110 -0,121 -0,140 -0,534 -0,578 0,353 0,456

s.d.

0,092 0,148 0,060 0,054

m.v.

0,276 0,307 0,089 0,094 -0,114 -0,113 -0,522 -0,565 0,365 0,401

s.d. m. v.

0,096 0,018 0,064 0,029 0,021 0,011 0,118 0,078 0,160 0,046 0,279 0,289 0,090 0,095 -0,121 -0,125 -0,526 -0,572 0,369 0,384

s.d.

0,093 0,074 0,062 0,065

0,022

0,026

0,029

0,023

0,020

0,020

0,032

0,098

0,099

0,090

0,025

0,130

Table 2 CHAOS ANALYSIS OF BLOOD PRESSURE SIGNAL LYAPUNOV

h

x2

SPECTRUM ^3

A,4

^5

Kolmogorov Entropy

Five intervals

NR

NR

NR

NR

NR

NR

of data m.v.

0,557 0,250

0,032

-0,213 -0,705

0,838

s.d.

0,036 0,019

0,023

0,043

0,033

0,033

m.v.

0,561 0,251

0,012

-0,215 -0,687

0,824

s.d.

0,025 0,004

0,006

0,030

0,025

0,015

m.v.

0,577 0,252

0,040

-0,232 -0,695

0,868

s.d.

0,027 0,011

0,013

0,015

0,008

0,051

m. v.

0,553 0,259

0,018

-0,220 -0,704

0,829

s.d. m. v.

0,016 0,006 0,570 0,246

0,009 0,009 0,012 -0,249

s.d.

0,011 0,002

0,006

0,005

0,006 -0.706

0,018 0,827

0,018

0,018

0,118

0,121

0,085

0,105 0,106

0,144 0,196

0,152 0,198

0,135 0,154 0,139

58 Fig.l

la-ii-OI DV WON'I KESP

Li

1 !

i « V-

•v.' \ X X\f V\J \ \ \ \ M V \ r \ N v \ V \ \ W M V- > H*-»

Acknowledgements Authors wish to thank Ms Anna Maria Papagni for her technical assistance. References 1. 2. 3.

Akselrod S, Gordon D, Ubel FA, Shannon DS, Borger AC, Choen RJ. Science 1981;213:220-225 Babloyantz A, Destexhe A. Is the normal heart a periodic oscillator. Biological Cybernetics 1988; 58: 203-211 Badii R, Politi A. Dimensions and Entropies in Chaotic Systems. Springer Berlin, 1986

59 4.

5. 6. 7.

8. 9.

Guevara MR, Glass L, Shrief A. Phase-locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. Science 1981; 214: 1350-1353 Kitney RI, Fulton T, Mc Donald AH, Likens DA. J. Biomed. Eng. 1985; 7: 217-225 Kitney RS, Rompelman O. The study of heart rate variability. Oxford University Press, 1980 Madwed JB, Albrecht P, Mark RG, Cohen RJ. Low-frequency oscillations in arterial pressure and heart rate: a simple computer model. Am. J. Phys. 1989; 256: H1573-H1579 Schreiber T. Interdisciplinary application of non linear time series methods. Physics Report 1999; 308: 1-64 Streen MD. Nature 1975; 254: 56-58

60

COMPUTER ANALYSIS OF ACOUSTIC RESPIRATORY SIGNALS A. VENA, G.M. INSOLERA, R. GIULIANI**', T. FIORE Department of Emergency and Transplantation, Bari University, Policlinico Hospital, Piazza Giulio Cesare, 11, 70124 Bari- Italy. (*)Center of Innovative Technologies for Signal Detection and Processing, Bari, Italy. e-mail: antonvenaCalvahoo.com G. PERCHIAZZI Department of Clinical Physiology, Uppsala University Hospital, S-75185 Uppsala,

Sweden

Evaluation of breath sounds is a basic step of patient physical examination. The auscultation of the respiratory system gives direct information about the structure and function of lung tissue that is not to be achieved with any other simple and non-invasive method. Recently, the application of computer technology and new mathematical techniques has supplied alternative methodologies to respiratory sounds analysis. We present a new computerized approach to analyzing respiratory sounds.

1

Introduction

Acoustic respiratory signals have been the subject of considerable research over the last years, however their origin is still not completely known. It is now generally accepted that, during respiration, turbolent motion of a compressive fluid in larger airways with rugous walls (trachea ad bronchi) generates acoustic energy [5]. This energy is transmitted through the airways and lung parenchima to the chest wall that represent a non-stationary system. [1,4]. Pulmonary diseases induce anatomical and functional alterations in the respiratory system; changes in the quality of lung sounds (loudness, length and frequency) are often directly correlated to pathological changes in the lung. The traditional method of auscultation is based on a stethoscope and human auditory system; however, due to the poor response of the human auditory system to lung sounds (low frequency and low sign-to-noise ratio) and the subjective character of the technique, it is common to find different clinical description of the respiratory sounds. Lung-sounds nomenclature has long been unclear: until last decades were used the names derived from originals given by Laennec [10] and translated into english by Forbes [2]. In 1985, the International Lung Sounds Association (I.L.S.A.) has composed an international standard classification of lung sounds that include fine and coarse crackles, wheezes and rhonchi: each of these terms can be described acoustically [13].

61

The application of computer technology and recent advancements in signal processing have provided new insights into acoustic mechanisms and supplied new measurements of clinical importance from respiratory sounds. Aim of this study is to develop a system for acquisition and elaboration of respiratory acoustic signals: this would provide an effective, non-invasive and objective support for diagnosis and monitoring of respiratory disorders. 2

Respiratory Sounds

Lung sounds in general are classified into three major categories: "normal" (vesicular, bronchial and bronchovesicular breath sounds), "abnormal" and "adventitious" lung sounds. Vesicular breath sounds consist of a quiet and soft inspiratory phase followed by a short, almost silent expiratory phase. They are low pitched and normally heard over most lung fields of a healthy subject. These sounds are not generated by gas flow moving through the alveoli (vesicles) but are the result of attenuation of breath sound produced in the larger bronchi. Bronchial breath sounds are normally heard over the trachea and reflect turbulent airflow in the main-stem bronchi. They are loud, high-pitched and the expiratory phase is generally longer than the inspiratory phase with a tipical pause between the phases. Bronchial sounds heard over the thorax suggest lung consolidation and pulmonary disease. Bronchovesicular breath sounds are normally heard on both sides of the sternum in the first and second intercostal spaces. They should be quieter than the bronchial breath sounds and increased intensity of these sounds is often associated with increased ventilation. Abnormal lung sounds include the decrease/absence of normal lung sounds or their presence in areas where they are normally not heard (bronchial breath sounds in peripheral areas where only vesicular sounds should be heard). This is characteristic of parenchima consolidation (pneumonia) that transmit sound from the lung bronchi much more efficiently than through the air-filled alveoli of the normal lung. The term "adventitious" (adventitious lung sounds) refers to extra or additional sounds that are heard over normal lung sounds and their presence always indicates a pulmonary disease. These sounds are classified into discontinuous (crackles) or continuous (wheezes) adventitious sounds. Crackles are discontinuous, intermittent and nonmusical noises that may be classified as "fine" (high pitched, low amplitude, and short in duration) and "coarse" (low pitched, higher in amplitude, and long in duration). Crackles are generated by fluid in the small airways or by sudden opening of closed airways. Their presence is often associated with inflammation or infection of the small bronchi, bronchioles, and alveoli, with pulmonary fibrosis, with heart failure and many other cardiorespiratory disorders. Wheezes are continuous (since their duration is much longer than that of crackles), lower-pitched and musical breath sounds, which are superimposed on the normal lung sounds. They originate by air moving through small airways narrowed by constriction or

62

swelling of airway or partial airway obstruction. They are often heard (during expiration, or during both inspiration and expiration) in patients with asthma or other obstructive diseases. Other respiratory sounds are: rhonchi (continuous sounds that indicate partial obstruction by thick mucous in the bronchial lumen, oedema, spasm or a local lesion of the bronchial wall); stridor (high-pitched harsh sound heard during inspiration and caused by obstruction of the upper airway); snoring (acoustical signals produced by a constriction in the upper airway, usually during sleep) and pleural rubs (low-pitched sounds that occur when inflamed pleural surfaces rub together during respiration). 3

Review of literature

Many studies focused on the acoustic properties of normal lung sounds in healthy subjects [6,7] and their changes with airflow [8]. In 1996 Pasterkamp et al., by using the Fast Fourier Transform (FFT), analysed and described the lung sound spectra in normal infants, children, and adults [14]. At the end of the 1980's, normal and pathological lung sounds were displaied and studied in the time and frequency domain [15]. Various works investigated the characteristics of crackles due to asthma, chronic obstructive pulmonary diseases (COPD), heart failure, pulmonary fibrosis, and pneumonia [16,12,9]. In 1992 Pasterkamp and Sanchez indicated the significance of tracheal sounds analysis in upper airway obstructions [17]. Malmberg et al. analysed changes in frequency spectra of breath sounds during histamine challange test in adult asthmatics subjects [11]. In the last years, the application of Wavelet Transform has demonstrated the possibility of elaborating properly non-stationary signals (such as crackles); by comparing the ability of Fourier and Wavelet based techniques to resolve both discrete and continuous sounds, many studies concluded that the wavelet-based methods had the potential to effectively process and display both continuous and discrete lung sounds [3]. 4

Signal acquisition and processing methods

Lung sounds transmitted through the respiratory system can be acquired by an equipment able to convert the acoustic energy to electrical signal. Then, the next elaboration phase, by using of specific mathematical transformations, returns a sequence of data that allows to study the features of each signal. In this study, respiratory sounds were picked up over the chest wall of normal and abnormal subjects by an electronic stethoscope (Electromag Stethoscope ES120, Japan).

63

The sensor was placed over the bronchial regions of the anterior chest (second intercostal space on the mid clavicular line), the vesicular regions of the posterior chest (apex and base of lung fields, bilaterally) and the trachea at the lower part of the neck, 1-2 cm to the right of the medline. Sounds were amplified, low-pass filtered and recorded in digital format (Sony Minidisc MZ-37, Japan) using a rate of sampling of 44.1 Khz. and 16 bit quantization. The signal was transferred to a computer (Intel Pentium 500 Mhz. Intel Corp., Santa Clara, CA, USA) and then analyzed by a specific Fourier Transform based spectral analysis software (CoolEdit pro 1.0, Syntrillium Software Corp., Phoenix, USA). Because of the clinical necessity to correlate the acoustic phenomenon to the phases of the human respiration, a method of analysis dedicated to the time/frequency plane was applied: the STFT (Short Time Fourier Transform). It provided "spectrograms" related to different respiratory acoustic patterns that, according to the intensity and frequency changes in the time domain, were analyzed offline. The spectrogram shows, in a three-dimensional coordinate system, the acoustic energy of a signal versus time and frequency. We studied normal breath sounds (vesicular and tracheal) from healthy subjects without pulmonary diseases and adventitious lung sounds (crackles and wheezes) from patients with pneumonia and COPD (chronic obstructive pulmonary disease), spontaneously breathing. The signals were examinated for artifacts (generally emanating from defective contact between sensor and chest wall or background noise) and contaminated segments were excluded from further analysis. 5

Results

Normal breath sounds (vesicular and tracheal) showed typical spectra with a frequency content extending up to 700 Hz {vesicular sounds) and 1600 Hz {tracheal sound). Generally, at a frequency below 75-100 Hz there are artefacts from the heart and the muscle sounds. Inspiratory amplitude was higher than expiratory amplitude for vesicular sounds and lower than expiratory amplitude for tracheal sounds (fig. 1, fig. 2).

Fig. 1 Vesicular sound

Fig. 2 Tracheal sound

64

Discontinuous adventitious sounds (crackles) appeared as non-stationary explosive end-inspiratory noise with a frequency content extending beyond 1000 Hz; their duration was less than 200 msec (fig. 3). Continuous adventitious sounds (wheezes) appeared as expiratory spectral densities, harmonically related, at 300 Hz, 600 Hz and 1200 Hz; their duration was longer than 200 msec. (fig. 4).

Fig. 3 Crackles

6

Fig. 4 Wheezes

Conclusions

In this study significant changes in averaged frequency spectra of breath sounds were demonstrated passing from healthy to sick lungs. Moreover, this processing method was able to classify abnormal patterns into different pathology-related subgroups. Implementation of this technology on a breath-to-breath basis, will provide an useful tool for continuous bed-side monitoring by a computerized auscultation device which can record, process and display the respiratory sound signals with sophisticated visualization techniques. Future perspectives for respiratory sound reasearch include the building of miniaturized systems for non-invasive and real-time monitoring; the application of multimicrophone analysis to evaluate regional distribution of ventilation; respiratory sounds databases; remote diagnosis systems; automatic recognition systems of acoustic respiratory patterns by artificial neural networks.

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7

References 1. Cohen A. Signal Processing Methods for Upper Airways and Pulmonary Dysfunction Diagnosis. IEEE Engineering In Medicine And Biology Magazine, (1990). 2. Forbes J. A Treatise of the Diseases of the Chest, 1st ed. Underwood. London, (1821). 3. Forren J.F., Gibian G. Analysis of Lung Sounds Using Wavelet Decomposition, (1999). 4. Fredberg J.J. Acoustic determination of respiratory system properties. Ann. Biomed. Eng. 9 (1981) pp. 463-473. 5. Gavriely N. Breath Sounds Methodology. Boca Raton, FL: CRC Press, Inc., (1995). 6. Gavriely N., Nissan M., Rubin A.E., Cugell D.W. Spectral characteristics of chest wall breath sounds in normal subjects. Thorax 50 (1995) pp. 12921300. 7. Gavriely N., Herzberg M. Parametric representation of normal breath sounds. J. Appl. Physiol. 73(5) (1992) pp. 1776-1784. 8. Gavriely N., Cugell D.W. Airflow effects on amplitude and spectral content of normal breath sounds. J. Appl. Physiol. 80(1) (1996) pp. 5-13. 9. Kaisla T., Sovijarvi A., Piirla P., Rajala H.M., Haltsonen S., Rosqvist T. Validated Methods for Automatic Detection of Lung Sound Crackles. Medical and Biological Engineering and Computing 29 (1991) pp. 517-521. 10. Laennec R.T.H. De I'auscultation mediate ou traite du diagnostic de maladies des poumons et du coeur, fonde principalement sur ce nouveau moyen d'exploration. Brosson et Chaude. Paris, (1819). 11. Malmberg L.P., Sovijarvi A.R.A., Paajanen E., Piirila P., Haahtela T., Katila T. Changes in Frequency Spectra of Breath Sounds During Histamine Challange Test in Adult Asthmatics and Healthy Control Subjects. Chest 105 (1994) pp. 122-132. 12. Munakata M., Ukita H., Doi I., Ohtsuka Y., Masaki Y., Homma Y., Kawakami Y. Spectral and Waveform Characteristics of Fine and Coarse Crackles. Thorax 46 (1991) pp. 651-657. 13. Pasterkamp H., Kraman S.S., Wodicka G.R. Respiratory Sounds. Advances Beyond the Stethoscope. Am J Respir Crit Care Med 156 (1997) pp. 974987. 14. Pasterkamp H., Powell R.E., Sanchez I. Lung Sound Spectra at Standardized Air Flow in Normal Infants, Children, and Adults. Am J Respir Crit Care Med 154 (1996) pp. 424-430. 15. Pasterkamp H., Carson C , Daien D., Oh Y. Digital Respirasonography. New Images of Lung Sounds. Chests (1989) pp. 1505-1512.

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16. Piirila P., Sovijarvi A., Kaisla T., Rajala H.M., Katila T. Crackles in Patients with Fibrosing Alveolitis, Bronchiectasis, COPD, and Heart Failure. Chest 99(5) (1991) pp. 1076-1083. 17. Pasterkamp H., Sanchez I. Tracheal Sounds in Upper Airway Obstruction. Chest 102 (1992) pp. 963-965.

67

T H E I M M U N E SYSTEM: B CELL B I N D I N G TO MULTIVALENT A N T I G E N

Gyan Bhanot IBM Research, Yorktown Hts., NY 10598, E-mail: [email protected]

USA

This is a description of work done in collaboration with Yoram Louzoun and Martin Weigert at Princeton University. Experiments in the late 80's by Dintzis etal revealed puzzling aspects of the activation of B-Cells as a function of the valence (number of binding sites) and concentration of presented antigen. Through computer modeling, we are able to explain these puzzles if we make an additional (novel) hypothesis about the rate of endocytosis of B-Cell receptors. The first puzzling result we can explain is why there is no activation for low valence (less than 10-20). The second is why the activation is limited to a small narrow range of antigen concentration. We performed a computer experiment to model the B-Cell surface with embedded receptors diffusing in the surface lipid layer. We presented these surface receptors with antigen with varying concentration and valence. Using experimentally reasonable values for the binding and unbinding probabilities for the binding sites on the antigens, we simulated the dynamics of the binding process. Using the single hypothesis that the rate of endocytosis of bound receptors is significantly higher than that of unbound receptors, and that this rate varies inversely as the square of the mass of the bound, connected receptor complex, we are able to reproduce all the qualitative features of the Dintzis experiment and resolve both the puzzles mentioned above. We were also able to generate some testable predictions on how chimeric B-Cells might be non-immunogenic.

1

Introduction

This paper is a description of work done in collaboration with Yoram Louzoun and Martin Weigert at Princeton University*. I begin with a brief introduction to the human immune system and the role of B and T Cells in it 2 . Next, I describe the B-Cells receptor/antibody and how errors in the coding for the light chains on these receptors can result in chimerical B-Cells with different light chains on the same receptor or different types of receptors on the same cell. After this, I describe the Dintzis experiments 3 ' 4 , 5 and the efforts to explain these experimental results using the concept of an Immunon 6 - 7 . There is also analytic work by Perelson 8 using rate equations to model the binding and activation process. This is followed by a description of our computer modeling experiment, its results and conclusions 1 .

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2

Brief Description of H u m a n Immune System

The human immune system 2 , on encountering pathogen, has two distinct but related responses. There is an immediate response, called the Innate Response and there is also a slower, dynamic response, called the Adaptive Response. The Innate Response, created over aeons by the slow evolutionary process, is the first line of defense against bacterial infections, chemicals and parasites. It comes into effect immediately and acts mostly by phagocytosis (engulfment). The Adaptive Response is evolving even within an individual, is slower in its action (with a latency of 4-7 days) but is much more versatile. This Adative Response is created by a complex process involving cells called lymphocytes. A single microliter of fluid in the body contains about 2500 lymphocytes. All cellular components of the Immune System arise in the bone marrow from hematopoietic stem-cells, which differentiate to produce the other more specialized cells of the immune system. Lymphocytes derive from a lymphoid progenitor cell and differentiate into two cell types called the B-Cell and the TCell. These are distinguished by their site of differentiation, the B-Cells in the bone marrow and the T-Cell in the thymus. B and T Cells both have receptors on their surface that can bind to antigen (pieces of chemical, peptides, etc.) An important difference between B and T Cell receptors is that B-Cell receptors are bivalent (have two binding areas) while T-Cell receptors are monovalent (with a single binding area). In the bone marrow, B-Cells are presented with self antigen, eg. pieces of the body's own molecules. Those B-Cells that react to such self antigen are killed. Those that do not are released into the blood and lymphatic systems.T-Cells on the other hand are presented with self antigen in the thymus and are likewise killed if they react to it. Cells of the body present on their surface pieces of protein from inside the cell in special structures called the MHC (Major Histocompatibility Complex) molecules. MHC molecules are distinct between individuals and each individual carries several different alleles of MHC molecules. T-Cells are selected in the thymus to bind to some MHC of self but not to any self peptides that are presented on these MHC molecules. Thus, only T-Cells that might bind to foreign peptides presented on self MHC molecules are released from the thymus. There are two types of T-Cells, distinguished by their surface proteins. They are called CD8 T-Cells (also called killer T-Cells) and CD4 T-Cells (also called helper T-Cells). When a virus infects a cell, it uses the cell's DNA/RNA machinery to replicate itself. However, while this is going on, the cell will present on its surface pieces of viral protein on MHC molecules. CD8 T-Cells in the surrounding medium are programmed to bind strongly to such MHC molecules presenting

69 non-self peptides. After they bind to the MHC molecule, they send a signal to the cell to commit suicide (apoptose) and then unbind from the infected cell. Also, once activated in this way, the CD8 T-Cell will replicate aggressively and seek out other infected cells to send them the suicide signal. The CD4 T-Cells on the other hand, recognize viral peptides on B-cells and macrophages (specialized cells which phagocytose or engulf pathogens, digest them and present their peptide pieces on MHC molecules). The role of the CD4 T-Cell, when it binds in this way, is to signal the B-Cell and macrophages to activate and proliferate. B-Cell that are non-reactive to self antigens in the bone marrow are released into the blood and secondary lymphoid tissue. They have a life time of about three days unless they successfully enter lymphoid follicles, germinal centers or the spleen and get activated by binding to antigen presented to them there. Those that have the correct antibody receptors to bind strongly to viral peptide (antigen), will become activated and will start to divide, thereby producing multiple copies of themselves with their specific high affinity receptors. This process is called 'clonal selection' as the clone which is fittest (binds most strongly to presented antigen) is selected to multiply. The B-Cells that bind to antigen will also endocytose their own receptors with bound antigen and present it on their surface on MHC-II molecules for an activation signal from CD4 T-Cells. Once a clone is selected, the B-Cells also mutate and proliferate to produce variations of receptors to achieve an even better binding specificity to the presented antigen. B-Cells whose mutation results in improved binding will receive a stronger activation signal from the CD4 T-Cells and will out-compete the rest. This process is called 'affinity maturation'. Once the optimum binding specificity B-cells are produced, they are released from the germinal centers. Some of these differentiate into plasma cells which release large numbers of antibodies (receptors) with high binding affinity for the antigen. These antibodies mark the virus for elimination by macrophages. Some B-Cells go into a latent phase (become memory B-Cells) from which they may be activated if the infection recurs. It is clear from the above discussion that there are two competing pressures in play when antigen binds to B-Cells. One pressure is to maximize the number of surface bound receptors, until a critical threshold is reached when the BCell is activated and will proliferate. The other pressure is to endocytosis the receptor-antigen complex followed by presentation of the antigen peptide on MHC-II molecules, binding to CD4 T-Cells and an activation signal from that binding. To function optimally, the immune system must carefully balance these two processes of binding and endocytosis.

70

Unbound receptors on the surface of B-Cells are endocytosed at the rate of about one receptor every half hour. However, the binding and activation of B-Cells happens in a time scale of a few seconds to a minute (for references to many of the details of the numerical values used in this paper, refer to the references in 1 ) . If endocytosis is to compete with activation, as it must for the process described above to work, then bound receptors must be endocytosed much more frequently than once every half hour. Since there is no data available on the exact rate of endocytosis for bound receptors, we made the assumption in our simulation that the probability of endocytosis of a single B-Cell receptor bound to antigen is of the same order of magnitude as the probability of binding of antigen to the receptor. There is a strong probability that multiple receptors are linked by bound antigen before they are endocytosed. We make the reasonable assumption that the probability of endocytosis of the receptor-antigen cluster is inversely proportional to the square of the mass of the cluster. Let us now discuss, in a very simplified way, the structure of the B-Cell receptor/antibody. The B-Cell receptor is a Y shaped molecule consisting of three equal sized segments, connected by disulfide bonds. The antigen binding sites are at the tip of the arms of the Y. These binding sites are made up of two strands (heavy and light) each composed of two regions, one which is constant and another which is highly variable, called the constant and variable regions respectively. The process that forms the antibody first creates a single combination of the heavy and light chains (H,L) sections and then combines two such (H,L) sections by disulfide bonds to create the Y shaped antibody. In diploid species, such as humans, whose DNA strands come from different individuals, there are four ways to make the (H,L) combinations using genes from either of the parent DNA strands. Thus if the parent types make HI, LI, and H2, L2 respectively, in principle, it would be possible to make four combinations: (H1,L1), (H2,L2), (H1,L2) and (H2,L1). The classical dogma in immunology is allelic exclusion, which asserts that, in a given B-Cell, when two strands of (H,L) fuse to form a receptor, only the same (H,L) combination is always selected. This will ensure that for a given B-Cell, all the receptors are identical. However, sometimes this process does not work and B-Cells are found with both types of light chains in receptors on the same cell 9 . It turns out that there are two distinct types of light chains called K and A.Normally in humans the ratio of B-Cells with n or A chains is 2:1 with each cell presenting either a KK or a AA light chain combination.However, as mentioned above, sometimes allelic exclusion does not work perfectly and BCells present KX receptors or the same cell presents receptors of mixed type,a combination of some which are KK, some which are AA and some which are KA.

71

A given antigen will bind either to the A or the re chain, or to neither, but not to both. Thus areAB-Cell receptor is effectively monovalent. Furthermore, a B-Cell with mixedrereand AA receptors would effectively have fewer receptors available for a given antigen. It is possible to experimentally enhance the probability of such genetic errors and study the immunogenicity of the resulting B-Cells. This has been done in mice. The surprising result from such experiments is that chimerical B-Cells are non-immunogenic 9 . We shall attempt to explain how this may come about as a result of our assumption about endocytosis.

3

The Dintzis Experimental Results and the Immunon Theory

Dintzis etal 3 ' 4 , 5 did an in-vivo (mouse) experiment using five different fluoresceinated polymers as antigen (Ag). The results of the experiment were startling. It was found that to be immunogenic, the Ag mass had to be in a range of 105 —106 Daltons (1 Dalton = 1 Atomic Mass Unit) and have a valence (number of effective binding sites) greater than 10-20. Antigen with mass or valence outside this range elicited no immune response for any concentration. Within this range of mass and valence, the response was limited to a finite range of antigen concentration. A model based on the concept of an Immunon was proposed to explain the results 6 ' 7 . The hypothesis was that the B-Cell response is quantized, ie. to trigger an immune response, it is necessary that a minimum number of receptors be connected in a cluster cross linked by binding to antigen. This linked cluster of receptors was called an Immunon and the model came to be called the 'Immunon Model'. However, a problem immediately presents itself: Why are low valence antigens non immunogenic? Why can one not form large clusters of receptors using small valence antigen? The Immunon model had no answer for this question. Subsequently, Perelson et al 8 developed mathematical models (rate equations) to study the antigen-receptor binding process. Assuming that B-Cell response is quantized, they were able to show that at low concentration, because of antigen depletion (too many receptors, too little antigen), an Immunon would not form. However, the rate equations made the flaws in the Immunon model apparant. They were not able to explain why large valence antigen were necessary for an immune response nor why even such antigen was tolerogenic (non-immunogenic) at high concentration.

72

4

Modeling the B-Cell Receptor Binding to Antigen: Our Computer Experiment

The activation of a B-cell is the result of local surface processes leading to a cascade of events that result in release of antibody and/or presentation of antigen. The local surface processes are binding, endocytosis and receptor diffusion. Each of these is governed by its own time and length scales, some of which are experimentally known. To model B-cell surface dynamics properly, the size of the modeled surface must be significantly larger than the largest dynamic length scale we wish to model and the time steps used must be smaller than the smallest dynamic time scale. Further, the size of the smallest length scale on the modeled surface must be smaller than the smallest length scale in the dynamics. The size of a B-cell receptor is 3 nm and this is the smallest surface feature we will model. The size of a typical antigen in our simulation is 5-40 nm. The diffusion rate of receptors is of the order of D = 10x1 - 5~ 10 cm 2 /s and the time scale for activation of a cell is of the order of a few seconds to a few tens of seconds (r ~ 100s). Hence the linear size of the surface necessary in our modeling is L > \TDT ~ 1 — 1\im. This is the maximum distance that a receptor will diffuse in a time of about 100s. We choose a single lattice spacing to represent a receptor. The linear size of our surface was chosen to be 1000 lattice units which approximately represents a physical length of 3 — 4/xm. The affinity of receptor-hapten binding is lO 5 ]^^ 1 for a monovalent receptor. The affinity of a bivalent receptor depends on the valence of the antigen and on the distribution of haptens on the antigen. The weight of a single hapten is a few hundred Daltons. Hence, the ratio of the on-rate to the off-rate of a single receptor hapten pair is ~ 100 - 1000. We choose an on-rate of 0.2 and an off rate of 0.001 in dimensionless units. Our unit of time was set to 0.05 milli second. This was done by choosing D in dimensionless units to be 0.1 which means that the effective diffusion rate is (0.1x(3nm)7(0.05ms) ~ 2.0xl0- 1 0 cm 2 /s. The affinity of chimeric B-Cell receptors was set lower because they bind to DNA with a lower affinity. For them, we used an on-rate of 0.1 and an off-rate of 0.01. The cell surface was chosen to have periodic boundary conditions as this simplifies the geometry of the modeling considerably. The size of our cell surface is equivalent to 20% of a real non-activated B-cell.A B-cell typically has 50000 receptors on its surface.Hence, we modeled with 10000 receptors initially placed on random sites of the lattice. In each time step, every receptor was updated by moving it to a neighboring site (if the site was empty) with a probability D = 0.1. Receptors that are bound to antigen were not allowed to

73

move. At every time step, receptors which have free binding sites can bind to other haptens on the antigen or to any other antigen already present on the surface. They can also unbind from hapten to which they are bound. Once an antigen unbinds from all receptors, it is released within 5 time steps on average. Once every 20 time steps, the receptors were presented with new antigen at a constant rate which was a measure of the total antigen concentration. We varied this concentration rate in our modeling. The normal rate of endocytosis of unbound receptors is once every 1/2 hour. If this is the rate of endocytosis for bound receptors also, it will be too small to play a role in antigen presentation. Thus we must assume that a bound receptor has an higher probability of being endocytosed compared to an unbound receptor. A receptor can bind to two haptens and every antigen can bind to multiple receptors. This cross linking leads to the creation of large complexes. We assume that the probability to enodcytose a receptor-antigen complex is inversly proportional to the square of its mass. The mass of the B cell receptor is much higher than the mass of the antigens, so, when computing the mass of the complex we can ignore the mass of the antigen. We thus set the endocytosis rate only as a function of the number of bound receptors. The rate of endocytosis for the entire complex was chosen to be inversly proportional to the square of the number of receptors in the complex. More specifically, we set the probability to endocytose an aggregate of receptors to be 0.0005 divided by the square of the number of receptors in the aggregate. For chimeric B-cells we reduced the numerator in this probability by a factor of 100. 5

Results

The results of our computer study are shown in figure (1), where the solid line shows the number of bound receptors after 10 seconds of simulation as a function of antigen valence. The data are average values over several simulations with different initial positions for receptors and random number seeds. The dashed line shows the number of endocytosed receptors. One observes a clear threshold below which the number of bound surface receptors stays close to zero followed by a region where the number of bound receptors increases and flattens out. This establishes that we can explain the threshold in antigen valence in the Dintzis experiment. The reason for the threshold is easy to understand qualitatively. Once an antigen binds to a receptor, the probability that its other haptens bind to the other arm of the same receptor or to one of the other receptors present in the vicinity is an exponentially increasing function of the number of haptens. Also, once an antigen is multiply bound in a complex, the probability of all

74

the haptens unbinding is an exponentially decreasing function of the number of bound haptens. Given that receptors once bound may be endocytosed, low valence antigen bound once will most likely be endocytosed or will unbind before it can bind more than once (ie. before it has a chance to form an aggregate and lower its probability to endocytose.) As the valence increases, the unbinding probability decreases and the multiple binding probability increases until it overcomes the endocytosis rate. Finally, for high valence, one will reach a steady state between the number of receptors being bound and the number endocytosed in a given unit of time. In figures (2) and (3), we show the number of bound receptors (solid line) and endocytosed receptors (dashed line) as a function of the antigen concentration for two different values of valence. Figure (2) has data for high valence (20) and figure (3) for low valence (5). It is clear that for high valence, there is a threshold in concentration below which there are no bound receptors (no immune response) followed by a range of concentration where the number of bound receptors increases followed by a region where it decreases again. The threshold at low concentration is easy to understand. It is caused by antigen depletion (all antigen that binds is quickly endocytosed). The depletion at high concentration comes about because of too much endocytosis, which depletes the pool of available receptors. For low valence (figure (3)), there is no range of concentrations where any surface receptors are present. The reason is that the valence is too low to form aggregates and lower the rate of endocytosis and is also too low to prevent unbinding events from happening fast enough. Thus all bound receptors get quickly endocytosed. The high rate of endocytosis for high concentration probably leads to tolerance, as the cell will not survive such a high number of holes on its surface. Figures (1), (2) and (3) are the major results of our modeling. They clearly show that the single, simple assumption of an increased rate of endocytosis for bound receptors and reasonable assumptions about the way this rate depends upon the mass of the aggregated receptors is able to explain both the low concentration threshold for immune response as well as the high concentration threshold for tolerogenic behavior in the Dintzis experiment. It can also explain the dependence of activation on valence, with a valence dependent threshold (or alternately, a mass dependent threshold) for activation. Now consider the case of chimeric B-Cells. It turns out that these cells bind to low affinity DNA but are not activated 9 . DNA has a high valence and is in high concentration when chimeric B-Cells are exposed to it. To model the interaction of these B-Cells, we therefore used a valence of 20 and lowered the binding rate. We considered two cases:

75

Case 1: The B-cell has KK and AA receptors in equal proportion. This effectively halves the number of receptors since antigen will bind either to the KK receptor or the AA receptor but not to both. Figure (4) shows the results of our modeling for this case. Note that the total number of bound receptors is very low. This is due to the low affinity. However, the endocytosis rate is high, since receptors once bound will be endocytosed before they can bind again and lower their probability of endocytosis. Thus in this case we would expect tolerogenic behavior because of the low number of bound receptors. Case 2: The K and A are on the same receptor. This means that the receptor is effectively monovalent since antigen that binds to one of the light chains will not, in general, bind to the other. In a normal bivalent receptor, the existance of two binding sites creates an entropy effect where it becomes likely that if one of the sites binds, the other binds as well. The single binding site on the K\ receptors means that antigen binds and unbinds, much like the case of the T-Cell. Thus, although the number of bound receptors at any given time reaches a stady state, the endocytosis rate is low, since receptors do not stay bound long enough to be endocytosed. Figure (5) shows the results of the modeling which are in agreement with this qualitative picture. The non-immunogenicity of K\ cells would come about because of the low rate of endocytosis and consequent lack of T-Cell help. Our modeling thus shows that for chimeric receptors, non-immunogenicity would arise from subtle dynamical effects which alter the rates of binding and endocytosis so that either activation or T-Cell help would be compromised. These predictions could be tested experimentally.

76 Number of bound and endocitosed receptors as a function of valence

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77 Low valence antigen

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References 1. Y. Louzoun, M. Weigert and G. Bhanot, "A New Paradigm for B Cell Activation and Tolerance", Princeton University, Molecular Biology Preprint, June 2001. 2. C. A. Janeway, P. Travers, M. Walport and J. D. Capra, "Immunobiology - The Immune System in Health and Disease", Elsevier Science London and Garland Publishing New York, 1999. 3. R. Z. Dintzis, M. Okajima, M. H. Middleton, G. Greene, H. M. Dintzis, "The Immunogenicity of Soluble Haptenated Polymers is determined by Molecular Mass and Hapten Valence", J. Immunol. 143:4, Aug. 15, 1989. 4. J. W. Reim, D. E. Symer, D. C. Watson, R. Z. Dintzis, H. M. Dintzis, "Low Molecular Weight Antigen Arrays Delete High Affinity Memory B cells Without Affecting Specific T-cell Help", Mol. Immunol., 33:17-18, Dec. 1996. 5. R. Z. Dintzis, M. H. Middleton and H. M. Dintzis, "Studies on the Immunogenicity and Tolerogenicity of T-independent Antigens", J. Immunol., 131, 1983. 6. B. Vogelstein, R. Z. Dintzis, H. M. Dintzis, "Specific Cellular Stimulation in the Primary Immune Response: a Quantized Model", PNAS 79:2, Jan.

79 1982. 7. H. M. Dintzis, R. Z. Dintzis and B. Vogelstein, "Molecular Determinants of Immunogenicity, the Immunon Model of Immune Response", PNAS 73, 1976. 8. B. Sulzer, A. S. Perelson, "Equilibrium Binding of Multivalent Ligands to Cells: Effects of Cell and Receptor Density", Math. Biosci. 135:2, July, 1996; ibid. "Immunons Revisited: Binding of Multivalent Antigens to B Cells", Mol. Immunol. 34:1, Jan. 1997. 9. Y. Li, H. Li and M. Weigert, "Autoreactive B Cells in the Marginal Zone that Express Dual Receptors", Princeton University Molecular Biology Preprint, June 2001.

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S T O C H A S T I C MODELS OF I M M U N E S Y S T E M A G I N G L. MARIANI, G. TURCHETTI Department of Physics, Via Irnerio 46, 40126 Bologna, Italy Centro Interdipartimentale L. Galvani, Universita di Bologna, Bologna, Italy E-mail: [email protected] [email protected] F. LUCIANI Max Planck Institute for Compex Systems, Noetnitzer 38, Dresden, E-mail: [email protected]

Germany

The Immune System (IS) is devoted to recognition and neutralization of antigens, and is subject to a continuous remodeling with age (Immunosenescence). The model we propose, refers to a specific component of the IS, the cytoxic T lymphocytes and takes into account the conversion from virgin (ANE) to memory and effector (AE) phenotypes, the virgin cells injection by thymus and the shrinkage of the overall compartment. The average antigenic load as well as the average genetic properties fix the the parameters of the model. The stochastic variations of the antigenic load induce random fluctuations in both compartments, in agreement with the experimental data. The results on the concentrations are compatible with a previous simplified model and the survival curves are in good agreement with independent demographic data. The rate of mortality, unlikely the Gomperz law, is zero initially and asymptotically, with an intermediate maximum, and allows to explain the occurrence of very long living persons (centenarians).

1

Biological Complexity

The Immune System (IS) preserves the integrity of the organism, continuously challenged by internal and external agents (antigens). The large variety of antigens ranging from mutated cells and parasites to viruses, bacteria and fungi requires a rapid and efficient antagonistic response of the organism. At the top of the philogenetic tree the evolution has developed a specific (clonotipic) immunity which cooperates with the ancestral innate immunity to control the antigenic insults ^h The innate system has an arsenal of dendritic cells and macrophagi with a limited number of receptors capable of recognizing and neutralizing classes of antigens. With the appearance of vertebrates the increase of complexity stimulated the development of a system based on two new types of cells, B and T lymphocytes, with three distinct tasks: to recognize the antigens, to destroy them and to keep track of their structure through a learning process. This kind of immunological memory is the key for a more efficient response to any subsequent antigenic insult caused by an antigen that the organism has already experienced (this is the basis of vaccination). The specific response is based on a variety of memory cells which are activated, by specific

81

molecules of the antigen, presented by the APC (antigen presenting cells) to their receptors. There are two main T cells compartments: the virgin cells, which are produced (with B lymphocytes) in the bone marrow but maturate all their surface receptors in the thymus, and the memory cells, which are activated by the antigenic experience and preserve the information. The memory cells specific to a given antigen form a clone which expands with time, subject to the constraint that total number of cells remains almost constant with a small decrease with age (shrinkage of IS). The virgin cells instead, after reaching a maximum in the early stage of life, decrease continuously since the IS is not able to compensate their continuous depletion due to various biological mechanisms (conversion into memory cells, progressive inhibition of the thymic production, peripheric clonal competition) I2-3!. The systems with self-organizing hardware, cognitive and memory properties and self replicating capabilities are by definition complex. The immune and nervous systems exhibit these features at the highest degree of organization and can be taken as prototypes of complex systems. Indeed the specific (clonotipic) immune system has a hardware hierarchically organized , capable of receiving, processing and storing signals (from its own cytochines and immunoglobulines network and from the environment), and to create a memory, self replicating via the DNA encoding, which allows a long term evolutionary memory. For this reason the mathematical modeling has been particularly intensive on the IS. Since this system exhibits a large number of space time scales, modeling is focused either on specific microscopic phenomena with short time scales or on large scale aspects with long time scales ranging from a few weeks (acute antigenic response) to the entire lifespan. 2

T lymphocytes

We will focus our attention on the dynamics of the T cells populations on a long time scale disregarding the detailed microscopic behavior which is certainly very relevant on short time scales. The virgin T lymphocytes developed by the thymus have a large number of receptors TCR, built recombining genie sequences. This large set of variants (up to 1016 in humans), known as T cell repertoire, allows the recognition by steric contact of the antigen fragments presented by the APC (Antigen Presenting Cells), which degrade the proteins coming from the englobed antigens via proteolytic activities and show the peptides resulted from the cleavage to the surface molecules MHC I 4 ' 5 !. Other stimuli, such as the cytokines t 1 !, determine the differentiation into effector and memory cells and their proliferation (clone expansion). The memory cells, unlikely the effector ones, are long lived and show a sophisticated

82

| T helper ]

ANE (Virgin)

/*™\

AE (memory+effector)

[ T Cytotoxic]

ANE CVirgin)

AE (memory+effector)

Figure 1: Markers of Virgin and Memory plus Effector T lymphocytes. Schematic organization of main T cell pool: CD4+ (Helper) and CD8+ (cytotoxic) lymphocytes.

cognitive property allowing a more efficient reaction against a new insult of an experienced antigen. The T lymphocytes are split into two groups: the cytotoxic and helper T cells. The former attack and destroy cells infected by intra-cellular antigens, such as virus and kind of bacteria, the latter contribute to extracellular antigenic response, not described here. They are labeled by CD8 + (cytotoxic) and CD4 + (helper) according to the surface markers used to identify them. Each group is further split into virgin, effector and memory cells, whose role has been outlined, and are identified by some others surface markers, see figure 1. We are interested in the dynamics of two populations, the Antigen Not Experience (ANE) virgin T cells, and Antigen Experienced (AE) effector and memory T cells, which are identified by the CD95~ and CD95 + surface markers respectively. 3

Modeling immunosenescence

In this note we propose a mathematical model to describe the time variation of the ANE and AE T cells compartments due to the antigenic load and to the remodeling of the system itself. The antigenic load has sparse peaks of high intensity (acute insults) and a permanent low intensity profile with rapid random variations (chronic antigenic stress). In a previous work I 6 ' a simple model for the time evolution AE and ANE T cells concentrations was proposed on the basis of Franceschi's theory of immunosenescence, which sees the entire IS undergoing a very deep reshaping during the life span. The exchange between the compartments were considered due to antigen stimulated conversion and to reconversion due to secondary stimulation. The average antigenic load contributed to define these conversion rates jointly with a genetic average. The deterministic part of the model described the decrease of the ANE CD8 + T

83

cells concentration in agreement with experimental data, while the stochastic forcing, describing the chronic stress I 7 ' 8 !, allowed to obtain individual histories. The spread about the mean trajectory was also compatible with the data on T cells concentration and allowed to obtain survival curves in good agreement with independent demographic data, starting from the hypothesis that the depletion of the ANE T cells compartments is a mortality marker. The present model is intended to introduce some improvements by taking into account the remodeling of the immune system with age and is formulated on the ANE and AE populations rather than on the concentrations. Moreover the antigenic load is introduced on both the ANE and AE variation rates with an adjustable mixing angle. The complete model, that will be described in detail in the next section, has several parameters, but the relevant point is that if we neglect the remodeling, and compute the concentration, the results are very similar to the original ones; moreover the data on the AE T cells are fairly well reproduced t 3 '. The introduction of the remodeling effects shows that a further improvement still occurs especially for the early stage were the simplified model was not adequate. The last part is dedicated to the survival curves obtained from the model. A very important difference is found with respect to the classical Gomperz t 9 l survival law: the rate of mortality vanishes initially and asymptotically whereas it increases exponentially in the Gomperz law. This results, which explains the presence of very long lived individuals (centenarians), supports the biological hypothesis (depletion of ANE T cells compartment) of Franceschi's theory. 4

Mathematical model and results

The mathematical model is defined by dV _ -aV dt ~ dM _ dt

aV

1+

+ fi e~xt

-/3M + /3M 7

M+

. +

—

+ ecos2(6>) f(i)

2//,N/-/^ 2

(*)^)

(1)

where V denotes the number of ANE (virgin) CD8 + cells and M the number of AE (effector +memory) CD8+ cells, M+ = M if M > 0 and M+ = 0 if M < 0. The parameter a gives the conversion rate of virgin cells due to primary antigenic insult whereas {3 is the reactivation rate of memory cells due to secondary antigenic stimulus, which has an inhibitory effect on the virgin cells. In the primary production of AE cells we have taken the conversion and reconversion terms proportional to (1 + ^M+)~l, in order to take into

§ o

84

500

0

Memory

o

c

8 „o

o

° (ft)

°

___

•

,^-V--" o o ^-^"O 0

Jfcg^~t- --I -« -200

-100 0

Time (years)

120

0

Time (years)

120

Figure 2: Comparison of the model with experimental data for the virgin (ANE) and memory plus effector (AE) CD8+ T cells for the following parameters a = 0.025, /? = 0.01, e = 15, 0 = 35°, 7 = 0.004, A = 0.05, ^ = 15 and V(0) = 50. The curves are (V(t)) + fcoy (t) and (M{i)) + kaM{t) with k = - 2 , 0 , 2 .

account the shrinkage of the T cells compartment. The term fie~xt describes the production by thymus which is assumed to decay exponentially. Finally e£(£), where

«(*)> = o

(at)at')) = s(t-t')

(2)

is the contribution of stochastic fluctuations to the conversion rates. The mixing angle 0 gives the weight of this term on the ANE and AE compartments. The results are compared with experimetal data in figure 2. 4-1

Simplified model

It was previously studied in the deterministic version and is obtained from (1) by setting 7 = y, = e = 0. Since M + V is constant the same equations were satisfied by the concentrations v = V/(V + M) and m = M/(V + M) and the stochastic equation for v was t 10 ' 6 ! dv = -{a-fi)v-P ~dl

+ eZ{t)

(3)

where e ~ e/(V r (0)+M(0)), and was solved with initial condition v(0) = 1. The deterministic model obtained from (1) setting e = 0 can be solved analytically and we consider two simple cases: /? — 7 = 0 describing the effect of thymus and /3 = /j, — 0 describing the effect of shrinkage.

85 e=o

9=0

9=45

1000

9=45

500

'" 8 \

' 0

?o

X

f

F

1

^N * ^* >.•" °" " o^9— . . . ^~Time (years)

. . - • - • "

°

,..'' & ,—

s

•—r

^

/u.

S

•f°

t

°

Time (years)

M

° ^-stS^s.

r-.'J*-.

•?nn

120

Virgin

o

^iv

120

-mn °

Time (years)

12

°

°

/ ^ " o o° *

..ta-..

- f

*S- * B

8«

Time (years)

12

°

Figure 3: Comparison with CD95 data ' 2 1 of virgin (ANE) and memory plus effector (AE) populations for the model without shrinkage and thymus for two different mixing angles. The parameters are a - 0.02, /3 = 0.005, £ = 10, V(0) = 400 and 6 ~ 0 (left figures) and 9 = 45° (right figures). The curves are (V{t)) + kav(t) and (M(t)) + kaM(t) with k = -2,0,2.

4-2

Analytic solutions without noise

The deterministic solution of the model without thymus and shrinkage 7 = fi = 0, for initial conditions V(0) and M(0) = 0, reads (V(t)) = V0

(M(t)) = V0a

/3-a

e(0-a)t

_ J

0-a

(4)

and the T cell population is conserved M(t) + V(t) = V(0). The deterministic solution with no shrinkage 7 = 0 can be analytically obtained t 11 !. Since 0 < P < a choosing for simplicity ,5 = 0 one has (V(t)) =

V(0)e-at+v

e-at

_

e-Xt

A—a

(M(t)) = V(0)-(V(t))

+

?-{l-e-»)

(5) The solution with shrinkage and no thymic term n = 0, choosing for simplicity /? = 0, reads (V(t))=V(0)e

-at

(M(t)) = 7 - 1 ( [1 + 2 7 V(0)(1 - e-at)

Y1/2

- 1 ) (6)

The graph of (V(t)} (5) exhibits a peak at t = (log A — loga)/(A — a) if 1/(0) = 0. The peak disappears when the thymus contribution vanishes. Conversely (M(t)) is monotonic increasing but the thymus enhances its value. The shrinkage reduces considerably the increase of (M(t)) whereas it does not affect (V(t)). The stochastic term generates a family of immunological histories. Their spread is measured by the variance. In figure 3 we show the effect of the mixing angle for the same set of parameters chosen for the simplified model.

86 Thymus

Shinkage

Shinkage

500

Virgin

Virgin

§

Thymus

fa

X

inn

-100

Time (years)

120

Time (years)

120

^-^ ••.,

°

Time (years)

" " - —

120

Time (years)

12

°

Figure 4: Comparison with CD95 data I 2 J of virgin (ANE) and memory plus effector (AE) populations for the model with a = 0.025, /9 = 0.01, e = 15, 6 = 35°. On the left side we consider the contribution of thymus with A = 0.05, /x = 15, 7 = 0 and V(0) = 50 . On the right side the contribution of shrinkage is shown for 7 = 0.004, /i = 0 and V(0) = 400. The curves are (V(t)) + kav(t) and (M(t)) •+- kaM{t) with k - - 2 , 0 , 2 .

When 6 grows, the rms spread av of V decreses, whereas the rms spread aM of M increases. The separate effect of thymus and shrinkage is shown in figure 4, for the same parameters as figure 2. 5

Survival curves

The simplified model with noise given by equation (3), corresponds to the Ornstein-Uhlembeck process. The probability density, satisfies the FokkerPlanck equation and has an expilict solution (v)(t)

+ (1 - t;oo)e- t / T

p(v,t) =

1 ^2Trai(t)

exp ~'r [

(v-(v)(t)[ 2a2 (t)

(7) where T = (a - 0)~\ Voo = -0T and a2(t) = \ e2 T (1 - e~2tlT). In figure 5 we compare the results of the model with demographic data. Assuming that the depletion v = 0 of the virgin T cells compartment marks the end of life it is possible to compute from (7) the survival probability up to age t r+00

S(t)

2 du

Jx(t)

x(t) =

a(t)

(8)

Neglecting the thymus and shrinkage effects the concentrations obtained from equation (1) are close to values of the simplified model if d = 0. Indeed when 7 = /x = 0 we have M(t) + V{t) = V(0) + M(0) + ew(t) where w(t) denotes the Wiener process. Setting v = V/{V + M) and V = V0 + eVx and M = M0 + * — 0. The lower integration end point can be written as x(t) = C

exp^y^ Vl - e- 2 «/ T 1

H(f

1/2

(v* - Uoo)

(10)

where tt is the death age {v(tr)} = u* namely e~'*/ T = (u* — Voo)/(l — ^oo)- In figure 5 we fit the demographic data of human males using (8) and the values of the parameters are close to the ones obtained from the fit of CD8 + T cells concentrations.

6

Comparison with Gomperz law

Making the simplifying assumption t* = T we obtain a survival probability depending on two parameters C, T

88

just as the Gomperz law, which is defined by ^

= -RSG

f

= £

->

50(t)=exp(-C0(e^0-l))

(12)

Our mortality rate is not monotonically increasing, as for Gomperz law, but decreases after reaching a maximum, see figure 5. It is better suited to describe the survival of human populations in agreement with demographic data. This property, due to the randomly varying antigenic load on the organism, explains the occurrence of very long lived persons (centenarians). We notice that x(t) oc - i - 1 / 2 as t ->• 0 and x(oo) = C so that -C2/2

Jim S(t) = 1

e

lim S(t) =

—7=r

(13)

We notice that 5(+oo) > 0 means nonzero probability of indefinite survival. However 5(+oo) ~ 1 0 - 3 for C = 3 and it is below 1 0 - 6 for C = 5 our law imposes to fix a reasonable lower bound on C. We further notice that

«*H

f

C

t=T

T

'

V2TT(1 -

e- 2 )

(14)

The meaning of the parameters is obvious: T is the age at which the survival probability is exactly 50% and the slope of the curve there is proportional to C. We can say that C measures the flatness of the graph of S(t). For the mortality rate R = —S/S we have the following asymptotic behavior

7

Conclusions

We consider the long time behavior of the CD8 + virgin T cells and CD8 + antigen experienced T cells compartments and the remodeling of the IS system. The stochastic variations of the antigenic load determine a spread in the time evolution of the cells number, in agreement with experiments. The results are compatible with a previous simplified model for the virgin T cells concentrations and provides survival curves compatible with demographic data. The effect of thymus and remodeling improves the description of the early stage for the virgin T cells and late stage of antigen experienced T cells. 8

Acknowledgments

We would like to thank Prof. Franceschi for useful discussions on the immune system and ageing.

89

9

References 1. A Lanzavecchia, F.Sallustio Dynamics of T lymphocytes Responses: Intermediates,Effector and Memory Science 290, 92 (2000) 2. F. Fagnoni, R. Vescovini, G. Passeri, G. Bologna, M. Pedrazzoni, G. Lavagetto, A. Casti, C. Franceschi, M. Passeri & P. Sansoni, Shortage of circulating naive CD8 T cells provides new insights on immunodeficiency in aging. Blood 95, 2860 (2000) 3. F. Luciani, S. Valensin, R. Vescovini, P. Sansoni, F. Fagnoni, C. Franceschi, M. Bonafe, G. Turchetti, Immunosenescence: The Stress Theory AStochastical Model for CD8+ T cell Dynamics in Human Immunosenescence: implications for survival and longevity, J. Theor. Biol. 213,(2001) 4. A. Lanzavecchia, F. Sallusto, Antigen decoding by T lymphocytes: from synapse to fate determination, Nature Immunology 2, 487 (2001) 5. G.Pawelec et al, T Cells and Aging, Frontiers in Bioscience 3, 59 (1998) 6. F. Luciani, G. Turchetti, C. Franceschi, S. Valensin, A Mathematical Model for the Immunosenescence, Biology Forum 94, 305 (2001). 7. C. Franceschi, S. Valensin, M. Bonafe, G. Paolisso, A. I. Yashin, D. Monti, G. De Benedictis, The network and the remodeling theories of aging: historical background and new perspectives., Exp. Gerontol. 35, 879 (2000) 8. C. Franceschi, M. Bonafe, S. Valensin. Human immunosenescence: the prevailing of innate immunity, the failing ofclonotipic immunity, and the Riling of immunological space. Vaccine. 18,1717(2000) 9. B. Gompertz On the nature of the function expressive of the law of human mortality, and on the new mode of determining the values of life contingencies. Philos. Trans. R. Soc. London 115, 513 (1825) 10. F. Luciani Modelli hsico-matematici per la memoria immunologica e l'immunosenescenza , Master thesis, Univ. Bologna (2000) 11. L. Mariani Modelli stocastici deU'immunologia: Risposta adattativa, Memoria e Longevita' Master thesis, Univ. Bologna (2001) 12. L.A. Gavrilov, N. S. Gavrilova The Biology of Life Span: a quantitative approach ( Harwood Academic Publisher,London, 1991) 13. L. Piantanelli, G. Rossolini, A. Basso, A. Piantanelli, M. Malavolta, A. Zaia, Use of mathematical models of survivorship in the study of biomarker of aging: the role of Heterogeneity, Mechanism of Ageing and Development 122, 1461 (2001)

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NEURAL NETWORKS AND NEUROSCIENCES

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93

ARTIFICIAL NEURAL NETWORKS IN NEUROSCIENCE

N. ACCORNERO, M. CAPOZZA Dipartimento di Scienze Neurologiche, Universita di Roma LA SAPIENZA

We present a review of the architectures and training algorithms of Artificial Neural Networks and their role in Neurosciences.

1.

Introduction: Artificial Neural Networks

The way an organism possessing a nervous system behaves depends on how the network of neurons making up that system functions collectively. Singly, these neurons spatially and temporally summate the electrochemical signals produced by other cells. Together they generate highly complex and efficient behaviors for the organism as a whole. These operational abilities are defined as "emergent" because they result from interactions between computationally simple elements. In other words, the whole is more complex than the sum of its parts. Our understanding of these characteristics in biological systems comes largely from studies conducted with artificial neural networks early in the 1980s [1]. Yet the biological basis of synaptic modulation and plasticity were perceived by intuition 40 years earlier by Hebb, and the scheme for a simple artificial neuronal network, the perceptron, was originally proposed by Rosenblatt [2] and discussed by Minsky [3] in the 1960s. An artificial neural network, an operative model simulated electronically (hardware) or mathematically (software) on a digital processor, consists of simple processing elements (artificial neurons, nodes, units) that perform algorithms (stepwise linear and sigmoid functions) on the sum or product of a series of numeric values coming from the various input channels (connections, synapses). The processing elements distribute the results of the output connections multiplying them by the single connection "weights" received from the other interconnected processors. The final complex computational result therefore depends on how the processing units function, on the connection weights, and on how the units are interconnected (the network architecture). To perform a given task (training or learning), an artificial net is equipped with an automated algorithm that progressively changes at least one of these individual

94 computational elements (plasticity) almost exactly as happens in a biological neuronal network.

CONNECTIONS PLASTICITY

F(l)

Z^-—^F(')*P2 F(I)*P3 (l)*P4

SIMULATION

HARDWARE

SOFTWARE

Figure 1 : Comparison between a biological neuron and an artifical neuron, and a hardware-software simulation. The functions of the processing elements (units) and the network architecture are often pre-determined: the automated algorithms alter only the connection weights during training. Other training methods entail altering the architecture, or less frequently, the function of each processing unit. The architecture of a neural network may keep to a pre-determined scheme (for example with the processing elements, artificial neurons, grouped into layers, with a single input layer, several internal layers, and a single output layer). Otherwise it starts from completely chance connections that are adjusted during the training process.

95

A R T I F I C I A L NEURAL NETWORKS TOPOLOGY NETWORK

CLUSTER

LAYERED

Figure 2 Variable architecture of artificial neural networks. Network training may simply involve increasing the differences between the various network responses to the various input stimuli (unsupervised learning) so that the network automatically identifies "categories" of input [4, 5, 6]. Another training method guides the network towards a specific task (making a diagnosis or classifying a set of patterns). Networks designed for pattern classification are trained by trial and error. Training by trial and error can be done in two ways. In the first, an external supervisor measures the output error then changes the connection weights between the units in a way that minimizes the error of the network (supervised learning) [7, 8]. The second training method involves selective mechanisms similar to those underlying the natural selection of biological species — a process that makes random changes in a population of similar individuals, then eliminates those individuals having the highest error and reproduces and interbreeds those with the lowest error. Reiterating the training examples leads to genetic learning of the species [9].

96

0> J

SUPERVISED

I "

BEITA RlJt.ES

M

ERROR 8ACIC-PMOPASATION

P '.

UNSUPERVISED

TOWETlTlWe

UEARftHMG

GENETIC AiGORITHRSS l -HI I l in I i Hi

UtaHFUHlBlH CROSSOVER

Figure 3. Training modalities

The choice of training method depends on the aim proposed. If the network is intended to detect recurrent signal patterns in a "noisy" environment, then excellent results can be obtained with a system trained through unsupervised learning. If one aims to train a diagnostic net on known knowledge, or to train a manipulator robot on precise trajectories, then one should choose a multilayered network trained through unsupervised learning. If the net is designed for use as a model, that is to simulate biologic nervous system functions, then the ideal solution is probably genetic learning. Adding the genetic method to either of the other two methods will improve the overall results. In summary, biological and artificial neural networks are pattern transformers. An input stimulation-pattern produces an output pattern-response, especially suited to a given aim. To give an example from biology: a pattern of sensory stimuli, such as heat localized on the extremity of a limb, results in a sequence of limb movements that serve to remove the limb from the source of heat. A typical example of an artificial network is a system that transforms a pattern of pathologic symptoms into a medical diagnosis.

97

Input and output variables can be encoded as a vectorial series in which the value of a single vector represents the strength of a given variable. The power of vectorial coding becomes clear if we imagine how some biological sensory systems code the reality of nature. The four basic receptors located on the tongue (bitter-sweet-saltacid) allow an amazing array of taste sensations. If each receptor had only ten discrimination levels - and they certainly have more - we could distinguish as many as 10.000 different flavors. On this basis, each flavor corresponds to a point in a four-dimensional space identified by the 4 coordinates of the basic tastes. Similar vectorial coding could make up the input of an artifical neural network designed to identify certain categories of fruit. One or more internal (hidden) layers would transform this coding first into numerous arbitrary hidden (internal) codes, and ultimately into output codes that classify or recognize the information presented.

APPLE

BANANA

BITTER

SWEET

ACID 1

SALTY

CHERRY

« • INPUT VECTORIAL CODING

GRAPE

OUTPUT POSITIONAL CODING

Figure 4. A forward-layered neural network This network designed to recognize or diagnose can be generalized to a wide range of practical applications, from geological surveying to medicine and economic evaluation. This type of network architecture, termed "forward", because its connections all converge towards the output of the system, is able to classify any "atemporal" or "static" event. Yet reality is changeable, input data can change

98

rapidly, and the way in which these data follow one another can provide information that is essential for recognizing the phenomenon sought or for predicting how the system will behave in the future. Enabling a network to detect structure in a time series, in other words to encode "time", means also inserting "recurrent" connections (carrying output back to input). These connections relay back to the input unit the values computed by units in the next layer thus providing information on the pattern of preceding events. Changing of the connection weights during training is therefore also a function of the chain of events. The nervous system is rich in recurrent connections. Indeed, it is precisely these connections that are responsible for perceiving "time". If a "forward" network allows the input pattern to be placed in a single point of the multidimensional vector space, a recurrent network will evaluate this point's trajectory in time.

DYNAMIC RECOGNITION T I M E CODING

PREDICTION

#

Figure 5. Recurrent layered neural network

Advances in neural networks, the unceasing progress in computer science, and the intense research in this field have brought about more profound changes in technology than are obvious at first glance. Paradoxically, because these systems have been developed thanks to digital electronics and by applying strict

99 mathematical rules, the way artificial neural networks function runs counter to classic computational, cognitive-based theories. A distinguishing feature of neural networks is that knowledge is distributed throughout the network itself rather than being physically localized or explicitly written into the program. A network has no central processing unit (CPU), no memory modules or pointing device. Instead of being coded in symbol (algorithmic-mathematical) form, a neural network's computational ability resides in its structure (architecture, connection weights and operational units). To take an example from the field of mechanics, a network resembles a series of gears that calculates the differential between the rotation of the two input axles and relays the result to the output axle. The differential is executed by the structure of the system with no symbolic coding. The basic features of an artificial neural network can be summarized as follows: 1) Complex performance emerging from simple local functions. 2) Training by examples 3) Distributed memory, and fault tolerance. 4) Performance plasticity. Connectionist systems still have limited spread in the various fields of human applications, partly because their true potential remains to be discovered, and partly because the first connectionist systems were proposed as alternatives to traditional artificial-intelligence techniques (expert systems) for tasks involving diagnosis or classification. This prospect met with mistrust or bewilderment, due paradoxically to their inherent adaptiveness and plasticity, insofar as misuse of these qualities can lead to catastrophic results. Disbelievers also objected that the internal logic of artificial neural networks remains largely unknown: networks do not indicate how they arrive at a conclusion. These weaknesses tend to disconcert researchers with a determinist background (engineers, mathematicians and physicists) and drive away researchers in medical and biological fields, whose knowledge of mathematics and computer science is rarely sufficient to deal with the problems connectionist systems pose. 2.

Application of artificial neural networks to neuroscience

Neural networks now have innumerable uses in neuroscience. Rather than listing the many studies here, we consider it more useful to give readers some reference points for general guidance. For this purpose, though many categories overlap or remain borderline, we have distinguished three: applications, research, and modelling. Applications essentially use neural networks for reproducing, in an automated, faster or more economic manner, or in all three ways, tasks typically undertaken by human experts. The category of applications includes diagnostic neural networks (trained to

100 analyze a series of symptoms and generate a differential diagnosis, and even a refined one, and networks designed for pattern recognition (currently applied in clinical medicine for diagnostic electrocardiography, electromyography, electroencephalography, evoked potentials, and neuroophfhalmology) [10] Other networks are designed to segment images (for example, to identify anatomical objects present in radiologic images, and highlight them in appropriate colors or reconstruct them in three dimensions). In general, these networks have received supervised training. Of especial interest to neurologists are diagnostic networks trained to make a typically neurologic diagnosis such as the site of lesion. One of these nets visualizes the areas of probable lesion on a three-dimensional model of the brainstem, thus supplying a topographic, rather than a semantic diagnosis [11].

OPTONET FLOWCHART

autoMAtio visual Finlri ana lijsor output

I*

lncoMpi«t» eonsroous i-isht

Figure 6: Automated diagnosis of the visual field.

101

EMG -NET A NEURAL NETWORK FOR THE DIAGNOSIS OF SURFACE MUSCLES ACTIVITY

"wi m i P ' ' lfcO(.H." 0

(1 - k2)a

'-a 0 0 %a

\ 0

M =

(4) 0

V

IfaQlfa

0

(l-k2)a\J

where a is the (absolute) slope of the local tent maps and k2 is the diffusive coupling strength. The thermodynamic formalism formally proceeds by raising the (matrix) entries to the (inverse) temperature /3, and then focusing, as the dominating effect, on the largest eigenvalue as a function of the inverse temperature. For large network sizes, the latter converges towards fi(p,k) = (\(l-k2)a\)P

+ (ak2)0.

(5)

This expression explicitly shows that the network behavior is determined by two sources: by the coupling (fc2) and by the site instability (a). Using this expression of the largest eigenvalue, we obtain the free energy of our model as F(/3) = log((| a(l — k2) Q'3 + (ak2)@). From the free energy, the largest network Lyapunov exponent is derived as a function of the diffusive coupling strength k2 and the slope of the local maps a, according to the formula X=

-^F(/3,k2)\0=1,

(6)

- ' 1 - k2 log(| a(l - k2) |) + ak2 log(ak2) a | 1 — k2 I +ak2

(7)

which yields the final result

K{a,k2) —

Fig. 2a shows a contour plot of A n (a, k2), for identically coupled identical tent maps, over a range of {a, A;2}-values. In Fig. 2b, a cut through this contour plot is shown, at parameters that correspond to those used in the numerical simulations of the biologically motivated, variable coupled map lattice, displayed in Fig. 2c. The qualitative equivalence of the two approaches is easily seen. Numerical simulations of coupled parabola show furthermore that the displayed characteristic behavior is preserved even in the presence of non-hyperbolicities. As a function of the slope a of the local tent map (which corresponds to the local excitability K) and of the coupling strength k2, contour lines indicate the instability of the network patterns. As can be seen, due to the coupling, even for locally chaotic maps (a > 1), stable network patterns may evolve (often

114

in the form of statistical cycling, see [11])- Upon further increasing the local instability, chaotic network behavior of turbulent characteristics emerges.

Figure 2. a) Network Lyapunov exponent A n , describing the stability of patterns of a network of coupled tent maps, as a function of the absolute site map slope a and coupling k^. Contour lines of distance 0.25 are drawn, where the heavy line, indicating the location of zero Lyapunov exponents \ n — 0), separates stable networks (to the left) from unstable networks (to the right), b) Cut through the contour plot of a), at a = 1.25. c) Maximal site-Lyapunov exponent A m a x of a network of locked inhibitory site maps, as a function of the coupling £2- For the network, the local excitability is K = 0.5 for all sites, and Q is from the interval [0.8,0.85]. The behavior of this network closely follows the behavior predicted by the tent-map model. For some simulations, the peak at &2 = 1 is not so obvious. These cases are better modeled by a slightly modified behavior [4].

The stable patterns do not correspond to emergent macroscopic, comparable to synchronized, behavior. Therefore, in order to estimate the potential for synchronization, we need to concentrate on the parameter region where macroscopic patterns evolve, that is, on the statistical cycling regime. However, the parameter space that corresponds to this behavior, is very small, even for the tent map model. When we compare the model situation with our simulations from biologically motivated variable networks, we again observe that the overall picture provided by the tent map model of identical maps still applies. To show this in a qualitative manner, we compare the contour plot

115

of the tent map model with the numerically calculated Lyapunov exponent of the biological network, which shows the identical qualitative behavior. Based on our insight into the tent-map model behavior, we conclude that in the biologically motivated network, a notable degree of global synchronization would require a large subset of all binary connections to be in the chaotic interaction regime. This possibility, however, only exists for the inhibitory connections (excitatory connections are unable to reach this state [10]). Moreover, the part of the phase space on which the chaotic maps dwell, is rather small (although of nonzero measure, see [11]). It is then reasonable to expect that for the network including biological variability, statistical cycling is of vanishing measure, and therefore cannot provide a means of synchronizing neuron firing on a macroscopic scale. To phrase it more formally: this implies that by methods of self-organization, the network cannot achieve states of macroscopic synchronization. In addition, we also investigated whether Hebbian [13] learning rules acting on the weak connections between centers of stronger coupling could be a remedy for this lack of coherent behavior. Even with this additional mechanism, the model does not show macroscopic synchronization. The observation that the tent and the biological response site maps yield qualitative identical properties, has some additional bearing. In simple systems of nearest-neighbor coupled map lattices, it is found that the order parameter corresponding to the average phase would display a phase transition at high enough coupling strength, as the system is essentially equivalent to an Ising model at a finite temperature. This is qualitatively similar to our model, where a first order phase transition is observed at the coupling &2 = 1, for all values of the local instability.

Synchronization via thalamic input Layer IV's task is believed to be more centered around amplification and coordination, than on computation. Accurately modeling layer IV is more difficult, since, because of the smaller size of its typical neurons, it is difficult to measure in-vitro response profiles. Our model of the "amplifier" layer IV is based on biophysically detailed, variable model neurons that are connected in an all-toall fashion. This ansatz is partially motivated by the facts that layer IV is more densely connected than other layers and that natural, measured responses can easily be reproduced when using this setting. If synchronization - understood as an emergent, not a feed-forward property - is needed for computational and cognitive tasks, the question remains how this property is generated. In our simulations of biophysically detailed models of layer IV cortical architecture [3,14], we discovered a strong tendency of this layer to respond to stimula-

116

tion with coarse-grained synchronization. This synchronization is based on intrinsically non-bursting neurons that develop the bursting property, as a consequence of the recurrent network architecture and the feed-forward thalamic input. Detailed numerical simulations yield the result that, in the major part of the accessible parameter space, collective bursting emerges. That is, all individual neurons are collectivized, in the sense that, in spite of their individual characteristics, they all give rise to dynamics with very similar, on a coarse grained scale synchronized, characteristics (see Fig. 3). In fact, using methods of noise cleaning (noise, in this sense, are small variations due to the individual neuron characteristics), we find that the collective behavior can be represented in a four-dimensional model, having a strong positive (Ai ~ 0.5), a small positive, a zero and a very strong negative Lyapunov exponent. More explicitly, it is found that the basic behavior of the involved neuron types are identical and hyperchaotic [15]. The validity of the latter characterization has been checked by comparing the Lyapunov dimensions {(IKY ~ 3.5) with the correlation dimensions (d ~ 3.5). Moreover, different statistical tests have been performed to assess that noise-cleaning did not modify the statistical behavior of the system in an inappropriate way. As a function of the feed-forward input current, we observed an astonishing ability of the layer IV network to generate well-defined firing patterns.

i

i

i

i

i

i

i

i

i

kkk -80

-40

0 time lag [ms]

40

80

Figure 3. Coarse-grained synchronized activity of layer IV dynamics, where 80 excitatory and 20 inhibitory individual neurons are coupled in an all-to all fashion. The cross-correlogram between two excitatory neurons indicates strong synchronization.

117

When we compared the responses from the layer IV model with data from in vivo anesthetized cat (17 time series from 4 neurons of unspecified type from unspecified layers), we found corresponding behavior. Not only the Lyapunov exponents (generally hyperchaotic, Xmax ~ 0.8, the second exponent little above zero), correspond well to the simulated ones from the model {\max ~ 0.5, second exponent also little above zero). Also, the measured dimensions were in the range predicted by the model of layer IV; specific characteristic patterns found in vivo could be reproduced by our simulation model with ease. Of particular interest are step-wise structures found in the log-log-plots used for the evaluation of the dimensions [16] (see Fig. 1). However, as the majority of the measured in vivo neurons could not be attributed to layer IV, the natural hypothesis is that the other layers inherit these characteristics from the latter.

4

Complexity of network response and in vivo data

To investigate in more detail the relation between layer IV with the remaining layers, we calculated for the coupled map lattice model a recently proposed, biologically relevant complexity measure, Cs(l,0) [17]. To evaluate this quantity, we first calculated from the largest eigenvalue the free energy of the network, and from this quantity, C s ( l , 0 ) . We find that the highest complexities are situated in the area beyond the line separating negative and positive Lyapunov exponents (see Fig. 4, and compare with Fig. 2). Moreover, the area of highest complexity roughly coincides with the area where the model Lyapunov exponents agree with the in vivo measured exponents. This suggests that the natural system has the tendency of being tuned to a state of high complexity. These coincidences of modeling and experimental aspects lead us to believe that the ability of the network, to fire in well-separated characteristic time scales or in whole patterns, is not accidental, but serves to evoke corresponding responses by means of resonant cortical circuits. However, as has been mentioned above, not every neuron shares this property. In our recent studies of in vivo anesthetized cat data, we found, in evoked or spontaneous firing experiments, essentially three different classes of behavior. The neurons of the first class show no patterns in their firing at all. The neurons of the second class are able to pick up stimulation patterns and convert them into well-defined firing patterns. Neurons of the third class respond with smeared patterns that seem not to be compatible with the stimulation paradigms (for the first two classes see Fig. 1). With regard to the interspike distributions, for the first class, a long-tail behavior of the interspike distribution is characteristic. For the second class, a clean separation of the distribution into two regimes, dominated by individual interspike intervals and by compound pat-

118 2

1.5

k2

0.5

0 0

1

2

3

4

a Figure 4. Region in the parameter space (site map slope a and coupling £2), where the biologically relevant complexity measure C$(l,0) is maximal. The contour lines increase in steps of 0.1, starting from the line at the rigth upper corner with C s ( l , 0 ) = 0.1. From the comparison of Lyapunov exponents, it is suggestive that the measured biological neurons are tuned to working points of maximal complexity.

terns, respectively, is found. The characteristics of the last class indicate a mixture of the properties of the two other classes. In all cases, the behavior at long interspike interval times is governed, in the log-log plot, by a linear part, i.e., is long-tail. 5

Relevance of cortical chaos

Chaotic firing emerges from the proposed model, as well as from the in vivo data that we compare with, with nearly identical Lyapunov exponents and fractal dimensions. The agreement between Kaplan-Yorke and correlation dimensions [12] corroborates the consistency of the results obtained. The question then arises of what functional, possibly computational, relevance this phenomenon could be associated with? Cortical chaos essentially reflects the ability of the system to express its internal states (e.g., a result of computation) by choosing among different interspike intervals (ISI) or, more generally, among distinct patterns of firing. This mechanism can be viewed in a broader context. Chaotic dynamics is generated through the interplay of distinct unstable periodic orbits, where the system follows a particular orbit until, due to the instability of the orbit, the orbit is lost and the system follows another orbit, and so on. It is then natural to exploit this wealth of structures hidden within chaos, especially for technical applications. The task that needs to be solved to

119

do so, is the so-called targeting, and chaos control, problem: The chaotic dynamics first needs to be directed onto a desired orbit, on which it then needs to be stabilized, until another choice of orbit is submitted. From an informationtheoretic point of view, information content can be associated with the different periodic orbits. This view is related to beliefs that information is essentially contained in the patterns of neuronal firing. If well-resolved interspike intervals can be extracted from the spike trains of a neuron, the interspike lengths can directly be mapped onto symbols. A suitable transition matrix then specifies the allowed, and the forbidden, successions of interspike intervals. I.e., this transition matrix provides an approximation to the grammar of the natural system. In the case of collective bursting, it may be more useful to associate information content with firing patterns consisting of characteristic intermittent successions of spikes. In a broader context, the two approaches can be interpreted as realizations of a statistical mechanics description by means of different types of ensembles [18-19]. In the case of artificial systems or technical applications, strategies on how to use chaos to transmit messages, and more generally, information, are well developed. One basic principle used is that small perturbations applied to a chaotic trajectory are sufficient to make the system follow a desired symbol sequence containing the message [20]. This control strategy is based upon the property of chaotic systems known as "sensitive dependence on initial conditions" . Another approach, which is currently the focus of applications in areas of telecommunication, is the addition of hard limiters to the system's evolution [21-22]. This very robust control mechanism, can, due to its simplicity, even be applied to systems running at Giga-Hertz frequencies. It has been shown [23] that optimal hard limiter control leads to convergence onto periodic orbits in less than exponential time. In spite of these insights into the nature of chaos control, which kind of control measures should be associated with cortical chaos, however, is unclear. In the collective bursting case of layer IV, one possible biophysical mechanism would be a small excitatory post-synaptic current. When the membrane of an excitatory neuron is perturbed at the end of a collective burst with an excitatory pulse, the cell may fire additional spikes. Alternatively, at this stage inhibitory input may prevent the appearance of spikes and terminate bursts abruptly. In a similar way, also the firing of inhibitory neurons can be controlled. Another possibility, is the use of local recurrent loops to establish delay-feedback control [24]. In fact, such control loops could be one explanation for the abundantly occurring recurrent connections among neurons. The relevant parameters in this approach are the time delay of the re-fed signal, and the synaptic efficacy, where especially the latter seems biologically realistic. In addition to the encoding of information,

120

one also needs read-out mechanisms, able to decode the signal at the receiver's side. Thinking in terms of the encoding strategies outlined above, this would amount to the implementation of spike-pattern detection mechanisms. Besides simple straightforward implementations based on decay times, more sophisticated approaches, such as the recently discovered activity-dependent synapses [25-27], seem natural candidates for this task. Also the interactions of synapses, with varying degrees of short-term depression and facilitation, could provide the selectivity for certain spike patterns. Small populations of neurons, which (due to variable axonal, and synaptic potential propagation delays) achieve supra-threshold summation only for particular input spike sequences, is yet another possible mechanism.

6

Conclusion

We find that the origin of synchronized firing in the cortex - if its existence can be proven experimentally - would most likely be in layer IV, and be heavily based on recurrent connections and simultaneous thalamic feed-forward input. We expect that firing in patterns, in this layer, is able to trigger specific resonant circuits in other layers, where then the actual computation is done (which we propose to be based on the symbol set of an infinity of locked states [4]). It is a fact that the majority of ISI measurements from in vivo cat visual cortex neurons and simple statistical models of neuron interaction show emergent long-tail behavior, but this also might be the result of the interaction between different areas of the brain or a consequence of the input structure, or of a mixture of all of them. Long-tail interspike interval distributions are in full contrast to the current assumption of a Poissonian behavior that originates from the assumption of random spike coding. We propose that in the measurements that are compatible with the Poissonian spike train assumption, layer IV explicitly shuts down the long-range interactions via inhibitory connections or by pumping energy into new temporal scales that no longer sustain the ongoing activity. Long-tailed ISI distributions could, however, also be of relevance from the point of view of the tendency of the system of tuning itself to a state of maximal complexity Cs(l,0). At such a state, due to the structure of the measure, long range and slowly decaying correlations can be expected to dominate the dynamical behavior. Recently, it has been reported that noise can synchronize even coupled map lattices with variable lattice maps [28]. It will be worthwhile investigating whether this property also holds for our site maps (which we are willing to believe), and whether this finally can be attributed to a kind of mean-field activity generated by the noise. To what extent the assumptions made are the

121

relevant ones will have to be explored in continued work in the future, from the biological, as well as from the mathematical side.

References

1. Chaos and Noise in Biology and Medicine, eds. M. Barbi and S. Chillemi (World Scientific, Singapore, 1998). 2. J.K. Douglass, L. Wilkens, E. Pantazelou, and F. Moss, Nature 365, 337340 (1993). 3. R. Stoop, D. Blank, A. Kern, J.-J. v.d. Vyver, M. Christen, S. Lecchini, and C. Wagner, Cog. Brain Res., in press (2001). 4. R. Stoop, L.A. Bunimovich, and W.-H. Steeb, Biol. Cybern. 83, 481-489 (2000). 5. C. Von der Malsburg in Models of Neural Networks II, eds. E. Domany, J. van Hemmen, and K. Schulten, 95-119 (Springer, Berlin, 1994). 6. W. Singer in Large-Scale Neuronal Theories of the Brain, eds. C. Koch and J. Davis, 201-237 (Bradford Books, Cambridge MA, 1994). 7. R. Stoop, K. Schindler, and L.A. Bunimovich, Acta Biotheoretica 48, 149-171 (2000). 8. R. Stoop in Nonlinear Dynamics of Electronic Systems, eds. G. Setti, R. Rovatti, G. Mazzini, 278-282 (World Scientific, Singapore, 2000). 9. F.C. Hoppensteadt and E.M. Izhiekevich, Weakly Connected Neural Networks (Springer, New York, 1997). 10. R. Stoop, K. Schindler, and L.A. Bunimovich, Nonlinearity 13, 1515-1529 (2000). 11. J. Losson and M. Mackey, Phys. Rev. E 50, 843-856 (1994). 12. R. Stoop and R F . Meier, J. Opt. Soc. Am. B 5, 1037-1045 (1988); J. Peinke, J. Parisi, O.E. Roessler, and R. Stoop, Encounter with Chaos (Springer, Berlin, 1992). 13. D. Hebb, The Organization of Behavior (Wiley and Sons, New York, 1949). 14. D. Blank, PHD thesis (Swiss Federal Institute of Technology ETHZ, 2001). 15. O.E. Roessler, Phys. Lett. A 7 1 , 155-159 (1979). 16. A. Celletti and A. Villa, Biol. Cybern. 74, 387-393 (1996). 17. R. Stoop and N. Stoop, submitted (2001). 18. R. Stoop, J. Parisi, and H. Brauchli, Z. Naturforsch. a 46, 642-646 (1991). 19. C. Beck and F. Schloegel, Thermodynamics of Chaotic Systems: An Introduction (Cambridge University Press, Cambridge, 1993).

122 20. S. Hayes, C. Grebogi, E. Ott, and A. Mark, Phys. Rev. Lett. 73, 17811784 (1994). 21. N. Corron, S. Pethel, and B. Hopper, Phys. Rev. Lett. 84, 3835-3838 (2000). 22. C. Wagner and R. Stoop, Phys. Rev. E 63, 017201 (2000). 23. C. Wagner and R. Stoop, J. Stat. Phys. 106, 97-107 (2002). 24. K. Pyragas, Phys. Lett. A 170, 421-428 (1992). 25. L. Abbott, J. Varela, K. Sen, and S.B. Nelson, Science 275, 220-224 (1997). 26. M.V. Tsodyks and H. Markram, Proc. Natl. Acad. Sci. USA 94, 719-723 (1997). 27. A. M. Thomson, J. Physiol. 502, 131-147 (1997). 28. C. Zhou, J. Kurths, and B. Hu, Phys. Rev. Lett. 87, 098101 (2001).

123

SELECTIVITY P R O P E R T Y OF A CLASS OF E N E R G Y B A S E D L E A R N I N G RULES IN P R E S E N C E OF N O I S Y SIGNALS A.BAZZANI, D.REMONDINI Dep. of Physics

and Centro Interdipartimentale v.Irnerio 46 40126 Bologna, and INFN sezione di Bologna E-mail:

Galvani Univ. of ITALY [email protected]

Bologna,

N.INTRATOR Inst,

for Brain

and Neural Systems,

Brown

Univ., Providence

02912 RI,

USA

G.CASTELLANI DIMORFIPA

and Centro Interdipartimentale di Sopra 50, 4OO64 Ozzano

Galvani, Univ. of Bologna, dell'Emilia, ITALY

v.

Tolara

We consider the selectivity property of a class of energy based learning rules with respect to the presence of clusters in the input distribution. These rules are a generalization of the BCM learning rule and use the distribution momenta of order > 2. The analytical results show that selective solutions are possible for noisy input data up to a certain signal to noise ratio and that the introduction of a bias in the input signal could improve the selectivity. We illustrate this effect with some numerical simulations in a simple case.

1

Introduction to the B C M neuron model

The BCM neuron 1 has been introduces to analyze the plasticity property of a biological neuron. In particular the model takes into account the LTP (Long Term Potentiation) 7 and LTD (Long Term Depotentiation) 4 phenomena observed in the visual neuron response under modification of the experience. In its simplest formulation the BCM model assumes that the neuron response c to an external stimulus d 6 R " is linear: c = m • d where m are the synaptic weights. The changing of the weights m is described by the equation m = ${c,0)d

(1)

where 8 is an internal threshold of the neuron and the typical shape of the function $ is given by figure 1. At each time the external signal d can be considered the realization of a random variable of given distribution. If the threshold 9 is fixed the equilibrium position c = 9a is unstable and only the LTD behavior is described by eq. (1). To overcome this difficulty one introduces the hyphotesis that the threshold 6 depends on the external envi-

124

Figure 1. Typical shape of the function (c,#) that defines the BCM neuron.

ronment0, according to = < c2 > ' - *

(2)

where < > denotes the average with respect to the input distribution. Using the definition (2), we get a stochastic differential equation whose integration should be explained. Let At the integration step we have m{t + At) - m(t) + $(m(£) • d, 6)d At;

(3)

if the realizations of d are independent in the limit At —> 0, one can prove that m(t) satisfies the average equation 6 m=< $((m(t)

-d,0)d>

(4)

which substitutes the initial equation (1). A simple class of possible functions $ is given by ${c,6)=c(c

.p-2 _ 0)

and the average equation (4) reads rh = d£/dm the energy function

£ =

< c p > _ « V 2q

" T h e external environment is assumed to be stationary.

(5) where we have introduced

(6)

125

The case p = 3 and q — 2 is commonly referred as the BCM case. Due to the presence of high order momenta (p > 3) the energy function (6) provides a complex non-supervised analysis of the data, performing exploratory projection pursuit that seeks for multi-modality in data distributions 5 . The existence of stable equilibria is related to the existence of local minima for the energy (6). A simple calculation shows that the condition p < 2q is necessary for the existence of stable non-trivial fixed points in the equation (4), that performs a LTP behavior. We are interested in stable solutions that select different clusters eventually present in the data distribution. A ideal selective solution should give a high output c when the input d belongs to a single fixed cluster and c = 0 if d belongs to the other clusters. A general approach to study the stability property of the equilibrium points gives the following Proposition 2 : the stable equilibrium points m* of eq. (4) are related to the local maxima i/* of the homogeneous function Tvp f(y) = -j-

y

G

Rn

(7)

constrained on the unit sphere yCy = 1 where T is the symmetric p-tensor associated to the p-momenta of the input distribution Tilt...,in =

ii,...,in

= l,...,n

(8)

and C is the metric defined by the second moment matrix CV, = < didj >; the correspondence is explicitly given byTO*= pf(y*)y*-

2

Selective equilibria for the B C M neuron

According to a previous result 3 the BCM neuron has a selectivity property with respect to n linear independent vectors Vj e R". Let the input signal rfbea discrete random variable that takes values among the vectors Vj with equal probability l/n and let Oj = y • Vj, a standard calculation shows that the function (7) can be written in the form

126

constrained on the unit sphere V - o2- = n. Let us suppose oi > 0b, then by differentiating eq. (9) we get the system ^ + o r = 0 * * = - * j=2,..,n J aoj aoj o\ Then the critical points are computed from the equations o

r

0>r

2

-°r

2

)=°

j=2,.-,n

(10)

(ii)

It is easy to check the existence of a local maximum Oj = 0, j = 2, ..,n and O! = y/n so that the BCM neuron is able to select only the first vector i>i among all the possible input vector. According to the relation Oj = y • Vj the vectors y defined by the equilibrium solutions are directed as the dual base of the input vector Vj. We call this property the selectivity property of the BCM neuron 3 . We observe that in the value of selective solutions o are independent from the lengths and the orientation of the input vectors v; this is not true for the corresponding outputs c = m • v of the neuron whose values depend on the choice of the signals v. However the numerical simulations show that the basin of attraction of the stable selective solutions has a strong dependence from the norms of the vectors v; this effect makes very difficult to distinguish signals of different magnitude and we assume that a procedure could be applied in order to normalized the input signals. The situation becomes more complicate when we consider the effect of an additive noise on the selectivity property of a BCM neuron. Let us define the random variable

j

where £ is a random vector that selects with equal probability one of the vector Vj and 77 is a gaussian random vector where < rjj > = 0 and < rjiTjj >= Sijcr2 jn i,j = 1,.., n c . For seek of simplicity we consider the case p = 3, q = 2, but the calculations can be generalized in a straightforward way. It is convenient to introduce the matrix V whose lines are the vectors v and the positive define symmetric matrix S = VVT where VT is the transposed matrix. The cubic function (7) can be written in the form

/(°) = ^ E o , 3 + ^ ( ° - ^ l o ) I > k

(13)

Due to the parity property of the function (7) we can restrict our analysis to the subspace oj > 0 j = 1,.., n c This choice allows to keep constant the signal to noise ratio varying the dimensionality n.

127

and it is constrained on the ellipsoid n

YJo)+a2{o-S~1o)=n

(14)

3= 1

After some algebraic manipulations oi+az(S

L

o)i =

2

—-2

we get the equation \

ii

i=i

s

i=i

(15)

,2 +2^(5^0), ^ o , + n - X : < j=i

j=i

According to our ansatz the r.h.s. of eq. (15) is of order 0(a2)/oi can estimate 2 (0{l) oi ~ cH I — ^ + aO(l)oi j

so that we

(16)

where we have defined a = max; = 2,.., n IS/i*! to take into account the leading term of (S~1o)i and 0(a2) to denote a term of order a2. Eq.(16) shows that we can have a selective solution only if a n: b = Y^k=2 PkVk where the coefficients (3k satisfy to the equation ( v i -J^PkVk)

-vi = 0

l = 2,..n

(20)

The solution exists since V2,.-.,vn are linearly independent; it is straightforward to verify that vi — b is also orthogonal to the vector b. If one introduces the bias b, the matrix S is diagonalized in two blocks (the first block is the element Su) and a = 0; therefore we expect an increase of the selectivity for the input signal v\. We observe that different bias are necessary to increase

129

the selectivity with respect to the different input signals Vj and an exhaustive analysis of the input space would require a neural network of n neurons with inhibitory connections. The introduction of a bias b changes the norms of the input vectors so that it is necessary to apply a normalization procedure that could decrease the signal to noise ratio; this may destroy the advantages of a bias. At the moment it is not available an efficient procedure which automatically computes the bias; this problem is at present under consideration. 4

Numerical simulations

In order to show the selectivity property of a BCM neuron and the effect of a bias we have considered a the plane case; the input distribution has been defined by eq. (12) where the vectors v\,V2 lie on the unit circle at different angles a. We have normalized each input vector d on the unit circle so that the effect of noise enters only in the phase; the noise level a has been varied in the interval [0,1]. We study two cases for the energy function (6): p = 3 and g = 2 that corresponds to the BCM neuron and p = 4 and g = 3 that simulates kurtosis-like learning rule for the neuron. The initial synaptic weights are choosen in a neighborhood of the origin near the vertical axis. To quantify the separability we introduce the quantity A = \m • v2\ - \m • vi\

V2a\\m\

(21)

that measures the distance between the projections of the signals v\ and V2 along the direction of the stable solution m*. When A is > 1 we can detect the presence of two clusters in the input distribution with high probability. In the fig. 2 we report A as a function of noise level a in the case of a separation angle a — 90° and a = 10°; we have used a statistics of 106 input vectors. We observe that in the case of the BCM neuron and a = 90 the selectivity decreases in a sudden way at a certain value of the noise level. This effect is due to the presence of a critical noise level ac (see eq. (17)) at which the selective solutions bifurcate in the complex plane. In the case of the kurtosis-like neuron the presence of a critical value ac is not detected in the figure 2; this is a peculiar property of the kurtosis energy and of the relation between the second and the fourth momenta of a gaussian distribution. This is illustrates in the fig. 3 where we have plotted the neuron outputs o = y -v\ (black curve) and o = y • v? (red curve) in the case a = 90°: the presence of a bifurcation for the BCM neuron (left plot) is clear. However in the case a = 10° the selectivity of the kurtosis-like neuron is lost very soon (fig. 2

130

0

0.2

0.4

0.6

0

Figure 2. Separability property for a BCM (black circles) and kurtosis-like (red squares) neuron; the left plot refers to a separation angle a = 90° between the input signals whereas the right plot refers to a separation angle a — 10°; we have used a statistics of 10 6 input vectors and the threshold A = 1 is also plotted.

0

0

0.2

0.4

0.6

0.8

1

0.5

0

0.2

0.4

0.6

0.8

1

Figure 3. Normalized neuron outputs o = y • VJ j = 1,2 for the selective solution in the BCM case (left plot) and in the kurtosis-like case (right plot); the separation angle between the input vectors is a = 90°.

right); this is the effect of the appearance of a stable non-selective solution that attracts the neuron. We have checked the effect of a bias b that selects the second input V2 in the case of a separation angle a = 10°. The results are plotted in figure 4. The fig. 4 shows that the introduction of a bias increases the selectivity both for the BCM and kurtosis-like neuron; both the neurons loose their selectivity at a noise level a ~ .6, but the BCM neuron performs a better separation of the input clusters.

131 100

10

A

0.1

0.01 0.001 0

0.2

0.4

0.6

a

0.8

1

0

0.2

0.4

0.6

0.8

1

a

Figure 4. Comparison of the separability property without (circles) and with (squares) a bias for a separation angle a = 10° between the input vectors: the left plot refers to the BCM neuron whereas the right plot to the kurtosis-like neuron.

5

Conclusions

The analytical study of the selectivity property of neurons whose learning rules depend on the input distribution momenta of order > 3, suggests that a better performance could be achieved by using a bias in the input data. A numerical simulation on a simple example shows that this prevision is correct also for noisy input data. However further studies are necessary to understand the effect of a low statistics in the input data since the bias could decrease the signal to noise ratio. Moreover an algorithmic procedure to compute the bias is not yet available. References 1. E. L. Bienenstock and L. N Cooper and P. W. Munro,Journal of Neuroscience 2, 32 (1982). 2. A. Bazzani, D.Remondini,N. Intrator and G. Castellani,submitted to Neural Computation , (2001). 3. C.Castellani, N.Intrator, H.Shouval and L.N.Cooper, Network:Comput.Neural Syst. 10, 111 (1999). 4. Dudek, S. M. and Bear, M. F. Proc. Natl. Acad. Sci. 89, 4363 (1992). 5. Friedman, J. H. J. Am. Stat. Ass. 82, 249 (1997). 6. N. Intrator and L. N Cooper,Neural Networks 5, 3 (1992). 7. A. Kirkwood, H.-K. Lee and M. F. Bear, Nature 375, 328 (1995).

132 Pathophysiology of Schizophrenia: fMRI and Working Memory

GIUSEPPE BLASI AND ALESSANDRO BERTOLINO

Universita degli Studi di Bari, Dipartimento

di Scienze Neurologiche

e

Psichiatriche

P.zza Giulio Cesare, 11 -70124 - Bari, Italy E-mail: [email protected]

Functional Magnetic Resonance (fMRI) is an imaging technique with high spatial and temporal resolution that allows investigation of in vivo information about the functionality of discrete neuronal groups during their activity utilizing the magnetic properties of oxy- and deoxy-hemoglobin. fMRI permits the study of normal and pathological brain during performance of various neuropsychological functions. Several research groups have investigated prefrontal cognitive abilities (including working memory) in schizophrenia using functional imaging. Even if with some contradictions, large part of these studies have reported relative decrease of prefrontal cortex activity during working memory, defined as hypofrontality. However, hypofrontality is still one of the most debated aspects of the patophysiology of schizophrenia because the results can be influenced by pharmacotherapy, performance and chronicity. The first fMRI studies in patients with schizophrenia seemed to confirm hypofrontality. However, more recent studies during a range of working memory loads showed that patients are hypofrontal at some segments of this range, while they are hyperfrontal at others. These studies seem to suggest that the alterations of prefrontal functionality are not only due to reduction of neuronal activity, but they probably are the result of complex interactions among various neuronal systems.

Functional Magnetic Resonance Imaging (fMRI) Like its functional brain imaging forebears single photon emission tomography (SPECT) and positron emission tomography (PET), fMRI seeks to satisfy a longterm desire in psychiatry and psychology to define the neurophysiological (or functional) underpinnings of the so-called 'functional' illnesses. For much of the last century, attempts to define the 'lesions' causing these illnesses, such as schizophrenia, major depression and bipolar disorder, have been elusive, leading to their heuristic differentiation from 'organic' illnesses, like stroke and epilepsy, with more readily identifiable pathogenesis. FMRI offers several advantages in comparison to functional nuclear medicine techniques, including low invasiveness, no radioactivity, widespread availability

133

and virtually unlimited study repetitions [49]. These characteristics, plus the relative ease of creating individual brain maps, offer the unique potential to address a number of long-standing issues in psychiatry and psychology, including the distinction between state and trait characteristics, confounding effects of medication and reliability [80]. Finally, the implementation of 'realtime' fMRI will allow investigators to tailor examinations individually while a subject is still in the scanner, promising true interactive studies or 'physiological interviews' [26]. The physical basis of fMRI is the blood oxygenation level dependent (BOLD) effect, that is due to oxygenation-dependent magnetic susceptibility of hemoglobin . Deoxyhemoglobin is paramagnetic, causing slightly attenuated signals intensity in MRI image voxel containing deoxygenated blood. During neuronal firing, localized increases in blood flow oxygenation and consequently reduced deoxyhemoglobin cause the MRI signal to increase. It is therefore assumed that these localized increases in BOLD contrast reflect increases in neuronal activity. The BOLD mechanism has been further clarified by more recent experiments. By using imaging spectroscopy, which allows selective measurement of both deoxyhemoglobin and oxyhemoglobin, Malonek and Grinvald [52] demonstrated that hemoglobin-oxygenation changes in response to neuronal activation are biphasic: an early ( = I T ^ T

(?)

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The weights are updated according to the gradient descent learning rule9:

Aw?fw =-r,-^- +aAwff

(8)

where E is the error function E

= \ E K " - °M]2'

4.

5

Applications to real sequence data

As an application of the CMC method, we report the analysis of a sample of 202 subjects from the Pacific area, known to generate 89 haplotypes and which have been also thoroughly studied by anthropological point of view (results of the

205

anthropological analysis can be found in Tommaseo et al.). The dataset consisted of 89 sequences of 71 variant sites taken from the mtDNA hypervariable region HVRI. The CMC results are shown in Fig. 4 as a phenogram illustrating a temporal evolution where the time scale is fixed in terms of the resolution parameter . New clusters originate at each branching point, their cardinality being described by both a number and a circle of decreasing radius with cluster size. The final classification has been obtained with a distance weighted by site variability and external parameters k, 9, respectively set at &=10 and (9=0.35. The optimal parameters have been selected by applying the above described resampling method on 100 resamplings randomly extracted from the original dataset with r =0.75 . The optimal value of 9, corresponding to the terminal branching of the phenogram represents the final classification to be compared with results from known techniques. Clustering results were found to be consistent with anthropological data. The same sample has been investigated with other two methods , widely used in DNA sequence classification, the Neighbor Joining method, and the Reduced Median Network (RMN). The NJ method generates a tree starting from an estimate of the genetic distance matrix, here calculated by Stationary Markov Model. As for CMC results, the NJ tree evidenced three main subdivisions. The largest group of sequences (49 haplotypes) - identified as Group I - is clearly distinguished from the two other clusters, Group II and Group III (Table 1). The RMN method was used for deeper insight of haplotype genetic relationships. It generates a network which harbours all most parsimonious trees. The resultant network (data not shown) is quite complex, as a consequence of the high number of haplotypes considered in the analysis, while its reticulated structure reflects the high rate of homoplasy in the dynamic evolution of the mtDNA HVRI region. The topological structure of the RMN also evidences three major haplotype clusters, which reflects the same "haplotype composition" shown by the NJ tree that has been constructed on the distance matrix computed by the Stationary Markov Model (Table 1).

6

Conclusions

In this paper we propose a novel distance method for phylogeny reconstruction and sequence classification that is based on the recently proposed CMC algorithm, as well as on a biologically motivated definition of distance. The main advantage of the algorithm relies in high effectiveness and low computational cost which make it suitable for analysis of large amounts of data in high dimensional spaces. Simulations on artificial datasets show that CMC

206 algorithm outperforms, on average, the well known NJ method, in terms of measured Symmetric Distance between the true tree and the reconstructed one. ref.

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CITAK_317 UNA^94

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MAPPI_302 ASMAT_397 UNA_93

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Table 1. Comparison of classification results obtained on the Pacific area data set by the CMC method (V=site variability distance , E=entropic distance), Neighbor Joining performed on Stationary Markov Model (S) distance matrix, Reduced Median Network (R) Since we are dealing with a distance method of general applicability , any prior biological information has to be coded in an ad hoc distance definition, in order to improve the reliability of sequence grouping. That is the rationale for the

207

introduction of site variability and entropy terms in distance measures that account for the dependency of classification on the different rates of variation occurring on sites. Performances obtained by applying both distance definitions on two population datasets have been compared with classification obtained using SMM and Reduced Median Network [4]. We found that our method performs as well as the two known techniques but at lower complexity and computational costs. Moreover, compared to RMN, the method has the main advantage of providing an easy reading and interpretation of results regardless the dataset size. Further investigations are currently carried on regarding the use of CMC method for phylogenetic inference and the possibility to perform divergence time estimate by relating internal node depths of CMC trees to the estimated number of substitutions along lineages. Acknowledgements This work has been partially supported by the MURSTPRIN99 and by "program Biotecnologie, legge 95/95 (MURST 5%)"- Italy. References 1. Anderson, S., A. T. Bankier, B. G. Barrell, M. H. L. de Bruijn, A. R. Coulson, et al., 1981 Sequence and organization of the human mitochondrial genome. Nature 290:457-465. 2. Angelini, L., F. De Carlo, C. Marangi, M. Pellicoro and S. Stramaglia, 2000 Clustering data by inhomogeneous chaotic map lattices. Phys. Rev. Letters 85(3): 554-557. 3. Angelini, L., F. De Carlo, M. Mannarelli, C. Marangi, G. Nardulli, M. Pellicoro, G. Satalino, S. Stramaglia, 2001 Chaotic neural network clustering: an application to landmine detection by dynamic infrared imaging. Optical Engineering Volume 40, Issue 12, pp. 2878-2884. 4. Bandelt, H. J., P. Forster, C. S. Bryan and M. B. Richards, 1995 Mitochondrial portraits of human population using median network. Genetics 141: 743-753. 5. Felsenstein, J., 1993 PHYLIP (Phylogeny Inference Package), Department of Genetics, University of Washington, Seattle. 6. Gilbert D.G., IUBio Archive for Biology Data and Software, USA. 7. Hasegawa, M. and Yano T., 1984 Maximum likelihood method of phylogenetic inference from DNA sequence data. Bulletin of the Biometric Society of Japan 5:1-7. 8. Hasegawa, M. and S. Horai, 1991 Time of the deepest root for polymorphism in human mitochondrial DNA. J. Mol. Evol. 32(l):37-42.

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9. 10. 11.

12. 13.

14.

15. 16.

17. 18. 19.

Lanave, C , G. Preparata, C. Saccone and G. Serio, 1984 A new method for calculating evolutionary substitution rates. J. Mol. Evol. 20:86-93. Levine, E. and E. Domany, 2000 Resampling Method For Unsupervised Estimation Of Cluster Validity. Preprint arXiv:physics/0005046, 18 May 2000. Manrubia, S.C. and C. Mikhailov, 1999 Mutual Synchronization and Clustering in Randomly Coupled Chaotic Dynamical Networks, Phys. Rev. E 60:15791589. Pagel, M., 1999 Inferring the historical patterns of biological evolution. Nature 401: 877-884. Pesole, G. and C. Saccone, 2001 A novel method to estimate substitution rate variation among sites in large dataset of homologous DNA sequences. Genetics 157(2):859-865. Rambaut, A. and Grassly, N. C. (1997) Seq-Gen: An application for the Monte Carlo simulation of DNA sequence evolution along phylogenetic trees. Comput. Applic. Biosci., 13: 235-238. Robinson, D. F., and L. R. Foulds. 1981. Comparison of phylogenetic trees. Math. BioSci. 53:131-147. Saccone, C , C. Lanave, G. Pesole and G. Preparata, 1990 Influence of base composition on quantitative estimates of gene evolution. Meth. Enzymol. 183:570-583. Saitou, N. and M. Nei, 1987 The neighbor-joining method: a new method for reconstructing phylogenetic trees. Mol. Biol. Evol. 4:406-425. Sole, R.V., S.C. Manrubia, J. Bascompte, J. Delgado and B. Luque, 1996 Phase transitions and complex systems. Complexity 4:13-26. Wiggins, S., 1990 Introduction to applied nonlinear dynamical systems and chaos. Springer, Berlin.

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F I N D I N G REGULATORY SITES F R O M STATISTICAL ANALYSIS OF N U C L E O T I D E F R E Q U E N C I E S I N T H E U P S T R E A M R E G I O N OF E U K A R Y O T I C G E N E S

a

M. Caselle" and P. Provero a ' 6 Dipariimento di Fisica Teorica, Universita di Torino, and INFN, sezione di Torino, Via P. Giuria 1, 1-10125 Torino, Italy. e-mail: [email protected], [email protected] Dipartimento di Scienze e Tecnologie Avanzate, Universita del Piemonte Orientale, 1-15100 Alessandria, Italy.

Dipartimento

F. Di Cunto and M. Pellegrino di Genetica, Biologia e Biochimica, Universita di Torino, Via Santena 5 bis, I-10100, Torino, Italy. e-mail: [email protected]

We discuss two new approaches to extract relevant biological information on the Transcription Factors (and in particular to identify their binding sequences) from the statistical distribution of oligonucleotides in the upstream region of the genes. Both the methods are based on the notion of a "regulatory network" responsible for the various expression patterns of the genes. In particular we concentrate on families of coregulated genes and look for the simultaneous presence in the upstream regions of these genes of the same set of transcription factor binding sites. We discuss two instances which well exemplify the features of the two methods: the coregulation of glycolysis in Drosophila melanogaster and the diauxic shift in Saccharomyces cerevisiae.

1

Introduction

As more and more complete genomic sequences are being decoded it is becoming of crucial importance to understand how the gene expression is regulated. A central role in our present understanding of gene expression is played by the notion of "regulatory network". It is by now clear that a particular expression pattern in the cell is the result of an intricate network of interactions among genes and proteins which cooperate to enhance (or depress) the expression rate of the various genes. It is thus important to address the problem of gene expression at the level of the whole regulatory network and not at the level of the single gene 1 ' 2 ' 3,4 ' 5 . In particular, most of the available information about such interactions concerns the transcriptional regulation of protein coding genes. Even if this is not the only regulatory mechanism of gene expression in eukaryotes it is certainly the most widespread one.

210

In these last years, thanks to the impressive progress in the DNA array technology several results on these regulatory networks have been obtained. Various transcription factors (TF's in the following) have been identified and their binding motifs in the DNA chain (see below for a discussion) have been characterized. However it is clear that we are only at the very beginning of such a program and that much more work still has to be done in order to reach a satisfactory understanding of the regulatory network in eukaryotes (the situation is somehow better for the prokaryotes whose regulatory network is much simpler). In this contribution we want to discuss a new method which allows to reconstruct these interactions by comparing existing biological information with the statistical properties of the sequence data. This is a line of research which has been pursued in the last few years, with remarkable results, by several groups in the world. For a (unfortunately largely incomplete) list of references s e e 2,3,4,5,6,7,8,9 j n p a r t i c u i a r the biological input that we shall use is the fact that some genes, being involved in the same biological process, are likely to be "coregulated" i.e. they should show the same expression pattern. The simplest way for this to happen is that they are all regulated by the same set of TF's. If this is the case we should find in the upstream" region of these genes the same T F binding sequences. This is a highly non trivial occurrence from a statistical point of view and could in principle be recognized by simple statistical analysis. As a matter of fact the situation is much more complex than what appears from this idealized picture. TF's not necessarily bind only to the upstream region. They often recognize more than one sequence (even if there is usually a "core" sequence which is highly conserved). Coregulation could be achieved by a complex interaction of several TF's instead than following the simple pattern suggested above. Notwithstanding this, we think that it is worthwhile to explore this simplified picture of coregulation, for at least three reasons. • Even if in this way we only find a subset of the TF's involved in the coregulation, this would be all the same an important piece of information: It would add a new link in the regulatory network that we are studying. • Analyses based on this picture, being very simple, can be easily performed on any gene set, from the few genes involved in the Glycolysis (the first example that we shall discuss below) up to the whole genome (this will be the case of the second example that we shall discuss). This "With this term we denote the portion of the DNA chain which is immediately before the starting point of the open reading frame (ORF). We shall characterize this region more precisely in sect.3 below.

211

feature is going to be more and more important as more and more DNA array experiment appear in the literature. As the quantity of available data increases, so does the need of analytical tools to analyze it. • Such analyses could be easily improved to include some of the features outlined above, keeping into account, say, the sequence variability or the synergic interaction of different TF's. To this end we have developed two different (and complementary) approaches. The first one (which we shall discuss in detail in sect.3 below) follows a more traditional line of reasoning: we start from a set of genes which are known to be coregulated (this is our "biological input") and then try to recognize the possible binding sites for the TF's. We call this approach the "direct search" for coregulating TF's. The second approach (which we shall briefly sketch in sect.4 below and is discussed in full detail in 1 0 ) is completely different and is particularly suitable for the study of genome-wide DNA array experiments. In this case the biological input is taken into account only at the end of the analysis. We start by organizing all the genes in sets on the basis of the overrepresented common sequences and then filter them with expression patterns of some DNA array experiment. We call this second approach the "inverse search" for coregulating TF's. It is clear that all the candidate gene interactions which we identify with our two methods have to be tested experimentally. However our results may help selecting among the huge number of possible candidates and could be used as a preliminary test to guide the experiments. This contribution is organized as follows. In sect.2 we shall briefly introduce the reader to the main features of the regulatory network (this introduction will necessarily be very short, the interested reader can find a thorough discussion for instance in n ) . We shall then devote sect.3 and 4 to explain our "direct" and "inverse" search methods respectively. Then we shall discuss two instances which well exemplify the two strategies. First in sect.5 we shall study the coregulation of glycolysis in Drosophila melanogaster. Second, in sect.6 we shall discuss the diauxic shift in Saccharomyces cerevisiae. The last section will be devoted to some concluding remarks. 2

Transcription factors

As mentioned in the introduction, a major role in the regulatory network is played by the Transcription Factors, which may have in general a twofold action on gene transcription. They can activate it by recruiting the transcription

212 machinery to the transcription starting site by binding enhancer sequences in the upstream noncoding region, or by modifying chromatine structure, but they can also repress it by negatively interfering with the transcriptional control mechanisms. The main point is that in both cases TFs act by binding to specific, often short DNA sequences in the upstream noncoding region. It is exactly this feature which allows TF's to perform a specific regulatory functions. These binding sequences can be considered somehow as the fingerprints of the various TF's. The main goal of our statistical analysis will be the identification and characterization of such binding sites. 2.1

Classification

Even if TF's show a wide variability it is possible to try a (very rough) classification. Let us see it in some more detail, since it will help understanding the examples which we shall discuss in the following sections. There are four main classes of binding sites in eukaryotes. • Promoters These are localized in the region immediately upstream of the coding region (often within 200 bp from the transcription starting point). They can be of two types: - short sequences like the well known CCAAT-box, TATA-box, GCbox which are not tissue specific and are recognized by ubiquitous TFs — tissue specific sequences which are only recognized by tissue specific TFs • Response Elements These appear only in those genes whose expression is controlled by an external factor (like hormones or growing factors). These are usually within lkb from the transcription starting point. Binding of a response element with the appropriate factor may induce a relevant enhancement in the expression of the corresponding gene • Enhancers these are regulatory elements which, differently from the promoters, can act in both orientations and (to a large extent) at any distance from the transcription starting point (there are examples of enhancers located even

213

50-60kb upstream). They enhance the expression of the corresponding gene. • Silencers Same as the enhancers, but their effect is to repress the expression of the gene. 2.2

Combinatorial regulation.

The main feature of TF's activity is its "combinatorial" nature. This means that: • a single gene is usually regulated by many independent TF's which bind to sites which may be very far from each other in the upstream region. • it often happens that several TF's must be simultaneously present in order to perform their regulatory function. This phenomenon is usually referred to as the "Recruitment model for gene activation" (for a review see x) and represents the common pattern of action of the TF's. It is so important that it has been recently adopted as guiding principle for various computer based approaches to detect regulatory sites (see for instance 4 ). • the regulatory activity of a particular TF is enhanced if it can bind to several (instead of only one) binding sites in the upstream region. This "overrepresentation" of a given binding sequence is also used in some algorithms which aim to identify TF's. It will also play a major role in our approach. 3

The "direct" search method

In this case the starting point is the selection of a set of genes which are known to be involved in the same biological process (see example of sect. 5). Let us start by fixing a few notations: • Let us denote with M the number of genes in the coregulated set and with gi, i = 1 • • • M the genes belonging to the set • Let us denote with L the number of base pairs (bp) of the upstream non coding region on which we shall perform our analysis. It is important to define precisely what we mean by "upstream region" With this term we denote the non coding portion of the DNA chain which is immediately before the transcription start site. This means that we do not consider as

214

part of this region the UTR5 part of the ORF of the gene in which we are interested. If we choose L large enough it may happen that other ORFs are present in the upstream region. In this case we consider as upstream region only the non coding part of the DNA chain up to the nearest ORF (even if it appears in the opposite strand). Thus L should be thought of as an upper cutoff. In most cases the length of the upstream region is much smaller and is gene dependent. We shall denote it in the following as L(g). • In this upstream region we shall be interested in studying short sequences of nucleotides which we shall call words. Let n be the length of such a word. For each value of n we have N = 4" possible words Wi,i = 1 • • • N. The optimal choice of n (i.e. the one which optimize the statistical significance of our analysis) is a function of L and M. We shall see some typical values in the example of sect.5 In the following we shall have to deal with words of varying size. When needed, in order to avoid confusion, we shall call A;-word a word made of k nucleotides. Let us call U the collection of upstream regions of the M genes p i , . . . QM- Our goal is to see if the number of occurrences of a given word W{ in each of the upstream regions belonging to U shows a "statistically significant" deviation (to be better defined below) from what expected on the basis of pure chance. To this end we perform two types of analyses. First level of analysis This first type of analysis is organized in three steps ^Construction of the "Reference samples". The first step is the construction of a set of p "reference samples" which we call Ri,i = 1, • • -p. The Ri are nonoverlapping sequences of LR nucleotides each, extracted from a noncoding portion of the DNA sequence in the same region of the genome to which the genes that we study belong but "far" from any ORF. From these reference samples we then extract for each word the "background occurrence probability" that we shall then use as input of the second step of our analysis. The rationale behind this approach is the idea that the coding and regulating parts of the genome are immersed in a large background sea of "silent" DNA and that we may recognize that a portion of DNA has a biological function by looking at statistical deviations in the word occurrences with respect to the background.

215

However it is clear that this is a rather crude description of the genome, in particular there are some obvious objections to this approach: — There is no clear notion of what "far" means. As we mentioned in the introduction one can sometimes find TF's which keep their regulatory function even if they bind to sites which are as far as ~ 50kb from the ORF — It is possible that in the reference samples the nucleotide frequencies reflect some unknown biological function thus inducing a bias in the results — It is not clear how should one deal with the long repeated sequences which very often appear in the genome of eukaryotes We shall discuss below how to overcome these objections. Background probabilities. For each word w we study the number of occurrences n(w, i) in the ith sample. They will follow a Poisson distribution from which we extract the background occurrence probability of the word. This method works only if p and LR are large enough with respect to the number of possible words N (we shall see in the example below some typical values for p and LR). However we have checked that our results are robust with respect to different choices of these background probabilities. • Significant words. From these probabilities we can immediately construct for each n-word the expected number of occurrences in each of the upstream sequences of U and from them the probabilities p(n, s) of finding at least one n-word simultaneously present in the upstream region of s (out of the M) genes. By suitably tuning L, s and n we may reach very low probabilities. If notwithstanding such a low probability we indeed find a n-word which appears in the upstream region of s genes then we consider this fact as a strong indication of its role as binding sequence for a TF's. We may use the probability p(n, s) as an estimate of the significance of such a candidate binding sequence. As we have seen the critical point of this analysis is in the choice of the reference sample. We try to avoid the bias induced by this choice by crossing the above procedure with a second level of analysis Second level of analysis

216

The main change with respect to the previous analysis is that in this case we extract the reference probabilities for the n-words from an artificial reference sample constructed with a Markov chain algorithm based on the frequencies of k-words with k £ .

A?;,.,

(l)

where To is the known native state, T is a trial structure that has the same length of T 0 . A s is the contact matrix of structure 5 , whose element Ay is 1 if residues i and j are in contact in the native state (i.e. their Ca separation is below the cutoff r = 6.5 A) and 0 otherwise. This symmetric matrix encodes the topology of the protein. The energy-scoring function of Eq. (1) ensures that the state of lowest energy is attained in correspondence of structures with the same contact map of To- This, in principle, may lead to a degenerate ground state since more than one structure can be compatible with a given contact matrix. In practice, however, unless one uses unreasonably small values of r, the degenerate structures are virtually identical. In fact, for r K, 6.5 A the number of distinct contacts is about twice the protein length; this number of constraints nicely matches the number of degrees of freedom of the peptide (two dihedral angles for each non-terminal CA), thus avoiding both under- and over-constraining the ground states. The introduction of this type of topology-based folding models can be traced back to the work of Go and Scheraga 9 . For a long time, the interesting property of these systems was the presence of an all-or-none folding process, that is the finite-size equivalent of first-order transitions in infinite systems. This is illustrated in the example of Fig. 1 where we have reported energy and specific heat of the model applied to the target protein 1HJA; the code refers to the Protein Data Bank tag. The plotted data were obtained through stochastic (Monte Carlo) equilibrium (constant-temperature) samplings. It is interesting to note the presence of a peak, that can be identified with the folding transition of the model system. At the peak, about 50 % of the native structure (measured as the fraction of formed native contacts 16,17 ) is formed, consistently with analogous results on different proteins 15 . It is, however, possible to investigate the equilibrium properties of the system in finer

237

1hja -20 -30

-50 -60

100

80

d<E>/dT

60 40 20 0

Figure 1. Plots of the energy (top) and specific heat (bottom) as a function of temperature for protein lhja. The curves were obtained through histogram reweighting techniques.

detail, for example examining the probabilities of individual native contacts to be formed at the various temperatures. Naturally, at high temperatures all contacts will be poorly formed, while at sufficiently low temperatures they will all be established. It is then tempting, and physically appealing, to draw an analogy between this progressive establishment of native structure and the one observed in a real folding process. However, in principle, the equilibrium properties of our model need not parallel the dynamical ones of the real system. Thus, it was a striking surprise when we established that, indeed, a qualitative and even quantitative connection between the two processes could be drawn 10 . In the past years other groups have used similar or alternative techniques to elucidate the role of the native state topology in the folding process n. 12 . 13 . 14 ^ confirming the picture outlined here. An initial validation of this strategy was carried out by considering two target proteins, chymotrypsin inhibitor and barnase, that have been widely investigated in experiments. For each of them we generated several hundred structures having about 40 % native content. It turned out that the most frequent contacts shared by the native conformation of 2ci2 with the others involved the helical-residues 30-42 (see Fig. 2). Contacts involving such

238

residues were shared by 56% of the sampled structures. On the other hand, the rarest contacts pertained to interaction between the helix and /?-strands and between the /3-strands themselves. A different behaviour (see Fig. 2) was found for barnase, where, again, for overlap of « 40%, we find many contacts pertaining to the nearly complete formation of helix 1 (residues 8-18), a partial formation of helix 2, and bonds between residues 26-29 and 29-32 as well as several non-local contacts bridging the /^-strands, especially residues 51-55 and 72-75.

Figure 2. Ribbon plot (obtained with RASMOL) of 2ci2 (left) and barnase (right). The residues involved in the most frequent contacts of alternative structures that form « 40% of the native interactions are highlighted in black. The majority of these coincide with contacts that are formed at the early stages of folding.

Both this picture, and the one described for CI2 are fully consistent with the experimental results obtained by Fersht and co-workers in mutagenesis experiments 18 ' 19 . In such experiments, the key role of an amino acid at a given site is probed by mutating it and measuring the changes in the folding and equilibrium characteristics. By measuring the change of the folding/unfolding equilibrium constant one can introduce a parameter, termed Rvalue, which is zero if the mutation is irrelevant to the folding kinetics and 1, if the change in folding propensity mirrors the change in the relative stability of the folded and unfolded states (intermediate values are, of course, possible). Ideally, the measure of the sensitivity to a given site should be measured as a suitable susceptibility to a small perturbation of the same site (or its environment).

239 Unfortunately, this is not easily accomplished experimentally, since substitution by mutation can be rarely regarded as a perturbation. Notwithstanding this difficulty, from the analysis of the Rvalues obtained by Fersht, a clear picture for the folding stages of CI2 and barnase emerges. In both cases, the crucial regions for both proteins are the same as those identified through the analysis of contact formation probability reported above. This provides a sound a posteriori justification that it is possible to extract a wealth of information about the sites involved in crucial stages of the folding process. Despite the fact, that such sites are determined from the analysis of their crucial topological role with respect to the native state with no input of the actual protein composition, they correlate very well with the key sites determined experimentally. A striking example is provided in the following subsection, which focuses on an enzyme encoded by the HIV virus. It the following we shall show that from the mere knowledge of the contact map of the enzyme, one can isolate a handful of important sites which correlate extremely well the key mutating sites determined in clinical trials of anti-AIDS drugs. 2.1

Application to HIV-1 protease: drug resistance and folding Pathways

To further corroborate the validity of the proposed model in capturing the most delicate folding steps we consider an application to an important enzyme, the protease of the HIV-1 virus (pdb code laid), which plays an essential role in the spreading of the viral infection. Through extensive clinical trials 20 , it has been established that there is a well-defined set of sites in the enzyme that are crucial for developing, through suitable mutations, resistance against drugs and which play a crucial role in the folding process 21 . To identify the key folding sites we looked for contacts whose establishment further enhances the probability of other contacts to be formed. A possible criterion to identify such contacts is through their contribution to the overall specific heat. At a fixed temperature, T, the average energy of the system described by the Hamiltonian (1) can be written as:

(E(T)) = -J2^PiAT)

(2)

'J

where the pij(T) is the equilibrium probability of residues i and j to be in contact. Hence, the specific heat of the system is:

c; m = Ǥ23>__ EA jj*affi.

(3,

240

Thus, contribution of the various contacts to the specific heat will be then proportional to how rapidly the contact is forming as temperature is lowered. The contacts relevant for the folding process, will be those giving the largest contribution to Cv at (or above) the folding transition temperature. Armed with this insight, we can use this deterministic criterion to rank the contacts in order of importance. Our simulations on the protease of HIV-1 21 , are based on an energyscoring function that is more complex than Eq. (1). As usual, amino acids are represented as effective centroids placed on Ca atoms, while the peptide bond between two consecutive amino acids, i, i + 1 at distance r ^ + i is described by the anharmonic potential adopted by Clementi et al. 22 , with parameters a = 20, 6 = 2000. The interaction among non-consecutive residues is treated again in Go-like schemes9 which reward the formation of native contacts with a decrease of the energy scoring function. Each pair of non-consecutive amino acids, i and j , contributes to the energy scoring function by an amount:

Votf? »j

+ 5V1(l-ArHj)(j^j

,

(4)

where ro = 6.8A, r^- denotes the distance of amino acids i and j in the native structure and A r ° is the native contact matrix built with an interaction cutoff, r, equal to 6.5 A. Vo and \\ are constants controlling the strength of interactions (VQ = 20, V\ = 0.05 in our simulations). Constant temperature molecular dynamics simulations were carried out where the equations of motion are integrated by a velocity-Verlet algorithm combined with the standard Gaussian isokinetic scheme 23,21 . Unfolding processes can be studied within the same framework by warming up starting from the native conformation (heat denaturation). The free-energy, the total specific-heat, Cv, and contributions of the individual contacts to Cv were obtained combining data sampled at different equilibrium temperatures with multiple histogram techniques 24 . The thermodynamics quantities obtained through such deconvolution procedures did not depend, within the numerical accuracy, on whether unfolding or refolding paths were followed. The contacts that contribute more to the specific heat peak are identified as the key ones belonging to the folding bottleneck and sites sharing them as the most likely to be sensitive to mutations. Furthermore, by following several individual folding trajectories (by suddenly quenching unfolded conformations below the folding transition temperature, Tf0u) we ascertained that all such

241

dynamical pathways encountered the same kinetic bottlenecks determined as above. For the ft sheets, the bottlenecks involve amino acids that are typically 3-4 residues away from the turns - specifically, residues 61, 62, 72, 74 for ft, 10, 11, 12, 21, 22, 23 for ft and 44, 45, 46, 55, 56, 57 for ft. At the folding transition temperature, T/ 0 w, the formation of contacts around residues 30 and 86 is observed. The largest contribution to the specific heat peak is observed from contacts 29-86 and 32-76 which are, consequently, identified as the most crucial for the folding/unfolding process, and denote this set as the "transition bottleneck", (TB). Such sites are physically located at the active site of HIV-1 PR, which is targeted by anti AIDS drugs 25 . Hence, within the limitations of our simplified approach, we predict that changes in the detailed chemistry at the active site also ruin key steps of the folding process. To counteract the drug action, the virus has to perform some very delicate mutations at the key sites; within a random mutation scheme this requires many trials (occurring over several months). The time required to synthesize a mutated protein with native-like activity is even longer if the drug attack correlates with several bottlenecks simultaneously. This is certainly the case for several anti-AIDS drugs. Indeed Table 1 summarizes the mutations for the FDA approved drugs 20 . In Table 2, we list the sites taking part to the three most important contacts in each of the four bottlenecks TB, ft, ft and ft. Remarkably, among the first 23 most crucial sites predicted by our method, there are 6 sites in common with the 16 distinct mutating sites of Table 1. The relevance of these matches can be assessed by calculating the probability of occurrence by chance. By using simple combinatorial calculations, it is found that the probability to observe at least 6 matches with the key sites of Table 1 by picking 12 contacts at random among the native ones is approximately 1 %. This result highlights the high statistical correlation between our prediction and the evidence accumulated from clinical trials. In conclusion, the strategy presented here, which is entirely based on the knowledge of the native structure of HIV-1 protease, allows one both to identify the bottlenecks of the folding process and to explain their highly significant match with known mutating residues 21 . This and similar approaches should be applicable to identify the kinetic bottlenecks of other viral enzymes of pharmaceutical interest. This could allow a fast development of novel inhibitors targetting the kinetic bottlenecks. This is expected to dramatically enhance the difficulty for the virus to express mutated proteins which still fold efficiently into the same native state with unaltered functionality.

242

Name

Point Mutations

Bottlenecks

R T N 26.27

20, 33, 35, 36, 46, 54, 63, 71, 82, 84, 90 30, 46, 63, 71, 77, 84, 10, 32, 46, 63,71, 82, 84 10, 46, 48, 63, 71, 82, 84, 90 46, 63, 82, 84

TB, 01,02,03

NLF28 I N D 29.30 S Q V 29.30,31

APR32

TB, 0 2 , 0 3 TB, 01,02,03

TB,0i,0 2 ,03 TB, 0 2 , 03

Table 1. Mutations in the protease associated with FDA-approved drug resistance 2 0 . Sites highlighted in boldface are those involved in the folding bottlenecks as predicted by our approach. Pi refers to the bottleneck associated with the formation of the i-th /3-sheet, whereas T B refers to the bottleneck occurring at the folding transition temperature Tf0id (see next Table).

Bottleneck TB 0i 02

ft

Key sites 22, 29, 32, 76, 84, 86 10,11,13,20,21,23 44,45,46,55,56,57 61,62,63,72,74

Table 2. Key sites for the four bottlenecks. For each bottleneck, only the sites in the top three pairs of contacts have been reported.

3

Optimal shape of a compact polymeric chain

Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest packing fraction; only recently has it been proved 33 ' 34 that the answer for infinite systems is a succession of tightly packed triangular layers, as conjectured by Kepler several centuries ago. This problem has had a profound impact in many areas ranging from the crystallization and melting of atomic systems, to optimal packing of objects and subdivision of space 33,34,35,36,37 The close-packed hard sphere problem is simply stated: given N hard spheres of radius R, how should we arrange them so that they can be fit in the box with smallest possible side, IP. Interestingly, the role of R and L can be reversed in the following alternative, but equivalent, formulation: given a set of N points inside a box of side L, how should we arrange them so that the spheres centred in them have the (same) maximum radius, R? Also in this second case, as in the first one, the spheres are not allowed to self intersect or

243

cross the box boundaries. Here we study an analogous problem, that of determining the optimal shapes of closely packed compact strings. This problem is a mathematical idealization of situations commonly encountered in biology, chemistry and physics, involving the optimal structure of folded polymeric chains. Biopolymers like proteins have three dimensional structures which are rather compact. Furthermore, they are the result of evolution and one may think that their shape may satisfy some optimality criterion. This naturally leads one to consider a generalization of the packing problem of hard spheres to the case of flexible tubes with a uniform cross section. The packing problem then consists in finding the tube configuration which can be enclosed in the minimum volume without violating any steric constraints. As for the "free spheres" case, also this problem admits a simple equivalent re-formulation that we found more apt for numerical implementation. More precisely we sought the curve which is the axis, or centerline, of the thickest tube (the analog of the sphere centers in the hard sphere packing problem) that can be confined in the pre-assigned volume 38 . The maximum thickness associated to a given centerline is elegantly defined in terms of concepts recently developed in the context of ideal knot shapes 3 9 ' 4 0 ' 4 1 , 4 2 ' 4 3 ' 4 4 . The thickness A denotes the maximum radius of a uniform tube with the string passing through its axis, beyond which the tube either ceases to be smooth, owing to tight local bends, or it self-intersects. The presence of tight local bends is revealed by inspecting the local radius of curvature along the centerline. In our numerical attempt to solve the problem, our centerline was represented as a succession of equidistant beads. The local radius of curvature was then measured as the radius of the circumcircle going through three consecutive points. Remarkably, the same framework can be used to deal with the non-local restrictions to the maximum thickness occurring when two points, at a finite arclength separation, come in close approach. In this case one can consider the smallest radius of circles going through any non-local triplet of points. When both local and non-local effects are taken into account, one is naturally lead to define the thickness of the chain by considering all triplets of particles and selecting the smallest among all the radii 42 . For smooth centerlines, an appreciable reduction of the complexity of the algorithm can be found by considering only triplets where at least two of the points are consecutive 42 . Besides this intrinsic limitations to the thickness, one also needs to consider the extrinsic ones due to the presence of a confining geometry. In fact, the close proximity of the centerline to the walls of the confining box may further limit the maximum thickness.

244

As for the packing of free spheres, also the present case is sensitive to the details of the confining geometries when the system is finite. An example of the variety of shapes resulting from the choice of different confining geometries is given in Fig. 3.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3. Examples of optimal strings. The strings in the figure were obtained starting from a random conformation of a chain made up of N equally spaced points (the spacing between neighboring points is defined to be 1 unit) and successively distorting the chain with pivot, crankshaft and slithering moves. A stochastic optimization scheme (simulated annealing) is used to promote structures that have larger and larger thickness. Top row: optimal shapes obtained by constraining strings of 30 points with a radius of gyration less than R. a) R = 6.0, A = 6.42 b) R = 4.5, A = 3.82 c) R = 3.0, A = 1.93. Bottom row: optimal shapes obtained by confining a string of 30 points within a cube of side L. d) L = 22.0, A = 6.11 e) L = 9.5, A = 2.3 f) L - 8.1, A = 1.75.

In order to reveal the "true" bulk solution one needs to adopts suitable boundary conditions. The one that we found most useful and robust was to replace the constraint on the overall chain density with one working at a local level. In fact, we substituted the fixed box containing the whole chain, with the requirement that any succession of n beads be contained in a smaller box of side /. The results were insensitive (unless the discretization of the chain was poor) to the choice of n, I and even to replacing the box with a sphere etc.

245

The solutions that emerged out of the optimizaton procedure were perfectly helical strings, corresponding to discretised approximations to the continuous helix represented in Fig. 4b, confirming that this is the optimal arrangement. In all cases, the geometry of the chosen helix is such that there is an equality of the local radius of curvature (determined by the local bending of the curve) and the radius associated with a suitable triplet of non-consecutive points lying in two successive turns of the helix. In other words, among all possible shapes of linear helices, the one selected by the optimization procedure has the peculiarity that the local radius of curvature is equal to the distance of successive turns. Hence, if we inflate uniformly the centerline of this helix, one observes that the tube contacts itself near the helix axis exactly when successive turns touch. This is a feature that is observed only for a special ratio c* = 2.512... of the pitch, p, and the radius, r, of the circle projected by the helix on a plane perpendicular to its axis. As this packing problem is considerably more complicated that the hard spheres one, we have little hope to prove analytically that, among all possible three-dimensional chains, the helix of Fig. 4b is the optimally packed one. However, if we assume that the optimal shape is a linear helix, it is not too difficult to explain why the "magic" ratio we can explain why the "magic" ratio p/r = c* is observed. In fact, when p/r > c* the local radius of curvature, given by p = r{\ + p2 /(2wr)2), is smaller than the half of the distance of closest approach of points on successive turns of the helix (see 4a). The latter is given by the first minimum of 1/2-^2 - 2cos(2?rt) +p2t2 for t > 0. Thus A = p in this case. On the other hand, if p/r < c*, the global radius of curvature is strictly lower than the local radius, and the helix thickness is determined basically by the distance between two consecutive helix turns: A ~ p/2 if p/r 1 in the 'local' regime and / < 1 in the 'non-local' regime. In our computer-generated optimal strings, the value of / averaged over all sites in the chain differed from unity by less than a part in a thousand. It is interesting to note that, in nature, there are many instances of the appearance of helices. It has been shown 10 that the emergence of such motifs in proteins (unlike in random heteropolymers which, in the melt, have structures conforming to Gaussian statistics) is the result of the evolutionary pressure

246

(a)

(b)

(c)

Figure 4. Maximally inflated helices with different pitch to radius ratio, c. (a) c = 3.77: the thickness is given by the local radius of curvature, (b) c = 2.512...: for this optimal value the local and non-local radii of curvature match, (c) c = 1.26: the maximum thickness is limited by non-local effects (close approach of points in successive turns). Note the optimal use of space in situation (b), while in cases (a) and (c), empty space is left between the turns or along the helix axis.

exerted by nature in the selection of native state structures that are able to house sequences of amino acids which fold reproducibly and rapidly 38 and are characterized by a high degree of thermodynamic stability 17 . Furthermore, because of the interaction of the amino acids with the solvent, globular proteins attain compact shapes in their folded states. It is then natural to measure the shape of these helices and assess if they are optimal in the sense described here. The measure of / in a-helices found in naturally-occurring proteins yields an average value for / of 1.03±0.01, hinting that, despite the complex atomic chemistry associated with the hydrogen bond and the covalent bonds along the backbone, helices in proteins satisfy optimal packing constraints. An example is provided in Fig. 5 where we report the value of / for a particularly long a-helix encountered in a heavily-investigated membrane protein, bacteriorhodopsin.

247

1c3w (first helix)

3.4 3.2 3 2.8 2.6 2.4

Ft local R non-local

11

21

11 Helix site Figure 5. Top. Local and non-local radii of curvature for sites in the first helix of bacteriorhodopsin (pdb code lc3w). Bottom. Plot of / values for the same sites.

This result implies that the backbone sites in protein helices have an associated free volume distributed more uniformly than in any other conformation with the same density. This is consistent with the observation 10 that secondary structures in natural proteins have a much larger configurational entropy than other compact conformations. This uniformity in the free volume distribution seems to be an essential feature because the requirement of a maximum packing of backbone sites by itself does not lead to secondary structure formation 5 ' 6 . Furthermore, the same result also holds for the helices appearing in the collagen native state structure, which have a rather different geometry (in terms of local turn angles, residues per turn and pitch 45 ) from average a-helices. In spite of these differences, we again obtained an average / = 1.01 ± 0.03 very close to the optimal situation. 4

Conclusions

In summary, we have shown that topology-based models can lead to a vivid picture of the folding process. In particular, they allow not only the overall

248

qualitative characterization of the rate-limiting steps of the folding process, but also to pinpoint crucial sites that, for viral enzyme, should be targetted by effective drugs. We have carried out a successful validation of this strategy against data for clinical trials on the HIV-1 protease. We have then addressed the question of whether there exists a simple variational principle accounting for the emergence of secondary motifs in natural proteins. A possible selection mechanism has been identified in terms of optimal packing requirements. The numerical evidence presented here support unambiguously the fact that, among all three-dimensional structures with uniform thickness, the ones that make the most economic use of the space are helices with a well-defined geometry. Strikingly, the optimal aspect ratio is precisely the same that is observed in helices of naturally-occurring proteins. This provides a hint that, besides detailed chemical interactions, a more fundamental mechanism promoting the selection and use of secondary motifs in proteins is associated with simple geometric criteria 38 ' 46 . Acknowledgments Support from INFM, Murst Cofin 1999 and Cofm 2001 is acknowledged. References 1. D. Baker, Nature 405, 39, (2000). 2. C. Chothia, Nature 357, 543, (1992). 3. Pauling L., Corey R. B. and Branson H. R., Proc. Nat. Acad. Sci. 37, 205 (1951). 4. Hunt N. G., Gregoret L. M. and Cohen F. E., J. Mol. Biol. 241, 214 (1994). 5. Yee D. P., Chan H. S., Havel T. F. and Dill K. A. J. Mol. Biol. 241, 557 (1994). 6. Socci N. D., Bialek W. S. and Onuchic J. N. Phis. Rev. E 49, 3440 (1994). 7. Anfinsen C. Science 181, 223 (1973). 8. P. G. Wolynes J. N. Onuchic and D. Thirumalai, Science 267, 1619, (1995) 9. N. Go & H. A. Scheraga, Macromolecules 9, 535, (1976). 10. C. Micheletti, J. R. Banavar, A. Maritan and F. Seno, Phys. Rev. Lett. 82, 3372, (1999). 11. Galzitskaya 0 . V. and Finkelstein A. V. Proc. Natl. Acad. Sci. USA 96, 11299 (1999).

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12. Munoz V. , Henry E. R., Hofrichter J. and Eaton W. A. Proc. Natl. Acad. Sci. USA 95, 5872 (1998). 13. Aim E. and Baker D. Proc. Natl. Acad. Sci. USA 96, 11305 (1999). 14. Clementi C , Nymeyer H. and Onuchic J. N., J. Mol. Biol., in press (2000). 15. Lazaridis T. and Karplus M. Science 278, 1928 (1997). 16. Kolinski A. and Skolnick J. J. Chem. Phys. 97, 9412 (1992). 17. Sali A., Shakhnovich E. and Karplus M. Nature 369, 248 (1994). 18. Fersht A. R. Proc. Natl. Acad. Sci. USA 92, 10869 (1995). 19. Itzhaki L. S., Otzen D. E. and Fersht A. R., J. Mol. Biol. 254, 260 (1995). 20. Ala P.J, et al. Biochemistry 37, 15042-15049, (1998). 21. Cecconi F., Micheletti C , Carloni P. and Maritan A. Proteins: Str. Fund Gen., 43, 365-372 (2001). 22. Clementi C , Carloni P. and Maritan A. Proc. Natl. Acad. Sci. USA 96, 9616 (1999). 23. Evans D. J., Hoover W. G., Failor B. H., Moran B., Ladd A. J. C. Phys. Rev. A 28, 1016 (1983). 24. Ferrenberg A. M. and Swendsen R. H. Phys. Rev. Lett. 63, 1195 (1989). 25. Brown A. J., Korber B. T., Condra J. H. AIDS Res. Hum. Retroviruses 15, 247 (1999). 26. Molla A. et al. Nat. Med. 2, 760 (1996). 27. Markowitz M. et al. J. Virol. 69, 701 (1995). 28. Patick A. K., et. al. Antimicrob. Agents Chemother. 40, 292 (1996) . 29. Condra J.H. et al. Nature 374, 569 (1995). 30. Tisdale M. at al. Antimicrob. Agents Chemother. 39, 1704 (1995). 31. Jacobsen H. et al. J. Infect. Dis. 173, 1379 (1996). 32. Reddy P. and Ross J. Formulary 34, 567 (1999). 33. Sloane N. J. A. Nature 395, 435 (1998). 34. Mackenzie D. Science 285, 1339 (1999). 35. Woodcock L.V. Nature 385, 141 (1997). 36. Car R. Nature 385, 115 (1997). 37. Cipra B. Science 281, 1267 (1998). 38. A. Maritan, C. Micheletti, A. Trovato and J. R. Banavar, Nature 406, 287, (2000). 39. Buck G. and Orloff J. Topol. Appl. 61, 205 (1995). 40. Katritch V., Bednar J., Michoud D., Scharein R.G., Dubochet J. and Stasiak A., Nature 384, 142 (1996). 41. Katritch V., Olson W. K., Pieranski P., Dubochet J. and Stasiak A. Nature 388, 148 (1997).

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42. Gonzalez O. and Maddocks J. H. Proc. Natl. Acad. Sci. USA 96, 4769 (1999). 43. Buck G., Nature 392, 238 (1998). 44. Cantarella J., Kusner R. B. and Sullivan J. M., Nature 392, 237 (1998). 45. Creighton T. E., Proteins - Structures and Molecular Properties, W.H. Freeman and Company, New York (1993), pag. 182-188. 46. A. Maritan, C. Micheletti and J. R. Banavar, Phys. Rev. Lett. 84, 3009, (2000).

251

T H E PHYSICS OF M O T O R P R O T E I N S G. L A T T A N Z I International

School for Advanced Studies (S.I.S.S.A.) and INFM, 34013 Trieste, Italy E-mail: [email protected]

via Beirut,

2~4,

via Beirut,

2~4,

A. M A R I T A N International

School for Advanced Studies (S.I.S.S.A.) 34013 Trieste, Italy The Abdus Salam International Center for Theoretical 34100 Trieste, Italy

and INFM, Physics,

Strada

Costiera

11,

Motor proteins are able to transform the chemical energy of ATP hydrolysis into useful mechanical work, which can be used for several purposes in living cells. The paper is concerned with problems raised by the current experiments on motor proteins, focusing on the main question of conformational changes. A simple coarse-grained theoretical model is sketched and applied to the motor domain of the kinesin protein; regions of functional relevance are identified and compared with up-to-date information from experiments. The analysis also predicts the functional importance of regions not yet investigated by experiments.

1

Introduction to the biological problem

The increasing precision in the observation of single cells and their components can be compared to the approach of one of our cities by air 1 : at first we notice a complex network of urban arteries (streets, highways, railroad tracks). Then, we may have a direct look at traffic in its diverse forms: trains, cars, trucks and buses traveling to their destinations. We do not know the reason for that traffic, but we know that it is essential to the welfare of the entire city. If we want to understand the rationale for every single movement, we need to be at ground level, and possibly drive a single element of the traffic flow. In the same way, biologists have observed the complex network of filaments that constitute the cytoskeleton, the structure that is responsible also for the mechanical sustain of the cell. Advances in experimental techniques have finally cast the possibility of observing traffic inside the cell. This transport system is of vital importance to the functioning of the entire cell; as ordinary traffic jam, or a dafect in the transportation network of a city, can impair its organized functionmg, occasional problems in the transport of chemical components inside the cell can be the cause of serious cardiovascular diseases or neurological disorders.

252

The study of the transportation system and its molecular components is therefore of great relevance to medicine. Recent advances in single molecule experiments 2,3 allowed us to be spectators at the ground level for the first time, i.e. to observe the single molecular elements of the traffic flow inside the cells. These components are called protein motors. 1.1

Fuel: ATP

The fuel for such motors is ATP (adenosine triphosphate). ATP is an organic molecule, formed by the nucleotide adenine, ribose sugar and three phosphate bonds, held together by two high energy phosphoanhydride bonds 4 . Removal of a phosphate group from ATP leaves ADP (adenosine diphosphate) and an inorganic molecule of phosphate, Pj, as in the hydrolysis reaction: ATP + H20—>ADP

+ Pi+H+.

(1)

This reaction corresponds to a release of 7.3 kcal/mol of free energy. Indeed, under standard chemical conditions, this reaction requires almost one week to occur 5 , but it is accelerated, or catalyzed, by proteins. These proteins are called motors, since they are able to transduce one form of energy (chemical) into useful mechanical work. 1.2

Characteristics of motor proteins

Protein motors are very different from our everyday life motors: first, they are microscopic, and therefore they are subject to a totally different physical environment, where, for instance, thermal agitation has a strong influence on their motion. In addition, their design has been driven by evolutionary principles operating for millions of years, and therefore they are optimized to have a high efficiency, and specialized to the many different purposes required in the functioning of living cells. Our everyday motors usually operate with temperature differences, therefore, no matter how clever we are in designing the motor, its efficiency is always limited by the Carnot theorem 6 . This is no longer true for motor proteins. Indeed any temperature difference, on the length scale of proteins (the nanometer) would disappear in a few picoseconds, therefore they are isothermal machines, operating at the constant temperature of our body. They are not limited by Carnot theorem, and their efficiency could be rather close to 1, meaning that they are able to convert chemical energy almost entirely into useful work.

253

1.3

Different families, different tasks

Most molecular motors perform sliding movements along tracks using the energy released from the hydrolysis of ATP, to make macroscopic motion, such as muscle contraction and maintain cell activities. Among these, the most important families are: myosins, kinesins and dyneins. The study of myosin dates back to 1864. It is usually found in bundles, as in the thick filaments in our muscle cells and is extremely important for muscle contraction. Kinesin, discovered in 1985, is a highly processive motor, i.e. could take several hundred steps on a filament called microtubule without detaching 7 ' 8 , whereas muscle myosin was shown to execute a single "stroke" and then dissociate. Kinesins form a large superfamily, and the individual superfamily proteins operate as motor molecules in various cell types with diverse cargoes. Given that transportation requirements are particularly demanding and complex in neurons, it is not a surprise that the highest diversity of kinesins is found in brain 1 . The discovery of kinesin only partially explained how membrane vesicles are transported in cells. Some movements, such as retrograde axonal transport, are in the direction opposite to this kinesin-dependent movement. Thus there must be a second group of motor proteins responsible for this motility. Such a group exists; it is composed of the dyneins, a superfamily of exceptionally huge proteins. In neurons, kinesin and dynein motor molecules have not only been involved in intracellular axonal and dendritic transport, but also in neuronal pathfinding and migration. Given the various fundamental cellular functions they serve in neurons, such mechanisms, if defective, are expected to contribute to onset or progression of neurological disorders. But these are not the only motor proteins. Another important track is the DNA. And specific machines move upon DNA filaments, unzip them and copy them in RNA. 1.4

Structure

Until 1992, it appeared as though kinesin and myosin had little in common 9 . In addition to moving on different filaments, kinesin's motor domain is less than one-half the size of myosin and initial sequence comparisons failed to reveal any important similarities between these two motors. Their motile properties also appeared to be quite different. In the last few years of research, however, the crystal structures of kinesin have revealed a striking similarity to myosin, the structural overlap pointing

254

to short stretches of sequence conservation 10,11 . This suggested that myosin and kinesin originated from a common ancestor. The opportunity to study and compare numerous kinesin and myosin motors provides a valuable resource for understanding the mechanism of motility. Because kinesin and myosin share a similar core structure and evolutionary ancestry, comparison of these motors has the potential to reveal common principles by which they convert chemical energy into motion 9 . Members of each family have similar motor domains of about 30 — 50% identical residues that can function as autonomous units. The proteins are differentiated by their nonmotor or tail domains. The motor domains, also called head domains, have no significant identity in amino acid sequence, but they have a common fold for binding the nucleotide (ATP). Adjacent to the head domain lies the highly a-helical neck region. It regulates the binding of the head domain by binding either calmodulin or calmodulin-like regulatory light chain subunits (called essential or regulatory light chains, depending on their function). The tail domain contains the binding sites that determine whether the tail binds to the membrane or binds to other tails to form a filament, or attaches to a cargo. Motor proteins are composed of one or two (but also rarely three) motor domains, linked together by the neck regions, which form the neck linker part of the motor. To understand how hydrolysis of ATP is coupled to the movement of a motor head along filaments, we need to know the three-dimensional structure of the head domain. An important feature in the structure of the myosin head is the presence of two clefts on its surface. One cleft is bound to the filament, while the other contains the ATP binding site. The clefts are separated by 3.5 nm, a long distance in a protein. The presence of surface clefts provides a mechanism for generating large movements of the head domain: we can imagine how opening or closing of a cleft in the head domain, by binding or releasing ATP, causes the head domain to pivot about the neck region, so that a change in the conformation of the protein may occur. 1.5

Conformational change: experiments

Conformational changes have been detected using advanced experimental techniques such as Fluorescence Resonance Energy Transfer12 (FRET); FRET determines the distance between two probes on a protein, called donor (D) and acceptor (A). When the emission spectrum of the donor fluorophore and the excitation spectrum of the acceptor fluorophore overlap, and they are located close to each other (in the order of nanometers), the excited energy of

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the donor is transferred to the acceptor without radiation, resulting in acceptor fluorescence. When the donor and acceptor are far apart, the donor fluoresces. It is therefore possible to determine the distance between the donor and acceptor fluorophores attached to two different sites on a protein by monitoring the color of the fluorescence. For myosin, it has been shown 13 that fluorescence intensities of the donor and acceptor vary spontaneously in a flip-flop fashion, indicating that the distance between the donor and acceptor changes in the range of hundreds of angstroms; that is, the structure of myosin is not stable but instead thermally fluctuates. These results suggest that myosin can go through several metastable states, undergoing slow transitions between the different states. 1.6

The problem of conformational change

The ATP binding pocket is a rather small region in the myosin (or kinesin) motor domain. Yet, the information that ATP is bound to this well localized site, can be transferred to very distant regions of the domain, so that the entire protein may undergo a conformational change. FRET can be used to monitor the distance between parts of the motor domain, but it is not possible to probe all of the regions, because of time and budget limitations. The identification of possible targets for FRET experiments would be therefore a task for theoretical modeling. A theoretical model would be of great help in answering some of the following questions: which parts are expected to undergo the largest displacements in a conformational change? Which are sensible to ATP binding? Which are important for the biological function of the protein? Which are responsible for the transfer of information? 2

Gaussian Network Model ( G N M )

Many theoretical models have been proposed for the analysis of protein structure and properties. The problem with protein motors is that they are huge proteins, whose size prevents any attack by present all-atoms computer simulations. To make things worse, even under an optimistic assumption of immense computer memory to store all necessary coordinates, the calculations needed would show only few nanoseconds of the dynamics, but the conformational rearrangements usually lie in the time range of milliseconds. Therefore, a detailed simulation of the dynamics is not feasible and furthermore it is not guaranteed that all the details can shed light on the general mechanism. Yet, in recent years, dynamical studies have increased our appreciation of

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the importance of protein structure and have shed some light on the central problem of protein folding14'15'16. Interestingly, coarse grained models proved to be very reliable for specific problems in this field. The scheme is as follows. Proteins are linear polymers assembled from about 20 amino acid monomers or residues. The sequence of aminoacids (primary structure) varies for different molecules. Sequences of amino acid residues fold into typical patterns (secondary structure), consisting mostly of helical (a helices) and sheetlike (/? sheets) patterns. These secondary structure elements bundle into a roughly globular shape (tertiary structure) in a way that is unique to each protein (native state). Therefore, the information on the detailed sequence of amino acids composing the protein, uniquely encodes its native state. Once the latter is known, one may forget about the former (this is the topological point of view). The GNM is a recently developed simple technique which drives this principle to its extreme. It has been applied with success to a number of large proteins 17 and even to nucleic acids 18,19 . 2.1

Theory

Bahar et al. 20 proposed a model for the equilibrium dynamics of the folded protein in which interactions between residues in close proximity are replaced by linear springs. The model assumes that the protein in the folded state is equivalent to a three dimensional elastic network. The nodes are identified with the Ca atoms" in the protein. These undergo Gaussian distributed fluctuations, hence the name Gaussian Network Model. The native structure of a given protein, together with amplitudes of atomic thermal fluctuations, measured by x-ray crystallography, is reported in the Brookhaven Protein Data Bank 21 (PDB). Given the structure of a protein, the Kirchhoff matrix of its contacts is defined as follows:

r-=t- = | ~ l 3

*

l i f r

\0

r c

(2)

ifry>rc

[Z)

«-

N

r « = - 2_^ Ti/t

(3)

where the non-zero off-diagonal elements refer to residue pairs i and j "Carbon atoms in amino acids are numbered with Greek letters: for each residue there is at least one carbon atom, Ca, but there could be also additional carbon atoms, called Cg,

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that are connected via springs, their separation r^ being shorter than a cutoff value rc for inter-residue interactions. The diagonal elements are found from the negative sum of the off-diagonal terms in the same row (or column); they represent the coordination number, i.e. the number of individual residues found within a sphere of radius rc. The Kirchhoff matrix is conveniently used 22 for evaluating the overall conformational potential of the structure: V = |ARTrAR.

(4)

Here A R is the iV-dimensional vector whose elements are the 3 dimensional fluctuation vectors A R j of the individual residues around their native position, while 7 represents a free parameter of the model. The cross-correlations between residue fluctuations are found from the simple Gaussian integral:

(AR; • ARj) = - J - f(AHi

• ARj)<J ( ^

A R ; j e - ' ^ ' ^ r f j A R } (5)

where the integration is carried over all possible fluctuation vectors A R , Zjv is the partition function, and 5 (J2i ARj) accounts for the constraint of fixing the position of the center of mass. This integral can be calculated exactly, yielding: (ARi.ARj) = ^ I [ r -

1

]i.

(6)

where T _ 1 is the inverse of F in the space orthogonal to the eigenvector with zero eigenvalue. This inverse can be expressed in a more elegant way as a sum over all non-zero eigenvalues Xk and eigenvectors ii* of T, so that (ARi • ARj) can be finally expressed as the sum over the contributions [ARj • ARj]^, of the individual modes: