11 and 21 phase entrainment in a system of two coupled limit cycle oscillators

J. Math. Biology (1984) 20:133-152

Journal of

Mathematical Biology 9 Springer-Verlag 1984

1:1 and 2:1 phase entrainment in a system of two coupled limit cycle oscillators W. L. Keith and R. H. Rand Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA

Abstract. A model of a pair of coupled limit cycle oscillators is presented in order to investigate the extent of, and the transition between, 1 : 1 and 2:1 phase entrainment, a phenomenon exhibited by numerous diverse biological systems. The mathematical form of the model involves a flow on a phase torus given by two coupled first order nonlinear ordinary differential equations which govern the oscillators' phase angles (i.e. their respective positions around their limit cycles). The regions corresponding to 1 : 1 and 2 : 1 phase entrainment in an appropriate parameter space are determined analytically and numerically. The bifurcations occurring during the transition from 1 : 1 to 2:1 phase entrainment are discussed. Key words: Oscillators - - limit cycles - - phase entrainment - - bifurcations

Introduction Recent research on problems in bio-oscillation theory have utilized simplified models of limit cycle oscillators (Cohen et al. [1], Guevara and Glass [7], Ermentrout and Kopell [2], Hoppensteadt and Keener [8], Rand et al. [12]). In this paper we extend previous work by proposing a model of two limit cycle oscillators which takes into account both 1 : 1 and 2:1 phase entrainment. We briefly cite the following examples of biological systems which have exhibited 2:1 phase entrainment: In the vertebrate heart disorder known as cardiac arrhythmias, every other pulse generated by the sinuatrial node pacemaker is ineffective in driving the ventricular rhythm. The process by which a normal heart becomes diseased may be viewed as a transition from 1 : 1 to 2 : 1 phase entrainment (Guevara and Glass [7]). In experiments on the neurobiology of swimming in the fish Icthyomyzon unicuspis (the lamprey), Cohen et al. [1] reported a transition between 1 : 1 and 2:1 phase entrainment as follows: An isolated spinal cord undergoing phase locked "fictive swimming" was lesioned, resulting in the observation that the two

134

W.L. Keith and R. H. Rand

portions of the cord on either side of the lesion became 2:1 phase entrained (whereas before the lesion the two portions were 1 : 1 phase entrained). In this p a p e r we shall be interested in certain aspects of the general problem of 1 : 1 versus 2:1 phase entrainment in a system of two oscillators. In particular we wish to address the question of the extent of 1 : 1 versus 2 : 1 entrainment as a function of the uncoupled frequencies and the coupling between the oscillators. In addition we shall discuss the nature of the transition between these two entrained states, and will show that in our model a complicated sequence of bifurcations accompanies this transition. Model

We follow Cohen et al. [1] and postulate a general biological oscillator which exhibits a unique stable limit cycle oscillation. Since we wish to make the model as general as possible, we make no assumptions about the containing space, except that the number of dimensions (and hence the number of physical variables needed to describe the system) be greater than one. In order to describe the oscillation mathematically, we parameterize the limit cycle by its phase 0 chosen to run from 0 to 2 ~ radians over the course of a complete cycle. Moreover we assume that the phase O(t) flows uniformly in time t: 0=~o

implies O ( t ) = t o t + O ( O )

(mod2~-).

(1)

Here m o d 2~r means that we do not distinguish between values of 0 which differ by multiples of 2~- (since the limit cycle is topologically a circle). In order to model the interaction between two such oscillators, we assume that the limit cycle in an individual oscillator continues to persist when perturbed by feedback from another oscillator (i.e. we assume that the limit cycle is structurally stable). If 01 and 02 represent the respective phases of the two oscillators, we write: 01 -=0) 1 +h1(Ol, 02) (2) 02 = w2 +h2(01, 02)

(3)

where hi(01, 02) represents the influence of oscillator 2 on oscillator l, and similarly for h2(Ol, 02). Since each 0~ is defined m o d 2~r, we further assume that the functions h~(01, 02) are 21r periodic in their arguments. We may envision the resulting flow as taking place on a two dimensional torus which may be represented as a square with opposite sides identified (see Fig. 1). In Cohen et al. the functions hi were chosen thus: 01 = ~ol - a sin(01 - 02)

(4)

02 = o)2+a s i n ( 0 1 - 02)

(5)

where a is a coupling coefficient. By subtracting these equations it is easy to show that if Itol-WEI < 12al this system exhibits two l : l phase locked motions, one stable and one unstable: 9 [wl-w2~ Ol(t) - 02(t) = a r c s l n ~ , ] .

(6)

135

1 : 1 and 2 : 1 phase entrainment in two coupled oscillators

(o) Oz

Oi

(b) O, Fig. 1. The 0i, 02 phase torus (c) can be represented by a square with opposite sides identified, as in (a). Fig. l(b) shows the intermediate step in which the left and right sides of (a) have been joined, giving a cylinder. Fig. l(c) results from joining the top and bottom ends of (b). The curve L represents a possible limit cycle

2rr

2~~ '

/c)

These motions represent limit cycles on the phase torus, see Fig. 2. N o t e that when plotted on a square with identified sides as in Fig. 2, these phase locked motions consist o f straight line segments with slope unity. N o t e also that these limit cycles on the phase torus in the c o u p l e d system are distinct from and should not be c o n f u s e d with, the limit cycles in the space of physical variables in the d y n a m i c s o f the original oscillators which were parameterized by 01 and 02. In the case that [Wl-w21> 12=1, no such 1:1 phase locked limit cycles exist, and the m o t i o n is said to "drift". In order to m o d e l 2 : 1 phase locking, we m o d i f y Eqs. (4), (5) as follows: 0~ = W l - / 3 s i n ( 0 1 - 2 0 2 )

(7)

02 = w2+/3 sin(01-202).

(8)

Here /3 is a coupling coefficient. In the case that ]wl-2w21 12/31, Eqs. (7), (8) do not admit 2:1 phase locked limit cycle solutions, but rather exhibit drift. In Appendix E both sets of Eqs. (4), (5) and (7), (8) are derived from a perturbation method applied to two coupled van der Pol oscillators. In order to obtain a model which exhibits both I : 1 and 2:1 phase locked motions, we combine the systems previously discussed in the following form: 01 = (.01 --OL

sin(01- 0 2 ) - f l s i n ( 0 , - 2 0 2 )

02 = o92 + a sin(01 - 02) +/3 sin(01-202).

(10) (11)

For the special cases/3 = 0 and a = 0 , Eqs. (10), (11) reduce to Eqs. (4), (5) and (7), (8) respectively. Although these limiting cases can be derived from coupled van der Pol oscillators (see Appendix E), we propose the present model of coupled oscillators as independent of any derivation based on van der Pol oscillators, but rather as being of interest in its own right without a more elementary derivation. In a similar spirit, Hoppensteadt and Keener [8] presented a study of phase locking !n radial isochron clocks based on the system: 01=l 02 = ~o + e

~ tn, n~

am,, sin(m01 - nO2) co

(with am., = a . . . . V m ~ n). Note the similarity in the coupling term between their model and our posited Eqs. (10), (11). In order to reduce the number of parameters involved in this study, we set /3 = 1 - a in what follows: 01 = w l - a sin(01 - 02) - ( 1 - a ) sin(01-202)

(12)

02 = ~o2+a sin(01- 02) +(1 - a ) sin(0~- 202).

(13)

We shall be interested in the dynamics of Eqs. (12), (13) as a is tuned from 0 to 1 for various values of Wl and w2. In particular we wish to investigate for which values of a the system behaves in a 1 : 1 or 2:1 manner. Before proceeding with the analysis of this model, we wish to distinguish between the kind of 1 : 1 (or 2: 1) behavior exhibited by Eqs. (12), (13) in the case that a = 1 (or cr = 0), called phase locking, and a more general kind of

1: 1 and 2: l phase entrainment in two coupled oscillators

137

277-

Ol

/ Fig. 4. A l : 1 phase entrained motion

0

27r

01

Fig. 5. A 2:1 phase entrained motion

0

02

:// 0a

277"

interaction called phase entrainment. While a 1 : 1 phase locked m o t i o n involves a linear relationship between 01 and 02 (Eq. (6)), and plots as a straight line (Fig. 2), a 1 : 1 phase entrained m o t i o n will be defined to be a periodic m o t i o n for which 01 advances by 2~- whenever 02 advances by 2zr. That is, a m o t i o n will be said to be 1 : 1 phase entrained if

011o2=0.-011o2-0.+2= 27r for all 0". A 1 : 1 phase entrained m o t i o n is s h o w n in Fig. 4. Note that 1 : 1 phase locked motions are special cases of 1 : 1 phase entrained motions in which a linear relationship exists between 01 and 02 for all time. Similarly we define a 2 : 1 phase entrained motion to be one for which

011o~_o*-011o2-o*+2~=4~ for all 0". That is, 01 advances by 4~- w h e n e v e r 02 advances 2~r. See Fig. 5. The importance o f phase entrained motions lies in the fact that Eqs. (12), (13) admit 1 : 1 and 2 : 1 phase entrained motions for a wide range o f parameter values, while these equations only exhibit 1 : 1 or 2:1 phase locked motions for a = 1 or 0, respectively.

Analysis of the model In the previous section we showed that the two oscillator model given by Eqs. (12), (13) exhibits 1 : 1 and 2:1 phase locking when a = 1 and 0 respectively. In A p p e n d i x A we show that this model also exhibits a 3 : 2 phase locked motion

138

w . L . Keith and R. H. R a n d 091

211 2

a

092 Fig. 6. Phase locked regions in 0)1, 0)2, o~ parameter space. The planes c~ = 0, 1 contain regions of 2:1 and 1 : 1 locking, respectively, s h o w n shaded. The plane a = 89contains the indicated line on which 3 : 2 locking occurs

8I

a--O.O~/ e2

a=.5/~ 91 a=.594~ ~

/a =.590

/~ ~~a=.400 a=.593

Fig. 7. Flow on the 01, 02 phase torus during transition from 2:1 to 1 : 1 phase entrained motions. As c~ increases from 0.39 to 0.4, a complicated sequence of bifurcations occurs which is conjectured to involve p: q phase entrained motions for all rational p/q between 1 and 2. Here, we display the 2: 1, 3 : 2, 4 : 3, 5 : 4, and 1 : 1 cases. Note that w h e n t~ = 0 we have a 2: 1 phase locked motion

1 : 1 and 2:1 phase entrainment in two coupled oscillators

139

to I

57 -2

I 0

~

,

I:1

la

Z 2

Fig. 8. 1 : 1 and 2:1 phase entrainment regions in Wl, c~ space with w2 = 1. The dashed line represents the boundary of the 1 : 1 region and intersects the line a = 1 at w1= 3 (see Appendix A)

w h e n a = 89 a n d t h a t no o t h e r p h a s e l o c k e d m o t i o n s are possible. See Fig. 6 w h e r e the r e g i o n s o f p h a s e l o c k i n g are d i s p l a y e d in Wl, w2, a p a r a m e t e r space. W e p a s s n o w f r o m c o n s i d e r a t i o n o f p h a s e l o c k e d m o t i o n s to p h a s e e n t r a i n e d m o t i o n s . To r e d u c e the n u m b e r o f p a r a m e t e r s , we set w2 = 1 in all t h a t follows. W e b e g i n b y p r e s e n t i n g the results f r o m t y p i c a l n u m e r i c a l i n t e g r a t i o n s o f Eqs. (12), (13). W e set w~ = oJ2 = 1 a n d vary c~ f r o m 0 to i. F o r 0 < a ~ 0.39, the system d i s p l a y s 2 : 1 p h a s e e n t r a i n m e n t , while for 0.40 ~ ~ < 1, 1 : i p h a s e e n t r a i n m e n t is o b s e r v e d . A c o m p l i c a t e d s e q u e n c e o f b i f u r c a t i o n s occurs in the t r a n s i t i o n region, 0.39 ~ a ~ 0.40, see Fig. 7. W e will r e t u r n to a d i s c u s s i o n o f these b i f u r c a t i o n s later. Since the a t t r a c t i o n onto a stable limit cycle b e c o m e s very w e a k as a a p p r o a c h e s a critical ( b i f u r c a t i o n ) value, the n u m e r i c a l i n t e g r a t i o n s m u s t be c a r r i e d o u t over very long time intervals. In o r d e r to a v o i d this difficulty, we d e v e l o p e d an a n a l y t i c a l - n u m e r i c a l t e c h n i q u e w h i c h m o r e efficiently yields the critical v a l u e o f c~ at w h i c h a p: q p h a s e e n t r a i n e d s o l u t i o n ceases to exist. T h e details o f this m e t h o d are given in A p p e n d i x B. The resulting 1:1 a n d 2 : 1 e n t r a i n m e n t r e g i o n s in the w~, a p l a n e are s h o w n in Fig. 8. W h e n a = 0 a n d 1, the t r a n s i t i o n curves for 2 : 1 a n d 1 : 1 e n t r a i n m e n t r e s p e c t i v e l y yield the critical v a l u e o f o9~ o b t a i n e d f r o m the p r e v i o u s p h a s e l o c k e d analysis.

2"0-

|

"

S t j

01

ir .."

I

Fig. 9. A 2 : 1 drift flow (3 distinct trajectories are shown)

0

02

Jt

2v"

140

w.L. Keith and R. H. Rand

In p a r t i c u l a r , for a = 1, the 2 : 1 e n t r a i n m e n t t r a n s i t i o n curves intersect at Wl =7, f o r m i n g a w e d g e s h a p e d region (Fig. 8). The p o i n t ~ol =7, a = 1 m a y b e a n a l y t i c a l l y s h o w n to c o r r e s p o n d to a 2 : 1 drift s o l u t i o n ( K e i t h [9]). (By a 2 : 1 drift s o l u t i o n we m e a n a situation in w h i c h the p h a s e torus is filled with 2 : 1 p h a s e e n t r a i n e d m o t i o n s , n o n e o f w h i c h are attracting, see Fig. 9.) W e s u p p l e m e n t e d this n u m e r i c a l w o r k in the n e i g h b o r h o o d o f this w e d g e s h a p e d r e g i o n with a p e r t u r b a t i o n m e t h o d . At the p o i n t Wl =~,7 o~ = 1, an exact s o l u t i o n o f the 2 : 1 drifting m o t i o n is k n o w n ( C o h e n et al. [1]). W e set e = 1 - a a n d ~o~ = 7 + he + O(62), a n d p e r t u r b e d off the e x a c t s o l u t i o n for small e. W e offer a b r i e f s u m m a r y o f the m e t h o d in A p p e n d i x C, a n d here note j u s t that the resulting values o f s l o p e A turn out to be 0 , - 3 . The resulting w e d g e s h a p e d region agrees closely with the p r e v i o u s n u m e r i c a l results, Fig. t0. N e x t we offer a p r o o f o f the existence o f 1 : 1 a n d 2 : 1 p h a s e e n t r a i n e d m o t i o n s b a s e d o n the P o i n c a r e - B e n d i x s o n t h e o r e m . T h e exact n a t u r e o f this p r o o f c o m p l e ments the a p p r o x i m a t e n a t u r e o f the p r e v i o u s n u m e r i c a l a n d p e r t u r b a t i o n results. A p p e n d i x D c o n t a i n s a s u m m a r y o f this work, w h i c h results in the regions o f 1 : t a n d 2 : 1 p h a s e e n t r a i n m e n t s h o w n in Fig. 11. N o t e t h a t the m e t h o d involves OAI

7/2

L~

0.w

Fig. 10. The wedge shaped region of 2:1 phase entrainment near a = 1. The dotted lines are the results from perturbation theory. The solid lines are the results from the numerical integration of the F equation

(OI

5

3 2:1 I 0

O.5

Ct

1.0

Fig. 11. The regions in the o)a, a plane (with o)2 = 1) in which the existence of limit cycles (corresponding to 1: 1 and 2 : 1 phase entrained motions) may be proven from the Poincare-Bendixson theorem. See Appendix D. The boundaries of the displayed regions are obtained analytically, and found to be straight line segments

1 : 1 and 2 : 1 phase entrainment in two coupled oscillators

141

only sufficient conditions on the existence o f phase entrained motions, so the associated regions in Fig. 11 lie inside the c o r r e s p o n d i n g numerically obtained regions s h o w n in Fig. 8. Let us n o w consider the nature o f the bifurcations which occur during the transition f r o m 1 : 1 to 2 : 1 phase entrainment. The analytical-numerical technique discussed in A p p e n d i x B and utilized to obtain the limiting parameter values for 1 : 1 a n d 2 : 1 entrainment in Fig. 8, was used again here. For a given ratio p/q, a differential equation was derived (see A p p e n d i x B) which exhibits periodic solutions only for those parameter values for which there is p: q phase entrainment. We numerically integrated this equation for nine different ratios p/q between 1 and 2, fixing wl = w2 = 1 and varying a (cf. Fig. 7). The results are shown in Fig. 12. E a c h value o f p/q corresponds to a plateau of phase entrainment, the size of the plateau being larger, the smaller the value o f q is. Since this flow on the torus T 2 has no singular points, a Poincare m a p on a cross section can be

2:1 7:4 *~5:3 3:2

9:8 4:5

5:4

,, 6:5,7:

I ~

,

i

F

0.595

0.5925

i

0.5955

w

0.594

0.5945

Fig. 12. Plateaus of p: q phase entrainment in the t~,p/q plane. (Note the breaks in the horizontal scale)

I

§

I

I §

I 2

......

I §

+ I

I +

o Fig. 13. The Cantor function

i

i--

g(o~) with 15 intervals shown

142

W.L. Keith and R. H. Rand

constructed. This m a p will be a diffeomorphism on S ~, and therefore one can apply the results of Denjoy [4] (cf. Guckenheimer and Holmes [6]) which show that the rotation number depends continuously on parameters (here a and ~ol). We conclude that all rational p/q between 1 and 2 are exhibited by this system. We also expect generically to find frequency plateaus for all rational rotation numbers. An example of function which exhibits this behavior is provided by the Cantor function, defined as follows: 1 OL< 2 we set g ( a ) = 8 9 for We consider the closed interval 0 0. See Fig. 8. Across this band, all rational p: q phase-entrained motions are conjectured to occur, for 1