leon glass et al- bifurcation and chaos in a periodically stimulated cardiac oscillator

8/3/2019 Leon Glass et al- Bifurcation and Chaos in a Periodically Stimulated Cardiac Oscillator

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Phys ica 7D (1983) 89-101N o r t h - H o l l a n d P u b l i s hi n g C o m p a n y

B I F U R C A T I O N A N D C H A O S I N A P E R I O D I C A L L Y S T I M U L A T E D C A R D I A CO S C I L L A T O RLeon GLASS, Michael R. GUEVARA and Alvin S H R I E RDepartment o f Physiology, M cG ill University, 3655 Dru mm ond Street , M ontreal, Quebec, Canada H3G 1Y 6a n dRafael P E R E ZInsti tuto de Fisica, Universidad Nacional de Mexic o, M exico City, Me xico

P e r i o d i c s t i m u l a t i o n o f a n a g g r e g a t e o f s p o n t a n e o u s l y b e a t i n g c u l t u r e d c a r d i a c c el ls d i s p l a y s p h a s e l o c k i n g , p e r i o d - d o u b l i n gb i f u r c a ti o n s a n d a p e r io d i c " c h a o t i c " d y n a m i c s a t d i f f e re n t v a lu e s o f t h e s t i m u l a t i o n p a r a m e t e r s . T h i s b e h a v i o r i s a n a ly z e d b yc o n s i d e r i n g a n e x p e r i m e n t a l l y d e t e r m i n e d o n e - d i m e n s i o n a l P o i n c a r 6 o r f i r s t r e t u r n m a p . A s i m p l i f i e d v e r s i o n o f t h ee x p e r i m e n t a l l y d e t e r m i n e d P o i n c a r 6 m a p i s p r o p o s e d , a n d s e v e r a l f e a t u r e s o f t h e b i f u r c a t i o n s o f t h i s m a p a r e d e s c r i b e d .

1 . I n t r o d u c t i o nCardiac dysrhythmias are abnormal cardiac

rhythms tha t often occur in diseased hearts. In whatfollows, we analyze biological and theoretical mod-els for the generation of cardiac dysrhythmias thatare associated with a lack of synchronization be-tween autonomous pacemaker sites in the heart.The theoretical techniques are also applicable to abroader range of problems dealing with the syn-chronization of nonlinear oscillators to a periodicinput.

In the normal human heart, the primary pace-making site is located in the sinoatrial node (SAN).From the SAN the cardiac impulse spreads se-quentially through the atrial musculature, the atrio-ventricular node (AVN), and specialized conduc-ting tissues to the ventricles [1 ]. Electrical excitationof the atria is followed 0.08-0.12 sec later by ex-citat ion of the ventricles. The electrochemicalevents responsible for the spread of excitation canbe monitored noninvasively by the electro-cardiogram (ECG) which is a record of the poten-tial differences between different points on the sur-face of the body. On the ECG, the P wave is

associated with the excitation of the atria, the QRScomplex with excitation of the ventricles, and the Twave with recovery of the excitability of the ventri-cles. Normal ly, excitation of cardiac tissue is associ-ated with mechanical contraction. Fig. IA shows anormal ECG. In a class of disorders called theatrioventricular (AV) heart blocks, there are abnor-malities in the relative timing of the atrial andventricular contractions. In first degree AV blockthe delay between the atrial and ventricular con-tractions is elevated above its normal maximalvalue. In second degree AV block, atrial con-tractions are not always followed by ventricularcontractions. The second degree AV blocks areoften periodic and are characterized by a ratiowhich gives the relative frequencies of atrial to ven-tricular contractions. Fig. 1B shows an example inwhich one cycle of 3:2 AV block (i.e. 3 atrial to 2ventricular contractions) is followed by five cyclesof 2:1 AV block. As well, second degree AVblock can show extremely irregular or fluctuatingrhythms ( fig . 1C) . In third degree AV block,there is an apparent lack of correlation between theatrial and ventricular rhythms, with separate pace-makers for each rhy thm (fig. 1D). Finally, electro-

0167-2789/83/0000-O000/$03.00 1983 North-Holland


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A

B

i i ! ~ i ! i ,

C

90 L. Glass et al./ B(/urcation and chaos in a periodically stimulate d cardiac oscillator

D

EFig. I. Electrocardiograms illust rating nor m al and pathological cardiac rhythms. (A) No rm al sinus rhythm . Each P wave is followeda f ixed t ime la te r ( the PR in te rva l) by a Q RS complex (1 : 1 ra tio in a t r ioventr icu la r conduct ion ) . (B) Second degree A V block wi thone 3 :2 W enckebach cyc le fo llowed by f ive consecutive 2 :1 cyc les . In the W enckebach ph enom enon , the PR in te rva l fo l lowing adrop ped ven tricular beat gradu ally increases until a P wave is no t followed b y a QRS complex (i .e . a ventri cular beat is dropped).(C) Second degree AV block sho wing segments with 2: 1, 4: 3, and 3 :2 co ndu ctio n ratios. (D) T hird degree heart block, No te thedissoc ia t ion of the a t r ia l and ven tr icu la r rhythms. (E) Sinus rhythm with a l te rna t ing PR in te rva ls (2 :2 cond uct ion) . The laddergramsbelow the t racings o f (B? (D ) schemat ica lly show the pa th of impulse format ion a nd con duct ion . The smal l dots in these d iagramsshow si tes of impulse genera t ion , a rrows show the spread of exc i ta t ion , while short dark bars show regions of b lock of conduc t ion .The sinoatrial node is at the top of the laddergram~ the ventricles at the bottom. The time between the heavier vertical l ines on theelectrocardiographic pape r is 0.20 sec. while the voltage difference between ad jacen t heavier h orizon tal l ines is 0.5 inV. Time in tervalsin seconds are ma rked o n some panels. (A ~( D ) show hu m an electrocardiograms reproduced with permission from [1], while (E) isfrom a cat reproduced with permission from [2].

c a r d i o g r a m s w i t h a l t e r n a t i n g P R i n t e r v a l s ( 2 : 2 a n d4 : 2 A V b l o c k ) h a v e b e e n o b s e r v e d i n a n i m a l s [ 2]a n d i n m a n [3 , 4 ]. F i g . I E s h o w s a n i n s t a n c e o f2 : 2 A V b l o c k p u b l i s h e d i n 1 9 10 .

A f u n d a m e n t a l h y p o t h e s i s u n d e r l y i n g o u r w o r kis t h a t c h a n g e s i n th e c a r d ia c r h y t h m c a n b ea s s o c i a te d w i t h b i f u r c a t i o n s i n t h e q u a l i t a t iv e d y -n a m i c s o f m a t h e m a t i c a l m o d e l s d e s c r i b in g g e n e r-a t i o n a n d c o n d u c t i o n o f t h e c a r d i a c i m p u l s e . T h e

e q u a t i o n s d e s c r i b i n g t h e e l e c t r i c a l a c t i v i t y o f t h eh e a r t a re c o m p l e x a n d p r e s u m a b l y v a r y f r o mi n d i v i d u a l t o in d i v i d u a l . H o w e v e r , t h e q u a l i t a t i v ef e a tu r e s o f t h e d y n a m i c s o f t h es e e q u a t i o n s f o r th eh e a r t , a n d i n p a r t i c u l a r t h e ir b i f u r c a t i o n s t r u c t u r e ,m a y w e l l b e s i m i l a r i n d i f f e r e n t i n d i v i d u a l s . I n d e e d ,c a r d i a c d y s r h y t h m i a s h a v e b e e n c la s s if ie d b y n o n -m a t h e m a t i c i a n s ( i. e. c a r d i o l o g i s t s ) o n t h e b a s i s o ft h e i r q u a l i t a t i v e d y n a m i c s [ 1 ] .


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L. Glass et al ./ Bifurcation and chaos in a periodically stimulated cardiac oscillator 91A n e a r l y t h e o r e t i c a l s t u d y o f c a r d i a c d y s -

r h y t h m i a s w a s m a d e b y v a n d e r P o l a n d v a n d e rM a r k i n 1 9 2 8 [ 5 ]. T h e y s h o w e d t h a t w h e n t h r e en o n l i n e a r o s c i l l a t o r s ( r e p r e s e n t i n g p a c e m a k e r s i t e so f t h e h e a r t ) a r e c o u p l e d t o g e t h e r , m a n y d i f f e r e n tr h y t h m s s im i l a r t o t h e A V b l o c k s c a n b e o b s e r v e da s t h e re l a t i v e f r e q u e n c i e s a n d c o u p l i n g p a r a m e t e r so f t h e o s c i l l a t o r s a r e v a r i e d . T h i s p i o n e e r i n g w o r kh a s s i n c e b e e n e x t e n d e d b y o t h e r r e s e a r c h e r s u s i n gana logue and d ig i t a l m ode ls [6 , 7 ] . As wel l , phys -i o l o g ic a l s tu d i e s h a v e p r o v i d e d a d d i t i o n a l e v i d e n c et h a t t h e c a n i n e h e a r t c o n t a i n s a s u b s i d i a r y p a c e -m a k e r s i t u a t e d j u s t b e l o w t h e A V N [8 , 9 ]. H o w -e v e r , m o s t c a r d i a c e l e c t r o p h y s i o l o g i s t s a s c r i b e t h eA V h e a r t b l o c k s t o b l o c k e d c o n d u c t i o n i n t h er e g i o n o f A V N [ 1 ] , w h e r e t h e r e i s a s s u m e d t o b ei m p u l s e c o n d u c t i o n b u t n o i m p u l s e g e n e r a t i o n .M a t h e m a t i c a l m o d e l s f o r t h i s s it u a t i o n s h o w c o r r e-s p o n d e n c e s w i t h t h e A V h e a r t b l o c k s b u t d o n o ts h o w p a t t e r n s w i t h a l t e r n a t i n g P R i n t e r v a l s[ 10 -1 2] . O u r e m p h a s i s i n t h e c u r r e n t w o r k i s o n t h ee f f e c t s o f p e r i o d i c s t i m u l a t i o n o f a n a u t o n o m o u sc a r d i a c o s c i l la t o r . T h i s s i t u a t i o n m a y b e a p p l i c a b l et o t h e A V h e a r t b l o c k s , a n d a l s o i s r e l e v a n t t o aw i d e r a n g e o f o t h e r s i t u a t i o n s , s u c h a s t h e a r t if i c ia lp a c i n g o f th e h e a r t a n d d y s r h y t h m i a s s u c h a sp a r a s y s t o l e .

A n a l y s i s o f t h e e f f e c t o f p e r i o d i c f o r c i n g o nr e l a x a t i o n o s c i l l a ti o n s o f th e v a n d e r P o l t y p e h a v eh a d a m a j o r i m p a c t o n m a t h e m a t i c s . E a r l y s t u d i e sd e m o n s t r a t e d b i s t a b i l i t y ( i n w h i c h o n e o f t w od i f f e ren t s t ab le os c i l l a t ing pa t t e rns i s pos s ib le , de -p e n d i n g o n t h e i n i t i a l c o n d i t i o n s ) a s w e l l a s a p e r i -o d i c d y n a m i c s [ 1 3 , 1 4 ]. T h i s o b s e r v a t i o n o f a p e r i-o d i c d y n a m i c s i n t h e 1 9 4 0 s p l a y e d a r o l e i n t h es u b s e q u e n t f o r m u l a t i o n o f t h e h o r s e s h o e m a p b yS m a l e ( se e t h e a c c o u n t i n [ i 5 ]) . M o r e r e c e n t s t u d i e so f p e r i o d i c f o r c i n g o f n o n l i n e a r o s c i l l a t o r s h a v es h o w n t h a t m u l t i s ta b i l i ty a n d a p e r i o d i c d y n a m i c sc a n b e a c c o u n t e d f o r b y c o n s i d e r a t i o n o f1 - d i m e n s i o n a l m ap s o f the c i r c le in to i t s e l f [ 16 -18].I t h a s a ls o b e e n s h o w n t h a t s u c h m a p s c a n d i s p l a ya p e r i o d i c d y n a m i c s t h a t r e s u l t f r o m a c a s c a d e o fp e r i o d - d o u b l i n g b i f u r c a t i o n s [ 1 9 - 2 3 ] .

R e c e n t e x p e r i m e n t a l s t u d i e s o n h y d r o d y n a m i c

[24-26], chem ica l [27 , 28 ] an d e lec t ro n ic [29 -31]s y s te m s h a v e s h o w n t h e p r es e n c e o f c o m p l e x d y -n a m i c b e h a v i o r s u c h a s q u a s i p e r i o d i c i t y , p e r i o d -d o u b l i n g b i f u r c a t i o n s , i n t e r m i t t e n c y , a n d c h a o s .T h e r e i s g r e a t i n t e r e s t i n a n a l y z i n g i n d e t a i l t h e" u n i v e r s a l " f e a t u r e s o f t h e t r a n s i t i o n s f r o m p e r i -o d i c t o c h a o t i c d y n a m i c s [ 3 2 - 3 6 ] .

I n o u r r e s e a r c h w e h a v e t r i e d t o d e v e l o p b i o l o g -i c a l m o d e l s f o r c a r d i a c d y s r h y t h m i a s a n d t o a n a -l y z e e x p e r i m e n t a l l y o b s e r v e d d y n a m i c s . I n s e c t i o n2 w e s h o w t h a t , u n d e r c e r t a i n a s s u m p t i o n s , t h ep e r i o d i c s t i m u l a t i o n o f a c a r d i a c o s c i l l a to r b y b r i e fc u r r e n t p u l s e s c a n b e r e d u c e d t o t h e a n a l y s i s o f aI - d i m e n s i o n a l m a p . I n s e c t i o n 3 w e c o n s i d e r t h ee f f e c t s o f p e r i o d i c s t i m u l a t i o n o f s p o n t a n e o u s l yb e a t i n g a g g r e g a t e s o f c el ls o b t a i n e d f r o m e m -b r y o n i c c h i c k h e a r t [ 3 7 ]. I n t h i s w o r k a c u r r e n tp u l se g e n e r a t o r i s a n a l o g o u s t o t h e S A N , w h e r e a st h e h e a r t c e l l a g g r e g a t e i s a n a l o g o u s t o t h e s u b -s i d ia r y p a c e m a k e r l o c a t e d b e l o w t h e A V N . A s t h ep e r i o d o f t h e s t i m u l a t i o n c h a n g e s , c o m p l e x b i f u r-c a t i o n s i n c l u d i n g p e r i o d - d o u b l i n g b i f u r c a t io n s c a nb e o b s e r v e d . I n s e c t i o n 4 w e i n v e s t i g a t e a1 - d i m e n s i o n a l m a p t h a t h a s b e e n p r o p o s e d a s as i m p l e m o d e l f o r p e r i o d i c f o r c i n g o f n o n l i n e a ro s c i l l a t o r s . T h i s m a p d e p e n d s o n 2 p a r a m e t e r sc o r r e s p o n d i n g t o t h e s tr e n g t h a n d f r e q u e n c y o f th ep e r i o d i c f o r c i n g .

2 . P u l s a t i l e s t i m u l a t i o n o f a c a r d i a c o s c i l l a t o r :t h e o r y

I n o r d e r t o a n a l y z e t h e b i f u r c a t i o n s o f p e r i o d -i c a l l y s t i m u l a t e d c a r d i a c o s c i l l a t i o n s m a t h e -m a t i c a l ly , a n u m b e r o f a s s u m p t i o n s a r e n e e d e d . I nt h i s s e c t i o n t h e m a i n a s s u m p t i o n s a r e e x p l i c i t l ys t a t e d . A s w e l l , th e r e s u l t i n g p r o p e r t i e s o f t h ed y n a m i c s a r e b r ie f l y d i s c u s se d . S i m i l a r a p p r o a c h e sh a v e b e e n p r e v i o u s l y e m p l o y e d [ 38 -4 0] . M o r e d e -t a i l s o n t h e m a t h e m a t i c a l a s p e c t s c a n b e f o u n d i n[41-43].Assumpt ion ( i) . A c a r d i a c o s c i l l a t o r u n d e r n o r m a lc o n d i t i o n s c a n b e d e s c r i b e d b y a s y s t e m o f o r d i -


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n a r y d i f f e r e n t i a l e q u a t i o n s w i t h a s i n g l e u n s t a b l es t e a d y s t a t e a n d d i s p l a y i n g a n a s y m p t o t i c a l l y s t a -b l e l i m i t c y c l e o s c i l l a t i o n w h i c h i s g l o b a l l y a t t r a c -t in g e x c e p t f o r a s e t o f s in g u l a r p o i n t s o f m e a s u r ez e r o .A s s u m e t h e l i m i t cy c l e h a s p e r i o d T o a n d t h a t w es t a r t w i t h i n i t i a l c o n d i t i o n s x ( t = O ) = x o , w i th x0b e i n g a n a r b i t r a r y p o i n t o n t h e l i m i t c y c l e . S e t t h ep h a s e o f x 0 t o b e z e r o . T h e n t h e p h a s e o f th e p o i n tx ( t ) i s de f i ne d t o be t /To ( m o d 1 ). T h u s , a p h a s eq ~(0 ~ ~ < 1 ) c a n b e a s s i g n e d t o e v e r y p o i n t o n t h el i m i t c y c l e . T h e e v e n t u a l p h a s e o f p o i n t s i n t h eb a s i n o f a t tr a c t i o n o f t h e c y cl e c a n n o w b e d e f in e d .L e t x ( t = 0 ) a n d x ' ( t = 0 ) b e t h e i n i t i a l c o n d i t i o n so f a p o i n t o n t h e cy c l e a n d a p o i n t n o t o n t h e c yc l er e s p e c t i v e ly , a n d x ( t ) , x ' ( t ) , b e t h e c o o r d i n a t e s o ft h e t r a j e c t o r i e s a t t i m e t . I f lim,+~+d(x(t) ,x ' ( t ) ) = 0 , w h e r e d i s t h e E u c l i d e a n d i s t a n c e , t h e nt h e e v e n t u a l p h a s e o f x ' ( t = 0 ) i s t h e s a m e a s t h ep h a s e o f x ( t = 0 ) . A l o c u s o f p o i n t s a l l w i t h t h es a m e e v e n t u a l p h a s e i s c a l le d a n i s o c h r o n . I f t h ee q u a t i o n s d e s c r i b i n g th e s y s t e m a r e o f d i m e n s i o n 2 ,t h e n t h e r e m u s t b e a t l e a s t o n e s i n g u l a r p o i n t a tw h i c h t h e e v e n t u a l p h a s e i s n o t d e f i n e d ( f i g . 2 ) . I ft h e s y s te m o f e q u a t i o n s is o f o r d e r g r e a t e r t h a n 2 ,t h e n t h e r e m u s t e x i s t a s et o f d i m e n s i o n /> 1c o n s i s ti n g o f p o i n t s w h e r e t h e e v e n t u a l p h a s e i s n o td e f i n e d [ 4 1 - 4 3 ] .

C o n s i d e r t h e e f f e c t o f d e l i v e r i n g a s t im u l u s s t a r t -i n g a t a t i m e w h e n t h e o s c i l l a t o r i s a t s o m e ( o l d )ph ase ~b (0 ~< ~b < i ) . In gen era l , a t the en d o f thep e r t u r b a t i o n , t h e o r b i t w i ll li e o n a d i f f e r e n t i s o -

//

x ] l i m i t c y c l e'@-.i s o c h r o n

I

Fig. 2. Phase space of a 2-dimensional nonlinear oscillator. Thesolid oriented closed curve is the limit cycle, while the dashedcurves represent isochrons. The point in the center is anequilibrium point, and therefore is phaseless and does not lie onany isochron.

c h r o n o f ( n e w ) p h a s e 0 . T h e n0 = g ( ~ ) , ( 1 )

w h e r e t h e f u n c t i o n g d e p e n d s o n t h e s ti m u l u sa m p l i t u d e a n d d u r a t i o n . T h e f u n c t i o n g is o f t e nc a l l e d t h e p h a s e t r a n s i t i o n c u r v e ( P T C ) [ 4 2 ] . N o t et h a t t h e r e c a n a l s o b e a s t i m u l u s w h i c h w i ll p e r t u r bt h e o s c i l l a t o r t o a p h a s e l e s s p o i n t .A ssumpt ion ( i i ) . F o l l o w i n g a s h o r t p e r t u r b a t i o n ,t h e t i m e c o u r s e o f t h e r e t u r n t o t h e l i m i t c y c l e i sm u c h s h o r t e r t h a n t h e s p o n t a n e o u s p e r i o d o fo s c i l l a t i o n o r t h e t i m e b e t w e e n p e r i o d i c p u l s e s .

T h i s a s s u m p t i o n a l l o w s o n e t o m e a s u r e t h e P T Cf o r t h e c a r d i a c p r e p a r a t i o n a n d u s e t h e e x p e r i -m e n t a l ly m e a s u r e d P T C t o c o m p u t e t h e e ff ec ts o fp e r i o d i c st i m u l a t io n . A s s u m e t h a t o n e o f t h e v a ri -a b l e s in t h e s y s t e m is e x p e r i m e n t a l l y m e a s u r a b l e .D e f i n e a r e f e r e n c e o r m a r k e r e v e n t t o b e a p a r t i c-u l a r p o i n t o n t h e w a v e f o r m ( e . g . t h e m a x i m u mv a l u e a tt a i n e d ). T h e t i m e b e t w e e n t w o c o n s e c u t i v em a r k e r e v e n t s o f th e s p o n t a n e o u s c y c le is th ep e r i o d T o o f t h e o s c i l l a ti o n . F i g . 3 A s h o w s t h es i t u a t i o n s c h e m a t i c a l l y . A s t i m u l u s ( w h i c h w e a s -s u m e i s a d e l t a f u n c t i o n ) i s d e l i v e r e d a t a t i m e 6f o l l o w i n g t h e p r e c e d in g s p o n t a n e o u s e v e n t . T h ed u r a t i o n T o f th e p e r t u r b e d c y c le is th e t i m e f r o mt h e e v e n t p r e c e d i n g t h e s t i m u l u s t o t h e e v e n ti m m e d i a t e l y s u b s e q u e n t t o t h e s t i m u l u s . N o t e t h a td u e t o f a s t r e l a x a t i o n b a c k t o t h e l i m i t c y c l e , t h ep o s t - s t im u l u s c y c le s a r e a p p r o x i m a t e l y o f d u r a t i o nT 0. T h u s , e s s e n t i a ll y a ll o f t h e e v e n t u a l p h a s e s h i f to c c u r s w i t h i n t h e p e r t u r b e d c y c l e . T h e r e f o r eT I T o = 1 + 4) -- 0, (2)w he re q~ = 6 / T o ( f i g . 3B) . S inc e T c a n be e xpe r i -m e n t a l l y m e a s u r e d f o r a g i v e n q~, a n d TO s k n o w n ,t h e P T C c a n b e d e t e r m i n e d ( f i g . 3 C ) .

N o w c o n s i d e r t h e e f f e c ts o f p e r i o d i c s t i m u l a t i o nw i th a t ime z be tw e e n suc c e s s ive s t imu l i . Ca l l i ng ~ b( m o d i ) t h e p h a s e o f t h e o s c il l a to r i m m e d i a t e l yb e f o r e t h e i t h s t i m u l u s w e o b t a i n


5/13

L . G l a s s e t a l . / B i f u r c a t i o n a n d c h a o s i n a p e r i o d i c a l l y s t i m u l a t e d c a r d i a c o s c i l la t o r 9 3A

s t i m

1 .5 I B ( i ),T/To . o ~0,5 (p

Y0 . 5 . ' ' " ""o 0 . 5q~I c ( i i )

. 0 0 0 . 5 1 , 0q ~Fig. 3. (A) Schematic diagrams illustrating the effect of per-turbation of a limit cycle oscillator by a single impulse. Thevertical lines represent the occurrence o f a marker eve nt (seetext). The spontaneous time between marker events (the periodof the limit cycle) is denoted by T0. A stimulus is delivered ata time 6 after a marker event, causing a change in the cyclelength from TO o T. The durations of the cycles following theperturbed c ycle are all very clo se to T0. (B) The normalizedperturbed interbeat interval T/To plotted as a function of theold pha se ~b = 6 /To. (C ) The new phase-old phase curve (phasetransition curve or PT C) computed from (B) using eq. (2). Inpart (i) of (B) and (C), the stimulus is weak enough to producetype 1 phase resetting, while in (ii) it is sufficiently strong toresult in type 0 behavior. The curves of (B) and (C) are takenfrom a very simple limit cycle model [19].

q ~* , ~b* . . . . q ~ * - i i s a c y c l e o f p e r i o d N . T h e c y c l ei s s t a b l e i f( ~ f v ) i = 4 ~ = N - I I / 6 1 f \ J (5 )

I f a n e x t r e m u m o f th e f u n c t i o n f is a p o i n t o n ac y c l e, th e s l o p e c o m p u t e d f r o m e q . (5 ) e q u a l s z e r o ,a n d t h e c y c l e i s c a l l e d a s u p e r s t a b l e c y c l e . A s t a b l ec y c le o f p e r io d N c o r r e s p o n d s t o s ta b le N : M p h a s el o c k i n g w i t h

NM = ~ [g (~b *) - ~b* + T To]. (6)i = 1

T h e r o t a t i o n n u m b e r is d e f in e d a s

p = lim q~' - q~o (7 )i~oo iF o r s t a b l e N : M p h a s e l o c k i n g th e r o t a t io n n u m -b e r i s r a t i o n a l , p = , M / N . I n s e c t i o n 3 w e u s e e q .( 3 ) t o p r e d i c t t h e p r o p e r t i e s o f t h e p e r i o d i c a l l ys t i m u l a te d c a r d i a c p r e p a r a t i o n . T h e o b s e r v a t i o n o fg o o d a g r e e m e n t b e t w e e n t h e t h e o r e t i c a l p r e d i c -t i o n s a n d e x p e r i m e n t a l o b s e r v a t i o n s p r o v i d e s ap o s t e r i o r i ju s t i f ic a t i o n f o r a s s u m p t i o n ( ii ). H o w -e v e r , a c a r e f u l e x p e r i m e n t a l a n d t h e o r e t i c a l a n a l -y s is o f th e c o n s e q u e n c e s o f t h e b r e a k d o w n o fa s s u m p t i o n ( ii ) h a s n o t y e t b e e n p e r f o r m e d .

, + , = f ( ~ , ) = g ( c k , ) + ~ T o , (3 )w h e r e g ( ~ b ) i s t h e e x p e r i m e n t a l l y d e t e r m i n e d P T Cf o r t h a t s t im u l u s s t r e n g t h a n d g ( ~ + j ) = g ( 4 ) ) + jf o r j a n i n t e g e r . E q . ( 3 ) r e p r e s e n t s a f ir s t r e t u r n o rP o i n c a r 6 m a p . S t a r t i n g f r o m a n i n i t i a l p h a s e ~ b0,o n e c a n i t e r a t e e q . ( 3 ) t o g e n e r a t e a s e q u e n c e ~ b0,(~1 = f ( ~ b 0 ) . . . . . ~ bs = f ( q ~ N - 1 ) = f N ( ~ b 0 ) . I f~ * ( m o d 1 ) = ~ b *( m o d l ),q~* (mo d 1 ) # ~b* (m od 1 ) , 1 ~< i < N , (4 )t h e n q ~ ' is s a i d t o b e a f ix e d p o i n t o f p e r i o d N a n d

A s s u m p t i o n ( i i i ) . T h e t o p o l o g i c a l c h a r a c te r i s ti c s o ft h e P T C c h a n g e i n s t e r e o t y p e d w a y s a s t h e s t i m u -l u s s t r e n g t h i n c r e a s e s .

A s a c o n s e q u e n c e o f a s s u m p t i o n (i) th e P T C isa c o n t i n u o u s m a p o f t h e u n i t c i rc l e i n t o i t se l fg : S ~ S I . T h e t o p o l o g i c a l d e g r e e ( o r w i n d i n gn u m b e r ) o f g is th e n u m b e r o f ti m e s 0 t r av e r s es t h eu n i t c i r c l e a s ~b t r a v e r s e s t h e u n i t c i r c l e o n c e . A tz e r o p e r t u r b a t i o n s t r e n g t h 0 = ~b s o t h a t b y c o n -t in u i ty t h e P T C is a m o n o t o n i c m a p o f d e g r e e 1 f o rs u f f i c ie n t l y s m a l l p e r t u r b a t i o n ( fi g. 3 C ( i) ) . A s t h es t i m u l u s s t r e n g t h i n c r e a s e s f r o m z e r o , o u r e x p e r i -m e n t a l s t u d i e s s e e m t o i n d i c a t e t h a t t h e P T C


6/13

94 L . G l a s s e t a l . / B ( / 'u r c a ti o n a n d c h a o s i n a p e r i o d i c a l ly s t i m u l a t e d c a r d i a c o s c i ll a t o r

r e m a i n s o f d e g r e e l , b u t u n d e r g o e s a t r a n s i ti o nf r o m a m o n o t o n i c t o a n o n - m o n o t o n i c f u n ct io n .W i n f r e e [ 4 3 ] h a s g i v e n e v i d e n c e t h a t a t h i g h s t i m -u l u s s t r e n g t h t h e P T C s f o r n e u r a l a n d c a r d i a co s c i ll a to r s c a n b e o f d e g r ee z e r o a n d t h e r e f o r en o n - m o n o t o n i c ( f i g . 3 C ( i i ) ) . W e c o n j e c t u r e t h a t a ss t i m u l u s s t r e n g t h i n c r e a s e s t h e P T C f o r b i o l o g i c a lo s c i l l a t o r s w i ll , p r o v i d e d a s s u m p t i o n ( i) is s a t is f ie d ,i n g e n er a l u n d e r g o t h e s e q u e n c e o f t r a n s i ti o n sd e g r e e 1 ( m o n o t o n i c ) ~ d e g r e e I ( n o n m o n o -t o n i c ) - - , d e g r e e 0 .W e h a v e p r e v i o u s l y c o n s i d e r e d t h e c as e o f a m o d e lo s c i l l a t o r i n w h i c h , a s t h e s t i m u l u s s t r e n g t h i si n c r e a s e d , t h e P T C u n d e r g o e s a d i r e c t t r a n s i t i o nf r o m d e g r e e 1 ( m o n o t o n i c ) t o d e g r e e 0 [ 19 ]. Ins e c t i o n 4 w e c o n s i d e r t h e p r o p e r t i e s o f a s i m p l em o d e l o f p h a s e l o c k i n g in w h i c h t h e P T C u n d e r -g o e s a t r a n s i ti o n f r o m d e g r e e 1 ( m o n o t o n i c ) t od e g re e 1 ( n o n - m o n o t o n i c ) .

3 . P u l s a t i l e s t i m u l a t i o n o f a c a r d i a c o s c i l l a t o r :e x p e r i m e n t [ 3 7 ]

A s a n a p p l i c a ti o n o f th e t h e o r y p r e s e n t e d i ns e c t i o n 2 , w e c o n s i d e r t h e e f f e c t s o f s i n g l e a n dp e r i o d i c s t i m u l a t io n o f s p o n t a n e o u s l y b e a t i n g a g -g r e g a t e s o f c el ls t a k e n f r o m t h e v e n t r i c l e s o f7 - d a y - o l d e m b r y o n i c c h i ck h e a r t . S p h e r o i d a l a g -g r e g a te s ( 1 0 0 - 2 0 0 # m i n d i a m e t e r ) o f e le c t ri c a ll yc o u p l e d c e ll s w e r e m a i n t a i n e d i n t is s u e c u l t u r e [ 44 ,4 5 ] . E a c h a g g r e g a t e b e a t s s p o n t a n e o u s l y w i t h ape r iod b e tw e e n 0 .4 a nd 1 .3 s e c . Ce l l s w i th in a s ing l ea g g r e g a t e a r e v e r y n e a r l y i s o p o t e n t i a l [ 4 6 ] . T h ev o l t a g e d i f f e r e n c e a c r o s s t h e c e l l m e m b r a n e ism e a s u r e d u s i n g a g l a s s m i c r o e l e c t r o d e f i l l e d w i t hK C I , i n s e r t e d i n t o o n e c e l l o f a b e a t i n g a g g r e g a t e .C u r r e n t p u l s e s a r e d e l iv e r e d t h r o u g h t h e s a m em i c r o e l e c t r o d e .

I n F i g . 4 a r e s h o w n t h e e f f e c ts o f b r i e f i n j e c t io nof c u r re n t a t d i f f e re n t pha se s (q~ = 6/To) o f t h ec a r d i a c c y c l e ( f o r t w o d i f f e r e n t c u r r e n t s t r e n g t h s ) .N o t e t h a t a t t h e h i g h e r c u r r e n t s t r e n g t h t h e r e i s a

, JL E M L I T , J E r, L

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F i g . 4 . T r a n s m e m b r a n e v o l t a g e r e c o r d e d f r o m a n a g g r e g a t e.T h e u p p e r m o s t t r a c in g s e a c h s h o w o n e c y c le o f s p o n t a n e o u sa c t i v i ty . T h e l e f t a n d f i g h t p a n e l s s h o w t h e e f f e c t s o f d e l i v e r i n g5 n A a n d 8 n A c o n s t a n t c u r r e n t p u l s e s ( 20 m s e c d u r a t i o n ) a td i f f e r e n t c o u p l i n g i n t e rv a l s . T h e c o u p l i n g i n t e rv a l 6 i s t h e t i m et o t h e b e g i n n i n g o f th e s t i m u l u s f r o m t h e u p s t r o k e o f t h ei m m e d i a t e l y p r e c e d i n g a c t io n p o t e n t i a l. T h e p e r t u r b e d i n t e r -b e a t i n t e rv a l is d e n o t e d b y T . N o t e t h a t a s t h e s t i m u l u s i n t e n s i t yi s i n c r e a se d f r o m 5 n A t o 8 h A , t h e t r a n s i t io n f r o m p r o -l o n g a t i o n t o s h o r t e n i n g o f t h e p e r t u r b e d c y c l e b e c o m e s m o r ea b r u p t a n d o c c u r s a t a s h o r t e r c o u p l i n g i n te r v a l. T h e i n t e r b e a ti n t e rv a l o f t h e c y c l e f o l l o w i n g t h e p e r t u rb e d c y c l e i s a l m o s te q u a l t o t h e c o n t ro l i n t e rb e a t i n t e rv a l i n t h e th r e e t r a c e s o f t h er i g h t - h a n d p a n e l t h a t s h o w s u c h c y c l e s .

v e r y r a p id c h a n g e i n t h e p e r t u r b e d i n t e r b e a t i n t er -v a l T f o r s t i m u l i d e l i v e r e d a t t im e s f r o m 1 5 0 m s t o1 7 0 m s i n t o t h e c y c l e . O n e e x p e c t s , b a s e d o nt h e o r e t ic a l c o n s i d e r a t i o n s t h a t t h e p l o t o f T v e rs u sq~ c a n b e d i s c o n t i n u o u s f o r s u f f i c i en t l y h i g h c u r r e n ts t r e n g t h s [ 42 ]. F r o m e x p e r i m e n t s , i t is d if f i cu l t ( a n dp e r h a p s i m p o s s i b l e ) t o e s t a b l i s h w h e t h e r s u c hc u r v e s a r e i n d e e d d i s c o n t i n u o u s .

U s i n g t h e T/To d a t a o b t a i n e d f r o m t h e s i n g l ep u l s e p e r t u r b a t i o n e x p e r i m e n t s ( f i g . 5 C ) a n d e q .( 2 ), it is p o s s i b l e t o c o n s t r u c t t h e r e t u r n m a p g i v e n


7/13

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Fig. 5. (A) Transmem brane poten tial from an aggregate as afunction of time, showing spontaneous electrical activity andthe effect of a 20 msec, 9 nA depolarizing pulse delivered at aninterval of 160 msec following the action potential upstroke. In(B to D), parts (ii) show results from this aggregate (aggregate2), while parts (i) are from aggregate 1, taken from a differentculture. (B) Membrane voltage as a function of phase 46,0 ~< 4' < 1. (C) P hase-resetting da ta, show ing the norm alizedlength T I T o of the perturbed cycle as a function of q$. (i) Pulsedura tion 40m sec, pulse amplitud e 5 nA; (i i) pulse duratio n20 m sec , pulse amp l i tude 9 nA. For approxim ately0.4 < 4' < 1.0 the action potential upstroke occurs during thestimulus a rt ifact and hence the perturbed cycle length can not beexactly determined. The dashed line represents a linear inter-pola tion that appro xim ates the data. D uring collection of thesedata , the average con trol interbeat intervals (_+1 stand arddeviation) w er e (i) F 0 = 515 +_5.7 msec and (i i)F 0 = 434 _+5.5 msec. (D) Poincar~ maps computed from eq. (3)and the da ta in fig. 5C; (i) r = 250 msec, (i i) ~ = 480 msec. Thedashed line represents a linear interpolation used in itera ting thePoincarb map; the solid l ine through the data points is a quart icfit for 0.22 < 4', < 0.37. R eprinted from [37] with pe rmission.

b y e q . ( 3) . E x a m p l e s o f r e t u r n m a p s f r o m t w od i f f e r e n t a g g r e g a t e s a r e s h o w n i n f i g . 5 D . U s i n ge q s . (3 ~ F (5 ), t h e e x p e r i m e n t a l l y d e r i v e d r e t u r n

m a p s c a n b e i t e ra t e d t o c o m p u t e t he r e s p o n s e t op e r i o d i c s t i m u l a t io n . T h e n u m e r i c a l l y p r e d i c t e dp h a s e l o c k i n g r e g i o n s ( f i g . 6 B ) a r e c o m p a r e d w i t he x p e r i m e n t a l o b s e r v a t i o n s ( f i g . 6 A ) . I n f i g . 6 C a r es h o w n t h e s t a b l e p h a s e s i n t h e p e r i o d - d o u b l i n gz o n e .

R e p r e s e n t a t i v e t ra c e s o f m i c r o e l e c t r o d e r e c o r d -i n g s a t d i ff e r e n t s t i m u l a t i o n f r e q u e n c i e s a r e s h o w nin fig . 7 . R e g u l a r a n d i r r e g u l a r d y n a m i c s t h e -o r e t i c a l l y p r e d i c t e d i n fi g. 6 w e r e o b s e r v e d . N o t i c et h a t i n f ig . 7 C ( i ) ( a g g r e g a t e 1 ) t h e r e i s a s p o n t a n e -o u s c h a n g e f r o m 1 :1 t o 2 : 2 p h a s e l o c k i n g a t as t i m u l a t i o n p e r i o d o f 55 0 m s . S t i m u l a t i o n o f th es e c o n d a g g r e g a t e a t a p e r i o d o f 4 90 m s p r o d u c e db o t h 4 : 4 a n d i r r e g u l a r p a t t e r n s ( f ig . 7 C ( ii ) a n df ig . 7 C ( ii i) re s p e c t i v e ly ) . C o m p u t a t i o n s s h o w t h a t2 : 2 a n d 4 : 4 p h a s e - l o c k i n g a s w e ll a s c h a o t i cd y n a m i c s a r e p r e d i c t e d i n th e r a n g e 4 6 0 - 4 9 0 m s f o rt h i s a g g r e g a t e ( f i g . 6 C ) .

T h e r e a r e s e v e r a l r e a s o n s t o e x p e c t d i s c r e p a n c i e sb e t w e e n t h e th e o r e t i c a l l y p r e d i c t e d a n d e x p e r i -m e n t a l l y o b s e r v e d d y n a m i c s . F i r s t , a s s u m p t i o n ( ii )i n s e c t i o n 2 i s n o t s t r i c t l y s a t i s f i e d s i n c e t h e r e c a nb e s l i g h t c h a n g e s i n t h e r a te o f t h e p r e p a r a t i o n f o rs e v e r a l b e a t s f o l l o w i n g t h e i n j e c t i o n o f a s in g l es t i m u l u s . T h e f o l l o w i n g t w o a d d i t i o n a l p h y s -i o lo g i c al c o n s i d e r a t i o n s a r e a l so i m p o r t a n t :i ) T h e r e i s b i o l o g i c a l " ' n o i s e " i n t r i n s i c t o t h e e x p e r i -m e n t a l p r e p a r a t i o n a s w e l l a s e n v i r o n m e n t a l " n o i s e "w h i c h i s n o t a c c o u n t e d f o r i n t h e t h e o r y . A l t h o u g ht h e f l u c t u a t i o n s i n i n t e r b e a t i n t e r v a l in t h e a b s e n c eo f s t im u l a t i o n a r e c o m p a r a t i v e l y s m a l l , s o m e o f t h ee x p e r i m e n t a l l y o b s e r v e d i r r e g u l a r p a t t e r n s m a yn o t b e d u e t o i n t ri n s i c a p e r i o d i c i t y in t h e d y n a m -i c a l s y s t e m i t s e l f , b u t r a t h e r t o t h e e f f e c t s o fb i o l og i c a l " n o i s e " g e n e r a t e d w i t h in t h e p r e p a r a -t io n o r e n v i r o n m e n t a l " n o i s e " a r i s in g f r o mf l u c t u a t i n g a m b i e n t c o n d i t i o n s . I n a d d i t i o n , s o m eo f t h e i r r e g u l a r i t y in d e t e r m i n i s t i c a l l y a p e r i o d i cp a t t e r n s ( s u c h a s t h a t s h o w n i n f ig . 7 C ( i ii ) w h i c his a s c r i b e d to " c h a o t i c " d y n a m i c s a r i s in g o u t o f ac a s c a d e o f p e r i o d - d o u b l i n g b i f u r c a ti o n s ) m a y w e llb e a c c o u n t e d f o r b y " n o i s e " . A d d i t i o n a l a n a l y s i s i sn e e d e d t o d e t e r m i n e th e e f fe c t o f " n o i s e " o n


8/13

96 L . G l a s s e t a l . / B i f u r c a t i o n a n d c h a o s i n a p e r i o d i c a l l y s t i m u l a t e d c a r d i a c o s c i l l a to r

I 0 0 3 0 0 5 0 0 7 0 0I 1 I i r I I

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5 5 0 5 5 0 5 7 0 4 5 0 4 7 0 4 9 0" 1 5 ( r e s e t )

Fig . 6 . Exper imenta l ly de te rmined and theore t i ca l ly computed responses to per iodic s t imula t ion of per iod z wi th the same pul sedura t ion s and ampl i tudes as in f ig . 5C. Par t s ( i ) r e fe r to agg rega te 1 , par t s ( i i) to aggrega te 2 . (A) Ex per imenta l ly de te rmined dynam ics :there a re three ma jor ph ase- locking regions (2 : 1 , 1 :1 , 2 :3) and three zones o f compl ica ted dy namics l abe l l ed :q /~ and 7 . (B)Theo re t i ca l ly predic ted dynamics ; note agreem ent wi th (A) . (C) Theore t i ca l ly predic ted dynam ics in zone/3 : curves give phase o r phasesin the cycle at which the st imuli fal l during 1:1, 2:2 and 4:4 locking; s t ippled regions show the range of phases in which the st imulusfa ll s dur ing i r regular dynam ics . R epr in ted f rom [37] wi th permiss ion .

"chaos" and thus to determine the extent to which"noise" is implicated in generating the irregulardynamics experimentally observed.

ii) Prolonged periodic stimulation of the aggregatechanges the properties o f the aggregate. Followinga long period of periodic stimulation with 1:1synchronization at frequencies higher than theintrinsic aggregate beating frequency, the cessationof stimulation leads to a transient slowing of thebeat rate below the control value ("overdrive sup-pression"). Conversely, after a period of "under-drive", "underdrive acceleration" occurs duringthe post stimulation recovery period [47]. Thespontaneous transition from 1:1 to 2:2 phaselocking observed in fig. 7C(i) during stimulation ata fixed frequency could arise from an increase in

the intrinsic frequency of the aggregate secondaryto underdrive.

Despite these considerations, there is goodagreement between theory and experiment. How-ever, probably as a consequence of "noise", wehave not been able to observe phase locking pat-terns which are theoretically computed to extendover small regions of parameter space [48-50].Consequently, experimental observation of theFeigenbaum constant, such as has been performedin periodically forced electronic oscillators [30, 31]and in experiments on turbulence [25] will beexceedingly difficult in this and other biologicalpreparations. On the other hand, biological prepa-rations may be ideal systems to analyze the effectsof "noise" in systems which would be predicted tobe deterministically "chaotic" in the absence ofnoise [51].


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L. Glass e t a l . / B i f urcat ion and chaos i n a per iodi ca l l y s t im ula t e d cardiac osc i l l a tor 97

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F i g . 7. R e p r e s e n t i v e t r a n s m e m b r a n e r e c o r d i n g s f r o m b o t ha g g r e g a t e s s h o w i n g t h e e f f e c ts o f p e r i o d i c s t i m u l a t io n w i t h t h es a m e p u l s e d u r a t i o n s a n d a m p l i t u d e s a s i n f i g . 5 C . ( A ) S t a b l ep h a s e - l o c k e d p a t t e rn s : ( i ) 2 : 1 ( a g g re g a t e 1 , r = 2 1 0 m s ) ; (i i) 1 : 1( a g g re g a t e 2 , ~ = 2 4 0 m s ) ; ( i ii ) 2 : 3 ( a g g re g a t e 2 , r = 6 0 0 m s ) .( B ) D y n a m i c s i n z o n e c t : i r r e g u l a r d y n a m i c s d i s p l a y i n g t h eW e n c k e b a c h p h e n o m e n o n ( a g g r e g a t e 1 , z = 28 0 m s ) . ( C ) D y -n a m i c s i n z o n e r : ( i) 1 : 1 p h a s e l o c k i n g s p o n t a n e o u s l y c h a n g i n gt o 2 : 2 p h a s e l o c k i n g ( a g g r e g a t e 1 , r = 5 5 0 m s ) . D u r i n g 2 : 2p h a s e l o c k i n g t h e r e a r e t w o d i s t in c t p h a s e s o f th e c y c l e a t w h i c ht h e s t i m u l i f a l l . ( i i ) 4 " 4 p h a s e l o c k i n g ( a g g re g a t e , 2 ,T = 4 9 0 m s ) . T h e r e a r e f o u r d i s t i n c t p h a s e s o f t h e c y c l e a t w h i c ht h e s t i m u l i f a l l . ( i i i ) I r r e g u l a r d y n a m i c s w i t h o n e a c t i o n p o t e n -t i a l in e a c h s t i m u l u s c y c l e ( a g g re g a t e 2 , z = 4 9 0 m s ) . T h e re i s an a r r o w r a n g e o f p h a s e s i n w h i c h t h e s t i m u li fa l l. ( D ) D y n a m i c si n z o n e 7 : i r r e g u l a r d y n a m i c s d i s p l a y i n g e s c a p e o r i n t e r p o l a t e db e a t s ( a g g r e g a t e 2 , z = 5 6 0 m s ) . R e p r i n t e d f r o m [ 3 7 ] w i t hp e r m i s s i o n .

4 . A m o d e l m a p [ 2 2 , 23]N u m e r i c a l c o m p u t a t i o n o f p h a se l o c k i n g u s in g

t h e e x p e r i m e n t a l l y d e t e r m i n e d P o i n c a r 6 m a p h a s

s h o w n c o m p l e x b i f u r c a t i o n s . I n t h i s s e c t i o n w ed i s c u s s t h e p o s s i b i l i t y t h a t a l a r g e t o p o l o g i c a l c l a s so f m a p s m i g h t d i s p l a y th e s a m e b i f u r c a t i o n s t ru c -t u r e a s t h e e x p e r i m e n t a l l y d e t e r m i n e d m a p s . S u c hb e h a v i o r m a y b e a n t i c i p a t e d s i nc e t h e b i f u r c a t io n sf o r t h e o n e p a r a m e t e r m a p o f th e i n t e rv a l i n t o i t se l fw i t h a s in g le m a x i m u m a r e l a r g e ly in d e p e n d e n t o ft h e f u n c t i o n a l f o r m o f t h e m a p [ 3 2 , 3 3 ] .

Cons ider the m ap g iven in eq . (3 ) where g (4~) i sa c o n t i n u o u s f u n c t i o n d e f i n e d o n t h e r e a l l i n e a n ddeg[g(~b) (mo d 1)] = 1 . A ssu m e th ere is a s inglem a x i m u m a t ( ] )ma x a n d a s i n g le m i n i m u m a t ~ b ~ , i nth e in te rv al [0, 1] (fig. 5 D(i)) . Le t ~b~ = H~(r) w he ret h e H ~ ( z ) a r e f u n c t i o n s f o u n d b y i t e r a t i n g e q . ( 3)f r o m ~b 0 = ~bma x. F o r j a n i n t e g e r , g(x +j) =g(x) +j, a n dHu ( j ) - HN(] - - l ) = N . ( 8 )T h e r e w i l l b e a s u p e r s t a b l e c y c l e f o r e a c h v a l u e o f

f o r w h i c h 4)0 ( m o d I ) = H u ( r ) ( r o o d 1 ). S in c eHu(z) ( m o d 1 ) e q u a l s a n y fi x e d v a l u e b e t w e e n 0 a n d 1a t l e a s t N t i m e s a s r v a r i e s f r o m j - 1 t o j , t h e r e w i llb e a m i n i m u m o f N s u p e r s ta b l e c y c l es a ss o c i a te dw i t h N d i f f e r e n t r o t a t i o n n u m b e r s o c c u r r i n g a t Nd i s t i n c t v a l u e s o f z . T h e i t e r a t e s o f t h e m i n i m u ma l s o g i v e r i s e t o s u p e r s t a b l e c y c l e s . C o n s e q u e n t l y ,there will be at least two values of z at which thereexist superstable cycles fo r each rational rotationnumber [ 2 2 ]. T h i s f i x e d p o i n t t h e o r e m d o e s n o td e p e n d o n t h e f u n c t i o n a l f o r m o f g .

T h e l - d i m e n s i o n a l , t w o p a r a m e t e r m a p~b,+ l =f(~ b~ ) = ~b~ + b si n 2rcq)~ + z, (9)w h e r e b i s a r e a l n u m b e r h a s r e c e n t l y b e e n d i s -c u s s e d b y s e v e r a l w o r k e r s a s a m o d e l f o r p e r i o d -ica l ly fo rce d no n l i ne ar os c i l l a to r s [22 , 23 , 35, 36 ,40 ]. F o r b < 1 /21 t, f (q$ i) i s a m on o t on ic fun c t io na n d i t i s k n o w n f r o m e a r l y s t u d ie s t h a t t h e r o t a t i o nn u m b e r i s a m o n o t o n i c f u n c t i o n o f ~ a n d t h a t o n l yp e r i o d i c a n d q u a s i p e r i o d i c d y n a m i c s e x i s t [ 5 2 ] . I na d d i t i o n , r e c e n t s t u d i e s h a v e d e s c r i b e d t h e a p p l i c a -t i o n o f r e n o r m a l i z a t i o n m e t h o d s t o a n a l y z e t h edy na m ics fo r 0 < b ~< l /2 rc [35 , 36 ] . W e p r im ar i ly


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98 L. Glass et al./ Bifurcation and chaos in a periodically stimulated cardiac oscillator

c o n s i d e r t h e d y n a m i c s f o r b > 1 / 2= [2 0 -- 23 ]. I n t h i sr e g i o n f (q S ;) is n o t m o n o t o n i c .

F o r e q . ( 9 ) i t i s s t r a i g h t f o r w a r d t o s h o w t h a tt h e r e e x i s t s a s t a b l e f i x e d p o i n t o f p e r i o d 1 a t

IO

~ b* = ~ s in I ( 10 ) o s

( c o r r e s p o n d i n g t o 1 : 1 p h a s e l o c k i n g ) f o r(1 __ .C)2 < b 2 < (1 .t .)2 q_ 7~ 2. ( l l )T h e s t a b l e p e r i o d 1 o r b i t a p p e a r s v i a a ta n g e n tb i f u r c a t i o n ( ~ f / O g p ) ~ , = 1 a t b = 1 - r a nd lo se s it ss t a b i l i t y v i a a p e r i o d - d o u b l i n g b i f u r c a t i o n( ? f / 8 ( o ) ~ , = - 1 at b = [(1 - z ) 2 + x 211"2. In ad d i-t i o n , f o r z = 1 i t i s e a s y t o c o m p u t e t h a t t h e r e i s af u r t h e r b i f u r c a t i o n t o t w o s t a b l e p e r i o d 2 o r b i t sw i t h r o t a ti o n n u m b e r 1 ( 2 : 2 p h a s e l o c k in g ) a tb = 0 .5 a n d a b i f u r c a t i o n t o t w o s t a b l e p e r i o d 4o r b i t s ( 4 : 4 p h a s e l o c k i n g ) a t b = ( 0 . 2 5 + 1/27t2) ~/2.F o r r = 1 .5 th e r e is a p e r i o d - d o u b l i n g b i f u r c a t i o nf r o m a p e r i o d 2 c y c l e ( 2 : 3 p h a s e l o c k i n g ) t o ape r iod 4 c yc le ( 4 : 6 ph a se loc k ing ) a t b = r t ~2 ~,'2.

F u r t h e r r e s u l t s o n t h e b o u n d a r i e s o f t h e p h a s el o c k i n g z o n e s w e r e o b t a i n e d b y n u m e r i c a l a n a l y s i s .T h e P o i n c a r 6 m a p i n e q . ( 9 ) w a s n u m e r i c a l l yi t e r a t e d f r o m d i f f e r e n t 4~0 a t m a n y p o i n t s i n t h e( b , z ) p a r a m e t e r s p a c e . T h e r e s u lt s a r e s h o w n i n fi g.8 . F o r m a n y r e g i o n s i n t h e p a r a m e t e r s p a c e t h e r ei s b i s t a b i l i t y i n t h a t o n e o f t w o s t a b l e c y c l e s i sa s y m p t o t i c a l l y r e a c h e d d e p e n d i n g o n t h e in i ti a lc o n d i t i o n q~ 0. I n a d d i t i o n , e v i d e n c e f o r c h a o t i cd y n a m i c s ( a p o s i t i v e L y a p u n o v n u m b e r ) w a sf o u n d [ 2 3 ] .

T h e I : 1 p h a s e l o c k i n g r e g i o n a r i s e s b y a t a n g e n tb i f u r c a ti o n a l o n g o n e b o u n d a r y a n d is l o s t v i a ap e r i o d d o u b l i n g b i f u r c a t i o n a l o n g t h e o t h e rb o u n d a r y . C o n s e q u e n t l y , s in c e t h e s l o p e o f t h ep e r i o d 1 c y c le is a c o n t i n u o u s f u n c t i o n o f b a n d ~ ,t h e r e m u s t b e a l o c u s o f p o i n t s w i t h p e r i o d 1 a l o n gw h i c h t h e s l o p e is e q u a l t o z e r o . A l o n g t h e l o c u s ,t he m a x i m u m ( o r m i n i m u m ) o f t he P o i n ca r 6 m a pw i l l b e o n t h e c y c l e a n d t h e c y c l e i s c a l l e d as u p e r s t a b l e c y c l e . T h i s w i l l o c c u r f o r

oio 15 20

TF i g . 8. L o c a l l y s t a b l e p h a s e l o c k i n g r e g i o n s fo r e q . (9 ). T h e l i n ea t b = 1 / 2 x s e p a ra t e s t h e r e g i o n s in w h i c h e q . ( 9) i s a m o n o t o n i cf u n c t i o n o n t h e u n i t c ir c le (b < 1 /2 x ) a n d a n o n m o n o t o n i cf u n c t i o n ( b > l / 2 x ) . T h e w i d t h s o f s o m e o f t h e p h a s e l o c k i n gz o n e s ( e. g . 3 : 4 , 2 : 3 ) b e c o m e s o n a r r o w a s b in c r e a s e s t h a t t h eb o u n d a r i e s h a v e b e e n c o l l a p s e d in t o a s i n g le l in e i n t h e d r a f t i n go f t h e f i g u re . I n t h e n o n - l a b e l l e d r e g i o n s a r e p h a s e - l o c k e d ,q u a s i p e r i o d i c a n d c h a o t i c d y n a m i c s . S l i g h tl y m o d i f i e d f r o m [ 23 ]w i t h p e r m i s s i o n .

( ! - - Z ) 2 q - 1 / 4/ l: 2 = / 9 2 , (12)w h e r e f o r ~ < i t h e p o i n t s d e f i n e d b y e q . (1 2 ) a r ed e r iv e d f r o m t h e m a x i m u m a n d f o r z > 1 t h ep o i n t s d e f i n e d b y e q . ( 1 2 ) a r e d e r i v e d f r o m t h em i n i m u m .

T h e l o ci o f t h e s u p e r s t a b l e c y c l e s a r e c a l le d t h es k e l e t o n [ 2 2 ] . I n f i g . 9 a r e s h o w n t h e b o u n d a r i e s o fth e N : M p h a s e l o c k i n g z o n e s (1 ~ < N ~< 5 ) f o r0 < b < 1 / 27 t a n d t h e a s s o c i a t e d s k e l e t o n f o rb > 1 / 2 m F i g . l 0 s h o w s t h e s k e l e t o n d e r i v e d f r o mthe m a x im um ~b . ... f o r N ~< 3 .

A b r a n c h o f t h e s k el e to n e x t e n d i n g d o w n w a r d st o b = 1/2 It i s c a l l e d a p r i m a r y b r a n c h , a n d a l lo t h e r b r a n c h e s a r e c a l l e d s e c o n d a r y . ( T h i s t e r m i -n o l o g y is d u e t o S . S h e n k e r w h o h a s i n d e p e n d e n t l yo b s e r v e d t h e f o l l o w i n g p r o p e r t i e s ) . S i n c e n o t w ob r a n c h e s o f t h e s k e l et o n d e r iv e d f r o m t h e m a x -i m u m c a n i n t e rs e c t , a n d t h e s k e l e t o n o f t h e 1 :1p h a s e l o c k i n g a s y m p t o t i c a l l y a p p r o a c h e s t h e li neb = 1 - r t h e s l o p e o f al l b r a n c h e s o f t h e s k e l e t o nd e r iv e d f r o m t h e m a x i m u m m u s t b e a s y m p t o t i c a l l y


11/13

L . G l a s s e t a l . / B ( f u r c a t i o n a n d c h a o s i n a p e r i o d i c a l l y s t i m u l a t e d c a r d i a c o s c i l l a t o r 99

1.0 1.25 15 175 207"Fig . 9 . The N : M ph as e l ock ing reg ions for 1 ~< N ~< 5 forb < 1 /2 z t a n d t h e a s s o c i a t e d s u p e r s t a b l e c y c l e s f o r b > l / 2 r t fo re q . (9 ) . F ro m [2 2 ] w i t h p e rm i s s i o n .

e q u a l t o - 1 f o r s u f f ic i e n t ly l a r g e v a lu e s o f b .F u r t h e r , a l l p r i m a r y b r a n c h e s d e r i v e d f r o m t h em a x i m u m m u s t m a i n t a i n t h e o r d e r in g g i ve n b ym o n o t o n i c a l l y i n c r e a s i n g r o t a t i o n n u m b e r s f o rf ixed b as ~ inc reas es ( f ig . 10) . F in a l ly , in f ig . 11 , wes h o w t h e s k e l e t o n o f th e p h a s e l o c k i n g z o n e s u p t op e r i o d 4 w i t h p = 1. L i m i t e d n u m e r i c a l a n a l y s i si n d i c a t e s a t o p o l o g i c a l l y e q u i v a l e n t s k e l e t o n a p -p e a r s i n o t h e r V - s h a p e d r e g i o n s f o r m e d b y t h ep r i m a r y b r a n c h e s o f t h e s k e le t o n f o r ea c h r a t i o n a lr o t a t i o n n u m b e r .

O n t h e b a si s o f t h e se n u m e r i c a l a n d a n a l y t ic a lr e s u l t s w e h a v e p r o p o s e d t h e f o l l o w i n g s t r u c t u r ef o r t h e s k e l e t o n a n d p h a s e l o c k i n g z o n e s o f e q . (9 )[ 22 ]. E a c h z o n e o f s t a b l e p h a s e l o c k i n g f o rb < 1 / 2z t e x t e n d s t h r o u g h b = 1 / 2 zt a n d t h e n s p l it si n t o t w o b r a n c h e s ( f i g s . 8 a n d 9 ) . I n t h e V - s h a p e dr e g io n o f th e e x t en s i o ns o f e a c h " A r n o l d t o n g u e "

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Fig . 10 . T he p er io d N sup ers t a b l e cyc l es for 1 ~< N ~< 3 der ivedf ro m q~ma~ fo r e q . ( 9 ). T h e re i s a s y m m e t r i c a l l y l o c a t e d s e t o fs u p e r s t a b l e c y c l e s f o u n d f r o m i t e r a t i o n o f (~min"

0.6 . 4;44 4:433 313b ~ 4~4 ~414

IIo0.7 1.0 1.3

7-F i g . 1 1 . S u p e r s t a b l e c y c l e s a s s o c i a t e d w i t h N : N p h a s e l o c k i n gp a t t e rn s , 1 ~< N ~ 4 ( ro t a t i o n n u m b e r p = 1 ). F ro m [2 2 ] w i t hp e r m i s s i o n .a r e p e r i o d - d o u b l i n g b i f u r c a t i o n s . T h e s k e l e t o n o ft h e p h a s e l o c k i n g z o n e s in e a c h o n e o f t h es eV - s h a p e d r e g i o n s is t o p o l o g i c a l l y e q u i v a l e n t t o t h eske l e to n fo r p = 1 ( fi g. 11 ), bu t w i th a d i f fe re n tr o t a t i o n n u m b e r .

T h e r e i s c o n s i d e r a b l e i n t e r e s t i n t h e t r a n s i t i o nf r o m q u a s i p e r i o d i c t o c h a o t i c d y n a m i c s i n p h y s i ca ls y s t e m s a n d m a t h e m a t i c a l m o d e l s [ 3 5 , 3 6 , 5 3 ] . I nt h e c u r r e n t c o n t e x t o n e q u e s t i o n i s , " W h a t h a p -p e n s t o t h e q u a s i p e r i o d i c o r b i t s ( i . e . t h o s e w i t hi r r a t io n a l r o t a t i o n n u m b e r ) k n o w n t o ex i s t w h e nb < 1 /2~ a s b pa s se s t h ro ug h t h e va lue b = 1 /2z t ? "F o r a m a p o f t h e c i r c le i n t o i ts e lf , f o r a s e t o fp a r a m e t e r s f o r w h i c h t h e r e i s b i s t a b i l i ty w i t h t w od i f f e r e n t r o t a t i o n n u m b e r s , t h e r e w i l l b e i n i t i a lc o n d i t i o n s w h i c h g i v e a l l i n t e r m e d i a t e r o t a t i o nn u m b e r s [ 17 ]. A c o n s e q u e n c e o f t h e b i s t a b i li t y o ft h e s in e m a p f o r b > l / 2 n is t h a t f o r b > l / 2 no r b i t s w i t h a g i v en i r r a ti o n a l r o t a t i o n n u m b e r w illb e p r e s e n t i n a w e d g e s h a p e d r e g i o n w h o s e t i p i son t he l i ne b = 1 /2z t. I t sh ou ld be p os s ib l e t od e s c r i b e t h e a p e r i o d i c o r b i t s a s s o c i a t e d w i t h a ni r r a t i o n a l r o t a t i o n n u m b e r u s i n g t e c h n i q u e s d e v e l -o p e d in s y m b o l i c d y n a m i c s a n d k n e a d i n g t h e o r y[16, 17 ] . H o w e v e r , i f t he re i s b i s t a b i l i t y i t is s ti ll no tk n o w n i f a l m o s t a ll p o i n t s w i ll a tt r a c t t o o n e o f t h et w o s t a b l e p e r i o d i c o r b i t s . F i n a l l y , t h i s d i s c u s s i o nh a s b e e n c o n c e r n e d w i t h a n a l y si s o f l - d i m e n s i o n a lm a p s . A c a r e f u l a n al y s is o f t h e b r e a k d o w n o fq u a s i p e r i o d i c d y n a m i c s in 2 - d i m e n s i o n a l m a p s h a ss h o w n a n o v e r l a p p i n g o f A r n o l d t o n g u e s s i m i la r totha t show n in f i g s . 8 a nd 9 [53 ] .


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10 0 L. Glass et al./ Bifurcation and chaos in a periodically stimulated cardiac oscillator

5. Conc lus ionsWe have shown that the effects of periodic

pulsatile stimulation of a cardiac oscillator can beanalyzed by consideration of a 1-dimensional mapwhich is obtained from experimental meas-urements. The dynamics in response to periodicstimulation are predicted by iterating this experi-mentally derived map and bear a close correspon-dence to the experimentally observed dynamics. Inparticular, stable phase locking, period-doublingbifurcations and aperiodic dynamics are all the-oretically computed and experimentally observed.The experimentally observed dynamics show pat-terns similar to many commonly observed cardiacdysrhythmias. Furthermore, this work gives anovel perspective to some of the aperiodic cardiacdynamics clinically observed as well as uncommonphase locked rhythms such as 2:2 and 4:2 AVblock. We have also analyzed the dynamics for asimple Poincar6 map (the sine map in eq. (9)) anddiscussed period-doubling bifurcations, bistabilityand chaos in this model system.

Experimentally observed transitions from peri-odic to chaotic dynamics in physical systems canoften be accounted for, at least qualitatively, by thebifurcations in simple 1- and 2-dimensional mapsunder parametric changes [26-29]. Unfortunately,in this work, it is not yet possible to compute fromfirst principles the underlying maps which give aphenomenological correspondence with theory.Consequently, the study of periodic forcing ofoscillations by short pulsatile inputs has advan-tages over other experimental techniques and maybe useful in other physical systems. There are manysimilarities between the experimentally measured1-dimensional map for the cardiac oscillator andthe maps derived from other experimental systems[26-29]. Clearly, further work is needed to carefullyprobe experimentally observed dynamics in phys-ical and biological systems and to clarify thebifurcation structure of maps which display quasi-periodic, periodic and aperiodic dynamics.

Analysis of the mathematics describing cardiacdysrhythmias has potentially rich rewards in the

diagnosis and treatment of heart disease. We hopethat our work will help stimulate interest in theseproblems by mathematicians and physicists.

A c k n ow le d gm e n t sThis research has been supported by grants from

the Natural Sciences and Engineering ResearchCouncil of Canada and the Canadian Heart Foun-dation. MRG is a recipient of a predoctoral train-eeship from the Canadian Heart Foundation. Wethank John Guckenheimer, James Sethna andScott Shenker for helpful conversations. Themanuscript was partially written while LG was inresidence at the Aspen Center for Physics.

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leon glass et al- bifurcation and chaos in a periodically stimulated cardiac oscillator

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