Er Men Trout 1991 Multiple

7/25/2019 Er Men Trout 1991 Multiple

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J. Math. Biol. (199l) 29:195-217

Journal o f

Mathematical

iology

Springer-Verlag 1991

Multiple pulse interactions and averaging

in systems of coupled neural oscillators

G B Ermen trout I and N Kopell 2

z Departm ent of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

2 Departm ent of Mathematics, Boston University, Boston , USA

Received November 28, 1989; received in revised form April 6, 1990

Abstract Osci l la tors coupled s t rongly a re capable of compl ica ted behavior

which may be pa thologica l for b io logica l cont ro l sys tems. Never the less , s t rong

coupl ing may be needed to prevent asynchrony. We discuss how some neura l

ne tworks may be des igned to achieve only s imple locking behavior when the

coupl ing i s s t rong. The des ign is based on the fac t tha t the m etho d o f averaging

produces equa t ions tha t a re capable only of locking or dr i f t , no t pa thologica l

complexi ty . Fur thermore , i t i s shown tha t osc i l la tors tha t in te rac t by means of

mult iple pulses per cycle , dispersed around the cycle , behave l ike averaged

equatio ns, even if the nu mb er o f pulses is small. We discuss the biological

in tu i t ion behind th is scheme, and show numer ica l ly tha t i t works when the

oscil la tors are tak en to be com posites, each unit o f which is gove rned b y a

wel l -known m ode l o f a neura l osc il lator . F ina l ly , we desc ribe numer ica l meth ods

for comput ing f rom equa t ions for coupled l imi t cyc le osc i l la tors the averaged

coupl ing func t ions of our theory .

Key words:

Oscil la t ions - N eur on s - Avera ging - N eura l c ircuits

1 Introduction

Many phys io logica l cont ro l sys tems a re governed by coupled neura l osc i l la tors

[ 1-4] . In these sys tems, i t is impo r tant to m ain ta in some degree of synchroniza -

t ion . In orde r to m ainta in synchrony , coupl ing be tween the osc i l lat ing uni ts mus t

be suff ic iently strong; w eak co upling can synch ronize onl y osci l la tors that are

very c lose in natural f requency. I f the dif ference between units is too great and

coupl ing i s weak, desyn chrono us behavior can a r ise , such as phase dr i f t [5]. On

the o ther hand, i f the coupl ing is too s t rong var ious pa tho logica l s i tua t ions a r ise .

Fo r exam ple , in [6] i t is show n th at s trong diffusive coup ling between chem ical

* Research partially supported by the National Science Foundation under grants DMS 8796235 and

DMS 8701405 and the Air Force Office of Scientific Research under University Research Contract

F 49620-C-0131 to Northeastern University


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196 G .B . Ermentrout and N. Kopell

osc i l l a to r s can l ead to chaos . S imi l a r ly , we have found tha t s t rong inh ib i to r i l y

co u p l ed n eu r a l o s c i l l a t o r s can s o me t i mes l ead t o co mp l ex d y n ami ca l b eh av i o r

[ u n p u b li s hed ] . I n a p r ev i o u s p ap e r [ 7 ] , w e s h o w ed t h a t m u t u a l ex c i t a to r y

coup l ing o f su f fi c ien t s t r eng th be tw een a p a i r o f l imi t cyc l e osc i l l a to r s can ac t t o

s top the osc i l l a ti on ( o sc i l l a to r dea th ) . T he m echan i sm d escr ibed in [7 ] can

o ccu r i n m o d e l s o f n eu r a l i n t e rac t io n s t h a t i n v o lv e ch emi ca l s y n ap s es ev en i f t h e

neura l o sc i l l a to r s a re i den t i ca l . We are t hus f aced wi th t he fo l lowing p rob lem:

H o w can w e ma i n t a i n s y n ch r o n y b e t w een p o s s i b l y d i s p a r a t e o s c i l l a t o r s w i t h o u t

s t r eng then ing the coup l ing so mu ch as t o cause loss o f osc i l l a ti on o r o the r

d y n ami c p a t h o l o g i e s ?

W e sho w in [7 ] t ha t o sc i l l a to r dea th c ann o t occ ur if t he osc i l l a to r s a re wea k ly

co u p l ed . Wh en co u p l i n g i s w eak , i n v a r i an t man i f o l d t h eo r y may b e u s ed t o

reduce the sys t em o f non l inear osc i l la to r s t o a se t o f equa t ions on a t o rus .

A v e r ag i n g t h eo r y may t h en b e u s ed t o o b t a i n eq u a t i o n s t h a t d ep en d o n l y o n t h e

differences of the pha ses o f the osc i l la to r s . Fo r a pa i r o f osc i l la to r s the ave raged

eq u a t i o n s h av e o n l y t w o p o s s i b l e b eh av i o r s : s y n ch r o n i za t i o n o r q u as i p e r i o d i c

osc i l la t i ons (phase d r i ft ) . The l a t t e r oc curs i f t he unco up le d f r equencies o f t he

osc i l l a to r s a re no t su f f i c i en t ly c lose t o one ano ther .

I n t h is p ap e r w e s h o w t h a t , ev en w i t h s t r o n g i n t e rac t io n s , t h e s y s t em m ay

b eh av e a s i f av e r ag ed , an d w e d es c ri b e s o me n eu r a l n e t w o r k s t h a t c an acco m -

p l ish thi s . The fund am enta l i dea is t ha t , i f i n t e rac t ions a re d i sper sed a ro un d the

cy c le o f th e o s c il la t o rs , t h e s y s t em can b e h av e a s t h o u g h t h e co u p l i n g w er e

av e r ag ed o v e r a cy c le , an d t h u s a l l o f th e a b o v e p a t h o l o g i ca l b eh av i o r is

p r ev en t ed . F o r s u ch a s y s t em, t h e co u p l i n g can b e s t r o n g en o u g h t o o v e r co me

man y f r eq u en cy i n h o mo g en e i t i e s t h a t co u l d o t h e r w i s e l e ad t o p h as e d r i f t .

The paper i s o rgan ized as fo l lows: i n Sec t . 2 , we rev i ew the der iva t ion o f

p h as e mo d e l s f r o m t h e f u l l s y s t em o f o s c i l l a t o r s . We t h en d i s cu s s h o w p h as e

eq u a t i o n s m o d e l li n g a p a i r o f co u p l ed o s c i ll a to r s m ay b e formally av e r ag ed , an d

wh en th is averag ing m etho d is va lid . In Sec t. 3 we sho w tha t , i f t he coup l ing

co n s is t s o f a s u m o f t e r ms each co r r e s p o n d i n g t o a p u ls e , an d i f t h e s e p u ls e s a r e

su f fi c ien t ly d i sper sed a rou nd the cyc l e , t hen the averag ing m eth od i s va l id fo r a

suff iciently large nu m be r o f pulses . W e also d iscuss the b io logica l in tu i t ion

b eh i n d t h e mu l t ip l e p u l s e co u p li n g , an d h o w s u ch co u p l in g i s n a tu r a l l y o b t a i n ed

w hen the osc i l l a to r is a com pos i t e o f neurons , poss ib ly wi th d i f f e ren t ce l l t ypes .

N u mer i ca l c a l cu l a t i o n s a r e d o n e i n S ec t . 4 t o s h o w t h a t ev en a s ma l l n u mb er o f

p u l s e s (e . g. 3 o r 4 ) m ay b e en o u g h t o r ad i ca ll y ch an g e t h e b eh av i o r o f t h e

co u p l ed sy s t em, f r o m o n e th a t u n d e r g o es o s c i l la t o r d ea t h i f t h e i n t e rac t io n i s

on ly by a s ing le pu l se t o one tha t phase- locks l i ke a sys t em whose coup l ing

depends on ly on the d i f f e rences on the phases .

Th e co m p o s i t e o s c i ll a to r s o f t h e n e t w o r k s o f S ec t. 3 h av e e l emen t s t h a t a r e

very t i gh t ly coup led compared to t he i n t e r -osc i l l a to r coup l ing . In Sec t . 5 , we

s h o w n u mer i ca l ly t h e r eq u i r em en t f o r t ig h t i n t r a -o s c i ll a to r co u p l i n g can b e

r e lax ed an d s till h av e t h e s y s t em b eh av e w i t h o u t p a t h o l o g y . Th i s i s d o n e b o t h f o r

p h as e mo d e l s ( r ed u ced f r o m f u l l eq u a t i o n s d e s c r i b i n g amp l i t u d es a s w e l l a s

p h as es ) a s w e l l a s th e f u ll eq u a t i o n s , an d a l lo w s t h e co n s t r u c t io n o f av e r ag i n g

ne two rks t ha t a re m ore phys i ca l ly r ea li s ti c and mo re genera l .

I n t h e ap p en d i x , w e co n s id e r n u mer i ca l me t h o d s f o r co m p u t i n g an ap p r o x i -

ma t i o n t o t h e av e r ag ed co u p l i n g f u n c t i o n s H ( w h i ch d ep en d o n l y o n t h e p h as e

di f ference ~b = 02 - 01) . H( t~) i s co m pu ted u nd er tw o al terna t ive hyp othes es . In

the f ir st , the limi t cyc l e has i n f in i te a t t r ac t ion b u t t he coup l ing need no t be

weak ; i n t he second , t he r a t e o f a t t r ac t ion to t he l imi t cyc l e i s f in ite , bu t t he


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Mu ltiple pulse interactions and averaging 97

c o u p l i n g is w e a k , I n t h e f ir s t c a se , o n e u s e s t h e n u m e r i c a l t e c h n i q u e s t o f ir s t r e d u c e

t o a p h a s e m o d e l a n d t h e n t o a v e r a g e . I n t h i s s i t u a t i o n , w h e n t h e a t t r a c t i o n i s

e x t r e m e l y s t r o n g , t h e d e p e n d e n c e o f th e lo c a l f r e q u e n c y o n t h e a m p l i t u d e s Y i i s

n o t v e r y i m p o r t a n t . I n t h e se c o n d c a s e, i n w h i c h th e a t t r a c t i o n n e e d n o t b e e x t r e m e l y

s t ro n g , t h e d e p e n d e n c e o f

d O i / d t

o n t h e a m p l i tu d e s m a y n o t b e ig n o r e d . T h i s is

a m o r e s u b t le c a lc u l a ti o n , w h i c h is d o n e i n tw o w a y s . W e a p p l y o u r m e t h o d t o

a cl as s o f n e u r a l e q u a t i o n s t h a t i n c lu d e t h e L e c a r - M o r r i s s y s t e m [8 ] a n d t h e

W i l s o n - C o w a n e q u a t io n s [9].

2 Redu ct ion to phase mo de l s and averag ing

2 . 1. R e d u c t i o n t o p h a s e m o d e l s

F o r c o m p l e t e n e s s , w e b r i e fl y r e v i e w th e r e d u c t i o n o f a p a i r o f c o u p l e d o s c i l la t o r s

t o a s y s te m o f e q u a t i o n s o n t h e t o m s . C o n s i d e r t h e c o u p l e d s y s t e m o f n o n l i n e a r

os c i l l a t o r s :

d X l / d t

= F I X , ) + f l G , X i , X 2 ),

d X 2 / d t = F 2 X 2 ) + f l G 2 X : , X , ) . 2 . 1 )

H e r e X j ~ R , f l i s a p a r a m e t e r d e t e r m i n i n g t h e s t re n g t h o f c o u p l i n g b e t w e e n t h e

t w o o s c i ll a to r s , G j i s t h e c o u p l i n g f u n c t i o n , a n d i t i s a s s u m e d t h a t

d X f l d t = F j ( X j )

h a s a s a s o l u t i o n a n o r b i t a l l y a s y m p t o t i c a l l y s t a b l e p e r i o d i c s o l u t i o n w i t h f re q u e n c y

c oj. I n [ 1 0 ] i t i s s h o w n t h a t t h e r e i s a c h a n g e o f c o o r d i n a t e s i n t h e n e i g h b o r h o o d

o f t h e l i m i t c y c l e s u c h t h a t , i f f l i s n o t t o o l a r g e , ( 2 . 1 ) c a n b e w r i t t e n a s

o g f I d O s / d t = 1 + q J ( Y s , O j ) + f l e j ( y j , Y k , O j , O k , f l) ,

( 2 . 2 )

d y J d t = a j ( y j , O j ) + 4 ( Y J, Y k , O j, O k , ) ,

j , k = 1 , 2 , j # k ( se e a l so t h e a p p e n d i x ) . H e r e q j ( 0 , 0 ) = a j ( 0 , 0 ) = 0 , a n d a ll

f u n c t i o n s a r e p e r i o d i c i n t h e i r 0 a r g u m e n t s . T h e c o o r d i n a t e s

Y i ~ R n - 1

a r e

t r a n s v e r s e t o t h e p e r i o d i c o r b i t s a n d 0 j ~ S t li es a l o n g t h e l im i t c y c l e o f t h e j t h

o s c i l l a t o r . I n a b s e n c e o f c o u p l i n g (/3 = 0 ) e a c h o s c i l l a t o r t r a v e r s e s i ts c y c l e w i t h

a p e r i o d o f 2rr/c 0j. T h e r e a r e t w o s i t u a ti o n s i n w h i c h w e c a n r e d u c e ( 2 . 2 ) f r o m a

2 n - d i m e n s i o n a l s y s t em t o a f lo w o n a t w o - d i m e n s i o n a l t o r u s . I f / / a n d IF I - F 2 1

a r e s u f f ic i en t ly s m a l l, w e c a n u s e i n v a r i a n t m a n i f o l d t h e o r y [ 1 1] t o c o n s t r u c t a t o r u s

y j (Oj , Ok)

w h i c h is i n v a r i a n t f o r (2 . 2 ). O n t h a t t o r u s , t h e e q u a t i o n s h a v e t h e f o r m

o ) f ' d O j / d t = 1 - t - / / h j O j , O k , / / ) ,

( 2 . 3 )

j , k = 1 , 2 , j ~ k . I f ](o j - o)k[ = 0 ( / / ) , t h e n a v e r a g i n g t h e o r y [ 12 ] i m p l i e s th a t t h e r e

i s a n a l m o s t i d e n t i t y c h a n g e o f c o o r d i n a t e s s u c h t h a t , i n t h e n e w v a r i a b l e s, ( 2 . 3 )

h a s t h e f o r m

( o f I d O J d t

= 1 + / / H j ( 0 k - - 0 j ) + 0 ( / /2 ) . ( 2 . 4 )

H e r e , / / j i s a 2 n - p e r i o d i c f u n c t i o n o f i ts a r g u m e n t s . I f q~ = 0 2 - 0 1, t h e f u n c t i o n s

/ / j a re d e t e r m i n e d f r o m h j b y

n ~ ~ ) = 1 / 2 ~ ) / /h 1 0 1 , O , + 4 ~ , // )

dO l

( 2 . 5 )

g 2 - 4 ,) = 1 / 2 r 0 / / h ~ O ~ ,O~ - ~ , / /) dO~ .


4/23

1 98 G . B . E r m e n t r o u t a n d N . K o p e l l

T h e s e c o n d s i tu a t i o n i n w h i c h 2 .2 ) c a n b e r e d u c e d t o a f lo w o n t h e t o m s i s

i f t h e a t t r a c t i o n t o t h e l i m i t c y c l e is s tr o n g s e e [7 ]). S t r o n g a t t r a c t i o n i m p l i e s

t h a t { yj} r e m a i n c l o s e t o 0 , s o th a t 2 . 2 ) re d u c e s t o

co) 1 d O j/dt = 1 + ~h: Oj, Ok, ~), 2 . 6 )

w h e r e n o w hj O j, Ok, 8) i s g iv en exp l i c i t l y by c0 j e j 0 , 0 , 0j , Ok, 8) .

I n t h e a b s e n c e o f t h e O /~2 ) t e r m s , 2 . 4 ) i s c a p a b l e o n l y o f p h a s e - l o c k i n g o r

p h a s e - d r if t . E q u a t i o n s o f t h e f o r m 2 .6 ) a r e m o r e g e n e r a l , a n d a r e c a p a b l e o f

c o n s i d e r a b l y m o r e c o m p l e x b e h a v i o r , i n c l u d i n g a v a r i e t y o f lo c k i n g p a t t e r n s a n d

o s c i l l a to r d e a t h [7 ]. T h e l a t t e r is v e r y e a s y t o o b t a i n , p a r t i c u l a r l y w h e n h i , h 2 a r e

d e r i v e d f r o m e x c i t a t o r y n e u r a l c o u p l i n g s e e [7 ]). I f t h e c o u p l i n g is su f f ic i e n tl y

s t r o n g th a t t h e a v e r a g in g t h e o r e m c a n n o t b e a p p li e d to p r o d u c e 2 .4 ) , w e a r e

c o n f r o n t e d b y t h e p r o b l e m s d i s c u s s e d i n t h e i n t r o d u c t i o n .

R e m a r k 2 . 1 . I n [7 ], w e s h o w t h a t f o r s o m e m o d e l s o f e x c i t a t o r y n e u r a l c o u p l i n g ,

t h e f u n c t i o n s hj Og, Ok, 8) h a v e t h e f o r m P j O k ) R j 0 i ) w h e r e P O) i s a pos i t i ve

p u l s e - l i k e f u n c t i o n a n d

R O)

i s a n a l o g o u s t o t h e p h a s e - r e s p o n s e c u r v e f o r a

f o r c e d o s c i l l a t o r . S e e [ 7] a n d [ 1 ].)

2 .2 . Averag ing

I f /~ i s s m a l l, a v e r a g i n g t h e o r y p r o v i d e s a n a l m o s t i d e n t i t y c h a n g e o f c o o r d i n a t e s

s u c h t h a t , i n t h e n e w c o o r d i n a t e s , t h e c o u p l i n g d e p e n d s o n l y o n t h e d i f fe r e n c e ~b

o f t h e p h a s e s . E v e n i f/ ~ i s n o t s m a l l , o n e m a y r e w r i t e 2 . 6 )

b y f o r m a l l y

a v e r a g i n g ,

l e a v i n g a r e m a i n d e r t e r m :

d O 1 / d t = (I )1 - ~ H I ( ( ~ ) + F 1 ( 0 1 , 0 2 ) , ( 2 . 7 )

dO2/dt = 0 )2 + H 2 - ~ b ) + F2 02 , 01 )

w h e r e ~b = 0 2 - 0 1, H I , / / 2 a r e a s in 2 .5 ) a n d

F 1 ( 0 1 , 0 2 ) = f l h l 0 1 , 0 2 , f l ) - H i ( 0 2 - 0 1 ) ,

F2 02, 01) = f lh2 02, O r, f l ) - H2 01 - 02).

L e t M = m a x [ F j 0 j , Ok,/~) I, J , k = 1 , 2 ; j ~ k . I f M ha pp en s t o be su f f i c i en t ly sm a l l

. 01 02

a s i s a u t o m a t i c a l l y t r u e i f / ~ is s m a l l ), t h e c o n s e q u e n c e s o f a v e r a g i n g t h e o r y f o r

s m a l l / ~ s t i l l h o l d . F o r e x a m p l e :

P r o p o s i t i o n 2 . 1 . S u p p o s e t h a t

d

= ~ 1 - ~ 2 + H , ~ ) - H 2 - ~ ) 2 . 8 )

has a hyperb o l i c f i xe d p o in t , wh ere c~ = 0 2 - 01 . T hen fo r M su f f i c i en t ly sma l l ,

2 . 7 ) has a hyperbo l i c l imi t cyc l e o f t he same s tab i l i t y charac t e r i s t i c s .

P r o o f . A c r i ti c a l p o i n t o f 2 . 8 ) c o r r e s p o n d s t o a p e r i o d i c s o l u t i o n o f

dO1/d t =co l + HI ~b), 2.9)

d O 2 / a t = o ~ + i - I 2 - e ~ ) .


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M ultiple pulse interactions and averaging 99

C o n s i d e r a P o i n ca r 6 s ec t i o n t h r o u g h t h a t p e r i o d i c s o l u t i o n an d t ak e t h e s ame

sect ion fo r (2 .7) . The n fo r M suff icient ly smal l , the Poinc ar6 m ap o f (2 .7) is st il l

def ined and i s c lose t o t ha t o f (2 .9 ) . The refo re , i t has a h ype rbo l i c f ixed po in t

wi th t he sam e s t ab i li t y charac t e r is t ics . [ ]

S ince M rep resen t s t he dev ia t ion f rom be ing averaged , we re fer t o i t as t he

m o d u l u s o f av e r ag eab i l it y . A s y s t em w i t h a lo w mo d u l u s can n o t u n d e r g o

osc i l l a to r d ea th , i .e . (2 .9 ) has n o c r i ti ca l po in t s . T hat is , by con t inu i ty we

h av e

P r o p o s i t i o n 2.2. Sup pos e tha t ] dO1/dt, dO2/dt)] o f (2 .9) i s b o u n d e d a w a y f r o m

zero. Th is is tru e fo r gen eric 091 , co2. Then fo r M suf f ic ient ly smal l , (2.7) has no

cr i t ical poin ts . [ ]

R e m a r k 2 . 2 .

Th e ab o v e p r o p o s i t i o n s a l s o w o r k f o r ch a i n s o f N o s c il la t o rs . T h e

k t h eq u a t i o n i s av e r ag ed o v e r Ok, w ith Ok+ l = q~k + Ok a n d Ok_ l = Ok -- ~k - 1.

The avera ged sys t em i s t hen a fun c t ion o f on ly the p hase d i f fe rences {q~k}. The

ma i n h y p o t h es i s , a s i n t h e ab o v e t h eo r em, i s t h a t t h e max i mu m o v e r t h e

N - t o r u s o f t h e d i ff e ren ce b e t w een t h e tr u e R .H .S . o f t h e eq u a t i o n an d t h a t o f

t h e av e r ag ed r i g h t - h an d s i d e ( co n s i d e r ed a s a f u n c t i o n o f {Ok }) be suff icient ly

small.

3 M u l t i p l e p u l s e s a n d th e r ed u c t i o n o f t h e m o d u l u s o f a v e r a g e a b i li t y

In t he l as t sec tion , we no te d tha t i f t he phase equ at ions (2 .7 ) a re c lose eno ugh

to the av erage d equat ions , t hen the i r behav io r is a lso s imi la r. H ere we sha l l

descr ibe som e ne two rks o f osc i l l a to r s wh ich exp lo i t t h is f ac t . These ne tw orks

are des igned so as t o d i s t r i bu t e t he impu l ses over t he cyc l e o f t he osc i l l a t i on ,

r a t h e r t h an co n cen t r a t i n g t h em n ea r o n e p a r t icu l a r p h as e o f t h e cy c le . A s w e

sha ll see , t hese ne tw orks beh ave l ike ave rag ed equa t ions , and hence avo id the

p r o b l ems d es c r i b ed ab o v e .

C o n s i d e r a p a i r o f o s c i ll a to r y u n it s an d a s s u me t h a t t h e s e u n it s a r e co m -

pose d , no t o f a s ing le pace m aker , bu t r a ther o f m osc i l la t ing subun i t s . (Th ese

subun i t s need no t be cap ab le o f osc i ll a ti ng in i so la t i on . ) Fo r exam ple , i t i s

be l i eved tha t t he segm enta l o sc i l la to r s o f t he l eech cen t ra l pa t t e rn gen era to r

cons i s t s o f a t l eas t fou r sub un i t s t h a t o sc i ll a te . (See [3 ] fo r a 197 8 ver s ion o f

the l oca l osc i l l a to r ; s i nce then , o ther osc i l l a t i ng componen t s have been d i scov-

e r ed ( M . N u s s b au m, p e r s . co mm. ) . ) We a s s u me f u r t h e r t h a t t h e s e s u b u n i t s

have f ixed phas e r e l a t i onsh ips t o o ne an o the r a nd do n o t a l l o sc i ll a t e i n phase ;

th i s can be accompl i shed us ing some inh ib i to ry coup l ing ( see Sec t . 5 ) . The

coup l ing wi th in a s ing le un i t i s assume d to be su f f ic i en tly s t rong so tha t , i f one

of t he un i ts i s phase- sh i f t ed b y a s timulus , t hen the o ther subun i t s fo l low

a l mo s t i mmed i a t e l y .

Co ns ider t he fo l lowing spec i a l exam ple , i n wh ich ea ch su bun i t o f an osc i l-

l a t i ng un i t synapses on ly on i t s ana logue in t he o ther osc i l l a to ry sys t em. These

assum pt ions a re i l l u s tr a t ed in F ig . 1 . Le t 01 den o te t he ph ase o f a f ixed

o s c il la t in g s u b u n i t i n o n e s y s t em an d 02 d en o t e t h e p h as e o f th e a n a l o g o u s

subun i t i n t he o ther sys t em. S ince we assume tha t a l l subun i t s i n each sys t em

have f ixed pha se r e l a t i ons , 01 de t e rm ines t he ph ases o f t he r ema in ing subun i t s :

01 q- ~1 . . . . , 01 ~ ~m - 1 W e a ssu m e a s imi lar re lat ionsh ip exis t s betw ee n 02 an d


6/23


F i g . 1 . A p a i r o f o s c i l l a ti n g n e t w o r k s ,

e a c h c o n s i s t i n g o f f o u r s u b u n i t s . T h e

p h a s e r e l a t i o n s h i p s w i t h i n a n e t w o r k a r e

f ixed , re l a t ive to G0 = 0 in each o sc i l l a to r

t h e o t h e r s u b u n i t s i n o s c i l l a t o r 2 . T h e e q u a t i o n s f o r t h i s n e t w o r k a r e

l m - I

d O l / d t = o ) 1 - ~ m i E-Z-O ~ 0 1 - [ - ~ i , 02 -~- ~1 ) ,

( 3 . 1 )

l m - I

d O 2 / d t = o ~2 + - - Y . h d O ~ + ~ , , O , + ~ , ) .

m i = 0

H e r e 4 0 = 0 . T h e s u m s i n (3 . 1 ) a ri s e o u t o f o u r a s s u m p t i o n s t h a t r e s e t t in g o n e

s u b u n i t i n s t a n t a n e o u s l y r e s e ts t h e o t h e r u n i t s b y a n i d e n ti c a l a m o u n t . T h e f a c t o r

1/m m u l t ip l y i n g t h e s e s u m s is a n o r m a l i z a t i o n s o t h a t t h e t o t a l i n fl u e n c e o f o n e

o s c il la ti n g s y s t e m o n t h e o t h e r h a s m a g n i t u d e i n d e p e n d e n t o f m . N o t e t h a t i f

t h e r e i s o n l y a s i n g l e u n i t o r i f a l l m u n i t s f i re s y n c h r o n o u s l y ( i .e . i = 0 ) ( 3 . 1 ) i s

t h e s a m e a s (2 . 6 ). T h u s , w e m a y v i e w e q u a t i o n s o f t h e f o r m ( 3 .1 ) a s g e n e ra l iz a -

t io n s o f o u r o r i g i n a l p h a s e m o d e l f o r t w o c o u p l e d o s c i ll a to r s . S u p p o s e m i s l a rg e ,

h ~ i s i n d e p e n d e n t o f i a n d t h e p h a s e s 4 ; h a v e a d e n s i t y r/( ~), i. e. t h e p r o b a b i l i t y

o f ~ l y in g b e t w e e n ~ a n d + d ~ is q( ~ ) d ~ . T h e n t h e s u m s i n ( 3 .1 ) c o n v e r g e t o

i n t e g r a l s a s m t e n d s t o i n f in i t y :

1 m 1 ~ 2 n

S m ~ - m i~ = o h l ( O l 'J - ~ i ' O 2 J - i ) ~ J o h ~ ( x ' x + O 2 - O l ) q ( X ' O l ) d X 3 . 2

I n p a r t i c u l a r, i f q ( ~ ) i s u n i f o r m , i .e ., t h e p h a s e s a r e d i s t r i b u t e d e q u a l l y a r o u n d t h e

c i rc l e, th e n q ( ~ ) = 1 / 2 n a n d ( 3 . 2 ) b e c o m e s

S _ _ 1 ~

h ~ ( x , x + O z - O O d x , ( 3 . 3 )

w h i c h i s j u s t t h e a v e r a g e d r i g h t - h a n d s i d e a s d e f in e d i n ( 2 .5 ) , ( 2 .9 ) . T h i s

c a l c u l a t i o n s h o w s :

P r o p o s i t i o n 3 . 1 . I f h~ and h~2 are independent o f i, and the puls es are un iform ly

dis t r ibuted around the cycle , then the mo dulus M tends to 0 as the num ber o f pulses

m tends to oo. Hence, the solut ions to ( 3 . 1 ) are close to those o f the averaged

equations ( 2 . 9 ) as the num ber o f subuni t s becom es large. []

Rem a r k 3 .1 .

E v e n a s m a l l n u m b e r o f p u l s e s ( m = 3 o r 4 ) c a u s e s a s i g n i fi c an t

d e c r e a se in t h e m o d u l u s o f a v e r a g ea b i li ty , a n d a d r a m a t i c c h a n g e i n t h e b e h a v i o r

o f t h e s y s t e m . T h i s i s d e t a i l e d n u m e r i c a l l y i n S e c t . 4 .

Remark 3 .2 . I n o r d e r t h a t M b e c o m e s m a l l, i t i s n o t n e c e s s a r y t h a t h ~ a n d h ~ b e

i n d e p e n d e n t o f i . I t s u ff i ce s t h a t t h e s e f u n c t i o n s b e s u f f i c ie n t l y s i m i l a r f o r d i f f e r-

e n t v a l u e s o f i . F u r t h e r m o r e , i t i s e a s y t o c o n s t r u c t e x a m p l e s i n w h i c h h ~ , h ~ a r e


7/23

M u l t i p l e p u l s e i n t e r a c t i o n s a n d a v e r a g i n g 2 01

i J O

o I _1 I I

a o . o

O~

l

t

m.

o

to

o

5

o

o l J

b

F i g . 2 a , b . M o d u l u s o f a v e r a g e a b i l i t y f o r p u l s e s o f d i f f e r e n t s tr e n g t h s a n d p h a s e s h i f ts f o r t h e

e q u a t i o n 01 = ~P O2)R 01)+ (1 - ct)P 02+ 1)R(01 + ~ l ) ; P O) = ( + os 0)4 , R O) = s in 0 . a ~1 = n ,

ct v a r i e s b e t w e e n 0 a n d . N o t e t h a t c t - - s t h e c a s e d e s c r i b e d i n t h e p r o p o s i t i o n , ~ = 0 c o r r e s p o n d s

t o a s i n g l e p u l s e ; b c t = , ~ v a r i e s b e t w e e n 0 a n d ~ . ~ , = 0 c o r r e s p o n d s t o a s i n g l e p u l s e a n d ~ l = n

i n t h e c a s e d e s c r i b e d i n t h e p r o p o s i t i o n

q u i t e d i f f e r e n t f o r d i f f e r e n t i, a n d t h e p u l s e s a r e n o t u n i f o r m l y d i s t r i b u t e d . F o r

e x a m p l e , c o n s i d e r t h e s i m p l e c a s e h ~ =

~ P O 2 ) R 0 1 ) ,

h i = ( 1 - - c 0 P ( 0 2 + ~ 1 )

R 0 1 + ~ 1 ) , w i t h P O ) l 1

( ~ + s c o s 0 ) 4 ,

R O ) =

s i n 0 . A s s e e n i n F i g . 2 , M i s

s i g n i f i c a n t l y d e c r e a s e d e v e n i f t h e t w o p u l s e s a r e q u i t e d i f f e r e n t i n s i z e ( ~ ) o r

n o t u n i f o r m l y d i s t r i b u t e d ( ~ l r ) .

R e m a r k 3 . 3 .

F i n a l l y , i t i s n o t n e c e s s a r y t h a t e a c h s u b u n i t o f a n o s c i l l a t o r

s y n a p s e o n l y o n i ts a n a l o g u e . I f s u b u n i t

I i )

o f o s c i l l a t o r 1 r e s p o n d s t o a p u l s e

f r o m s u b u n i t i o f o s c i l l a t o r 2 , t h e n s u c h a p u l s e c o n t r i b u t e s

e l ( 0 2 ~ ~ i ) g i ( o1 ~ - ~ i (i ))

t o th e e q u a t i o n f o r

dO~/dt.

T h e l a t t e r e x p r e s s i o n c a n b e w r i t t e n a s


8/23

202

G. B. Ermentrout and N. Kopell

i .e ., a t e r m o f t h e f o r m p r e v i o u s l y d i s c u s s e d , w i t h a d i f f e r e n t r e s p o n s e c u r v e . I t

is n o w c l e a r th a t t h e s a m e p r in c i p le s a l l o w o n e t o c o n s t r u c t a v e r a g i n g n e t -

w o r k s f r o m c o m p o s i t e o s c i ll a to r s , w i t h o u t r e q u i r in g t h a t t h e c o n n e c t i o n s

b e t w e e n t h e o s c i l l a t o r s l i n k o n l y a n a l o g o u s s u b u n i t s . O n e d o e s r e q u i r e t h a t t h e

e f fe c ti v e r e s p o n s e f u n c t i o n s / ~ h a v e t h e p r o p e r t y t h a t a v e r a g i n g t h e m l o w e r s

v a r i a t i o n f r o m t h e m e a n o v e r 0 , a s i n t h e p r e v i o u s e x a m p l e .

4 N u m e r i c a l r e s u l ts

I n t h is s e c ti o n, w e c o m p a r e P o i n c a r6 m a p s f o r a v e r a g e d s y s t e m s o f c o u p l e d

o s c i ll a t o rs t o o n e s w i t h m u l t i p le p u l s e , b u t u n a v e r a g e d c o u p l i n g . O n e o f o u r

m a i n c o n c l u s i o n s i s t h a t , e v e n f o r h i g h ly n o n l i n e a r a n d d i s c o n t i n u o u s m o d e l s

u s in g c o u p l in g b y f - f u n c t i o n s , t h e n u m b e r o f p u l se s r eq u i r e d f o r g o o d a g r e e m e n t

b e t w e e n t h e P o i n c a r 6 m a p s i s s m a l l ( e . g . , 3 o r 4 ) . W e a l s o c o m p a r e t h e a v e r a g e

p h a s e d i f fe r e n c e o v e r a c y c le f o r s o m e p h a s e - l o c k e d u n a v e r a g e d s y s t e m s w i t h t h e

c o n s t a n t p h a s e d i f f e r en c e o f t h e s a m e s y s t e m s w h e n a v e r a g e d . W e f in d t h a t t h e r e

i s l i t t l e d i f f e r e n c e e v e n i f t h e r e i s o n l y o n e p u l s e p e r c y c l e , i . e . , in the range where

these oscillators do not undergo death, th ey behave ve ry m uch l ike their averaged

counterparts.

W e s h a l l w o r k w i t h p h a s e e q u a t i o n s c o u p l e d b y p u s l e s , w h e r e t h e p h a s e

e q u a t i o n s a r e d e r i v e d b y a s s u m i n g t h a t t h e a t t r a c t i o n t o t h e li m i t cy c l e i s v e r y

l ar g e , a s se e m s t o b e t h e c a s e f o r t h e W i l s o n - C o w a n e q u a t i o n s [9 ]. ( S e e [7 ], 2,

R e m a r k 2 .1 .) I f t h e r e a r e m u l t i p le p u l s e s c e n t e re d a r o u n d p h a s e s

;, i = 1 . . . . m , th e e q u a t i o n s a r e

dOl/dt=COl + 1 ~ P O 2-i )R 0 1- i ) ,

i =

( 4 . 1 )

dO2/dt=o92+ 1 ~ P O , - i )R 0 2- i ) .

i =

W e s h a l l t a k e 4 , - =

2rci/m,

a n d

e o) = ( 0 . 5 + 0 . 5 c o s ( 0 ) ) o r 3 ( 0 ) , ( 4 . 2 )

R O)

= - ~[ sin (0 + r/) - sin(r/)] .

T h r o u g h o u t t h es e c o m p u t a t i o n s , w e h a v e s e t n = 4. U s i n g r e p e a t e d i n t e g r a ti o n

b y p a r t s , w e f i n d t h a t t h e a v e r a g e f u n c t i o n , a s d e f i n e d i n S e c t i o n 2 , i s

H( ~b) = I ' ~ I s i n ( - ~ b + r /) - n - -- ~1 s i n ( r / ) ]n ( 4 .3 )

w h e r e

I . = ~ P O ) d O -

1 . 3 . 5 . . . . . 2 n - 1

2 . 4 . 6 . . . . . 2 n

I n o u r f ir s t s e t o f s i m u l a t i o n s , w e fi x a ll p a r a m e t e r s ( n = 4 , r / = - t a n - ~ ( 0 . 5 ) ,

= 2 ) e x c e p t f o r m , t h e n u m b e r o f pu l se s , a n d c o m p a r e t h e P o i n ca r 6 m a p s o f

( 4 .1 ) w i t h t h o s e o f t h e a v e r a g e d e q u a t i o n s ( 2 .9 ) , w i t h o 9 1 = 1, 0 9 2 = 0 . 8 .

H ~ = / - / 2 = H , w i t h H g i v e n b y ( 4 .3 ) . F i g u r e 3 a s h o w s t h e P o i n c a r 6 m a p t a k e n a t

t h e 0~ = 0 s e c t i o n f o r P O) = 6 0) a n d m = 1 , a l o n g w i t h t h a t o f th e a v e r a g e d

e q u a t i o n ; t h e t w o a r e q u i t e d i s si m i la r . H o w e v e r , f o r a s f e w a s 4 p u l s e s p e r c y c le


9/23

M ultiple pulse interaction s and averaging 203

7r

2~

o ~ ~

i / / ~

o

a o ~ b o

2 r r

Fig. 3 a,b. Poincar6 m aps com pared to the averaged analogues dashed) for co1 = 1, co =0 .8,

P O ) = 6 0), R O) = -2[sin 0 + ~/) -sin ~/)], ~/= -tan- l ). a 1 Pulse per cycle; b 4 pulses per cycle

m = 4 ), t h e r e i s a q u a l i t a t i v e a n d q u a n t i t a t i v e s i m i l a r it y b e t w e e n t h e t w o , a s c a n

b e s e e n i n F i g . 3 b . T h i s r e s u l t i s t y p i c a l . F i g u r e 4 s h o w s t h e P o i n c a r 6 m a p s o f

4 . 1 ) f o r m = 1 a n d m = 4 , w i t h P O ) s m o o t h a s i n 4 .2 ) a n d R O) a s d e f i n e d i n

4 .2 ) a = 1, ~ / = - 0 . 5 ) , a l o n g w i t h t h a t o f t h e a v e r ag e d e q u a t i o n s f o r c o m p a r i -

s o n . A t m = 1 , t h e m a p i s q u i t e d i f f e r e n t a n d p h a s e s h i f t e d f r o m t h a t o f t h e

a v e r a g e , b u t f o r m = 4 , t h e m a p s a r e v e r y c lo s e . F o r h i g h e r v a l u e s o f m , t h e

P o i n c a r 6 m a p i s i n d i s ti n g u i s h a b le f r o m t h a t o f t h e a v e r a g e d e q u a t i o n s .

S u p p o s e t h a t w e c h o o s e ~ l a r g e e n o u g h s o t h a t t h e r e i s o s c i l l a t o r - d e a t h f o r

4 .1 ) wh en m = 1 i .e . ( 0 1 ( t ) , 0 2 ( / ) ) - - I ( 0 1 , 0 2) , a s t e a d y s t a te ) . I f w e t h e n a d d

m o r e p u l s e s i n c r e a s e m ) i t i s p o s s i b l e t o p r e v e n t o s c i l l a t o r d e a t h f o r t h e s y s t e m

o f o s c il la t o rs . A n a t u r a l q u e s t i o n i s: H o w m a n y p u l s e s a re r e q u i r e d t o p r e v e n t

o s c i l l a t o r d e a t h f o r a g i v e n p a i r o f o s c i l l a t o r s ? T h e a n s w e r t o t h i s q u e s t i o n

d e p e n d s o f c o u rs e o n t h e n a t u r e o f t h e f u n c t io n s , P O) , R O) , t h e s t r e n g t h o f

cou p l ing , ~ t, an d the f req uenc ie s e91 an d 0~2. F o r pu rp oses o f i l l u s t ra t ion , w e

2~

I

I

I I

/

I I

I I

I

0 0

a 0 2rr b 0 2rr

F i g . 4 a , b . S a m e a s F i g . 3 a , b , b u t P O)= + c o s 0 ) 4 . a 1 P u l s e p e r c y c l e ; b 4 p u l s e s p e r c y c l e


10/23


ab l e 1 . M a x i m u m c o u p l i n g

s t r e n g t h a n d p u l s e n u m b e r

m 0tm

1 2 .43

2 4 .88

3 8 .20

4 17.92

5 38 .08

c o n s i d e r t h e c a s e o f t w o i d e n t i c a l o s c i ll a t o r s a n d f ix r / = - 0 . 5 , c 01 , (.D

~--

1.0,

n = 4 , an d l e t 0t va ry . T ab le 1 show s the v a lues o f ~t a t w h ich a t l e a s t m pu l s e s

a r e r e q u i r e d t o p r e v e n t o s c i l l a t o r d e a t h . T h u s , i f 0t < 2 .4 3 , p h a s e - d e a t h c a n n e v e r

occur ; fo r 2 .43 < ~ < 4 .88 , a t l e a s t two pu l s e s a re requ i red to p reven t dea th , e t c .

I t i s c l e a r f r o m t h e t a b l e t h a t o n l y a f e w p u l s e s a r e r e q u i r e d t o p r e v e n t d e a t h ,

e v e n w i t h v e r y s t r o n g c o u p l i n g ( s t r o n g r e l a ti v e t o t h e s iz e o f t h e f r e q u e n c i e s o91

a n d 0 )2 ). W e n o t e t h a t w h e n at i s l a r g e , a n d i n t h e l o c k i n g r e g i m e , t h e P o i n c a r 6

m a p s f o r b o t h t h e a v e r a g e d e q u a t i o n s a n d E q s . ( 4 .1 ) a r e q u i t e f la t i n t h e se n s e

t h a t n o m a t t e r w h a t t h e i n i t i a l c o n d i t i o n i s , a f t e r o n e c y c l e , t h e t w o o s c i l l a t o r s

h a v e a l m o s t a z e r o p h a s e - d i ff e r e n c e ( t h u s , i f

A O)

i s t h e P o i n c a r 6 m a p ,

A O) ,~ 0

for a l l 0) .

I n t h e n e x t s e t o f n u m e r i c a l c a l c u l a ti o n s , w e c o m p a r e t h e a v e r a g e p h a se -

d i f f e r e n c e b e t w e e n 0 2 ( 0 a n d

01 0

w i t h t h e p h a s e - d i f f e r e n c e s o f t h e a v e r a g e d

e q u a t i o n s w h e n a s t a b l e p h a s e - l o c k e d s o l u t i o n t o ( 4 .1 ) e x i s ts . W e f i x 0)2 = 1 a n d

let 0)~ E (0, 1]. W e use P O ), R O) as in (4 .2 ) P O) s m o o t h , n = 4, r / = 0 .5 ) a n d

c o n s i d e r s e v e r al d i f f e r e n t v a lu e s f o r ~ t. W e w i l l fi n d t h e f o l l o w i n g b e h a v i o r : f o r

w e a k c o u p l i n g , a s t h e f r e q u e n c y 0)2 d e c r e a s e s t h e r e is a t r a n s i t i o n f r o m l o c k i n g

t o p h a s e - d r i f t ; f o r s t r o n g c o u p l i n g , t h e t r a n s i t i o n i s f r o m l o c k i n g t o o s c i l l a t o r -

d e a t h .

F o r any a v e r a g e d s y s t e m , t h e p h a s e - d i f f e r e n c e i s c o n s t a n t w h e n l o c k i n g

o c c u r s , a n d i f t h e m o d e l is o f t h e f o r m ( 4 .2 ) i t is e a s i ly f o u n d t h a t t h e

phase -d i f fe rence ~b = 0 2 - 01 be tw een the tw o ave r aged osc i l l a to r s s a t is f i es

sin(~b)

= / 0 ) 2 - 0 ) 1 ) ,

w h e r e x i s a c o n s t a n t d e p e n d i n g o n a l l o f th e p a r a m e t e r s . T h u s , i f s in (tk ) is

p lo t t ed a s a fu nc t ion o f 0)~ an d 0)2 f ixed a t 1 , a s t r a ig h t l ine i s ob ta ine d . W e

d e f i n e t h e a v e r a g e p h a s e - d i f f e r e n c e b e t w e e n t w o p h a s e - l o c k e d o c i U a t o r s b y

f [

4 > - - ~ o2 0 - Ol t ) dr ,

w h e r e T i s t h e p e r i o d o f t h e p h a s e - l o c k e d s y s t e m . I f 02 a n d 01 s a t i sf y a v e r a g e d

e q u a t i o n s o f t h e f o r m ( 2 .7 ) , t h e n ( 4 ~ ) = 4~ = 0 2 ( 0 - 0 1(t), a c o n s t a n t . I n t h e

f o l l o w i n g f i g u r es , w e p l o t s i n ( ( 4 ~ ) ) v e r s u s 0 )1 f o r m = 1 a n d 3 v a l u e s o f t h e

cou p l in g s t ren g th cc In F ig . 5a , e = 1 ; e i s su f f i c i en t ly sm a l l so tha t bo th the

a v e r a g e d a n d t h e u n a v e r a g e d e q u a t i o n s l o s e l o c k i n g a t a v a l u e o f 0)~ i n t h e

i n t e r v a l [ 0 . 5 , 1 ] . W e h a v e m a r k e d t h e c r i t i c a l v a l u e o f t h e f r e q u e n c y a t w h i c h

l o c k i n g is l o s t f o r t h e a v e r a g e d a n d u n a v e r a g e d s y s t em s . B e c a u s e e is s m a l l , i t

i s n o t s u r p r i s i n g t h a t t h e a v e r a g e p h a s e - d i f fe r e n c e s b e t w e e n t h e t w o o s c i l l a t o r s

a r e s i m i l a r f o r t h e a v e r a g e d a n d u n a v e r a g e d s y s t e m s .


11/23

,t

Multiple pulse interactions and averaging

0 . 5 ~ 0 . 7 0 . 8 0 . 9 1 . 0

to1

205

i I ~ - ~ ~

l

0 5 0 6 0 7 0 8 0 9 t 0

. w

r/)

3

0 . 7 5 0 , 8 0 O . B 5 0 . 9 0 0 . 9 5 1 . 0 0

co1

Fig. a- e. Sin(~b} vs. ~o for different

coupling strengths, w here (~b} = 1/

T S~ 02 0 - O1 t ) dr . Dashed curve shows

sin ~b for an averaged system, a Coup ling

strength = 1, arrows mark value of ~o, at

which locking is lost for single pulse

(~0.65) and averaged (~0.62) system;

b coupling strength = 2; e coupling

strength = 3

F i g u r e 5 b h a s p a r a m e t e r s i d e n t ic a l t o F i g . 5 a e x c ep t t h a t t h e c o u p l i n g is

t w i c e a s s t r o n g ( ~ = 2) . T h e c o u p l i n g i s s ti ll s u f fi c ie n t ly w e a k s o t h a t o s c i l l a t o r -

d e a t h d o e s n o t o c c u r f o r 0 . 5 ~< o91 ~< 1 . F i n a l l y , i n F i g . 5 c w e s h a l l s h o w t h e

a v e r a g e p h a s e - d i f f e r e n c e w h e n ~ = 3 . A t t h e c r i t i c a l v a l u e , o h ,,~ 0 . 7 5 , o s c i l l a t o r

d e a t h o c c u r s f o r ( 4 .1 ) s o t h a t t h e r e a r e n o p e r i o d i c s o l u t i o n s f o r s m a l l e r v a lu e s

o f c o 1

I n t h e l a s t s i m u l a t i o n , w e f ix a = 6 s o t h a t t h e r e i s o s c i l l a t o r - d e a t h f o r a n y

v a l u e o f a h ~< 1 w h e n m = 1. T h e c o u p l i n g i s n o w b y 4 p u l s e s ( m = 4 ), w h i c h

p r e v e n t s t h e o s c i l l a t o r d e a t h , a n d w e s h o w t h r e e c u r v e s i n F i g . 6 a s o~1 v a r i e s

f r o m 1 . 0 d o w n t o 0 . 1 . T h e s e c u r v e s a r e s i n ( ( ~ b ) ) , s in ( ~b m , .) , a n d s in (~ bm i,),

w h e r e ~ bm ,x ( r e s p . ~m in = m a x 0 2 0 - 0 1 t ) ( r e s p . m i n 0 2 ( 0 -

0 1 t ) ) .

W e

0


12/23

2 0 6

0.

I I 1 i ~~

0 3 0 5 0 6 0 8 1 0

G . B . E r m e n t r o u t a n d N . K o p e l l

F i g, , 6 . S a m e a s F i g . 5 a - c b u t c o u p l i n g -

s t r e n g t h = 6 , 4 p u l s e s p e r c y c l e s h o w i n g

s i n ~ . . . . s i n ~ b ~ n , s i n( q ~ ) , w h e r e

~ bm ax = m a x

0 2 0 - 0 1 0 ,

~ b m i n = m i n

O ~ t ~ T O ~ t ~ T

0 2 0 - - 0 1 t ) , ~ ) = I / T S ~ 0 2 ( 0 - - 0 1 ( t ) dt .

S m a l l d a s h e s a r e u s e d f o r f f m ~ x, l a r g e f o r

~ bm i , s o l i d f o r ( ~ )

5 Re al i s t i c ne twork s and s table d i str ibut ion of phases

In Sect. 3, we showed that if a network is able to stably distribute phases about

its cycle, then it is possible to obtain 1 : 1 locking and nearly averaged behavior

with strong interactions between oscillators. This averageability is shown to

prevent oscillator-death and other complex dynamical behavior. Two assump-

tions are implicit in the creation of these networks. First, we have assumed that

we can arrange coupling within the network so that the cells within the network

fire out of phase. Second, we have assumed that the interaction with another

similar network does not disrupt this firing pattern. That such networks are

realizable is not obvious. In this section, we show two examples, a phase model

and a neural model, that exhibit the desired properties: (i) within the network,

the oscillators are phase-shifted so that there is uniform dispersion of phases, and

(ii) when two such networks are coupled, it is possible to obtain stable 1:1

locking even though the two oscillations are far apart in frequency.

5 .1 . P h a s e m o d e l s

The most obvious way to set up a pattern of phases is to arrange the units of the

network in a geometric ring and attempt to form a travelling wave that cycles the

ring. If the wavelength is m, where m is the number of units in the ring, then

the phases must necessarily be a multiple of 2 r r / m apart. A wavelength of 0

corresponds to synchronous oscillations in the ring. Other wavelengths corre-

spond to nearest neighbor phase-lags of 2rc /k where k is a divisor of m. In [13],

it is shown that oscillators that are coupled in a ring with some inhibition have

as stable solutions waves of the desired form. Furthermore, if the inhibition is

strong enough, the synchronous solution of the ring is unstable--the phases

necessarily separate. (A more explicit definition of such inhibition is given

below.) The model analyzed in [13] has the form

Oj = 09 + ~ H Oi - O j , i - j ) , j = 0 . . . . . m . (5.1)

i O


13/23


S i n c e w e a r e s h o w i n g t h a t , e v e n i f th e i n t e r a c t i o n s a r e n o t v i a p h a s e - d i f f e re n c e s ,

t h e n e t w o r k s c a n a v e r a g e t o c r e a te si m i la r b e h a v i o r , w e d o n o t w a n t t o s t a r t

w i t h e q u a t i o n s o f t h e f o r m ( 5. 1) . I n s t e a d w e w i ll k e e p t h e r i n g g e o m e t r y a n d

c o n s i d e r a g e n e r a l i z a t i o n o f ( 5 . 1 ) :

m 1

O j = c o + ~ h O i , O j, i - j ) , j = 0 . . . . . m . (5 .2)

i = 0

W e h a v e i n m i n d t h e c l as s o f m o d e l s d e r iv e d in S e c t. 2 u n d e r t h e a s s u m p t i o n s o f

i n f i n i t e a t t r a c t i o n t o t h e l i m i t c y c le . F o r t h i s ty p e o f m o d e l ,

h O , , O j, i - j ) = P O , ) R O j ) w i - j ) , (5 .3)

w h e r e

P O i )

i s t h e i m p u l s e o f t h e o s c i l l a t o r i c a u s i n g t h e r e s p o n s e

R 0 i )

o n

o s c i l l a to r j w i t h w e i g h t w i - j ) . A b o u t R O j ) , w e a s s u m e t h a t R ' < 0 f o r a

n e i g h b o r h o o d o f 0 = 0 . S u c h a n R i s p r o d u c e d a s d i s c u s s e d i n S e ct . 2 f r o m

n e u r a l m o d e l s d e s c r i b i n g c o u p l i n g v i a e x c i t a t o r y i n t e r a c t i o n s , i . e . , o n e s i n w h i c h

t h e v o l t a g e i s i n c r e a s e d i n t h e t a r g e t n e u r o n . I n h i b i t o r y c o u p l i n g l e a d s t o

r e s p o n s e f u n c t i o n s R w i t h R ' > 0 n e a r 0 = 0 . T h u s , w e c a n m o d e l e x c i t a t o r y

( r e s p . i n h i b i t o r y ) c o u p l i n g u s i n g a f ix e d r e s p o n s e f u n c t i o n R s a t i s f y i n g R ' ( 0 ) < 0

a n d l e t t i n g t h e w e i g h t s w b e p o s i t i v e ( r e s p . n e g a t i v e ) .

E q u a t i o n ( 5. 3) i s d e r i v e d f r o m a r i n g o f id e n t ic a l n e u r o n s t h a t h a v e s t r o n g l y

a t t r a c t i n g l i m i t c y c le s a s s o l u t i o n s . P r e l i m i n a r y w o r k i n d i c a t e s f o r t h e n e t w o r k

( 5 .3 ) t h a t i f t h e c o u p l i n g i s e x c i t a t o r y , t h e n t h e i n - p h a s e s o l u t i o n

O j t ) = Oe t ) V i , j i s a s t a b l e s o l u t i o n t o ( 5 . 3 ) . W e m u s t a v o i d t h i s , f o r o t h e r w i s e

t h e o s c i l l a t o r s i n t h e r i n g a l l s y n c h r o n i z e a n d t h e n e t w o r k i s e q u i v a l e n t t o a

s i n g l e o s c i l l a t o r . T h u s , w e a s s u m e t h a t s o m e o f t h e c o u p l i n g i n t h e r i n g i s

i n h i b i t o r y .

W e co n s i d e r ( 5 .2 ) a n d ( 5 .3 ) c o u p l e d t o a n o t h e r s u c h r i n g w i t h e x c i t a t o r y

c o u p l i n g b e t w e e n r i n g s :

ao~ /at = Oil + R o~ ) w i - j ) P o ~ ) + ~ P o { ) ,

; (5.4)

O J z /d t = 092 + R O J 2 ) w i - - j ) P O i 2 ) + o ~ P O ~ ) ,

w h e r e

P u ) = (0.5 + 0.5 cos(u)) 4, R u ) = - ( s in (u + 0 .4 ) - - s in (0 .4 ) ) . (5 .5 )

E a c h o s c i l l a t o r r i n g 1 i s c o u p l e d t o n e i g h b o r s i n t h e r in g a n d t o i t s a n a l o g u e

o s c i l l a t o r in r i n g 2 . I n a b s e n c e o f c o u p l i n g b e t w e e n r i n g s (~ = 0 ), w e c h o o s e w k )

s o t h a t t h e r e i s a w a v e i n e a c h r i n g . I n t h e n u m e r i c a l e x p e r i m e n t w i t h r i n g

s t ruc tu re , we hav e cho sen m = 6 , w(0 ) = 0 , w(1 ) = w( - 1 ) = 0 .1,

w ( 2 ) = w ( - 2 ) = - 0 . 6 , a n d w ( 3 ) = - 0 . 2 , co l = 1 , a n d co2 = 0. 5.

I n t h e f i r s t n u m e r i c a l e x p e r i m e n t w e s i m p l y c o u p l e t w o o s c i l l a t o r s w i t h o u t

the r ing s t ruc tu re ( i .e . , we l e t w-= 0 ). F o r 0~ sm a l l t he re i s no lock ing , a s i s

expec ted s ince the f requenc ie s a re qu i t e d i f fe ren t . As ~ inc rea se s the re i s a

c o m p l i c a t e d s e q u e n c e o f n : m l o c k i n g p a t t e r n s f o r w h i c h m < n . T h a t i s, t h e r e

a r e a l w a y s m o r e c y c l e s o f t h e f a s t o s c i l l a to r 1 t h a n t h e r e a r e o f th e s l o w o s c i l l a to r

2 . A t a p a r t i c u l a r v a l u e o f ~ ( ~ ~ 1 .6 5), o s c i l l a to r 2 s t o p s ; t h a t is , i t n o l o n g e r

t r a v e r s e s a c o m p l e t e c y c le . O s c i l l a t o r 1 c o n t i n u e s t o t r a v e r s e t h e c y c le a n d t h i s

b e h a v i o r p e r s i s ts f o r a l l h i g h e r v a l u e s o f ~ . T h u s o s c i l l a t o r 1 a n d 2 d o n o t e v e r

l o c k i n a 1 : 1 m a n n e r .


14/23

2

2 8

2~

0 0 1

2rr

G. B. Ermentrout and N. KopeU

F i g . 7 . 0 v s . 0 f o r ~ = 2 . N o t e t h a t

t r a j e c t o r y is c l o s e t o s t r a i g h t li n e i .e .

0 - 0 is a l m o s t i n d e p e n d e n t o f t

2~

0 J ( t )

Fig. 8. 0 (t) vs. t. No te that the wav e structure is maintained even if the cou pling between rings is

mo derately strong

I n t h e s e c o n d e x p e r i m e n t w e u s e t h e r in g s t r u c t u r e . F o r ~ sm a l l, t h e r e i s a g a i n

n o l o c k i n g . A s w e i n c r e a s e ~ w e f i n d t h a t 1 : 1 l o c k i n g o c c u r s . I n F i g . 7 , w e s h o w

t h e p h a s e - p l a n e d i a g r a m o f 0 vs 0 f o r ~ = 2 . N o t e t h a t t h e t r a j e c t o r y i s n e a r ly

a s t r a i g h t l i n e , i n d i c a t i n g t h a t t h e s y s t e m i s b e h a v i n g a s i f i t w e r e a v e r a g e d . I n

F i g . 8 , w e s h o w t h e t i m e - d e p e n d e n c e o f t h e t r a j e c to r i e s f o r e a c h o f t h e p h a s e s .

T h i s fi g u re s h o w s t h a t ev e n w i t h s t r o n g c o u p l i n g b e t w e e n t h e r in g s , t h e p h a s e

r e l a t i o n s h i p s w i t h i n a r i n g a r e p r e s e r v e d . W e f u r t h e r n o t e t h a t t h e i n t r a - r i n g

c o u p l i n g i s s m a l l c o m p a r e d t o t h e c o u p l i n g b e t w e e n r i n g s ; t h u s w e d o n o t n e e d

t o a s s u m e t h a t t h e c o u p l i n g w i t h i n t h e r i n g i s m u c h s t r o n g e r .

F i n a l ly , o n e m i g h t a r g u e t h a t i t is t h e p r e s e n c e o f in h i b i t o r y c o u p l i n g w i t h i n

t h e r i n g r a t h e r t h a n t h e e x i s t e n c e o f p h a s e - s h i f t s t h a t i s r e s p o n s i b l e f o r p r e v e n t -

i n g th e o s c i l l a t o r d e a t h o f t h e c o u p l e d s y s te m . T o s ee i f t h is is t h e c a s e, w e


15/23

Mu lt ip le pulse in teract ions and averaging 2 9

c o n s i d e r t h e f o l l o w i n g v a r i a t io n o f 5 . 4 ) :

dOi/dt

= 3 ) 1

--~R Ol)Im~o= w i )P 01 )+ ~P 02 )] ,

5 . 6 )

]

O2/dt = o92 + R 0 2) w i ) P 0 2 ) ~ P 0 1)

i

E q u a t i o n 5 . 6 ) is j u s t t h e s y m m e t r i c f o r m o f 5 . 4 ), t h a t i s, 0 4 = 01 f o r a ll j a n d

s i m i la r ly f o r 0 ~. N o t e t h a t t h e s y m m e t r i c s o l u t i o n i s u n s t a b l e a s a s o l u t i o n t o

5 . 4 ) . ) A s i n t h e f i r s t c a s e , n u m e r i c a l e x p e r i m e n t s i n d i c a t e t h a t 1 : 1 l o c k i n g i s s ti ll

i m p o s s i b l e a n d a s e q u e n c e o f b i f u r c a t i o n s i m i la r to t h a t w i t h o u t t h e ri n g

s t r u c t u r e o c c u r s . F o r ~ l a r g e e n o u g h , o s c i l la t o r 2 is s t o p p e d .

W e h a v e d o n e s i m i l a r n u m e r i c a l e x p e r i m e n t s f o r d i f f e r e n t p a r a m e t e r s a n d

d i f fe r e n t n u m b e r s o f o s c il l a to r s i n t h e r in g a n d f i n d th e s a m e b e h a v i o r w i t h i n

s o m e li m i ts . O b v i o u s l y , i f ~ b e c o m e s t o o la r g e w e l l b e y o n d t h e v a l u e s t h a t y i e ld

: 1 l o c k i n g i n th e r i n g s) , t h e n o s c i l la t o r d e a t h c a n a n d d o e s o c c u r .

5.2. Neural net models

I n th i s s e ct io n , w e s tu d y t w o c o u p l e d r in g s o f W i l s o n - C o w a n n e u r a l e q u a t io n s :

d e ~ /d t = - e ~

+ f e f l l e E ~ - - f l ie I ~ + o ~ g J 2) ,

,

e ie - i iI - w k - j ) I ,

k = o 5 . 7 )

= + f e f l e e E 2 - - f l ie [ 2 +

d I { / d t = + f , . e , E { - 8 , ,1 { - w k - j ) I , j = 0 , . . . , 5

k =

H e r e E j a n d / j r e fe r t o p o p u l a t i o n s o f e x c i t a to r y a n d i n h i b i t o r y n e u r o n s . F o r

m = 1 a n d c o u p l i n g c o e f f i c i e n ts e = 0 t h e s y s t e m o f E l , I1 r e s p . E 2 , I2 ) f o r m s a n

o s c i ll a to r . T h e ju s t i f ic a t i o n f o r t h e f o r m o f 5 . 7 ) c a n b e f o u n d i n o t h e r a r t ic l es

[ 9 ] . C o u p l i n g

within

t h e r in g i s so l e y t h r o u g h t h e

inhibitory

n e u r o n s / j . C o u p l i n g

across t h e t w o r i n g s i s t h r o u g h t h e excitatory n e u r o n s . B o t h r i n g s a r e i d e n t i c a l

w i t h t h e e x c e p t i o n o f t h e p a r a m e t e r ,

flee,

w h i c h i s d i f f e r e n t i n t h e t w o r i n g s . T h i s

p a r a m e t e r h a s a l a r g e e f f e c t o n t h e i n t r in s i c f r e q u e n c y o f t h e o s c i l l a t o r s , s o i t i s

v a r i e d i n o r d e r t h a t t h e u n c o u p l e d s y s t e m s h a v e d i f f e r e n t f r e q u e n c i e s . T h e

p a r a m e t e r s u s e d i n o u r e x p e r i m e n t s a r e a s i n th e c a p t i o n t o F i g . 9. T h e r e a r e

t h r e e e x p e r i m e n t s a s i n t h e p h a s e m o d e l .

I n t h e f ir s t e x p e r i m e n t , w e a l lo w 5 . 7 ) t o e v o l v e f r o m r a n d o m i n it ia l

c o n d i t io n s . W e f in d a l a r g e r a n g e o f c o u p l i n g s t r e n g t h s e t h a t l e a d to 1 : 1

e n t r a i n m e n t w i t h n o d i s t o r ti o n i n t h e w a v e s h a p e s. F i g u r e 9 s h o w s t h e e x c i ta t o r y

v a r i a b l e s E~ t ) a n d E~ t ) . N o t e t h a t t h e f a s t e r o s c i l l a t o r i s s l i g h t l y a d v a n c e d . T h e

o t h e r c o m p o n e n t o s c i l l a t o r s h a v e i d e n t i c a l p r o f i l e s b u t t h e y a r e p h a s e - s h i f t e d . I n

F i g . 9 b , w e s h o w t h e c o m p l e t e s et o f w a v e s f r o m n e t w o r k 1. N o t e t h a t t h e p h a s e

r e l a t i o n s h i p s a r e n o t d i s t o r t e d b y t h e c o u p l i n g b e t w e e n t h e r i n g s .

I n t h e n e x t e x p e r i m e n t , w e t r y to e n t r a i n t h e tw o o s c i l la t o r s w i t h o u t

e x p l o i t i n g t h e r i n g s t r u c t u r e . F o r t h i s e x p e r i m e n t ,

w k ) - 0 .

W e f i n d t h a t f o r e

s m a l l , t h e r e i s n o l o c k in g ; r a t h e r t h e t w o o s c i l la t io n s u n d e r g o q u a s i - p e r i o d i c


16/23

G B E r m e n t r o u t a n d N . K o p e l l

2 1 0

tl.I

51 52

b 51 L 52

F i g . 9 . a . W i l s o n C o w a n n e t w o r k s c o u p l e d t o g e t h e r.

E t )

a n d

E t )

a r e p l o t t e d f o r o n e l o c k e d

c y c l e fl e~ = 1 2 fl eSe= 1 8 f li g = 0

file

= 2 0

f le i : 1 8 , W O ) = O , W 1 ) = W - - 1 = O . 1 ,

w(2) = w (- 2 ) = 2, w(3) = 0.3; f~(u) = ~(1 + tan h(u - )); ~(u) = ~ 1 + ta nh(u - 4)); ~ = 10. b E{ (t).

No te that the w aves persist after coupling

b e h a v i o r . A s ~t i n c r e a s e s , t h e r e a r e r e g i m e s o f p e r i o d i c n : m l o c k i n g f o r n # m ,

s e p a r a t e d b y q u a s i -p e r i o d i c a n d c h a o t i c b e h a v i o r . A t ~ ,~ 1 0, t h e s o l u ti o n s

a p p e a r t o b e c h a o t i c a n d a s ~ i n c re a s e s t h e y b e c o m e p e r i o d i c a n d b u r s t - l i k e

( sev e r a l sp i ke s r epe a t ed ) . A t c t ~ 10 .5 t h e r e i s 1 : 1 l ock i ng b u t t h i s pe r s i s t s o n l y

u n t i l 0 t ~ 1 0 .5 8 a t w h i c h p o i n t a l l o s c i ll a t io n s a r e s t o p p e d . T h e r e g i m e f o r 1 : 1

l o c k i n g is v e r y s m a ll a n d d e p e n d s a g r e a t d e a l o n t h e p a r a m e t e r s f l l and f l~e .

F i n a ll y , w e p e r f o r m t h e e x p e r im e n t a n a l o g o u s t o t h e t h i rd e x p e r i m e n t a b o v e

i n o r d e r t o c o n v i n c e o u r s e l v e s t h a t i t i s i n f a c t t h e i n t r a - r i n g p h a s e s h i f t s , n o t t h e

p r e s e n c e o f i n h i b i t o r y c o u p l i n g , t h a t a r e r e s p o n s i b le f o r o u r a b i l i ty t o o b t a i n 1 : 1

l o c k in g . T h u s , w e c o n s i d e r t h e m o d i f i e d v e r s io n o f (5 . 7 ) w h e r e

E~, I~)= E~ , I~)

a n d ( E ~, I ~ ) = ( E ~, I t ) . H e r e , w e

do

f i n d a s l i m r a n g e o f

c o u p l i n g s t r e n g t h s f o r w h i c h 1 : 1 l o c k i n g i s p o s s ib l e b u t t h e w a v e f o r m s a r e

s e v e r e l y d i s t o r t e d ; t h e f a s t e r o s c i l l a t o r h a s a s m a l l r e s i d u a l s p i k e t h a t a p p e a r s t o

b e a r e s i d u e o f a 2 : 1 l o c k e d s t a te .

T h e s e n u m e r i c a l re s u lt s a re o n l y p r e l i m i n a r y a n d b y n o m e a n s e x h a u s ti v e ;

o u r g o a l h e r e i s t o s h o w t h a t n e t w o r k s t h a t c a n p r o d u c e a v e r a g e d i n t e r a c t i o n s


17/23


a r e s i m p l y c o n s t r u c t e d . T h e s e i n t e r a c ti o n s a p p e a r t o m a k e i t m u c h e a si e r f o r t h e

t w o o s c i l la t o r s to e n t r a i n o v e r a l a r g e r a n g e o f c o u p l i n g s t r e n g t h s a n d i n t r in s i c

f r e q u e n c i e s w i t h o u t s i g n i f i c a n tl y d i s t o r t i n g t h e w a v e f o r m s o f t h e o s c i l la t i o n s . I n

a l a t e r p a p e r , w e w i ll s y s t e m a t i c a l l y e x p l o r e m o d e l s t h a t a r e v a r i a t i o n s o f ( 5 . 4)

a n d ( 5 . 7 ) .

Acknowledgement.

We w ish to tha nk L . Glass, whose resolute disbeliefprovoked us to b ack up some

of our claims.

Appendix Com putation of the averaged functions

W e d e s c r i b e in m o r e d e t a i l t h e c h a n g e o f c o o r d i n a t e s d i s c u s s e d i n S e c t. 2 i n o r d e r

t o o b t a i n e x p l i c i t i n f o r m a t i o n a b o u t t h e f u n c t i o n s i n ( 2 . 2 ) . W e t h e n u s e t h i s

i n f o r m a t i o n t o d e r iv e n u m e r i c a l t e c h n i q u es f o r o b t a i n i n g t h e a v e r a g e d e q u a t i o n s

( 2 .4 ) w h e n t h e r e is w e a k c o u p l i n g a n d t h e f u n c t i o n s

hk(Ok, Oj)

o f ( 2 .6 ) w h e n t h e

a t t r a c t i o n t o t h e l i m i t c y c le i s v e r y s t r o n g . W e w i l l s ee t h a t t h e c a s e o f w e a k

c o u p l i n g i s c o n s i d e r a b l y m o r e d i f fi c u lt , e s s e n t ia l l y b e c a u s e t h e p r o p e r t i e s o f t h e

e q u a t i o n s o f f t h e l i m i t c y c le s c a n a f f e ct t h e a v e r a g e d e q u a t i o n s , a n d w e d o t h a t

d e r i v a t i o n i n t w o w a y s . ( S e e [1 0] a n d t h e a p p e n d i x o f [1 4] f o r r e l a t e d c a l c u l a-

t i o n s .) F i n a l l y , w e a p p l y t h e m e t h o d s t o a c l a s s o f n e u r a l m o d e l s .

A 1 . Osc i l l a tor s w i th s t rong ly a t t rac t ing l imi t cyc l e s

C o n s i d e r t h e p a i r o f c o u p l e d o s c i l la t o r s ( 2 .1 ) . W e a s s u m e a s b e f o r e t h e

d X g / d t = F k ( X k )

h a s a n a s y m p t o t i c a l l y st a b l e l i m i t c y c le s o l u t i o n w h i c h w e s h a l l

ca l l

Uk(t/Ogk),

wh ere COk i s the f requ ency o f the l imi t cyc le an d

U k ( O )

is

2 n - p e r i o d i c i n i t s a r g u m e n t . L e t

Uk = Tk(Ok, Yk ) , Yk

~ R n - l , b e th e c o o r d i n a te

t r a n s f o r m a t i o n s . T h e s e t r a n s f o r m a t i o n s c a n b e c h o s en s o t h a t

uk( t ) = U~(Ok( t ) ) + M k(Ok( t ) ) yk ( t ) + O([y~

[ ) , (ml )

w h e r e

M k ( O )

i s an n x (n - - 1 ) ma t r ix and

M k ( O ) r M k ( O )

= 1(, _ ,) (, _ 1),

A2)

U ~ (O ) ~M ~ ( O ) = 0 1 ( n - 1 ~ .

I n t h e a p p e n d i x t o [ 7 ] i t w a s s h o w n t h a t

Oh, Yk

sa t i s fy

( D k I d O k / d t 1 + O f f l ( g k ) Z [ ( D k + ( D k ) r ) M k y k + G k] ,

( A 3 )

o 9 ; 1 d y k / d t = ( ( M k ) T D k M k + ( M k ) T M k ) Y k q - ( M k ) T G k .

( H e r e ~ o ; 1 a r is e s f r o m d i f f e r e n ti a t io n o f

Uk(Ok/~Ok)

w i t h r e s p e c t t o t i m e . ) W e w i l l

u s e ( A 3 ) a s t h e b a s i s f o r t h e r e s t o f t h e c a l c u l a t i o n s i n t h i s a p p e n d i x .

I n th e li m i t o f i n f i n it e a t t r a c t i o n t o t h e l im i t c y cl e, y k ~ 0 , a n d ( A 3 )

b e c o m e s ( 2 . 6 ) w i t h

h k ( O k , O j ) = o ~ O ; 1 ( O ~ ) ( V ~ ) ~ ( O ~ ) G ~ ( ~ : ~ ( O ~ ) , ~ 0A ). A4)

T h e s ig n if ic a n ce o f ( A 4 ) is t h a t i t a ll o w s u s t o c o m p u t e a n a p p r o x i m a t e p h a s e

m o d e l , w h i c h is m o r e a c c u r a t e t h e g r e a t e r t h e s tr e n g t h o f th e a t t r a c t i o n o f t h e

l i m i t c y c le . F u r t h e r m o r e , t h e a v e r a g e s i n ( 2 .7 ) , ( 2 .9 ) c a n b e q u i c k l y a n d e a s i ly


18/23

212

co mp u t ed w i t h o u t f i r s t p e r f o r mi n g t h e r ed u c t i o n t o a p h as e mo d e l :

' f 0

k((f f) = ~ 09k O kl( t ) (U 'k)T( t )G k(U a( t ) , U j( t 4- cp~) d t ,

whe re q~ = 0 - 0h. (See [7 ] fo r com pu ta t i on o f t he r e l evan t phas e mo del . )


( A 5 )

A 2 . W e a k c o u p li n g

I f t he coup l ing i s wea k , Eqs . (2 .1 ) ma y be d i r ec t ly averaged , i n s t ead o f

f ir s t r ed u ced t o a p h as e m o d e l an d t h en av e r ag ed . I t f o l l o w s f r o m i n v a r i an t

man i fo ld t heory t ha t when the coup l ing i s weak , t here a lways ex i s t s an i nvar i -

an t t o rus [11 ] . Ho we ver , un l ike t he f i rs t case , t he i nvar i an t t o rus i s no t neces -

sar il y the p ro du ct o f t he limi t cyc l es , and so am pl i t ude v ar i a t i ons m ay af fec t

t h e l o cal f req u en c ie s . Th e am p l i t u d e e f f ect s m ak e t h e ca l cu l a ti o n o f t h e

averaged func t ions H() much more d i f f i cu l t , s i nce we mus t dea l wi th t he

co m p l e t e n - d imen s i o n a l s y s t em in s t ead o f r es t ri c ti n g o u r av e r ag in g t o t h e

p h a s e co m p o n en t . O u r g o a l i n t h e rema i n d e r o f th is s ec t io n is t o d e r iv e a

genera l fo rm ula t ha t w i ll enab le us t o c a l cu l a t e t he phase e f f ec t s i n wea k ly

coup led osc i l l a to r s .

I n o r d e r t o s t u d y w eak co u p l i n g , w e mu s t a s s u me t h a t

F k

= F + O(e) where

~ 1 and the O(e) t e rms co n t r i bu t e smal l f r equency d i ff e rences be tw een the two

osc i l l a to r s . We rep l ace

G k

an d

eGg

an d ab s o r b t h e f r eq u en cy d i f f e r n ce t e r ms

in to t he eGk t e rms . We can assume wi th no l oss i n genera l i t y t ha t , t o l owes t

order , (ok = 1.

M e t h o d I .

W e r ep lace Y k b y

e S k ,

s ince

Y k

wi l l be 0 (8 ) on t he i nvar i an t t o rus .

E q u a t i o n ( A 3 ) b e c o m e s

dO k/d t = 1 + e [b (Ok)S + Q- ' (Ok ) (u ' )T (O k)G k(U (O k) ,

U(0j))] + O(e2),

( h 6 )

d s k / d t = a (O k )S k + M r ( O k ) G k ( U ( O k ) , U ( O b )) + 0 ( 8 ) .

w h e r e

b(O) = q

-I(U )T(D

-~ D T ) M ,

a(O ) = M T D M + ( M ' ) r M .

( A 7 )

N o t e t h a t

a ( O )

a n d

b ( O )

d ep en d o n l y o n t h e o r ig i n a l li m i t cy c l e an d d o n o t

de pe nd on the fo rm o f the coup l ing. T he solu t ion to (A6 ) i s Ok = t + (~k w here (~k

is cons t a n t t o l owe s t o rder . F r om (A6) , w e ob t a in equ a t ions fo r (~k and Sk:

d(~k/dt = e[b(t)sk(t; Zp) + Q- a ( t ) ( U ' ) r ( t ) G k ( U ( t ) , U ( t + ~))] + O(e2), (A 8a )

d s ~ / d t = a ( t ) s k (t ; q~) + n r ( t ) G k ( U ( t ) , U ( t

+ q~)) + O(e), (A 8b )

w here q~ = ~ - Ok O n the inva r iant m an i fold Sk = Sk(Ok, Oj) = sk(t , t + 2p) +

O(e) , wh en , t o l ow es t o rde r i n e , (~ is cons t an t . W e w r i te Sk( t ; ~ t o mak e t h i s

d ep en d en ce ex p l i c i t . W e s o l v e ( A 8 b ) f o r Sk(t; dp) and subs t i t u t e t h i s i n to (A8a) .

We are t hen in a pos i t i on t o average equat ion (A8a) , y i e ld ing

d t =


19/23


w h e r e

1 f f ~

k ( ~ ) = ~ b ( t )S k ( t ; (~ ) + 0 - ( t ) ( U ) r ( t ) G k ( U ( t ) , U ( t + q~-))d t .

T h e f un ctio n/-T k q~ ) c o n s i s ts o f t w o c o m p o n e n t s , o n e c o r r e s p o n d i n g t o t h e e f fe c ts

o f a m p l i t u d e o n t h e l o c a l fr e q u e n c i es . T h e s e c o n d c o m p o n e n t i s i d e n t i c a l t o

A 5 ) , s o t h e d i f f e r e n c e b e t w e e n t h e c o u p l i n g f u n c t i o n H k f o r s t r o n g l i m i t c y c l e s

a n d t h e t r u e c o u p l i n g f u n c t i o n / - 7 k i s the in teg ra l

f k ( ~ ) = ~ b ( t ) s k ( t ; 0~ ) d t . A 9 )

/-/k h a s t h e b e n e f i t t h a t i t d o e s n o t d e p e n d o n t h e c o o r d i n a t e m a t r i x

M k ,

s o t h a t

i t is ea s il y c o m p u t e d f o r a r b i t r a r y l i m i t c y cl es . W e n o w t u r n o u r a t t e n t i o n t o t h e

c o m p u t a ti o n o f ~ k .

F r o m a n u m e r i c a l s t a n d p o i n t , i t is v e r y w a s t e f u l to s o lv e f o r sk a n d s u b s t i tu t e

i n t o A 9 ) , s i n c e t h e v a l u e o f s k c h a n g e s e a c h t i m e t h e c o u p l i n g f u n c t i o n s a r e

c h a n g e d . F u r t h e r m o r e , i t is i m p r a c t ic a l t o m a i n t a i n t h e a r r a y s k i n m e m o r y s in c e,

f o r e a c h o f t h e n - 1 c o m p o n e n t s o f sk , o n e m u s t s t o r e K 2 n u m b e r s i f s k i s t o b e

eva lua te d a t K t imes an d K va lues o f 4~- Ins t e ad , w ha t i s de s i red i s a s ing le vec to r

f u n c t i o n U * ( t ) t h a t i s i n d e p e n d e n t o f t h e c o u p l i n g a n d i s s u c h t h a t

1 f 2 ~

k ( ~ ) = ~ U * r ( t ) G k ( U ( t ) , U ( t + q ~)) d t . A I O )

B e s id e s r e q u i r i n g l es s c o m p u t e r s t o r a g e , t h e o t h e r a d v a n t a g e o f U * i s t h a t i t n e e d

o n l y b e c a l c u l a t e d o n c e ; / - tk c a n t h e n b e c o m p u t e d f o r a r b i t r a r y t y p e s o f

c o u p l in g . W e w i ll n o w s h o w h o w s u c h a fu n c t i o n U * t ) c a n b e c o m p u t e d . L e t

E t ) s a t i s f y

d E / d t = a ( t ) E , E O) = I , _ 1) , - 1), A 11 )

i.e.,

E ( t )

i s t h e e x p o n e n t i a l o f

a( t ) .

L e t

Q ( t ) = b ( O E ( ~ ) d ~ . A 1 2 )

W e r e w r i t e A 9 ) a s

9 ~ k ( 6 ) = ~ b ( t ) E ( t ) E - l ( t ) s k ( t ; q~) d t

1 d / d t ( Q ( t ) ) E - l ( t )s k ( t ; ~ ) d t

2 ~

= - - 1 f2 ~

1 Q 2~r)E-x 2z0sk 2zr ; 6) - ~ J0 Q t ) d t ( E - l ( t ) s , ( t ; 4 7)) d t . A 1 3 )

2 g

N o t e t h a t Q t ) d o e s n o t d e p e n d o n t h e c o u p li n g a n d is a f u n c t i o n o n l y o f t h e

n o r m a l c o o r d i n a t e s a n d t h e li m i t c y c le . T o e v a l u a t e t h e la s t i n t e g r a l o f A 1 3 ) , w e

u s e t h e d e f i n i t i o n o f E ( t ) a n d A 8 b ) t o o b t a i n

d 1

d t [ E - ( t )s k ( t ; 6)] = E - l ( t ) M r ( t ) G k ( U ( t ) , U ( t + 6 ) ) . A1 4)

B e f o r e s u b s t i t u t i n g t h is i n t o A 1 3 ) , w e n e e d t o f i n d sk 2 1r; ~ . W e i n t e g r a t e A 1 4 )


20/23

214


an d use the fact tha t Sk(2rC; q~) = S k(0 ;~. Th us,

o

k 2 ~ ; ~ ) = [ I - e E r 0 1 - 1 e 2 ~ ) e-a t )M ~ t)Gk U t) , g t + ~ ) d t . A 1 5 )

Fina l ly , we subs t i t u t e (A14) and (A15) i n to (A13) t o ob t a in

~ g ) = ~ { a ; ~ ) [ I - e 2 ~ ) ] - I _ a t ) } e- ~ t )M ~ t l 6 ~ V t ) , V t + ~ ) ) d t

l f o : ~

- 2 ~ t r ( t ) G k ( U ( t ) , U ( t + q ~ )) d t . ( A 1 6 )

( N o t e t h a t w e h av e u s ed t h e f ac t t h a t

E ( t ) ,

E - ~ ( t ) a n d [ I - E ( t ) ] - ~ a ll c o m m u t e

in o rder t o s impl i fy (A16) . ) Th e ve c to r func t ion t /( t) is inde pend en t o f t he

coup l ing as r equ i red . Our func t ion U * ( t ) is

U * ( t ) = [ 0 - ~ ( t ) U ( t ) t l ( t ) ] . ( A 1 7 )

M e t h o d 2 . Th e ab o v e d e r i v a t io n o f t h e fu n c t i o n s / ~ k ( ~ h as a r a t h e r g eo me t r i c

f l av o r . A n o t h e r ap p r o ach i s t o u s e t h e F r ed h o l m a l t e r n a t i v e an d f o r ma l p e r t u r -

b a t i o n ex p an s i o n s . W e w i l l s k e t ch t h i s me t h o d b e l o w an d s h o w t h a t t h e r e s u l t s

a re equ iva l en t . Cons ider t he coup led equat ions :

d X k / d t = F ( X k ) + e G k (z ~ k, ~ . ( A 1 8 )

O n e s eek s s o l u t i o n s o f t h e f o r m

X k ( t ) = U ( t + O g ( z ) ) + e W g ( t , z ,

e) , where z =

e t

i s a s l ow t ime sca l e and W k ( t , Z , 5 ) is a s m o o t h f u n c t i o n o f it s co m p o n en t s .

Sub s t i t u t i on o f t h is fo rm ula i n to (A 18) y i e lds a t o rde r e

~ ( t + O k ) =-- [ d / d t - - D ( t + O g ) l W k ( t , z , O )

= - - U ( t + O k ) c 3 O k / a Z + G k ( U ( t + O k ) , U ( t + O j ) ) . ( A 1 9 )

Per iod i c so lu t i ons W k are sough t . The l i near equa t ion

.~q~(t)(( t) = f ( t)

has a per iod i c so lu t i on i f and on ly i f

o2~ Z * ( t ) f ( t ) d t = O ,

w h er e z * ( t ) i s t h e u n i q u e p e r i o d i c s o l u t i o n t o t h e ad j o i n t p r o b l em

[ d / d t - D T(t)]*(t) -- ~q *( t)z* (t) = 0,

f 2 ~ ( A 2 0 )

2 - -~ Z * ( t ) u ( t ) d t = 1 .

Th u s , ( A 1 9 ) h a s a s o l u t io n i f an d o n l y i f

1 fo~

Ok/ t3Z = f fI k( Oj - - Ok) - - ~ Z* ( t ) G k( U ( t ) , U ( t + Oj - Ok) ) d t . ( A 2 1 )

Eq u a t i o n ( A 2 1 ) h a s t h e s ame f o r m a s ( A 1 0 ) . W e p r o v e t h a t U * r ( t ) = Z*(t) is the

s o l u t io n t o ( A 2 0 ) an d t h u s s h o w u n i q u en es s o f U * b y v ir t u e o f t h e u n i q u en es s


21/23

M ultiple pu lse interactions and averaging 215

o f ;( *. Th i s a ls o i mp li e s t h a t H = ~ . W e s eek s o lu t i o n s x * ( t ) t o ( A 2 0 ) o f t h e f o r m

*( t) = U ' ( t ) ( ( t ) + M ( t ) z ( t ) .

( A 2 2 )

Subs t i t u t i on o f (A22) i n to (A20) l eads t o

U ( + U '( + M ' z + M z + D r U ' ( + D r M z

= 0 . (A23 )

W e mu l t i p l y ( A 2 3 ) b y U ' r an d u s i ng ( A 2 ) w e o b t a i n

U ' r U ( + 5 ( + ( U r M + U ' r D r M ) z + U ' r D r U ' ( = 0 . (A24 )

F r o m , r ,

- U k ) M k = ( U ' k ) r( D k ) r M k and the f ac t t ha t U = D U ' , w e s ee t h a t ( A 2 4 )

simplifies to

( U r U + u ' r u ) ( + 5~ = O.

Since ( U r U ' + U ' r U ) = ( u ' r u ' ) = 5 ' , w e h av e 5 ' ( + 5 ( ' = 0 . A l o n g w i t h t h e

norm al i za t i on con d i t i on o f (A20) t h i s impl i es

( ( t) = 5 - ' ( 0 - ( A 2 5 )

W e n o w m u l t i p l y ( A 2 3 ) b y M r an d , u s i n g ( A 2 ) , o b t a i n

z ' = - ( M r D r M + M T M ' ) z - ( M r U + M r D r u ' ) 5 - 1.

( A 2 6 )

Again us ing t he f ac t t ha t U = DU and a l so (A12) , w e see t ha t (A26) is j u s t

z = - a r ( t ) z - b r (t ). ( A 2 7 )

T h e f u n d a m e n t a l m a t ri x f o r z ' =

- a r ( t ) z

is

E - l ( t ) r

a n d

z ( t )

i s r equ i red t o be

2~r-periodic; thus

z (t ) = E - 1 ( 0

7 - { [ /_ E ( 2 r0 q - iQ(2r0 r _ Q r ( t ) } . (A 28)

C o mb i n i n g ( A 2 8 ) w i t h ( A 2 5 ) an d ( A 2 3 ) , w e s ee t h a t Z* ( t ) = U * r ( t ) as r equ i red .

[ ]

A3 . Avera ged coup ling fo r neura l mod e l s

W e n o w w i s h t o ap p l y t h e r e s u lt s o f S ec ts . A 1 an d A 2 t o t w o - d i me n s i o n a l n eu r a l

mod el s . The genera l r esu lt s in Sects . A1 an d A 2 o f t h i s append ix p rov ide

mach i n e r y f o r d e t e r mi n i n g t h e av e r ag ed f u n c t i o n ~ ( ~ b ) f o r a r b i t r a r y w eak l y

coup led osc i l l a to r s . Th i s t ask i s s t r a igh t fo rward once we have chosen coord ina t es

t o t r an s f o r m t h e s y s t em o f eq u a t i o n s i n t o eq u a t i o n s o f t h e f o r m ( A 3 ) . Th u s , w e

mu s t f i r s t p r o v i d e a me t h o d f o r co n s t r u c t i n g t h e ma t r i x M ( O ) which sat i sf ies the

cond i t i ons (A2) . In t he two-d imens iona l case , M ( O ) i s j u s t a two-d imens iona l

co l u mn v ec t o r .

Le t t h e co m p o n e n t s o f t h e li m i t- cy cl e f o r t h e t w o - d i men s i o n a l s y s tem b e

U(O)

- (ul (0), u2(0)). Th en

U'(O) = (u'l (0), u'2(O)).

W e l e t

5 0 ) = I I

u , o ) I I

2 -

+ u 2 o )

a n d c h o o s e

M ( O ) = ( u~ (O ), - u~ ( 0 )) r / x / - ~ .

Clear ly , th i s choice sat i sf ies (A2) .


22/23

H O)

216 G .B. Ermentrout and N. K opell

H e )

b

2~r

Fig. 10.

H O )

for Wils on-C owan equations, flee= 12, fl/ ,= 14,

flei =

18, fl~i=0;

f e U ) =

2X(l+tanh(u--)); f~ (u )= ~ l+ ta n h( u -6 )) , a excitatory-excitatory coupling; b inhibitory-

inhibitory coupling

We now use this M to construct H(~b) . Equations (A11) and (A12) are scalar

equations that are readily integrated using standard methods. This a l lows us to

easily construct the functions

U * O )

defined in (A17) and consequently determine

the averaged function ~(~b) for any two-dimensional osci l la tor .

We have writ ten a comput er p rogram for numerical calcula t ion of the

function

U * O )

which is then used to der ive the averaged function for any system

of weak ly coupled two -dime nsion al oscillators. In Fig. 10, we plot the average d

func t ions for two coupled Wilson-Cowan mode ls :

d E j / d t = - E j d - f e f l e e g ] . - - f l ie I j J r C r e E k ) ,

h29)

d l j / d t = - I j + f ~ f l e , E j - f l , , I j - C t ,I k ) ,

for j, k = 1, 2 j ~ k. In Fig. 10a, we assu me tha t 0ti = 0 and 0 < ~e ~ 1, i.e., we ak

exc itat ory -exc itat ory couplin g. In Fig. 10b, we assu me th at c(e = 0 and 0q is small

and posit ive , corresponding to weak inhibitory- inhibitory coupling. The slope a t

the origin determines the stability of the in-phase solutio n for two cou pled identical

oscil lators. Fro m this , one sees that w eak inhibitory- inhibitory coupling a lways

results in unstable in-phase solutions; the stable phase-difference is half a cycle.

Weak excitatory-excitatory coupling is more complicated; for identical oscillators,

the in-phase solution is not stable, and the stable solution has a small phase-differ-

ence. We have verified this by integrating the full equations (A29) for small *(e


23/23

Mu lt ip le pulse in te rac tions and averaging 217

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Er Men Trout 1991 Multiple