Synchronization of Pulse-Coupled Biological Oscillators
Author(s): Renato E. Mirollo and Steven H. Strogatz
Source: SIAM Journal on Applied Mathematics, Vol. 50, No. 6 (Dec., 1990), pp. 1645-1662
Published by: Society for Industrial and Applied Mathematics
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?
SIAM J. APPL. MATH.
Vol. 50, No. 6, pp. 1645-1662,December 1990
SYNCHRONIZATION
OF PULSE-COUPLED
RENATO E. MIROLLOt
AND
1990 SocietyforIndustrialand Applied Mathematics
010
BIOLOGICAL
OSCILLATORS*
STEVEN H. STROGATZt
Abstract.A simplemodel forsynchronousfiringof biological oscillatorsbased on Peskin'smodel of
thecardiacpacemaker[Mathematicalaspectsofheartphysiology,
CourantInstituteof MathematicalSciences,
New York University,New York, 1975, pp. 268-278] is studied. The model consistsof a population of
identicalintegrate-and-fire
oscillators.The couplingbetweenoscillatorsis pulsatile:whena givenoscillator
fires,it pulls the othersup by a fixedamount,or bringsthemto the firingthreshold,whicheveris less.
The main resultis thatforalmostall initialconditions,the populationevolves to a statein whichall
the oscillatorsare firingsynchronously.The relationshipbetween the model and real communitiesof
biological oscillatorsis discussed; examplesincludepopulationsof synchronously
flashingfireflies,
crickets
thatchirpin unison,electrically
synchronouspacemakercells,and groupsofwomenwhosemenstrualcycles
become mutuallysynchronized.
Keywords.synchronization,
biological oscillators,pacemaker,integrate-and-fire
AMS(MOS) subjectclassifications.92A09, 34C15, 58F40
1. Introduction.
Firefliesprovide one of the most spectacularexamples of synchronizationin nature [5], [6], [15], [20], [40], [48]. At nightin certainparts of
southeastAsia, thousandsof male fireflies
congregatein treesand flashin synchrony.
Recallingdisplayshe had seen in Thailand,Smith[40] wrote:"Imaginea treethirty-five
to fortyfeethigh. . ., apparentlywitha firefly
on everyleaf and all thefireflies
flashing
in perfectunison at the rate of about threetimesin two seconds, the treebeing in
completedarknessbetweenflashes.... Imagine a tenthof a mile of riverfrontwith
an unbrokenlineof[mangrove]treeswithfireflies
on everyleafflashingin synchronism,
the insectson the trees at the ends of the line actingin perfectunison withthose
between.Then,ifone's imaginationis sufficiently
vivid,he mayformsome conception
of thisamazingspectacle."
Mutualsynchronization
occursin manyotherpopulationsofbiologicaloscillators.
Examples includethe pacemakercells of the heart[23], [31], [34], [44]; networksof
neuronsin the circadian pacemaker[9], [33], [47]-[49] and hippocampus[45]; the
cells of the pancreas [38], [39]; cricketsthat chirpin unison [46];
insulin-secreting
and groups of women whose menstrualperiods become mutuallysynchronized[3],
and examples,see [47]-[49].
information
[30], [35]. For further
The mathematicalanalysis of mutualsynchronization
is a challengingproblem.
It is difficult
enoughto analyzethedynamicsof a singlenonlinearoscillator,let alone
a whole populationof them.The seminalworkin thisarea is due to Winfree[47]. He
simplifiedthe problemby assumingthatthe oscillatorsare stronglyattractedto their
limitcycles,so thatamplitudevariationscan be neglectedand onlyphase variations
need to be considered.Winfreediscoveredthatmutualsynchronization
is a cooperative
phenomenon,a temporalanalogue of the phase transitionsencounteredin statistical
physics.This discoveryhas led to a greatdeal of researchon mutualsynchronization,
especiallyby physicistsinterestedin the nonlineardynamicsof many-bodysystems
(see [8], [10], [29], [36], [37], [41]-[43] and referencestherein).
* ReceivedbytheeditorsAugust3, 1989; acceptedforpublication(in revisedform)December15, 1989.
t Departmentof Mathematics,Boston College, ChestnutHill, Massachusetts02167. The researchof
thisauthorwas supportedin partby National Science FoundationgrantDMS-8906423.
t DepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139.
The researchof thisauthorwas supportedin partby National Science FoundationgrantDMS-8916267.
1645
1646
R. E. MIROLLO
AND
S. H. STROGATZ
In mostof the previoustheoreticalworkon mutualsynchronization,
it has been
betweenoscillatorsare smooth.Much less workhas been
assumedthattheinteractions
done forthe case wherethe interactionsare episodic and pulselike.This case is of
sudden
specialimportanceforbiologicaloscillators,whichoftencommunicatebyfiring
impulses(see [13], [19], [33], [48, pp. 118-120]). For example,in the case of fireflies,
sees theflashof another,and respondsby
theonlyinteractionoccurswhenone firefly
shifting
its rhythmaccordingly[5], [20].
in a populationofpulse-coupled
This paper concernstheemergenceof synchrony
of the
oscillators.Our workwas inspiredby Peskin's model forself-synchronization
cardiacpacemaker[34]. He modeledthepacemakeras a networkof N "integrate-andfire"oscillators[2], [4], [17], [18], [24], [25], each characterizedby a voltagelikestate
variable xi, subjectto the dynamics
(1.1)
dxi
dt'=So- yxi,
0-Xi?1,
i==1,-
,N.
When xi = 1, the ith oscillator"fires"and xi jumps back to zero. The oscillatorsare
assumed to interactby a simpleformof pulse coupling:when a givenoscillatorfires,
whichever
it pulls all theotheroscillatorsup byan amountE, or pullsthemup to firing,
is less. That is,
(1.2)
xi(t)=1 oxj(t')=min(1,Xj(t)+?)
Vjoi.
Peskin [34] conjecturedthat "(1) For arbitraryinitial conditions,the system
synchronously.
(2) Thisremains
approachesa statein whichall theoscillatorsare firing
trueeven when the oscillatorsare not quite identical."He proved conjecture(1) for
the special case of N = 2 oscillators,underthe further
assumptionsof small coupling
strengthE and small dissipationy.
In this paper we studya more generalversionof Peskin's model and analyze it
forall N. Instead of thedifferential
equation (1.1), we assume onlythattheoscillators
rise towardthresholdwitha time-coursewhichis monotonicand concave down. We
do, however,retaintwo of Peskin'smostimportantassumptions:the oscillatorshave
identicaldynamics,and each is coupled to all the others.Our main resultis that,for
all N and foralmostall initialconditions,thesystemeventuallybecomessynchronized.
A corollaryis thatPeskin's conjecture(1) is trueforall N and forall E, y> 0. Our
methodsare elementaryand involvelittlemorethan considerationsof monotonicity,
concavity,etc.
In ? 2 we describeour model and analyzeit forthecase oftwo oscillators.Section
numberof oscillators.In ? 4 we relateour work
3 extendsthe analysisto an arbitrary
to previousresearchand discuss some possible applicationsand open problems.
2. Two oscillators.
2.1. Model. Firstwe generalizethe integrate-and-fire
dynamics(1.1). As before,
x
variable
which
is assumed to increase
each oscillatoris characterizedby a state
=
x
x
1.
When reachesthethreshold,theoscillator
monotonicallytowarda thresholdat
firesand x jumps back instantlyto zero, afterwhichthe cyclerepeats.
The new featureis that,insteadof (1.1), we assume onlythatx evolvesaccording
x
to =f(4)), wheref: [0, 1] -> [0, 1] is smooth,monotonicincreasing,and concavedown,
i.e., f'>0 and f"<0. Here / [0, 1] is a phase variable such that (i) do/dt= /T,
where T is the cycle period, (ii) 4 = 0 when the oscillatoris at its loweststatex = 0,
and (iii) 4 = 1 at the end of the cyclewhenthe oscillatorreachesthe thresholdx = 1.
Thereforef satisfiesf(0) = 0, f(1) = 1. Figure 1 shows the graphof a typicalf
OF BIOLOGICAL
SYNCHRONIZATION
1647
OSCILLATORS
x=
x100
00
1.0
FIG. 1. Graph of thefunction
f. The timecourse
oscillationis givenbyx
of theintegrate-and-fire
wherex is thestateand 4 is a phase variableproportional
to time.
Let g denote the inversefunctionf'- (which existssince f is monotonic).Note
thatg maps statesto theircorrespondingphases: g(x) = k.Because of the hypotheses
on f the functiong is increasingand concave up: g'> 0 and g"> 0. The endpoint
conditionson g are g(O) = 0, g(1) = 1.
Example. For Peskin's model (1.1), we find
=
9(x) =
C(1 - e-7+),
` I
n(
-x)
where C = 1 - e-. The intrinsic period T = y-1 In [SO/(SO - y)].
Now considertwooscillatorsgovernedbyf, and assume thattheyinteractby the
pulse-couplingrule (1.2). As shown in Fig. 2(a), the systemcan be visualized as two
pointsmovingto therightalong thefixedcurvex =f(4). Figure2(b) showsthesystem
Because
just beforea firingoccurs; Fig. 2(c) showsthe systemimmediatelyafterward.
the shiftedoscillator is confinedto the curve x =f(4b), the effectof the pulse is
between
tantamountto a phase shift.This is the onlyway thatthe phase difference
the oscillatorscan change-between firings,
both pointsmove withthe same constant
horizontalvelocityd+/ dt= 1/T.
2.2. Strategy.To prove thatthe two oscillatorsalwaysbecome synchronized,
we
firstcalculate the returnmap and then show that the oscillatorsare drivencloser
1.0C
x
0.0
1.0
(a)
(b)
(c)
rule(1.2).
FIG. 2. A systemof twooscillators
governedbyx =f(4), and interacting
bythepulse-coupling
(a) The state of thesystemimmediately
afteroscillatorA has fired.(b) The state of thesystemjust before
Thephase difference
betweentheoscillatorsis thesame as in (a). (c)
oscillatorB reachesthefiringthreshold.
The state of the systemjust afterB has fired.B has jumped back to zero, and the state of A is now
min (1, e +f(1-4O)).
1648
R. E. MIROLLO
AND
S. H. STROGATZ
is lockedinwhentheoscillators
each timethemap is iterated.Perfectsynchrony
together
have gottenso close togetherthatthe firingof one bringsthe otherto threshold.They
because theirdynamicsare identical.
remainsynchronizedthereafter
2.3. Returnmap and firingmap. The returnmap is definedas follows.Call the
two oscillatorsA and B, and let us strobethe systemat the instantafterA has fired
(Fig. 2(a)). Because A has just fired,its phase is zero. Let 'P denote the phase of B.
The returnmap R(4P) is definedto be the phase of B immediatelyafterthe nextfiring
of A.
To calculate the returnmap, observethataftera time 1 - 'P, oscillatorB reaches
threshold(Fig. 2(b)). During this time,A moves fromzero to an x-value given by
is
XA =f(1 -'P). An instantlater,B firesand XA jumps to E +f(1 - 4) or 1, whichever
less (Fig. 2(c)). If XA = 1, we are done-the oscillatorshave synchronized.Hence
The correspondingphase of A is g(8+f(1- 4)),
assume that xA=E+f(l-'P)<l.
as above.
whereg
We definethefiringmap h by
(2.1)
h(4P) =
g(?
+f(1 -
)).
the systemhas moved froman initialstate('A, PB) = (O, 0) to
Thus, afterone firing,
a currentstate ('A, 'B) = (h('P), 0). In otherwords,the systemis in essentiallythe
same state as when we started-but with 'P replaced by h(4P) and the oscillators
interchanged.Thereforeto obtain the returnmap R('P), we followthe systemahead
forone more iterationof h:
R(P) = h(h('P)).
(2.2)
A caveat about the domains of h and R: In the calculationleading to (2.1), we
occursafterthenextfiring.)
assumedthatE +f(1 - ') < 1. (Otherwisesynchronization
This assumptionis satisfiedforE E [0, 1) and 'P E (8, 1), where8 is definedby
8
=
-E).
1-g(i
Thus the domain of the map h, strictlyspeaking,is the subinterval(8, 1). Similarly,
the domain of R is the subinterval(8, h-'(8)). This intervalis nonemptybecause
8 < h-'(8) forE < 1, as is easily checked.
2.4. Dynamics.We will now show thatthereis a unique fixedpoint for R, and
thatthisfixedpointis a repeller.
1,for all 'P.
LEMMA 2.1. h'('P) <-1 and R'(')>
Proof It sufficesto show thath'('P) < -1, forall 'P, since R'('P) = h'(h('))h'('P).
From (2.1), we obtain h'('P) = -g'(? +f(1 - 'P))f'(1 - '). Since f and g are inverses,
the chain rule implies f'(1- -')
(2.3)
= [g'(f(1 - '))V1.
h'()-
=
g'f(1-'
Hence
))
Let u =f(1 -'P). Then (2.3) is of the form
(2.4)
h'=-
g( u)
g"> 0 and E > 0,so g'(? + u) > g'(u), forall u.Thisis wheretheconcavity
Byhypothesis,
hypothesison g is used. Finally,the hypothesisthat g'(u) >0 for all u impliesthat
0
h'<-1, as claimed.
OF BIOLOGICAL
SYNCHRONIZATION
OSCILLATORS
1649
2.2. Thereexistsa uniquefixedpointfor R in (8, h '(8)), and it is
PROPOSITION
a repeller.
Proof.To prove existence,it sufficesto finda fixedpoint for h, because (2.2)
impliesthatany fixedpoint forh is a fixedpoint forR. The fixedpoint equation for
h is
F(
(2.5)
O.
)-b-h()=
It is easy to checkthat
F(8) < O,
(2.6)
=1 - h'(4) > 2 > 0. Hence h has a unique fixed
and fromLemma 2.1 we have F'()
point
/
.
Since R(44)
(2.7)
F(h-'(8)) > O,
* and R'(4) > 1 by Lemma 2.1, we have
(4)
~~~~R
> 4) if 4) > +*,
R (0) < 4) if 4) < +8.
Hence the fixedpointforR is unique, and is a repeller. 0
The result(2.7) showsthatR has simpledynamics-fromany initialphase (other
than the fixedpoint), the systemis drivenmonotonicallytoward + = 0 or + = 1. In
otherwords, thesystemis alwaysdrivento synchrony.
2.5. Solvable example. By makinga convenientchoice forthe functionf(4), we
The special case concan gain furtherinsightinto the dynamicsof synchronization.
sideredhereillustratesa numberof qualitativephenomenathatoccur moregenerally.
Our criterionforchoosingf is thatthe firingmap h and thereturnmap R should
be as simple as possible. From Lemma 2.1, we know that h'(4) < -1. Suppose we
insistthat
(2.8)
W(4)=-A
of +. Then h and R would reduce to affinemaps:
whereA > 1 is independent
(2.9)
h(4) =-A ( -
(2.10)
R(4) = k2(4_ )) + 4).
) + 4)
Now we seek the functionf such that (2.8) is satisfied.Equation (2.4) dictates
theappropriatechoice off-its inversefunctiong mustsatisfythefunctionalequation
(2.11)
g'(--+ u)
g( )
A
Vu
u.
Equation (2.11) has solutionsof the form
(2.12)
g (u) = a
ebu
wherea and b are parametersand
(2.13)
A= e
(Note that(2.11) has moregeneralsolutionsthan (2.12), e.g., g'(u) = P(u) eb, where
for our
P(u) is any periodic functionwith period E. However (2.12) is sufficient
purposes.)
1650
R. E. MIROLLO
S. H. STROGATZ
AND
(2.12) and imposingtheendpointconditionsg(0) = 0 and g(1)
Afterintegrating
1, we find
g
(2.14)
=
bu
eb-1
=
1
The functionf = g-' is givenby
f(I)
(2.15)
b
in (1 +[eb
- 1]).
Thus we have a one-parameterfamilyof functionsf parametrizedby b. By our
earlierassumption,f is concave down; hence b > 0. Figure3 showsthe graphoff for
different
values of b. As b approacheszero,f approachesthe identitymap, indicated
by the dashed diagonal in Fig. 3; forlarge b,f risesveryrapidlyand thenlevels off.
Thus b measuresthe extentto whichf is concave down. In morephysicalterms,b is
analogousto theconductancey in the"leakycapacitor"model ( 1.1); botharemeasures
of the leakinessor dissipationin the dynamics.
implications.
The resultsabove have some interesting
emergesmorerapidlywhenthedissipation
Synchrony
(1) Rate ofsynchronization.
b or the pulse strengthE is large.We can estimatethe timeit takes forthe systemto
synchronize,startingfroman initialphase 40. Let kkdenote the kthiterateRk(Ck0),
and let Ak= 10k- 4k*ldenote the distance fromthe repellingfixedpoint /8. From
(2.10) we see thatAk growsexponentiallyfastin k:
2
Ak = AoA
= AO e2bk
Synchronyoccurswhen kkhas been drivento zero or one, and Ak 0(1); thenumber
of iterationsrequiredis
k
(2.16)
(( Eb n?)O
to theproductEb.
is inversely
proportional
Thus thetimetakento synchronize
1.0
0.8-
0.6-
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
b. The
FIG. 3. Graphsofthefunction
parameter
by(2.15),forthreevaluesofthedissipation
f defined
b = 0.
dasheddiagonallinecorresponds
to theidentity
map,forwhich
OF BIOLOGICAL
SYNCHRONIZATION
OSCILLATORS
1651
A similarresultwas foundby Peskin [34]. He used Taylor series expansionsto
approximatethe returnmap for the model (1.1) in the limitof small E and y. He
showed that the rate of convergenceto synchronydepends on the product ?y (to
between
is a "cooperativeeffect
lowestorderin E and y), and concludedthatsynchrony
the couplingand the dissipation;convergencedisappearswheneitherthe couplingor
thedissipationis removed."Our resultshowsthatthisproductdependenceholds even
if E and b are not small.
(2) Locationof thefixedpoint. For the special familyof functions(2.15), we can
findthe repellingfixedpoint /* explicitly,and therebysee how it depends on b and
E. Rewritingthe fixedpoint equation (2.5) as E =f(Gt*)-f(1 - 8*) and substituting
(2.15) forf we obtain
e b(l+e)_
(2.17)
b( bl )
-
1)
Figure4 showsthe graphof g*versusE forthreevalues of b. As E tendsto zero,
the fixedpoint always approaches g*= 2. This resultholds forgeneralf, as can be
seen from(2.5). Thus, in the limitof small coupling,the repellingfixedpointalways
occurs withthe oscillatorsat antiphase.
1.00
0.75-
0.50 0.00
0.25
0.50
0.75
1.00
FIG. 4. Dependenceof thefixedpoint0* on thepulse strength
e and thedissipationb, as givenby (2.17).
(3) Stabilitytypeofthefixedpoint.The eigenvalueA = ebe determinesthestability
typeof the fixedpoint /*.We have assumed throughoutthatE > 0 and b > 0; in this
case A> 1 and 0* is a repeller.It is worthnoting,however,that /*would be stabilized
if eitherE or b werenegative.This would occur fora systemwhichacceleratesup to
threshold(b < 0), or forone withinhibitory
coupling(e < 0). If both8, b < 0, thenthe
fixedpointis again a repeller.
3. Populationof oscillators.We turnnow to the general case of N oscillators.
Beforebeginningthe analysis,we give an intuitiveaccountof the way thatsynchrony
develops.
As the systemevolves,oscillatorsbegin to clump togetherin "groups" thatfire
at the same time.This gives rise to a positivefeedbackprocess,as firstdescribedfor
a different
systembyWinfree[47]-as a groupgetsbigger,itproducesa largercollective
pulse whenit fires,and therebytendsto bringotheroscillatorsto thresholdalong with
it.In thissensea largegrouptendsto growby"absorbing"otheroscillators.Absorptions
reduce the numberof groupsuntilultimatelyonly one group remains-at thatpoint
This scenariois illustratedby computersimulationin
the populationis synchronized.
? 3.4 below.
1652
R. E. MIROLLO
AND
S. H. STROGATZ
Our proof of synchronyhas two parts.The firstpart (Theorem3.1) shows that
foralmost all initialconditions,an absorptionoccurs in finitetime.The hypotheses
of Theorem3.1 are generalin thesense thattheyallow thegroupsto firewithdifferent
strengths,
correspondingto the different
numbersin each group.Afteran absorption
occurs,thereare N - 1 groups(or perhapsa smallernumberifseveraloscillatorswere
absorbed at the same time).
The second part of the proof (Theorem 3.2) rules out the possibilitythatthere
mightexistsetsof initialconditionsof positivemeasurewhich,aftera certainnumber
of absorptions,live foreverwithoutexperiencingthe finalabsorptionsto synchrony.
Taken together,Theorems3.1 and 3.2 show thatalmostall initialconditionslead to
eventualsynchrony.
In ? 3.1, we definethe statespace. The dynamicsare discussedin ? 3.2, usingthe
analogue ofthefiring
map h discussedabove. Afterdefining
thenotionof"absorptions"
more preciselyin ? 3.3, we presentsimulationsin ? 3.4 whichillustratethe way that
synchronyevolves. We state and prove the two parts of our main theoremin ? 3.5.
The argumentappears somewhattechnical,but is based on simpleideas involvingthe
volume-expansionpropertiesofthereturnmap. An exactlysolvableexampleis presented in ? 3.6.
3.1. State space. As before,we studythe dynamicsof the systemby "strobing"
it rightafterone of the oscillatorshas firedand returnedto zero. The state of the
systemis characterized by the phases
,
, * *,n
Thus the possible statesare givenby the set
(3.1)
of the remaining n
N - 1 oscillators.
S = {(Ol *** n) E Rns.t.0< l < 02 < ... < n < 1},
wherewe have indexedthe oscillatorsin ascendingorder.The oscillatorsare currently
labeled 0, 1,2, *, n, withthe conventionthat 00 = 0.
Because the oscillatorsare assumedto have identicaldynamicsand the coupling
is all to all, the flowhas a special property-itpreservesthe cyclic orderingof the
oscillators.The order cannot change between firings,because the oscillatorshave
of the functionf ensuresthatthe orderis
identicalfrequencies,and the monotonicity
maintainedaftereach firing.Hence the oscillatorsfirein reverseorderto theircurrent
index: the oscillatorcurrently
labeled n is the nextto fire,thenn -1, and so on. After
oscillatorn fires,we relabel it to zero, and relabel oscillatorj to j + 1, forall j < n.
This simpleindexingschemewould failiftheoscillatorshad different
frequencies;
then one oscillatorcould "pass" another,and the dynamicswould be more difficult
to analyze.The case of nonidenticalfrequenciesis relevantto realbiologicaloscillators
and is discussed brieflyin ? 4.
3.2. Firingmap. Let 0 = (1 ***,n) be the vectorof phases immediatelyafter
a firing.As in ? 2, we would like to findthe firingmap h,i.e., the map thattransforms
4 to the vectorof phases rightafterthe nextfiring.
To calculate h, note that the next firingoccurs aftera time 1 - n. During this
time,oscillatori has driftedto a phase 4i + 1 - On,where i = 0, 1,2, ... , n - 1. Thus
the phases rightbeforefiringare givenby the affinemap o-:Rn Rn definedby
0(O9l9..
49n)= (1- Ong49+ 1 Ong.. 49n-1+ 1 n)
(3.2)
(o71,
*2,
, nn)-
Afterthe firingoccurs,the new phases are givenby the map r: Rn Rn where
SYNCHRONIZATION
OF BIOLOGICAL
1653
OSCILLATORS
Togetherthe map
h
(3.4)
)
(o-
describesthe new phases of the oscillatorsafterone firing.
Note thatwe have implicitlyrelabeled the oscillators,so the image vectorh(4)
representsthephases of the oscillatorsformerly
labeled 0, 1,2, * *, n - 1. That is, the
originaloscillator0 has become 1, oscillator2 has become 3, * , and oscillatorn has
become oscillator0.
3.3. Absorptions.The set S is invariantunderthe affinemap o-,but notunderthe
map , because f(n)+ E+8- 1 is possible. When this happens,it means thatthe firing
of oscillatorn has also broughtoscillatorn -1 to thresholdalong withit. Thereafter
thetwooscillatorsact as one, because theirdynamicsare identicaland theyare coupled
in the same way to all the otheroscillators.We call such an eventan absorption.
Absorptionscomplicatemattersin two ways:
(1) Because of absorptions,thedomainof h is notall of S. The domainis actually
the set
S.e =
{ (dV
9.. * *
n) E: S
s-t-f(dVn-1
+ 1 - On) +
E<
1}9
or, equivalently,
(3-5)
Se.= {(1
..
9 n) E:S
s.t.
OVn
-
n-1 > 1
g(1l
)}
If 4 E: S - Se, an absorptionwill occur afterone firingof strength8.
(2) Absorptionscreate groupsof oscillatorsthatfirein unison witha combined
pulse strength
proportionalto the numberin the group.Equivalently,we can thinkof
a group as a single oscillatorwithan enhanced pulse strength.Thus we now must
allow forthe possibilityof different
pulse strengths
in the population. Howeverthis
turnsout to be easily handled-as will be seen in ? 3.5, the proofof synchrony
does
not require identical pulse strengths;it requires only that the pulse strengthsbe
nonnegativeand not all zero.
3.4. Numericalresults.To illustratethe emergenceof synchronization,
we now
presentthe resultsof a computersimulationof N= 100 oscillators.The systemwas
startedfroma random initial condition: the states xi, i= 1, ...,
*
N. were chosen
independentlyfroma uniformdistributionon [0, 1], and thenreindexedso thatthe
xi werein ascendingorder.(This reindexinginvolvesno loss of generalitysince each
oscillatoris coupled to all theothers.)The subsequentevolutionofthexi was governed
by (1.1) and (1.2), with SO = 2, y = 1, and E = 0.3.
Figure5 plotsthe numberof oscillatorsfiringas a functionof time.At firstthere
is littlecoherenceamongthe oscillators,and the systemorganizesitselfratherslowly.
Then synchronybuilds up in an acceleratingfashion,as expected by the positive
feedbackargumentgivenearlier.By t = 9 T the systemis perfectlysynchronized.
The slow initialbuildup of synchrony
is reminiscentof the observation[6] that
among southeastAsian fireflies,
synchronousflashing"builds up relativelyslowlyat
dusk in the displaytrees,whereeach male is being stimulatedby the lightfrommany
sources."In thepresentsystemas well,each oscillatorreceivesmanyconflicting
pulses
duringthe incoherentinitialstage.
Figure6 showsthe evolutionof thesystemin statespace. The systemwas strobed
immediatelyaftereach firingof oscillator i = 1. Afterthe firstfiring,some shallow
partsappear in thecurveof xi versusi, correspondingto oscillatorsin nearlythesame
state.By the seventhfiring,
these partshave become completely
flat-this means that
1654
R. E. MIROLLO
AND
S. H. STROGATZ
100
80-
number
of
oscillatorS60firing
40-
20 20
Lh
0
2T
4T
6T
8T
1OT
12T
time
FIG. 5. Numberof oscillators
firingas a functionof time,for thesystem(1.1) and (1.2), withN= 100,
of thenaturalperiodT of the
SO=2, y = 1, e = 0.3, and randominitialcondition.Timeis plottedin multiples
oscillators.Each period is dividedinto 10 equal intervals,and the numberof oscillatorsfiringduringeach
intervalis plottedvertically.
1st iterate
-~-
7thiterate
10thiterate
0.8-
0.6-
0.2-
0. l
0
20
40
60
80
100
oscillatorindex
6. The state of the systemafterthefirst,seventh,and tenthiterationsof the returnmap. Same
simulationas in Fig. 5. Theflatsectionsof thegraphscorrespond
to groupsof oscillatorsthatfirein unison.
FIG.
the correspondingoscillatorsare firingin unison. A dominantgrouphas emergedby
the tenthfiring;it also appears in Fig. 5 as the large,growingspike.
In the simulationdescribedabove, we used the followingconventions:
(1) If thefiring
of one oscillatorbroughtanotherto threshold,thelatteroscillator
was not allowed to fireuntilthe nexttimeit reached threshold.Anotherconvention,
perhapsmorenaturalbiologically,would be to let thelatteroscillatorfireimmediately
and possiblyignitea "chain reaction"of additional firing,
untilno otheroscillators
werebroughtto threshold.This alternativeconventionwould speed up the inevitable
of the system.Our main theoremstatedbelow is truein eithercase,
synchronization
but the notationbecomes more complicatedif chain reactionsare allowed.
of a synchronousgroupwas assumedto be the sum of the
(2) The pulse strength
individualpulse strengths.
As will be seen below, thisassumptionof additivity
is also
thatthe pulse strength
unnecessaryforthe proofof our main theorem.It is sufficient
SYNCHRONIZATION
OF BIOLOGICAL
1655
OSCILLATORS
of each groupbe nonnegative,withat least one grouphavinga pulse strength
greater
thanzero.
For the biological applicationswe have in mind,it is importantthatthetheorem
hold even if pulse strengths
are not additive.For example,it seems improbablethat
10 fireflies
flashingsimultaneously
would have 10 timestheeffect
thatone would have.
In themodels proposed by Buck and Buck [5], [6], the responseis proposedto be all
or none; the firefly's
pacemakeris assumed to be resetcompletelyby any flashabove
a certainstrength.
3.5. Main theorem.Suppose our N oscillatorsfirewithstrengths
El, * *
and assume not all Ei =0. Let rl,
, E
-0
be defined as in (3.3), corresponding to the
We reduce all indices modulo N, i.e., let ri= rj forj
, TN
pulse strengthsEl,
N,
i mod N, 1 ' j ' N. Then let hi= rioor.Let
(3.6)
Ai = {4
E
S s.t. 0
E
S81,hl(4)
E: S2,
h2h(4)
ES3,* * *, hi-lhi_2* ... h.(0)
E
S.
So Ai is theset of initialconditionsthatwill have at least i firings
beforean absorption
occurs. Let
(3.7)
A=n
i=l
Ai.
ThenA is theset of initialconditionsthatliveforeverwithoutany absorptions.
We now stateand prove the firstpart of our main result.
THEOREM
3.1. The set A has Lebesguemeasurezero.
Proof. A is measurable since it is a countable intersection of open sets. (In fact,
it is not hard to see thatA is closed.)
Considerthe returnmap
(3.8)
R = hNhNl**
*hi.
A is invariantunderthe map R, i.e.,
R(A)c A.
R is also one to one on its domain AN. Hence to show thatA has measurezero, it
suffices
to showthattheJacobiandeterminant
of R has absolutevalue greaterthanone.
From (3.8) and the definitionof hi,
N
H det(Dhi)
det(DR)=
i=l
(3.9)
N
H det (DTi) det (Do-).
=
i=l
The map o-is affineand satisfies o-N = I, so det (Do-) = ?1. (To see thato-N = I, note
that uN is the returnmap in the trivialcase when all Ei= 0; forthis noninteracting
systemthereturnmap is theidentity.)From(3.3) we findthatDri is a diagonal matrix,
and
n
det(DTi)|=
gD(f(f H
(k) + )f'(o-k)
k=l
Since f and g are inverses, the chain rule implies f'(o-k)
(3.10)
det (Dri)Ia
H
'
= [g'(f(0-k))-l,
so
1656
R. E. MIROLLO
AND
S. H. STROGATZ
By hypothesis,g"> 0 and g'> 0 so theright-hand
side of (3.10) is greaterthanor equal
to one, with equality if and only if Ei = 0. Hence det (Dri) > 1, unless Ei = 0. But by
assumptionEi $ 0 forat least one i. Hence Idet(DR)I > 1.
0
We turnnow to thesecond halfoftheproofof synchrony.
The argumentconcerns
the set of initialconditionswhich,aftera certainnumberof absorptions,live forever
We will show thatthisset has measurezero.
withoutreachingultimatesynchrony.
Beforedefiningthis set more precisely,we firstdiscuss the absorptionprocess.
For now,fixthepulse strengthE to ease the notation.Suppose 4 = (/1, *.. *, n) E Sn,
whereSn denotesthestatespace (3.1), previouslycalled S. (The dependenceon n now
becomes significant.)
Let o-(4) = (o-l, * , O-n).As above, let h denote the firingmap
correspondingto pulse strengthE:
g (f(aj)+
h(4) = r(o(0)))=(*
8),.
Absorptionoccurs if at least one of the coordinates o-j satisfiesf(o-j)+8?1, or
equivalently,o-j_ g(l - 8). Since the o-jincreasewithj, therewill be an index k such
that o-j_ g(1 - E) if and only ifj> k. Of course, k = n means no absorptionoccurs,
and k = 0 means perfectsynchrony
is achieved.When k < n,we say "4 getsabsorbed
by h to Sk." In thiscase we definethe image of 4 to be the point
(g(f(u1)
+ E),
+ E))
, g(f(k)
E Sk
the oscillatorsthatgetabsorbeddo not fire.
Note thataccordingto thisdefinition
This conventionagrees withthatused in the simulationshownin Figs. 5 and 6.
Assume
As above, we need to allow forthepossibilityof different
pulse strengths.
thatthe ith oscillator(or synchronousgroup of oscillators)fireswithstrength8i and
let hi denote the correspondingfiringmap (3.11) withE replaced by Ei.
Now we discuss the dynamicsunderiterationof the firingmaps. Assumethata
point 4 E: Sn getsabsorbed by h, to Sk. This means thatthe oscillatorscorresponding
to 82, 3, * * , En-k+l have been absorbedby the oscillatorcorrespondingto El. These
is El+ - - -+
oscillatorsnow forma synchronousgroupwhose combinedpulse strength
En-k+l- Afterthisabsorptionevent,the iterationproceeds on Sk, withE sequence
En-k+2,
En+l, E
+ '+
En-k+l -
Now we have a similarprocesson Sk to iterate.We continueuntilwe reachsynchrony
(k = 0) or get stuckforeverat some stage withk > 0.
DEFINITION.
Let B be the set of initialconditionsin Sn which,upon iterationof
the maps hi withE sequence El, 2, * * E En+l, neverachieve synchrony.
THEOREM
3.2. The set B has Lebesguemeasurezero.
induction
on n. The case n = 1 followsfromthe resultsof ? 2. Assume
Proof.By
for
all
the theoremis true
E sequences El, * , Ek on Sk, wherek < n. Let Br,kdenote
4
B
the set of E such that 4 survivesthe applicationsof hl,
, hri-, and thengets
absorbed by hr to Sk. Hence
B=Au
U
r_1
1_k'.n
Br,k
Let ,i denote Lebesgue measure.We alreadyknow that,i(A) = 0 fromTheorem
3.1. Hence it sufficesto show thatji (Brk) =0 foreach (r, k).
First we considerthe case where r = 1. Then Bl,k consistsof points which get
thesepointsmustbe absorbed into a set C in Sk
absorbed by h, to Sk. Furthermore,
forany problemon Sk, the set of pointswhichdo
of measurezero since, byinduction,
has measurezero.
not achieve synchrony
SYNCHRONIZATION
Let 4
and o-(4()=
E: Bl,k
(orl,
OF BIOLOGICAL
and let C be any set of measurezero in Sk. Write4)= (
,o-J). Then 4 is absorbed to
(g(f(o-1)
+
-1),
+
, g(f(o)
*.
1657
OSCILLATORS
F))
*,
),J
E C.
, o,k) E rj 1C, wherewe use thenotationrl forthemap on Sk now. Since
Hence (o-1,
,u(r11C) = 0. This means thatthe projectionof the set o-B1,k
rl is a diffeomorphism,
= 0. Since o- is also a
to Sk has measure zero. This is possible only if u(o-BBl,k)
=
diffeomorphism,
i (Bl,k) 0.
Now suppose r> 1. Consider any Br,k.We have
hrl1hr-2...
h,Brk c B1,k,-
Each hi is a diffeomorphism
on its domain. Hence p.(Br,k) = 0, forall k,r> 1.
0
3.6. Solvableexample. As in ? 2, we can obtainfurther
insight(and sharperresults)
if we assume thatf belongsto the one-parameterfamilyof functions(2.15). Then the
returnmap R becomes verysimple-it is givenby an affinemap whose linearpartis
a multipleof the identity,as we will now show.
In general,R is givenby
R = TNurN-1
where ri: R(3.11)
.T1,
R' is defined by (3.3) with E replaced by Ei:
ri(O*i,
,u-n) = (g(f(u)+
.0)g
Wu.)+
8i),
Now suppose thatg and f are givenby (2.14) and (2.15). Then
g(f(o,k) +
Ei) = ebio.k
+ constant,
and so
=
A I + constant,
where
Ai=e be,
(3.12)
Hence R reducesto
(3.13)
R = (A1
...
AN)I + constant,
because o- commuteswithI, and uN = I.
Remarks.(1) For this example,R is an expansionmap:
= (Al
JJR(0))-R(O)JJ
. .
AN)jj4)- Oj
>J11J-011V4),OE ANcR .
As before,we are assumingthatb > 0, Ei_ 0, forall i, and Ei? 0 forsome i.
An open questionis whetherR is always an expansionmap, giventhe original,
moregeneralhypotheseson the functionf
Theorem3.1 can be strengthened:theset A is (at most) a single
Furthermore,
repelling
fixedpoint.This followsfromthe observationsthatA is an invariantset and
R is an expansion.
(2) To gain some intuitionabout the repellingfixedpoint,suppose thatall the
pulse strengthsEi are verysmall, so that ri is close to the identitymap. Then the
repellingfixedpoint 4* is close to the fixedpoint of o-,namely,
4)* -(1,
N'
2, -,N-1).
1658
R. E. MIROLLO
AND
S. H. STROGATZ
In otherwords,the oscillatorsare evenlyspaced inphase. This generalizesthe earlier
resultthatforN = 2 and small E, the repellingfixedpoint occurs withthe oscillators
at antiphase.
4. Discussion. We have studied the emergenceof synchronyin a systemof
oscillatorswithpulse coupling.The systemgeneralizesPeskin'smodel
integrate-and-fire
[34] of the cardiac pacemaker by allowing more general dynamicsthan (1.1); we
assume only thateach oscillatorrises towardthresholdwitha time-coursewhich is
monotonicand concave down. Our main resultis thatfor all N and for almost all
initialconditions,the systemeventuallybecomes synchronized.
The analysisrevealsthe importanceof the concavityassumption(relatedto the
"leaky" dynamicsoftheoscillators)and thesignofthepulse coupling(theinteractions
are "excitatory").Like Peskin [34] we have found that synchronyis a cooperative
effectbetweendissipationand coupling-it does not occur unless both are present.
In retrospectit may seem obvious that synchronywould always emergein our
model. We have made some strongassumptions-theoscillatorsare identicaland they
are coupled "all to all." On the otherhand, we mightwell have imaginedthat the
systemcould remainin a stateof perpetualdisorganization,or perhapssplitinto two
Perpetualdisorganizationwould actuallyoccur
subpopulationswhichfirealternately.
ratherthanone whichis concave
ifthe oscillatorswereto followa lineartime-course,
and
the systemwould neversynwould
be
the
identity
the
return
map
down-then
chronize.
Winfree
In anycase, thebehaviorofoscillatorpopulationscan be counterintuitive.
made
of 71
machine"
in
a
"firefly
phenomena
surprising
[48] has described some
of
natural
frequencies.
narrow
distribution
a
oscillators
with
neon
electrically-coupled
Such oscillatorsare akin to those consideredhere,being based on a voltage which
accumulatesto a thresholdand then dischargesabruptly.Winfreefound thatwhen
the oscillatorswere coupled equally to one anotherthrougha commonresistor,the
no matterhow strongthe coupling!
systemneversynchronized,
4.1. Relationto previouswork.
4.1.1. Oscillatorpopulations.To put our workin contextwithpreviousresearch
levels of
on oscillatorpopulations,it is helpfulto distinguishamong threedifferent
phase locking,and frequencylocking.In this paper we
synchrony,
synchronization:
use the term"synchrony"in the strongestpossible sense: "synchrony"means "firing
in unison." Synchronyis possible in our model because the oscillatorsare identical.
neveroccursin real populationsbecause thereis alwayssome distribuTrue synchrony
of firingtimestion of naturalfrequencies,whichis thenreflectedin thedistribution
typicallythe fasteroscillatorsfireearlier.Nevertheless,some experimentalexamples
times
in thesense thatthespreadin firing
providea good approximationto synchrony,
is small comparedto the period of the oscillation.For example,this is the case for
the firingof heartpacemakercells [23], [31], [34], synchronousflashingof fireflies
amongwomen[30].
[5], [6], [20], chorusingof crickets[46], and menstrualsynchrony
in whichthe oscillatorsdo
"Phase locking"is a weakerformof synchronization
betweenany two oscillators
not necessarilyfireat the same time.The phase difference
is constant,but generallynonzero.Phase lockingarisesin studiesof oscillatorpopulations withrandomlydistributednaturalfrequencies[8], [10], [11], [29], [36], [37],
[42], [43], [47], as well as in wave propagationin the Belousov-Zhabotinskyreagent
[16], [28], [48], [50], [51] and in centralpatterngenerators[7], [26], [27].
"Frequencylocking"meansthattheoscillatorsrunat thesame averagefrequency,
but not necessarilywith a fixedphase relationship.If the coupling is too weak to
SYNCHRONIZATION
OF BIOLOGICAL
OSCILLATORS
1659
enforcephase locking,a systemwhichis spatiallyextendedmaybreakup intodistinct
oscillators.
plateaus [12], [16], [48] or clusters[36], [37], [42], [43] offrequency-locked
models. The oscillatorsin ourmodel obey "integrate-and4.1.2. Integrate-and-fire
fire"dynamics.This is a reasonable assumptionforbiological oscillators,whichoften
exhibitrelaxationoscillationsbased on the buildup and sudden dischargeof a membrane voltageor otheractivityvariable [17], [18]. Most previousstudiesof integrateand-fireoscillatorshave emphasizedthe dynamicsof a singleoscillatorin responseto
mode locking,and chaos [2],
periodic forcing,withspecial attentionto bifurcations,
oscilpopulationsof integrate-and-fire
[4], [24], [17]. Our concernis withinteracting
lators,forwhichmuch workremainsto be done.
of biological
4.1.3. Pulse coupling.The pulse coupling (1.2) is a simplification
a
an oscillator
causes
advances;
resetting
pulse
always
reality.It produces onlyphase
normal.
The
measured
to fireearlierthan
experimentally
phase-responsecurvesfor
of
cardiac
and
thechirpingofcrickets
thefiring
theflashingoffireflies,
pacemakercells,
all have morestructure
than(1.2) suggests(see [48, p. 119]). Otherauthorshavestudied
models which incorporatemore of the biological details of pulse coupling (see, for
example,the heartmodels of Honerkamp[21] and Ikeda, Yoshizawa, and Sato [22]
whichinclude absolute and relativerefractory
models of Buck
periods,or the firefly
and Buck [5], [6] whichinclude timedelay betweenstimulusand response).
A surprising
featureof pulse couplinghas recentlybeen discoveredby Ermentrout
models of neural oscillators,
and Kopell [13]. They found that in many different
excessivelystrongpulse coupling can cause cessation of rhythmicity-"oscillator
death"-and they also discuss the averagingstrategiesthat real neural oscillators
apparentlyuse to avoid such a fate.Note howeverthatthereis no oscillatordeath in
in the dynamics.
the simplemodel studiedhere,thanksto (unrealistic)discontinuities
The model consideredhere is similarto a model of
4.1.4. Synchronous
fireflies.
Photinus
firefly
synchronization
proposedby Buck [5]. In thecommonAmericanfirefly
pyralis,it appears that resettingoccurs exclusivelythroughphase advances. Buck
of thisspecies receivesa lightpulse near the end of its
postulatesthatwhen a firefly
cycle,its flash-control
pacemakeris immediatelyresetto threshold,as in our model.
However,in contrastto our model,pulses receivedduringthe earlierpartof the cycle
are assumed to have littleor no effect.
Buck [5] also assumesa linearincreaseof excitationtowardthreshold,in contrast
to the concave-downtimecoursein the presentmodel. He points out that Photinus
pyralisis "not usuallyobservedto synchronize"-thisis exactlywhatour modelwould
predictif the rise to thresholdwere actuallylinear! (Synchronywould also fail if the
timecoursewere concave-up.)
A more likelyexplanationforthe lack of synchronyin this species is that the
is too smallto overcomethevariabilityin flashingrate.This explanacouplingstrength
tion is supportedby the observation[5] that Photinuspyralishas a comparatively
foran individualmale,thecycle-to-cycle
variability
(standard
irregular
flashingrhythm;
deviation/meanperiod) is 1/20, comparedto 1/200formales of the Thai species
malaccae.
Pteroptyx
Finally,the simple model discussed here does not account forthose species of
fireflies
whichexhibitphase delays in responseto a lightflash[5], [20].
4.2. Directionsforfutureresearch.
4.2.1. Spatial structure.In its presentformthe model has no spatial structure;
each oscillatoris a neighborof all the others.How would the dynamicsbe affectedif
one replacedthe all-to-allcouplingwithmorelocal interactions,
e.g.,betweennearest
1660
R. E. MIROLLO
AND
S. H. STROGATZ
neighborson a ring,chain, d-dimensionallattice,or more general graph [1], [19],
[32]? Would the systemstillalways end up firingin unison,or would more complex
modes of organizationbecome possible?
By analogy withotherlarge systemsof oscillators,we expect that,systemswith
reducedconnectivity
shouldhave less tendencyto become synchronized
[8], [36], [42],
[43]. (A similarrule of thumbis well knownin equilibriumstatisticalmechanics:a
ringof Ising spins cannot exhibitlong-rangeorder,but higher-dimensional
lattices
can.) Thus a ringof oscillatorsis a leading candidatefora systemwhichmightyield
otherbehaviorbesides global synchrony.
But we have not observedany such behavior
in preliminary
numericalstudies-even a ringalways synchronizes.
We thereforesuspect that our systemwould end up firingin unison foralmost
all initialconditions,no matterhow theoscillatorswereinterconnected
(as long as the
interconnections
form a connected graph). If correct,this would distinguishour
pulse-coupledsystemfromdiffusively
coupled systemsof identicaloscillators,which
oftensupportlocally stable rotatingwaves [11], [14], [48].
In any case, it will be more difficult
to analyze our systemif the couplingis not
all to all, because we can no longerspeak of "absorptions."Recall thatin theall-to-all
case, a synchronoussubsetof oscillatorsremainssynchronousforever.This is not the
case withany othertopology-a synchronousgroup can now be disruptedby pulses
impingingon the boundaryof the group.
4.2.2. Nonidenticaloscillators.Anotherrestrictiveassumptionis that the oscillators are identical; then synchronyoccurs even with arbitrarily
small coupling. It
would be more realisticto let the oscillatorshave a randomdistributionof intrinsic
frequencies.Most of the workon mutualentrainment
of smoothlycoupled oscillators
deals withthis case [8], [10], [11], [29], [31], [36], [37], [42], [43], [47], [48], but
almostnothingis knownforthe case of pulse coupling.
One propertyof thepresentmodel can be anticipated:in a synchronous
population
thefastestoscillatorwouldset thepace. This idea oftencropsup in popular discussions
of synchronization,
and is widelyaccepted in cardiology,but it is knownto be wrong
forcertainpopulationsof oscillators[6], [9], [23], [26], [31], [48]; neverthelessit is
trueforthe presentmodel. To see this,imaginethatall the oscillatorshave just fired
and are now at x = 0. Then the firstoscillatorto reach thresholdis the fastestone.
Givenourassumptionthatthepopulationremainssynchronous,
all theotheroscillators
have to be pulledup to thresholdat thesame time.Thusthefrequencyofthepopulation
is thatof the fastestoscillator.
A periodicsolutionof thisformwould be possible ifthepulse strength
werelarge
enoughto pull theslowestoscillatorup to threshold.Even a weakerpulse mightsuffice
if it could set offa "chain reaction"of firingby otheroscillators.
Of course,such a synchronoussolutionmightnot be stable,and even if it were,
theremightbe otherlocally stable solutions.For instance,could a synchronousstate
coexist with arrhythmia?
Such bistabilitymighthave relevance for certainrhythm
disturbancesof the cardiac pacemaker.
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