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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 Stable URL: http://www.jstor.org/stable/2101911 . Accessed: 01/01/2011 11:48 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=siam. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected] Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend access to SIAM Journal on Applied Mathematics. http://www.jstor.org ? 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. 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