by Steven H. Strogatz and Ian Stewart

A subtle mathematical threadconnects clocks, ambling elephants,

brain rhythms and the onset of chaos

weight at the end of a string can takeany of an infinite number of closedpaths through phase space, dependingon the height from which it is released.Biological systems (and clock pendu­lums), in contrast, tend to have not onlya characteristic period hut also a char­acteristic amplitude. They trace a par­ticular path through phase space, andif some perturbation jolts them out oftheir accustomed rhythm they soon re­turn to their former path. If someonestartles you, say, by shouting, "Boo!",your heart may start pounding but soonrelaxes to its normal behavior.

Oscillators that have a standard wave­form and amplitude to which they re­turn after small pertUIbations are knownas limit-eycle oscillators. They incorpo­rate a dissipative mechanism to damposcillations that grow too large and asource of energy to pump up those thatbecome too smali.

THOUSANDS OF FIREFllES Dash in syn­chrony in this time exposure of a noctur­nal mating display. Each insect bas itsown rhythm, but the sight of its neigh·bors' lights brings that rhythm into har­mony with those around it. Such cou­plings among oscillators are at the heartof a wide variety of natural phenomena.

A:ingle oscillator traces out a sim­ple path in phase space. Whentwo or more oscillators are cou­

pled, however, the range of possible be­haviors becomes much more complex.The equations governing their behaviortend to become intractable. Each oscil­lator may be coupled only to a few im­mediate neighbors-as are the neuro­muscular oscillators in the smali intes­tine-or it could be coupled to ali theoscillators in an enormous communi­ty. The situation mathematidans find

are espedaliy conspicuous in livingthings: pacemaker cells in the heart; in­sulin-secreting cells in the pancreas; andneural networks in the brain and spinalcord that control such rhythmic behav­iors as breathing, running and che\ving.indeed, not all the oscillators need beconfined to the same organism: consid­er crickets that chirp in unison and con­gregations of synchronously flashingfireflies (see "Synchronous Fireflies," byJohn and 8isabeth Buck; SCIENTIFICAMERICAN, May 19761.

Since about 1960, mathematical bi­ologists have been studying simplifiedmodels of coupled oscillators that re­tain the essence of their biological proto­types. During the past few years, theyhave made rapid progress, thanks tobreakthroughs in computers and com­puter graphics, collaborations with ex­perimentalists who are open to theory,ideas borrowed from physics and newdevelopments in mathematics itself.

To understand how coupled oscilla­tors work together, one must first un­derstand how one oscillator works byitself. An oscillator is any system thatexecutes periodic behavior. A swingingpendulum, for example, returns to thesame point in space at regular inter­vals; furthermore, its velodty also risesand falis with (clockwork) regularity.

Instead of just considering an oscil­lator's behavior over time, mathemati­dans are interested in its motion throughphase space. Phase space is an abstract.space whose coordinates describe thestate of the system. The motion of apendulum in phase space, for instance,would be drawn by releasing the pen­dulum at various heights and then plot­ting its position and velocity. These tra­jectories in phase space turn out to beclosed curves, because the pendulum,like any other oscillator, repeats thesame motions over and over again.

A simple pendulum consisting of a

Coupled Oscillatorsand Biological Synchronization

102 SCIENTIFIC AMERICAN December 1993

STEVEN H. STROGATZ and IAN STEW·ART work in the middle ground betweenpure and applied mathematics. studyingsuch subjects as chaos and biological os­cillators. Strogatz is associate professorof applied mathematics at the Massachu­setts Institute of Technology. He earnedhis doctorate at HaIVard University witha thesis on mathematical models of hu­man sleep-Wake cydes and received hisbachelor's degree from Princeton Univer­sity. Stewart is professor of mathematicsand director of the InterdisdplinaIy Math·ematical Research Programme at the Uni·versity of Warwick in England. He haspublished more than 60 books, includ­ing three mathematical comic books inFrench, and writes the bimontWy "~'Iath­

cmatical Recreations" column for Scien­tifiC American.

I n February 1665 the great Dutchphysidst Christiaan Huygens, inven­tor of the pendulum clock, was con­

fined to his room by a minor illness.One day, with nothing better to do, hestared aimlessly at two clocks he badrecently built, which were hanging sideby side. Suddenly he noticed somethingodd: the two pendulums were swingingin perfect synchrony.

He watched them for hours, yet theynever broke step. Then he tried disturb­ing them-within half an hour they re­gained synchrony. Huygens suspectedthat the clocks must somehow be influ­endng each other, perhaps through tinyair movements or imperceptible vibra·tions in their common support. Sureenough, when he moved them to oppo­site sides of the room, the clocks grad­ualiy fell out of step, one losing five sec­onds a day relative to the other.

Huygens's fortuitous observation ini­tiated an entire subbranch of mathe­matics: the theory of coupled oscilla­tors. Coupled oscillators can be foundthroughout the natural world, but they

EO I £66l "oqWiJJaa NV:mmw :lIJI.I.N3DS

IO~ SC!ENTlnC AMERICAN December 1993

r

age drops instantly to zero-this pat­tern minlics the firing of a pacemakercell and its subsequent return to base­line. Then the voltage starts rising again,and the cycle begins anew.

A ilistinctive feature of Peskin's mod­el is its physiologically plausible formof pulse coupling. Each oscillator affectsthe others orily when it fires. It kickstheir vollage up by a fixed amount; ifany cell's voltage exceeds the thresh­old, it fires immeiliately. With these nilesin place, Peskin stated two provocativeconjectures: first, the system would al­ways eventually become synchronized;second, it would synchronize even ifthe oscillators were not quite identical.

When he tried to prove his conjec­tures, Peskin ran into technical road­blocks. There were no established math­ematical procedures for handling arbi­trarily large systems of oscillators. Sohe backed off and focused on the sim­plest possible case: two identical oscilla­tors. Even here the mathematics wasthorny. He restricted the problem fur­ther by allowing orily infinitesimal kicksand infinitesimal leakage through theresistor. Now the problem became man­ageable-ror this spedal case, he provedhis first conjecture.

Peskin's proof relies on an idea intro­duced by Henri Poincare, a virtuosoFrench mathematician who lived in theearly 1900s. Poincare's concept is themathematical equivalent of strobo­scopic photography, Take two identicalpulse-coupled oscillators, A and B, and

attached. Dutch physicist Christiaan Huygens invented thependulwn clock and was the first to observe this phenom­enon, inaugurating the study of coupled oscillators.

to handle mathematically because it in­troduces discontinuous behavior intoan otherwise continuous model and sostymies most of the standard mathe­matical techniques.

Recently one of us (StTOgatz), along\I;th Renato E. Mirollo of Boston Col­lege, created an idealized mathematicalmodel of fireflies and other pulse-cou­pled oscillator systems. We proved thatunder certain circumstances, oscillatorsstarted at different times will always be­come synchronized Isee "8ectronic Fire­flies," by Wayne Garver and Frank Moss,"The Amateur Scientist," page 1281.

Our work was inspired by an earlierstudy by Charles S. Peskin of New YorkUniversity. In 1975 Peskin proposed ahighiy schematic model of the heart'snatural pacemaker, a cluster of about10,000 cells called the sinoatrial node.He hoped to answer the question of howthese cells synchronize their indMdualelectrical rh\'!hms to generate a normalheartbeat.

Peskin modeled the pacemaker as alarge number of identical oscillators,each coupled equally strongly to all theothers. Each oscillator is based on anelectrical circuit consisting of a capaci­tor in parallel with a resistor. A constantinput current causes the voltage acrossthe capacitor to increase steadily. As thevoltage rises, the amount of current pas­sing through the resistor increases, andso the rate of increase slows down.When the voltage reaches a threshold,the capacitor discharges, and the volt-

PENDULUM CLOCKS placed near each other soon becomesynchronized (above) by tiny coupling forces transmittedthrough the air or by vibrations in the wall to which they are

easiest to describe arises when each os­cillator affects all the others in the sys­tem and the force of the coupling in­creases with the phase difference be­t'veen the oscillators. In this case, theinteraction between two oscillators thatare moving in synchrony is minimal.

Indeed, synchrony is the most famil­iar mode of organization for coupledoscillators. One of the most spectacu­lar examples of this kind of couplingcan be seen along the tidal rivers of Ma­laysia, Thailand and ew Guinea, wherethousands of male fireflies gather intrees at night and flash on and off inunison in an atlempt to attract the fe­males that cruise overhead. When themales arrive at dusk, their llickerings areuncoordinated. As the night deepens,pockets of synchrony begin to emergeand grow. Eventually whole trees puJ­sate in a silent, hypnotic concert thatcontinues for hours.

Curiously, even though the fireflies'display demonstrates coupled oscilla­tion on a grand scale, the details of thisbehavior have long resisted mathemat­ical analysis. Fireflies are a paradigm ofa "pulse coupled" oscillator system: tlleyinteract only when onc sees the suddenflash of another and shifts its rhythmaccordingly. Pulse coupling is commonin biology-consider crickets chirpingor neurons communicating via electri­cal spikes called action potentials-butthe impulsive character of the couplinghas rarely been included in mathemati­cal models. Pulse coupling is awkward

SCIENTIFIC AMERlCAN December 1993 105

POSITION

PHASE PORTRAIT

VELOCITY

eral description of the onset of oscilla­tion. He started by considering systemsthat have a rest point in phase space (asteady state) and seeing what happenedwhen one approximated their motionnear that point by a simple linear func­tion. Equations describing certain sys­tems behave in a peculiar fashion asthe system is driven away from its restpoint. Instead of either returning slow­ly to equilibrium or moving rapidly out·ward into instability, they oscillate. Thepoint at \vhich this transition takes placeis termed a bifurcation because the sys­tem's behavior splits into two branch­es-an unstable rest state coexists witha stable oscillation_ Hopf proved thatsystems whose linearized form under­goes this type of bifurcation are Iimit­cycle oscillators: they have a preferredwaveform and amplitude. Stewart andGolubitsk'Y showed that Hopf's idea canbe c"'<:tended to systems of coupled iden­tical oscillators, whose states undergobifurcations to produce standard pat­terns of phase locking.

For example, three identical oscillatorscoupled in a ring can be phase-lockedin four basic patterns. All oscillators canmove synchronously; successive oscilla­tors around the ring can move so thattheir phases differ by one third; two os­cillators can move synchronously whilethe third moves in an unrelated manner(except that it oscillates with the sameperiod as the others); and two oscilla­tors may be moving half a phase out ofstep, while the third oscillates t\\oce asrapidly as its neighbors.

The strange half-period oscillationsthat occur in the fourth pattern were a

PERJODIC MOTION can be representedin tenns of a time series or a phase por­trait. The phase portrait combines posi­tion and velocity, thus showing the entirerange of states that a system can dis­play. Any system that undergoes peri­odic behavior, no matter how complex,will eventually trace out a closed curvein phase space.

TIME SERIES

:~'POSITION:.~ I II I I II I I J

I I I 1~---{,~ '~'---""r-TIME

I I I II I I II I J

I I I '"I I I VELOCITY I

Synchrony is the most obvious caseor a general effect called phaselocking: many oscillators tracing

out the same pattern but not necessari­ly in step. When two identical osciJIatorsare coupled, there are exactly two pos­sibilities: synchrony, a phase differenceof zero, and antisynchrony, a phase dif­ference of one half. For example, whena kangaroo hops across the Australianoutback, its powerful hind legs oscillateperiodically, and both hit the ground atthe same instant. When a human nmsafter the kangaroo, meanwhile, his legshit the f:,'Tound alternately. If the nehvorkhas more than t\vo oscillators, the rangeof possibilities increases. In 1985 one ofus (Stewart), in collaboration with Mar­tin Golubitsl,y of the University of Hous­ton, developed a mathematical classifi­cation of the patterns of networks ofcoupled oscillators, follOWing earlierwork by James C. Alexander of the Uni­versity of Maryland and Ciles Auchmu­ty of the University of Houston.

The classification arises from grouptheory (which deals "oth symmerries ina collection of objects) combined withHopf bifurcation (a generalized descrip­tion of how oscillators "switch on"). In1942 Eberhard Hopf established a gen-

it is not inevilable. Indeed, coupled os­dilators often fail to synchronize. Theexplanation is a phenomenon knownas s)"lnmetry breaking, in which a singlesymmetric state-such as synchrony­is replaced by several less symmetricstates that together embody the origi­nal symmetry. Coupled oscillators area rich source of symmetry breaking.

chart their evolution by taking a snap­shot every time A fires.

What does the series of snapshotslook l.ike? A has just fired, so it alwaysappears at zero voltage. The voltage of·B, in contrast, changes from one snap­shot to the next. By solving his circuitequations, Peskin found an explicit butmessy formula for the change in B'svoltage between snapshots. The formu­la revealed that if the voltage is lessthan a certain critical value, it will de­crease until it reaches zero, whereas ifit is larger, it will increase. In either case,B \vill eventually end up synchronizedwith A.

There is one exception: if B's volt­age is precisely equal to the critical volt­age, then it can be driven neither up nordown and so stays poised at critical­ity. The oscillators fire repeatedly abouthalf a cycle out of phase from eachother. But this equilibrium is unstable,like a pencil balancing on its point. Theslightest nudge tips the system tQ\vardsynchrony.

Despite Peskin's successful analysisof the two-oscillator case, the case ofan arbitrary number of oscillators elud­ed proof for about 15 years. In 1989Strogatz learned of Peskin's work in abook on biological oscillators by ArthurT. Winfree of the University of Arizona.To gain intuition about the behavior ofPeskin's model, Strogatz wrote a com­puter program to simulate it for anynumber of identical oscillators, for anykick size and for any amount of leakage.The results were unambiguous: the sys·tern always ended up firing in unison.

Excited by the computer results, Stro­gatz discussed the problem with Mitollo.They reviewed Peskin's proof of the two­oscillatar case and noticed that it couldbe clarified by using a more abstractmodel for the individual oscillators. Thekey feature of the model turned out tabe the slmving upward curve of voltage(or its equivalent) as it rose toward thefiring threshold. Other characteristicswere unimportant.

Mitollo and Strogatz proved that theirgeneralized system always becomessynchronized, for any number of oscil­lators and for almost all initial condi­tions. The proof is based on the notionof "absorption"-a shorthand for theidea that if one oscillator kicks anotherover threshold, they "oil remain syn­chronized forever. They have identicaldynamics, after all, and are identicallycoupled to all the others. The two wereable to show that a sequence of absorp­tions eventually locks all the oscillatorstogether.

Although synchrony is the simpleststate for coupled identical oscillators,

1

surprise at first, even to Stewart andGolubitsky, but in fact the pattern oc­curs in rcaltife. A person using a walk­ing stick moves in just this manner:right leg, stick, left leg, stick, repeat. Thethird osciIJator is, in a sense, driven bythe combined effects of the other two:every time one of them hits a peak, itgives the third a push. Because the firsttwo oscillators are predsely antisynchro­nallS, the third oscillator peaks twicewhile the others each peak once.

The theory of symmetrical Hopf bi­furcation makes it possible to classifythe patterns of phase locking for manydifferent networks of coupled oscilla­tors. Indeed, Stewart, in collaborationwith James j. Collins, a biomedical en­gineer at Boston University, has beeninvestigating the striking analogies be­tween these patterns of phase lockingand the symmetries of animal gaits,such as the trot, pace and gallop.

Quadruped gaits closely resemble thenatural patterns of four· oscillator sys­tems. When a rabbit bounds, for exam­ple, it moves its front legs together,then its back legs. There is a phase dif­ference of zero bet\veen the two frontlegs and of one half between the frontand back legs. The pace of a giraffe issimilar, but the front and rear legs oneach side are the ones that move to­gether. When a horse trots, the lockingoccurs in diagonal fashion. An amblingelephant lifts each foot in turn, ,,1thphase differences of one quarter at eachstage. And young gazelles complete thesymmetry group with the prank, a four­legged leap in which all legs move insynchrony [see "Mathematical Recrea­tions," by Ian Stewart; SClEN1l1'IC AMER­ICAN, April 1991).

More recently, Stewart and Collinshave extended their analysis to thehexapod motion of insects. The tripod

SYMMETRY BREAKING governs the waysthat coupled oscillators can behave. Syn­chrony is the most symmetrical singlestate, but as the strength of the cou­pling between oscillators changes, otherstates may appear. Two oscillators cancouple in either synchronous or antisyn­chronous fashion (a, b), correspondingroughly to the bipedal locomotion of akangaroo or a person. Three oscillatorscan couple in four ways: synchrony (e),each one third of a cycle out of phasewith the others (d), two synchronousand one with an unrelated phase (e) orin the peculiar rhythm of two oscilla­tors antisyncbronous and the third run­ning twice as fast (n. This pattern is alsothe gait of a person walking slowly withthe aid of a stick.

/

b TWO OUT OF SYNCHRONY

a TWO IN SYNCHRONY

d THREE ONE THIRD OUT OF PHASE

C THREE IN SYNCHRONY

e TWO IN SYNCHRONY AND ONE WILD

f TWO OUT OF SYNCHRONY AND ONE TWiCE AS FAST

106 SCIENTIIX AMERICAN December 1993

d

close to their limit cycles at all times.This insight allowed him to ignore var­iations in amplitude and to consideronly their variations in phase. To incor­porate differences among the oscillators,Winfree made a model that captured theessence of an oscillator community byassuming that their natural frequenciesare distributed according to a narrowprobability function and that in otherrespects the oscillators are identical. Ina final and crucial simplification, he as­sumed that each oscillator is influencedonly by the collective rhythm producedby all the others. In the case of fireflies,for example, this would mean that eachfirefly responds to the collective nashof the whole population rather than toany individual firefly.

To visualize Winfree's model, imag­ine a swarm of dots running around acircle. The dots represent the phases ofthe oscillators, and the circle representstheir common limit cycle. If the oscilla­tors were independent, all the dots wouldeventually disperse over the circle, andthe collective rhythm would decay tozero. Incoherence reigns. A simple rulefor interaction among oscillators can re­store coherence, however: if an oscilla­tor is ahead of the group, it slows downa bit; if it is behind, it speeds up.

In some cases, this corrective cou­pling can overcome the differences innatural frequency; in others (such asthat of Gonyaulax), it cannot. Winfreefound that the system's behavior de­pends on the width of the frequencydistribution. If the spread of frequenciesis large compared ",th the coupling, thesystem always lapses into incoherence,just as if it were not coupled at all. Asthe spread decreases below a criticalvalue, part of the system spontaneous­ly "freezes" into synchrony.

Synchronization emerges coopera­tively. If a few oscillators happen tosynchronize, their combined, coherentsignal rises above the background din,exerting a stronger effect on the others.

ahead, and the slower ones fall behind (h, c). A simple cou­pling force that speeds up slower oscillators and slows downfaster ones, however, can keep them all in phase (d).

c

SCIENTtFlC AMEIUCAN December 1993 107

cies depends on the strength of thecoupling among them. If their inter­actions are too weak, the oscillatorswill be unable to achieve synchrony. Theresult is incoherence, a cacophony ofoscillations. Even if started in unisonthe oscillators will gradually drift outof phase, as did Huygens's pendulumclocks when placed at opposite ends ofthe room.

Colonies of the bioluminescent algaeGonyaulax demonstrate just this kindof desynchronization. J. Woodland Hast­ings and his colleagues at Harvard Uni­versity have found that if a tank full ofGonyaulax is kept in constant dim lightin a laboratory, it exhibits a circadianglow rhythm with a period close to 23hours. As time goes by, the waveformbroadens, and this rhythm graduallydamps out. It appears that the individualcells continue to oscillate, but they driftout of phase because of differences intheir naturai frequencies. The glow ofthe algae themselves does not maintainsynchrony in the absence of light fromthe sun.

In other oscillator communities thecoupling is strong enough to overcomethe inevitable differences in natural fre­quency. Polymath Norbert Wiener point­ed out in the late 1950s that such os­cillator communities are ubiquitous inbiology and indeed in all of nature.Wiener tried to develop a mathematicalmodel of collections of oscillators, buthis approach has not turned out to befn.tltful. The theoretical breakthroughcame in 1966, when Winfree, then agraduate student at Princeton Univer­Sity, began exploring the behavior oflarge populations of limit-cycle oscil­lators. He used an inspired combina­tion of computer simulations, mathe­matical analysis and experiments on anarray of 71 electrically coupled neon­tube oscillators.

Winfree simplified the problem tre­mendously by pointing out that if oscil­lators are weakJy coupled, they remain

ba

gait of a coclcroach is a very stable pat­tern in a ring of six oscillators. A trian­gle of legs moves in synchrony: frontand back left and middle right; thenthe other three legs are lifted with aphase difference nf nne half.

Why do gaits resemble the naturalpatterns of coupled nscillatnrs in thisway? The mechanical design of animallimbs is unlikely to be the primary rea­son. Limbs are nor passive mechanicaloscillators but rather complex systemsof bone and muscle controlled by equal­ly complicated nerve assemblies. Themostllkely source of this concordancebetween nature and mathematics is inthe architecture of the circuits in thenervnus system that control locomotion.Biologists have long hypothesized theexistence of networks of coupled neu­rons they call central pattern genera­tors, but the hypothesis has always beencontroversial. Nevertheless, neurons of­ten act as oscillators, and so, if centralpattern generators exist, it is reason­able to expect their dynamics to resem­ble those of an oscillator network.

Moreover, symmetry analysis solvesa significant problem in the central-pat­tern generator hypothesis. Most ani­mals employ several gaits-horses walk,trot, canter and gallop-and biologistshave often assumed that each gait re­quires a separate pattern generator.Symmetry breaking, however, impliesthat the same central-pattern generatorcircuit can produce all of an animal'sgaits. Only the strength of the couplingsamong neural oscillators need vary.

so far our analysis has been lim­ited to collections of oscillatorsthat are all strictly identical. That

idealization is convenient mathemat­ically, but it ignores the diversity thatis always present in biology. In any realpopulation, some oscillators will alwaysbe inherently faster or slower. The be­havior of communities of oscillatorswhose members have differing frequen-

NONIDENTICAL OSCIllATORS may start out in phase with oneanother (as shown on circle a, in which 360 degrees mark oneoscillation), but they lose coherence as the faster Does move

120

of an alignment of molecuies or elec­tronic spins in space.

The analogy to phase transitionsopened a new chapter in statistical me­chanics, the study of systems composedof enormous numbers of interactingsubunits, In 1975 Yoshiki Kuramoto ofKyoto University presented an elegantreformulation of Winfree's model, Ku­ramoto's model has a simpler mathe­matical slructure that allows it to be an­alyzed in great detail. Recently Strogatz,along with Mirella and Paul C. Matthewsof the University of Cambridge, foundan unexpected connection bet\veen Ku­ramoto's model and Landau damping,a puzzling phenomenon that arises inplasma physics when electrostatic wavespropagate through a highly rarefied me­dium. The connection emerged when we

24

Wben additional oscillators are pulledinto the synchronized nudeus, they am­plify its signal, This positive feedbackleads to an accelerating outbreak of syn­chrony. Some oscillators nonetheless re­main unsynchronized because their fre­quencies are too far from the value atwhich the others have synchronized forthe coupling to pull them in.

In developing his description, Win­free discovered an unexpected link be­tween biology and pbysics. He saw thatmutual synchronization is strikinglyanalogous to a phase transition suchas the freeZing of water or the sponta­neous magnetization of a ferromagnet.The width of the oscillators' frequencydistribution plays the same role as doestemperature, and the alignment of oscil­lator phases in time is the counterpart

~ 72 00TIME IN CONSTANT CONDITIONS (HOURS)

GONYAUlAX luminescent algae (top) change the intensity of their glow accordingto an internal clock that is affected by light. If they are kept in constant dim ligbt,the timing of the glow becomes less precise because the coupling among individu­al organisms is insufficient to keep them in sync (bottom)_

A sceptical appraisal ofHawking's bestselling

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RJRTIlER READING

THE TIMING OF BIOLOGICAL CLOCKS.Arthur T. Winfree. SCientific Americanlibrary, 1987.

fROM Q.OCKS TO CHAOS: 1H:E RHYI1-1MSor liFE. Leon Glass and Michael C.Mackey. Princeton University Press,1988.

SYNCHRONIZATION OF PuLsE-COUPL£DBIOLOGICAL OSCILLATORS. Renata E.MiraDa and Steven H. Strogatz in SlAMJournal on Applied Machematics, Val.50, No.6, pages 1645-1662; Decem­ber t990.

COuPL£D ONUNEAR OSCIllATORS BE­LOW TIlE SYNCHRONlZATION 1)-1RESH­OW: RElAxATION BY GENERAUZEDlANDAU DAMPING. Steven H. Strogatz,Renata E. MirolJa and Paul C. Matthewsin Physical Review Leners, Vol. 68, No.18, pages 2730-2733; May 4, 1992.

COUPlED NONUNF.AR OSClLLATORS AND

TIlE SYMMETR.IES OF ANIMAL GAITS.J. J. Collins and Ian Stewart in Journalof Nonlinear Science, Vol 3, No.3,pages 349-392; July 1993.

studied the decay to incoherence in os­cillator communities in which the fre­Quency distribution is too broad tosupport synchrony. The loss of coher­ence, it turns out, is governed by thesame mathematical mechanism as thatcontrolling the decay of waves in such"collisionless" plasmas.

T he theory of coupled osdllatorshas come a long way since Huy­gens noticed the spontaneous

synchronization of pendulum clocks.Synchronization, apparently a very nat­ural kind of behavior, turns out to beboth surprising and interesting. It isa problem to understand, which is notan obvious consequence of symmetry.Mathematidans have turned to the the­ory of symmetry breaking to classifythe general patterns that arise whenidentical, ostensibly symmetric oscilla­tors are coupled. Thus, a mathematicaldisdpline that has its most visible rootsin particle physics appears to governthe leap of a gazelle and the ambling ofan elephant. Meanwhile techniques bor­rowed from statistical mechanics illumi­nate the behavior of entire populationsof oscillators. It seems amazing thatthere should be a link between the vio­lent world of plasmas, where atoms rou­tinely have thelr electrons stripped off,and the peaceful world of biological os­dilators, where fireflies pulse silent­ly along a riverbank. Yet there is a co­herent mathematical thread that leadsfrom the simple pendulum to spatialpatterns, waves, chaos and phase tran­sitions. Such is the power of mathemat­ics to reveal the hidden unity of nature.

AT YOURNEWSSTAND

DECEMBER 28

COMINGIN THE

JANUARYISSUE...

Wetland Ecosystems in thelandscape: Fluctuating

Water levels, Biodiversityand Notional Policy

Cyanobacteria and TheirSecondary Metabolites

Breaking Intractability

Animal Sexuality

World languages andHuman Dispersal

TRENDS: CANCEREPIDEMIOLOGY

Timothy M. Beardsley

SEARCHING FORSTRANGE MAnER

Henry J. CrawfordLawrence Berkeley

LaboratoryCarsten Greiner

Duke University

THE FIRST DATANETWORKS

Gerard J. HolzmannAT&TBell Laboratories

Bjorn PehrsonThe Royo/lnstitute of

Technology, Stockholm

ALSO INJANUARY...

SCIENTIFICAMERICAN