Coupled Oscillators and Biological Synchronization A subtle mathematical thread connects clocks, ambling elephants, brain rhythms and the onset ofchaos

by Steven H. Strogatz and Ian Stewart

n February 1665 the great Dutch are espedaliy conspicuous in living weight at the end of a string can take physidst Christiaan Huygens, inven­ things: pacemaker cells in the heart; in­ any of an infinite number of closed I tor of the pendulum clock, was con­ sulin-secreting cells in the pancreas; and paths through phase space, depending fined to his room by a minor illness. neural networks in the brain and spinal on the height from which it is released. One day, with nothing better to do, he cord that control such rhythmic behav­ Biological systems (and clock pendu­ stared aimlessly at two clocks he bad iors as breathing, running and che\ving. lums), in contrast, tend to have not only recently built, which were hanging side indeed, not all the oscillators need be a characteristic period hut also a char­ by side. Suddenly he noticed something confined to the same organism: consid­ acteristic amplitude. They trace a par­ odd: the two pendulums were swinging er crickets that chirp in unison and con­ ticular path through phase space, and in perfect synchrony. gregations of synchronously flashing if some perturbation jolts them out of He watched them for hours, yet they fireflies (see "Synchronous Fireflies," by their accustomed rhythm they soon re­ never broke step. Then he tried disturb­ John and 8isabeth Buck; SCIENTIFIC turn to their former path. If someone ing them-within half an hour they re­ AMERICAN, May 19761. startles you, say, by shouting, "Boo!", gained synchrony. Huygens suspected Since about 1960, mathematical bi­ your heart may start pounding but soon that the clocks must somehow be influ­ ologists have been studying simplified relaxes to its normal behavior. endng each other, perhaps through tiny models of coupled oscillators that re­ Oscillators that have a standard wave­ air movements or imperceptible vibra· tain the essence of their biological proto­ form and amplitude to which they re­ tions in their common support. Sure types. During the past few years, they turn after small pertUIbations are known enough, when he moved them to oppo­ have made rapid progress, thanks to as limit-eycle oscillators. They incorpo­ site sides of the room, the clocks grad­ breakthroughs in computers and com­ rate a dissipative mechanism to damp ualiy fell out of step, one losing five sec­ puter graphics, collaborations with ex­ oscillations that grow too large and a onds a day relative to the other. perimentalists who are open to theory, source of energy to pump up those that Huygens's fortuitous observation ini­ ideas borrowed from physics and new become too smali. tiated an entire subbranch of mathe­ developments in mathematics itself. matics: the theory of coupled oscilla­ To understand how coupled oscilla­ ingle oscillator traces out a sim­ tors. Coupled oscillators can be found tors work together, one must first un­ ple path in phase space. When throughout the natural world, but they derstand how one oscillator works by A: two or more oscillators are cou­ itself. An oscillator is any system that pled, however, the range of possible be­ executes periodic behavior. A swinging haviors becomes much more complex. pendulum, for example, returns to the The equations governing their behavior STEVEN H. STROGATZ and IAN STEW· ART work in the middle ground between same point in space at regular inter­ tend to become intractable. Each oscil­ pure and applied mathematics. studying vals; furthermore, its velodty also rises lator may be coupled only to a few im­ such subjects as chaos and biological os­ and falis with (clockwork) regularity. mediate neighbors-as are the neuro­ cillators. Strogatz is associate professor Instead of just considering an oscil­ muscular oscillators in the smali intes­ of applied mathematics at the Massachu­ lator's behavior over time, mathemati­ tine-or it could be coupled to ali the setts Institute of Technology. He earned dans are interested in its motion through oscillators in an enormous communi­ his doctorate at HaIVard University with phase space. Phase space is an abstract. ty. The situation mathematidans find a thesis on mathematical models of hu­ space whose coordinates describe the man sleep-Wake cydes and received his bachelor's degree from Princeton Univer­ state of the system. The motion of a sity. Stewart is professor of mathematics pendulum in phase space, for instance, would be drawn by releasing the pen­ and director of the InterdisdplinaIy Math· THOUSANDS OF FIREFllES Dash in syn­ ematical Research Programme at the Uni· dulum at various heights and then plot­ chrony in this time exposure ofa noctur­ versity of Warwick in England. He has ting its position and velocity. These tra­ published more than 60 books, includ­ nal mating display. Each insect bas its jectories in phase space turn out to be own rhythm, but the sight of its neigh· ing three mathematical comic books in closed curves, because the pendulum, French, and writes the bimontWy "~'Iath­ bors' lights brings that rhythm into har­ like any other oscillator, repeats the cmatical Recreations" column for Scien­ mony with those around it. Such cou­ tifiC American. same motions over and over again. plings among oscillators are at the heart A simple pendulum consisting of a ofa wide variety ofnatural phenomena.

102 SCIENTIFIC AMERICAN December 1993 EO I £66l "oqWiJJaa NV:mmw :lIJI.I.N3DS PENDULUM CLOCKS placed near each other soon become attached. Dutch physicist Christiaan Huygens invented the synchronized (above) by tiny coupling forces transmitted pendulwn clock and was the first to observe this phenom­ through the air or by vibrations in the wall to which they are enon, inaugurating the study of coupled oscillators. easiest to describe arises when each os­ to handle mathematically because it in­ age drops instantly to zero-this pat­ cillator affects all the others in the sys­ troduces discontinuous behavior into tern minlics the firing of a pacemaker tem and the force of the coupling in­ an otherwise continuous model and so cell and its subsequent return to base­ creases with the phase difference be­ stymies most of the standard mathe­ line. Then the voltage starts rising again, t'veen the oscillators. In this case, the matical techniques. and the cycle begins anew. interaction between two oscillators that Recently one of us (StTOgatz), along A ilistinctive feature of Peskin's mod­ are moving in synchrony is minimal. \I;th Renato E. Mirollo of Boston Col­ el is its physiologically plausible form Indeed, synchrony is the most famil­ lege, created an idealized mathematical of pulse coupling. Each oscillator affects iar mode of organization for coupled model of fireflies and other pulse-cou­ the others orily when it fires. It kicks oscillators. One of the most spectacu­ pled oscillator systems. We proved that their vollage up by a fixed amount; if lar examples of this kind of coupling under certain circumstances, oscillators any cell's voltage exceeds the thresh­ can be seen along the tidal rivers of Ma­ started at different times will always be­ old, it fires immeiliately. With these niles laysia, Thailand and ew Guinea, where come synchronized Isee "8ectronic Fire­ in place, Peskin stated two provocative thousands of male fireflies gather in flies," by Wayne Garver and Frank Moss, conjectures: first, the system would al­ trees at night and flash on and off in "The Amateur Scientist," page 1281. ways eventually become synchronized; unison in an atlempt to attract the fe­ Our work was inspired by an earlier second, it would synchronize even if males that cruise overhead. When the study by Charles S. Peskin of New York the oscillators were not quite identical. males arrive at dusk, their llickerings are University. In 1975 Peskin proposed a When he tried to prove his conjec­ uncoordinated. As the night deepens, highiy schematic model of the heart's tures, Peskin ran into technical road­ pockets of synchrony begin to emerge natural pacemaker, a cluster of about blocks. There were no established math­ and grow. Eventually whole trees puJ­ 10,000 cells called the sinoatrial node. ematical procedures for handling arbi­ sate in a silent, hypnotic concert that He hoped to answer the question of how trarily large systems of oscillators. So continues for hours. these cells synchronize their indMdual he backed off and focused on the sim­ Curiously, even though the fireflies' electrical rh\'!hms to generate a normal plest possible case: two identical oscilla­ display demonstrates coupled oscilla­ heartbeat. tors. Even here the mathematics was tion on a grand scale, the details of this Peskin modeled the pacemaker as a thorny. He restricted the problem fur­ behavior have long resisted mathemat­ large number of identical oscillators, ther by allowing orily infinitesimal kicks ical analysis. Fireflies are a paradigm of each coupled equally strongly to all the and infinitesimal leakage through the a "pulse coupled" oscillator system: tlley others. Each oscillator is based on an resistor. Now the problem became man­ interact only when onc sees the sudden electrical circuit consisting of a capaci­ ageable-ror this spedal case, he proved flash of another and shifts its rhythm tor in parallel with a resistor. A constant his first conjecture. accordingly. Pulse coupling is common input current causes the voltage across Peskin's proof relies on an idea intro­ in biology-consider crickets chirping the capacitor to increase steadily. As the duced by Henri Poincare, a virtuoso or neurons communicating via electri­ voltage rises, the amount of current pas­ French mathematician who lived in the cal spikes called action potentials-but sing through the resistor increases, and early 1900s. Poincare's concept is the the impulsive character of the coupling so the rate of increase slows down. mathematical equivalent of strobo­ has rarely been included in mathemati­ When the voltage reaches a threshold, scopic photography, Take two identical cal models. Pulse coupling is awkward the capacitor discharges, and the volt- pulse-coupled oscillators, A and B, and

I O~ SC!ENTlnC AMERICAN December 1993

r chart their evolution by taking a snap­ it is not inevilable. Indeed, coupled os­ eral description of the onset of oscilla­ shot every time A fires. dilators often fail to synchronize. The tion. He started by considering systems What does the series of snapshots explanation is a phenomenon known that have a rest point in phase space (a look l.ike? A has just fired, so it always as s)"lnmetry breaking, in which a single steady state) and seeing what happened appears at zero voltage. The voltage of· symmetric state-such as synchrony­ when one approximated their motion B, in contrast, changes from one snap­ is replaced by several less symmetric near that point by a simple linear func­ shot to the next. By solving his circuit states that together embody the origi­ tion. Equations describing certain sys­ equations, Peskin found an explicit but nal symmetry. Coupled oscillators are tems behave in a peculiar fashion as messy formula for the change in B's a rich source of symmetry breaking. the system is driven away from its rest voltage between snapshots. The formu­ point. Instead of either returning slow­ la revealed that if the voltage is less ynchrony is the most obvious case ly to equilibrium or moving rapidly out· than a certain critical value, it will de­ or a general effect called phase ward into instability, they oscillate. The crease until it reaches zero, whereas if Slocking: many oscillators tracing point at \vhich this transition takes place it is larger, it will increase. In either case, out the same pattern but not necessari­ is termed a bifurcation because the sys­ B \vill eventually end up synchronized ly in step. When two identical osciJIators tem's behavior splits into two branch­ with A. are coupled, there are exactly two pos­ es-an unstable rest state coexists with There is one exception: if B's volt­ sibilities: synchrony, a phase difference a stable oscillation_ Hopf proved that age is precisely equal to the critical volt­ of zero, and antisynchrony, a phase dif­ systems whose linearized form under­ age, then it can be driven neither up nor ference of one half. For example, when goes this type of bifurcation are Iimit­ down and so stays poised at critical­ a kangaroo hops across the Australian cycle oscillators: they have a preferred ity. The oscillators fire repeatedly about outback, its powerful hind legs oscillate waveform and amplitude. Stewart and half a cycle out of phase from each periodically, and both hit the ground at Golubitsk'Y showed that Hopf's idea can other. But this equilibrium is unstable, the same instant. When a human nms be c"'<:tended to systems of coupled iden­ like a pencil balancing on its point. The after the kangaroo, meanwhile, his legs tical oscillators, whose states undergo slightest nudge tips the system tQ\vard hit the f:,'Tound alternately. If the nehvork bifurcations to produce standard pat­ synchrony. has more than t\vo oscillators, the range terns of phase locking. Despite Peskin's successful analysis of possibilities increases. In 1985 one of For example, three identical oscillators of the two-oscillator case, the case of us (Stewart), in collaboration with Mar­ coupled in a ring can be phase-locked an arbitrary number of oscillators elud­ tin Golubitsl,y of the University of Hous­ in four basic patterns. All oscillators can ed proof for about 15 years. In 1989 ton, developed a mathematical classifi­ move synchronously; successive oscilla­ Strogatz learned of Peskin's work in a cation of the patterns of networks of tors around the ring can move so that book on biological oscillators by Arthur coupled oscillators, follOWing earlier their phases differ by one third; two os­ T. Winfree of the University of Arizona. work by James C. Alexander of the Uni­ cillators can move synchronously while To gain intuition about the behavior of versity of Maryland and Ciles Auchmu­ the third moves in an unrelated manner Peskin's model, Strogatz wrote a com­ ty of the University of Houston. (except that it oscillates with the same puter program to simulate it for any The classification arises from group period as the others); and two oscilla­ number of identical oscillators, for any theory (which deals "oth symmerries in tors may be moving half a phase out of kick size and for any amount of leakage. a collection of objects) combined with step, while the third oscillates t\\oce as The results were unambiguous: the sys· Hopf bifurcation (a generalized descrip­ rapidly as its neighbors. tern always ended up firing in unison. tion of how oscillators "switch on"). In The strange half-period oscillations Excited by the computer results, Stro­ 1942 Eberhard Hopf established a gen- that occur in the fourth pattern were a gatz discussed the problem with Mitollo. They reviewed Peskin's proof of the two­ TIME SERIES oscillatar case and noticed that it could be clarified by using a more abstract :~'POSITION: model for the individual oscillators. The .~ I I key feature of the model turned out ta I I I I be the slmving upward curve of voltage I I I J (or its equivalent) as it rose toward the ~---{,~I I I'~'---""r-TIME1 firing threshold. Other characteristics I I I I were unimportant. I I I I Mitollo and Strogatz proved that their I I J I I I '" generalized system always becomes I I I VELOCITY I synchronized, for any number of oscil­ lators and for almost all initial condi­ tions. The proof is based on the notion PHASE PORTRAIT of "absorption"-a shorthand for the VELOCITY idea that if one oscillator kicks another over threshold, they "oil remain syn­ chronized forever. They have identical PERJODIC MOTION can be represented dynamics, after all, and are identically in tenns of a time series or a phase por­ trait. The phase portrait combines posi­ coupled to all the others. The two were tion and velocity, thus showing the entire POSITION able to show that a sequence of absorp­ range of states that a system can dis­ tions eventually locks all the oscillators play. Any system that undergoes peri­ together. odic behavior, no matter how complex, Although synchrony is the simplest will eventually trace out a closed curve state for coupled identical oscillators, in phase space.

SCIENTIFIC AMERlCAN December 1993 105 a TWO IN SYNCHRONY surprise at first, even to Stewart and Golubitsky, but in fact the pattern oc­ curs in rcaltife. A person using a walk­ ing stick moves in just this manner: right leg, stick, left leg, stick, repeat. The third osciIJator is, in a sense, driven by the combined effects of the other two: every time one of them hits a peak, it gives the third a push. Because the first two oscillators are predsely antisynchro­ nallS, the third oscillator peaks twice while the others each peak once. b TWO OUT OF SYNCHRONY The theory of symmetrical Hopf bi­ furcation makes it possible to classify the patterns of phase locking for many different networks of coupled oscilla­ tors. Indeed, Stewart, in collaboration with James j. Collins, a biomedical en­ gineer at Boston University, has been investigating the striking analogies be­ tween these patterns of phase locking and the symmetries of animal gaits, such as the trot, pace and gallop. Quadruped gaits closely resemble the natural patterns of four·oscillator sys­ tems. When a rabbit bounds, for exam­ ple, it moves its front legs together, C THREE IN SYNCHRONY then its back legs. There is a phase dif­ ference of zero bet\veen the two front legs and of one half between the front and back legs. The pace of a giraffe is similar, but the front and rear legs on each side are the ones that move to­ gether. When a horse trots, the locking d THREE ONE THIRD OUT OF PHASE occurs in diagonal fashion. An ambling elephant lifts each foot in turn, ,,1th phase differences of one quarter at each stage. And young gazelles complete the symmetry group with the prank, a four­ legged leap in which all legs move in synchrony [see "Mathematical Recrea­ / tions," by Ian Stewart; SClEN1l1'IC AMER­ ICAN, April 1991). More recently, Stewart and Collins e TWO IN SYNCHRONY AND ONE WILD have extended their analysis to the hexapod motion of insects. The tripod

SYMMETRY BREAKING governs the ways that coupled oscillators can behave. Syn­ chrony is the most symmetrical single f TWO OUT OF SYNCHRONY AND ONE TWiCE AS FAST state, but as the strength of the cou­ pling between oscillators changes, other states may appear. Two oscillators can couple in either synchronous or antisyn­ chronous fashion (a, b), corresponding roughly to the bipedal locomotion of a kangaroo or a person. Three oscillators can couple in four ways: synchrony (e), each one third of a cycle out of phase with the others (d), two synchronous and one with an unrelated phase (e) or in the peculiar rhythm of two oscilla­ tors antisyncbronous and the third run­ ning twice as fast (n. This pattern is also the gait of a person walking slowly with the aid of a stick.

106 SCIENTIIX AMERICAN December 1993

1 a b c d

NONIDENTICAL OSCIllATORS may start out in phase with one ahead, and the slower ones fall behind (h, c). A simple cou­ another (as shown on circle a, in which 360 degrees mark one pling force that speeds up slower oscillators and slows down oscillation), but they lose coherence as the faster Does move faster ones, however, can keep them all in phase (d). gait of a coclcroach is a very stable pat­ cies depends on the strength of the close to their limit cycles at all times. tern in a ring of six oscillators. A trian­ coupling among them. If their inter­ This insight allowed him to ignore var­ gle of legs moves in synchrony: front actions are too weak, the oscillators iations in amplitude and to consider and back left and middle right; then will be unable to achieve synchrony. The only their variations in phase. To incor­ the other three legs are lifted with a result is incoherence, a cacophony of porate differences among the oscillators, phase difference nf nne half. oscillations. Even if started in unison Winfree made a model that captured the Why do gaits resemble the natural the oscillators will gradually drift out essence of an oscillator community by patterns of coupled nscillatnrs in this of phase, as did Huygens's pendulum assuming that their natural frequencies way? The mechanical design of animal clocks when placed at opposite ends of are distributed according to a narrow limbs is unlikely to be the primary rea­ the room. probability function and that in other son. Limbs are nor passive mechanical Colonies of the bioluminescent algae respects the oscillators are identical. In oscillators but rather complex systems Gonyaulax demonstrate just this kind a final and crucial simplification, he as­ ofbone and muscle controlled by equal­ of desynchronization. J. Woodland Hast­ sumed that each oscillator is influenced ly complicated nerve assemblies. The ings and his colleagues at Harvard Uni­ only by the collective rhythm produced mostllkely source of this concordance versity have found that if a tank full of by all the others. In the case of fireflies, between nature and mathematics is in Gonyaulax is kept in constant dim light for example, this would mean that each the architecture of the circuits in the in a laboratory, it exhibits a circadian firefly responds to the collective nash nervnus system that control locomotion. glow rhythm with a period close to 23 of the whole population rather than to Biologists have long hypothesized the hours. As time goes by, the waveform any individual firefly. existence of networks of coupled neu­ broadens, and this rhythm gradually To visualize Winfree's model, imag­ rons they call central pattern genera­ damps out. It appears that the individual ine a swarm of dots running around a tors, but the hypothesis has always been cells continue to oscillate, but they drift circle. The dots represent the phases of controversial. Nevertheless, neurons of­ out of phase because of differences in the oscillators, and the circle represents ten act as oscillators, and so, if central their naturai frequencies. The glow of their common limit cycle. If the oscilla­ pattern generators exist, it is reason­ the algae themselves does not maintain tors were independent, all the dots would able to expect their dynamics to resem­ synchrony in the absence of light from eventually disperse over the circle, and ble those of an oscillator network. the sun. the collective rhythm would decay to Moreover, symmetry analysis solves In other oscillator communities the zero. Incoherence reigns. A simple rule a significant problem in the central-pat­ coupling is strong enough to overcome for interaction among oscillators can re­ tern generator hypothesis. Most ani­ the inevitable differences in natural fre­ store coherence, however: if an oscilla­ mals employ several gaits-horses walk, quency. Polymath Norbert Wiener point­ tor is ahead of the group, it slows down trot, canter and gallop-and biologists ed out in the late 1950s that such os­ a bit; if it is behind, it speeds up. have often assumed that each gait re­ cillator communities are ubiquitous in In some cases, this corrective cou­ quires a separate pattern generator. biology and indeed in all of nature. pling can overcome the differences in Symmetry breaking, however, implies Wiener tried to develop a mathematical natural frequency; in others (such as that the same central-pattern generator model of collections of oscillators, but that of Gonyaulax), it cannot. Winfree circuit can produce all of an animal's his approach has not turned out to be found that the system's behavior de­ gaits. Only the strength of the couplings fn.tltful. The theoretical breakthrough pends on the width of the frequency among neural oscillators need vary. came in 1966, when Winfree, then a distribution. If the spread of frequencies graduate student at Princeton Univer­ is large compared ",th the coupling, the o far our analysis has been lim­ Sity, began exploring the behavior of system always lapses into incoherence, ited to collections of oscillators large populations of limit-cycle oscil­ just as if it were not coupled at all. As sthat are all strictly identical. That lators. He used an inspired combina­ the spread decreases below a critical idealization is convenient mathemat­ tion of computer simulations, mathe­ value, part of the system spontaneous­ ically, but it ignores the diversity that matical analysis and experiments on an ly "freezes" into synchrony. is always present in biology. In any real array of 71 electrically coupled neon­ Synchronization emerges coopera­ population, some oscillators will always tube oscillators. tively. If a few oscillators happen to be inherently faster or slower. The be­ Winfree simplified the problem tre­ synchronize, their combined, coherent havior of communities of oscillators mendously by pointing out that if oscil­ signal rises above the background din, whose members have differing frequen- lators are weakJy coupled, they remain exerting a stronger effect on the others.

SCIENTtFlC AMEIUCAN December 1993 107 /1. -

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A sceptical appraisal of Hawking's bestselling A Brief History of Time revealing a major fallacy 24 ~ 72 00 120 TIME IN CONSTANT CONDITIONS (HOURS) GONYAUlAXluminescent algae (top) change the intensity of their glow according to an internal clock that is affected by light. Ifthey are kept in constant dim ligbt, /;1 the timing of the glow becomes less precise because the coupling among individu­ H4sLG al organisms is insufficient to keep them in sync (bottom)_ H4WKII'fG ERRED? Wben additional oscillators are pulled of an alignment of molecuies or elec­ into the synchronized nudeus, they am­ tronic spins in space. A SCepticol OPPl"O;sal of his bestseUing 'A Srief History plify its signal, This positive feedback The analogy to phase transitions of Time' • revealing Q major stil'nlifi( folrocy. leads to an accelerating outbreak of syn­ opened a new chapter in statistical me­ chrony. Some oscillators nonetheless re­ chanics, the study of systems composed GERHARD KRAUS main unsynchronized because their fre­ of enormous numbers of interacting Inl,oduCliOn by quencies are too far from the value at subunits, In 1975 Yoshiki Kuramoto of Profcuo, Jon Bocycn. which the others have synchronized for Kyoto University presented an elegant the coupling to pull them in. reformulation of Winfree's model, Ku­ 'There is a need for fresh In developing his description, Win­ ramoto's model has a simpler mathe­ free discovered an unexpected link be­ matical slructure that allows it to be an­ ideas and this work consists tween biology and pbysics. He saw that alyzed in great detail. Recently Strogatz, of precisely that.' mutual synchronization is strikingly along with Mirella and Paul C. Matthews PROFESSOR JAN BOEYENS analogous to a phase transition such of the University of Cambridge, found as the freeZing of water or the sponta­ an unexpected connection bet\veen Ku­ neous magnetization of a ferromagnet. ramoto's model and Landau damping, ~ The width of the oscillators' frequency a puzzling phenomenon that arises in distribution plays the same role as does plasma physics when electrostatic waves JANUS PUBLISHING COMPANY temperature, and the alignment of oscil­ propagate through a highly rarefied me­ £14.99 168pp Hardback Out Now lator phases in time is the counterpart dium. The connection emerged when we Available at all good bookshops studied the decay to incoherence in os­ cillator communities in which the fre­ Quency distribution is too broad to PUBLISH SCIENTIFIC support synchrony. The loss of coher­ ence, it turns out, is governed by the AMERICAN same mathematical mechanism as that controlling the decay of waves in such YOUR BOOK COMING "collisionless" plasmas. Since 1949 more than 15,000 authors have chosen the vantage IN THE he theory of coupled osdllators has come a long way since Huy­ Press subsidy publishing program. JANUARY T gens noticed the spontaneous synchronization of pendulum clocks. You are invited to send for a free illustrated ISSUE... Synchronization, apparently a very nat­ guidebook which explains how your book can ural kind of behavior, turns out to be be produced and promoted. Whether your sub­ THE FIRST DATA both surprising and interesting. It is ject is fiction, non· a problem to understand, which is not fiction or poetry, NETWORKS an obvious consequence of symmetry. scientific, scholar­ Gerard J. Holzmann Mathematidans have turned to the the­ Iy, specialized AT&TBell Laboratories ory of symmetry breaking to classify (even controver­ the general patterns that arise when sial), this hand­ Bjorn Pehrson identical, ostensibly symmetric oscilla­ The Royo/lnstitute of some 32-page tors are coupled. Thus, a mathematical brochure will show Technology, Stockholm disdpline that has its most visible roots you how to ar· in particle physics appears to govern range for prompt SEARCHING FOR the leap of a gazelle and the ambling of an elephant. Meanwhile techniques bor­ subsidy publica· STRANGE MAnER rowed from statistical mechanics illumi­ tion. Unpublished Henry J. Crawford nate the behavior of entire populations authors will find this booklet valuable and infor­ Lawrence Berkeley of oscillators. It seems amazing that mative. Foryourfree copy, write to: Laboratory there should be a link between the vio­ VANTAGE PRESS,lnc. Dept. F·53 lent world of plasmas, where atoms rou­ 516 W. 34th St., New York, N.Y. 10001 Carsten Greiner tinely have thelr electrons stripped off, Duke University and the peaceful world ofbiological os­ dilators, where fireflies pulse silent­ TRENDS: CANCER ly along a riverbank. Yet there is a co­ herent mathematical thread that leads EPIDEMIOLOGY from the simple pendulum to spatial Want to Timothy M. Beardsley patterns, waves, chaos and phase tran­ brush upon sitions. Such is the power of mathemat­ ics to reveal the hidden unity of nature. a foreign ALSO IN language? JANUARY... RJRTIlER READING With Audio·Forum's THE TIMING OF BIOLOGICAL CLOCKS. intennediate and Arthur T. Winfree. SCientific American advanced materials, it's Wetland Ecosystems in the library, 1987. easy to maintain and landscape: Fluctuating fROM Q.OCKS TO CHAOS: 1H:E RHYI1-1MS sharpen your foreign-language skills. Water levels, Biodiversity or liFE. Leon Glass and Michael C. Besides intenncdiate and advanced Mackey. Princeton University Press, audio·cassctte courses-most developed and Notional Policy 1988. for the U.S. State Department-we ofTer SYNCHRONIZATION OF PuLsE-COUPL£D foreign-language mystery dramas, Cyanobacteria and Their BIOLOGICAL OSCILLATORS. Renata E. dialogs recorded in Paris, games, music, Secondary Metabolites MiraDa and Steven H. Strogatz in SlAM and many other helpful materials. And if Journal on Applied Machematics, Val. you want to learn a new language, we 50, No.6, pages 1645-1662; Decem­ have beginning courses for adults and Breaking Intractability ber t990. for children. COuPL£D ONUNEAR OSCIllATORS BE­ We offer introductory and advanced LOW TIlE SYNCHRONlZATION 1)-1RESH­ Animal Sexuality materials in most ofthe world's languages: OW: RElAxATION BY GENERAUZED lANDAU DAMPING. Steven H. Strogatz, French, German, Spanish, Italian, World languages and Renata E. MirolJa and Paul C. Matthews Japanese, Mandarin, Greek, Russian, Human Dispersal in Physical Review Leners, Vol. 68, No. Arabic, Korean, and others. 238 courses 18, pages 2730-2733; May 4, 1992. in 86 languages. Our 22nd year. COUPlED NONUNF.AR OSClLLATORS AND For FREE 56-page catalog call TIlE SYMMETR.IES OF ANIMAL GAITS. 1-800-662-5180 or write: AT YOUR J. J. Collins and Ian Stewart in Journal NEWSSTAND of Nonlinear Science, Vol 3, No.3, aUDla·,:arwm" pages 349-392; July 1993. Room 3005, 96 Broad St. DECEMBER 28 r.,,;I....r,J rTn~A'l'7 (')0" A"'L070d