Sync, page 23
Steven Strogatz:
Well, of course you should come to me.
And after two pages of generous advice, he closed with:
Do keep in touch: You sound interesting.
Art Winfree
That was a dream come true. By then Winfree had become my hero. But he was in a biology department, and a graduate degree in biology was not in my plans—math was my subject. So how about a summer job with him? I sheepishly raised that possibility. Two weeks later, a reply arrived:
12-10-81
5 min after receiving yours of 12-1-81
Dear Steven——
This week a pile of $ fell on me so yes, I can provide a summer salary [ . . . ]
There is plenty of space in my lab and 2 Apple computers w/ various wonderful attachments. [ . . . ] I will be working at topol. puzzles about 3-D twisted + knotted waves in Zhabotinsky’ soup, + “moonlighting” applications to cardiac muscle (My Scient. Amer. article on sudden cardiac death will fill you in this spring.) I would be super-delighted to enlist your partnership in these endeavors. I think we could learn a lot together.
I will not encourage [ . . . ] or [ . . . ] or anybody else to offer you a position until you decline this one. I hope you won’t.
Impulsively,
Art Winfree
Winfree’s research agenda, stated in idiosyncratic code in his letter to me, was ahead of what everyone else was thinking about. Of course he was well outside the mainstream of normal science, with its tendency toward narrow specialization and its emphasis on reductionism, drilling down to smaller and smaller units of inquiry—he wasn’t thinking about single genes or quarks or neuron channels. But he was even outside the chaos revolution, which all of its practitioners felt was in the vanguard, but which was, in fact, already reaching maturity and about to give way to the next great trend: the study of nonlinear systems composed of enormous numbers of parts. Later christened as “complexity theory,” this movement would come to seem like a natural outgrowth of chaos, in some ways its flip side—instead of focusing on the erratic behavior of small systems, complexity theorists were fascinated by the organized behavior of large ones. Winfree’s earliest work on spontaneous synchronization of biological oscillators had already touched on that theme. By now it had matured in several ways.
For example, his letter mentioned his plans to work on “3-D twisted + knotted waves.” The key phrase here is 3-D. No one had ever looked into the behavior of self-sustained oscillators interacting in three-dimensional space. As we have seen earlier, when theorists first started analyzing the dynamics of oscillator populations, they ignored space altogether and concentrated on time alone, on rhythms in step, with no regard for how the oscillators were situated geographically. The breakthroughs of Wiener, Kuramoto, Peskin, and even Winfree himself had been restricted to the simplest possible case of all-to-all coupling, where each oscillator affects every other one equally. Global coupling was always recognized as nothing more than an expedient first step—it was the quickest way into the jungle of many-oscillator dynamics. There was no spatial structure to worry about; every oscillator is a neighbor to every other. Once that case was in hand, the next step up on the theoretical ladder was to consider oscillators arranged in a one-dimensional chain or ring. As you might expect, now something new can happen, something beyond pure synchrony: Waves of activity can propagate steadily from one oscillator to the next. In fact, in oscillator models with local coupling, waves turned out to be more common than sync. That makes intuitive sense from our own experience as fans at a football game: In a huge stadium, it’s a lot easier to start “the wave” and keep it going than it would be to get the whole crowd standing up and sitting down simultaneously. When a few mathematicians tried to climb even higher, to two-dimensional sheets of oscillators, they had to hold on for dear life. The analysis became almost intractable. So when Winfree decided to keep climbing—to go three-dimensional—no one else followed.
The reason for thinking about such questions, of course, is that most real oscillators are coupled locally, not globally. The intestine is a long tube of oscillating nerve and muscle cells, segmented into rings that squeeze rhythmically, but choreographed so that waves of digestion travel in the right direction, from the stomach to the anus. Each ring of oscillatory tissue is coupled electrically to its nearest neighbors on either side, making the intestine effectively a one-dimensional chain of oscillators. The stomach is something like a two-dimensional bag of neuromuscular oscillators, in the sense that its cells churn rhythmically and interact mainly with their neighbors along the surface of the stomach wall. And the heart is a thick, three-dimensional collection of dictatorial oscillatory cells (the pacemaker cells in the sinoatrial node and their subordinates) and subservient “excitable” cells that obey their commands; if triggered by a strong enough electrical stimulus, they fire once and return to rest, awaiting the next triggering pulse. When the heart is functioning normally, the pacemaker generates a wave of electrical excitation that spreads along specialized conduction fibers to the pumping chambers (the ventricles), causing them to contract and pump blood to the rest of the body.
In pathological cases, however, the excitable cells can mutiny and sustain a wave of their own, a rotating electrical tornado that fends off the incoming signals from the pacemaker. Cardiologists had known for decades that such “rotating action potentials,” or “circus movements,” could lead to tachycardia (abnormally fast heartbeat) and then degenerate into the lethal arrhythmia called ventricular fibrillation, where the heart muscle writhes helplessly, twitching and quivering but not pumping any blood. Every year, hundreds of thousands of apparently healthy people—people with no prior history of heart disease—die suddenly when their hearts fall into this pernicious mode of organization. When Winfree mentioned “ ‘moonlighting’ applications to cardiac muscle” in his letter, he was referring to these strange electrical tornadoes. He wanted to find out why they start, how they behave, and what could be done about them. Once the basic science was understood, he hoped, it should enable the design of defibrillators that are gentler than today’s crude devices, which burn the heart in order to save it.
In 1981, nonlinear dynamics had certainly not advanced to the stage where it could predict the behavior of such rotating waves in three dimensions. There was no hope of calculating their evolution in time, their lashing about, their swirling patterns of electrical turbulence. Even if the calculations were possible (assisted by a supercomputer, perhaps), any such attempt would be premature, since one wouldn’t know how to interpret the findings. In fact, no one even knew what a mug shot of one of these shadowy villains might look like. (They’d never been seen directly by cardiologists.) So Winfree felt that the first step was to learn how to recognize them, to anticipate their features in his mind’s eye; he would worry about their modus operandi later.
For the study of shapes in three dimensions, a coarser mathematics was needed, one that didn’t care about time but only about space. When Winfree mentioned “topol. puzzles,” he was referring to the branch of mathematics called topology, the study of continuous shape, a kind of generalized geometry where rigidity is replaced by elasticity. It’s as if everything is made of rubber. Shapes can be continuously deformed, bent, or twisted, but not cut—that’s never allowed. A square is topologically equivalent to a circle, because you can round off the corners. On the other hand, a circle is different from a figure eight, because there’s no way to get rid of the crossing point without resorting to scissors. In that sense, topology is ideal for sorting shapes into broad classes, based on their pure connectivity. Winfree’s plan was to use topology to classify the kinds of waves one might encounter in three-dimensional fields of excitable cells. Knowing what was possible, he’d know what to look for in later experiments and would have a hope of recognizing what would otherwise seem like bizarre, alien structures.
When I arrived at Winfree’s lab on a muggy day in June 1982, he was engrossed in some paperwork, sitting alone at a lab bench with his shirt wide open. I was a little embarrassed by the informality—my dad had accompanied me on the cross-country drive from Connecticut to Indiana, and this was his first look at my new hero—but Winfree disarmed us with his unbuttoned friendliness. Soon my dad took his leave, and it was just Winfree and me alone in his lab, with its beakers and Bunsen burners and razor blades everywhere. (I later found out that razor blades were his tool of choice for cutting. He’d happily shout “Zzzzupp!” whenever he wielded one to slice a piece of wire or millipore filter paper.)
The lab was quiet. No grad students or postdocs. But I was prepared for that—in earlier correspondence, when I’d asked who else would be working with us, Winfree wrote back, “Now I could make up tales about the other students + co-workers. But truth to tell, I have none. Maybe I am away too much to form relationships, maybe I have body odor, dunno . . . but population density = 1 in my lab. You will be a singular event. Does that undermine your confidence?”
We had only three months to work together, so I needed to learn quickly. Winfree felt I should get my hands dirty: no math or computers for a while. My first project was an experiment on what Winfree called Zhabotinsky soup, a chemical reaction that supports waves of excitation remarkably like the electrical waves that trigger the heartbeat. But it’s much simpler than a real heart—it’s not even alive—and it has no muscles or motion of any kind. It’s an idealized arena for exploring excitable wave propagation in its purest form. In that way, it plays the same role for heart waves that fruit flies play for genetics: a convenient simplification that captures the essence of more complicated phenomena.
Normally, the most amusing outcome you can hope for in a chemistry experiment is a puff of smoke or a noxious odor. In comparison, Zhabotinsky soup offers nonstop entertainment. When brewed according to its original recipe, it acts like a spontaneous oscillator, the chemical analog of pacemaker cells. It changes colors back and forth, rhythmically alternating between sky blue and rusty red dozens of times, before eventually relaxing to equilibrium about an hour later. At the molecular scale, the performance would appear even more impressive, if only we could see it: trillions of coupled oscillators, hoofing in perfect sync, the largest line dance ever assembled.
In its new, more subtle recipe, the reaction is excitable. At first it looks disappointingly inert. The oscillations are gone. But if you pour a thin layer of the red soup into a petri dish and then prick it with a silver wire or a hot needle, it suddenly launches a blue circular wave that expands and spreads like a grassfire. This is a chemical wave, a pulse of propagating excitation in which the reaction switches from a reduced state to an oxidized one. Once the wave has passed, the reaction reverts to quiescence and turns red again, just as grass eventually grows back after a grass fire. (This analogy is not perfect, however. The chemicals recover more rapidly than the prairie; a second wave can follow right behind.)
Chemical waves are completely different from the waves studied in traditional physics courses, like sound waves or the ripples on a pond. When a chemical wave spreads by diffusion, the surface of the liquid does not bob up and down. It remains motionless. What moves is a pattern of excitation, a kind of chemical contagion. Nor do these waves weaken like sound or ripples as they travel away from their origin. Each patch of the medium provides a fresh source of energy that refuels the wave, preventing it from damping out.
Now suppose you detonate two chemical waves at two different points in the petri dish. The blue circles expand and creep toward each other. When they collide, they do not interpenetrate or add up: They annihilate. And they do so for the same reason that onrushing grass fires snuff each other out: Neither can burn through the other’s ashes. In this metaphor, the ashes correspond to a region of exhaustion, a refractory zone in the wake of the wave. The chemical medium needs time to recover before it can become excited again.
In many ways, this chemical medium behaves like the human sexual response. Sexual arousal and recovery depend on the properties of nerve tissue, which, like Zhabotinsky soup, belongs to a general class of systems called excitable media. A neuron has three states: quiescent, excited, and refractory. Normally a neuron is quiescent. With inadequate stimulation, it shows little response and returns to rest. But a sufficiently provocative stimulus will excite the neuron and cause it to fire. Next it becomes refractory (incapable of being excited for a while) and finally returns to quiescence. The parallels with chemical waves extend to action potentials, the electrical waves that propagate along nerve axons. They too travel without attenuation, and when two of them collide, they annihilate each other. In fact, all of these statements are equally true of electrical waves in another excitable medium: the heart. That’s the beauty of this abstraction—the qualitative properties of one excitable medium hold for them all. They can all be studied in one stroke. The family resemblance among Zhabotinsky soup, nerve tissue, and heart muscle persists right on down to the structure of the mathematical equations that govern their nonlinear dynamics. The analogy runs deep.
But Zhabotinsky soup offers a number of advantages, especially for a beginning experimenter. No animals need to be sacrificed. There’s no confusing anatomy, like the intricate tangle of neural networks or the twisted-fiber architecture of the heart muscle. Best of all, the waves are visible to the naked eye and they move slowly, so there’s no need for any elaborate recording equipment. In contrast, the visualization of waves on the heart remains a formidable technical challenge to this day, even for labs with huge budgets, requiring voltage-sensitive dyes, multielectrode arrays, and other state-of-the-art technology.
With the help of Zhabotinsky soup, scientists have begun to unravel the secrets of wave propagation in excitable media. In particular, it was in Zhabotinsky soup that a new kind of wave was discovered: a rotating, self-sustaining wave shaped like a spiral. Although its geometry is graceful, its consequences are destructive. Rotating spiral waves on the heart are the culprits behind tachycardia and, in the worst case, ventricular fibrillation followed by sudden cardiac death.
The discovery of Zhabotinsky soup and its remarkable spiral waves is a tale of dogma, disappointment, and ultimate vindication. Of course, Zhabotinsky soup is not its real name—that’s just what Winfree always called it. Today it’s known as the BZ reaction, for Belousov and Zhabotinsky, the Russian scientists who invented it and refined it, respectively.
In the early 1950s Boris Belousov was trying to create a test-tube caricature of the Krebs cycle, a metabolic process that occurs in living cells. When he mixed citric acid and bromate ions in a solution of sulfuric acid in the presence of a cerium catalyst, he observed to his astonishment that the mixture became yellow, then faded to colorlessness after about a minute, then returned to yellow a minute later, then became colorless again, and continued to oscillate dozens of times before finally reaching equilibrium after about an hour.
Nowadays it comes as no surprise that chemical reactions can oscillate spontaneously; such reactions have become a standard demonstration in chemistry classes. But in Belousov’s day, his discovery was so radical that no one would believe it. It was thought that all solutions of chemical reagents must go monotonically to equilibrium, because of the laws of thermodynamics. Journal after journal brushed off Belousov’s paper. One editor even salted his rejection letter with a snide remark about Belousov’s “supposedly discovered discovery.”
Dejected, Belousov resolved never to share his breakthrough with his chemist colleagues. He did publish a brief abstract in the obscure proceedings of a Russian medical meeting, but hardly anyone noticed it until years later. Nevertheless, rumors about his amazing reaction circulated among Moscow chemists in the late 1950s, and in 1961 a graduate student named Anatol Zhabotinsky was assigned by his adviser to look into it. Zhabotinsky confirmed that Belousov had been right all along, and brought this work to light at an international conference in Prague in 1968, one of the rare occasions that Western and Soviet scientists were allowed to meet. At that time there was fervent interest in biological and biochemical oscillations, and the BZ reaction was seen as a manageable model of those more complex systems.
The analogy to biology turned out to be surprisingly close. In early 1970, A. N. Zaikin and Zhabotinsky found propagating waves of excitation in thin, unstirred layers of BZ reaction. The waves resembled concentric circles, and they annihilated upon collision, just like electrical waves in neural or cardiac tissue. They even seemed to emanate from something analogous to pacemakers, randomly scattered points that belched waves spontaneously.
After learning of this work, Winfree wrote to Zhabotinsky (whom he’d met two years earlier as a fellow grad student at the Prague conference) to ask whether he’d ever seen any other wave patterns besides concentric rings. Winfree had observed spiral waves in his own lab experiments on a certain kind of fungus, but that was a far more complex system composed of living creatures with circadian clocks. He wondered if spirals could also occur in Zhabotinsky’s much simpler chemical system. He doubted it on mathematical grounds; he thought he could prove that the waves had to be closed rings. But still no reply from Zhabotinsky. The mail from the Soviet Union was maddeningly slow in those days, especially between scientists (national security agencies at both ends were probably busy steaming open the envelopes). Winfree couldn’t bear the suspense. He concocted Zaikin and Zhabotinsky’s recipe for himself, and sure enough, spirals popped up everywhere. Winfree had no way of knowing it, but Zhabotinsky had also seen them in his 1970 thesis work, and Valentin Krinsky in Puschino had anticipated them in any excitable medium, heart muscle included. Spiral waves are now recognized to be a pervasive feature of all chemical, biological, and physical excitable media.
Boris Belousov would be pleased to see what he started.
In 1980, he, Zhabotinsky, and three other scientists were awarded the Lenin Prize, the Soviet Union’s highest medal, for their pioneering work on oscillating reactions. But it wasn’t much consolation—Belousov had died 10 years earlier.
