The Rhythm of Life : Nature's Numbers Chapter 7

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Nature is nothing if not rhythmic, and its rhythms are many and varied. Our hearts and lungs follow rhythmic cycles whose timing is adapted to our body's needs. Many of nature's rhythms are like the heartbeat: they take care of themselves, running "in the background." Others are like breathing: there is a simple "default" pattern that operates as long as nothing unusual is happening, but there is also a more sophisticated control mechanism that can kick in when necessary and adapt those rhythms to immediate needs. Controllable rhythms of this kind are particularly common-and particularly interesting-in locomotion. In legged animals, the default patterns of motion that occur when conscious control is not operating are called gaits.




Until the development of high-speed photography, it was virtually impossible to find out exactly how an animal's legs moved as it ran or galloped: the motion is too fast for the human eye to discern. Legend has it that the photographic technique grew out of a bet on a horse. In the 1870s, the railroad tycoon Leland Stanford bet twenty-five thousand dollars that at some times a trotting horse has all four feet completely off the ground. To settle the issue, a photographer, who was born Edward Muggeridge but changed his name to Eadweard Muybridge, photographed the different phases of the gait of the horse, by placing a line of cameras with tripwires for the horse to trot past. Stanford, it is said, won his bet. Whatever the truth of the story, we do know that Muybridge went on to pioneer the scientific study of gaits. He also adapted a mechanical device known as the zoetrope to display them as "moving pictures," a road that in short order led to Hollywood. So Muybridge founded both a science and an art.

Most of this chapter is about gait analysis, a branch of mathematical biology that grew up around the questions "How do animals move?" and "Why do they move like that?" To introduce a little more variety, the rest is about rhythmic patterns that occur in entire animal populations, one dramatic example being the synchronized flashing of some species of fireflies, which is seen in some regions of the Far East, including Thailand. Although biological interactions that take place in individual animals are very different from those that take place in populations of animals, there is an underlying mathematical unity, and one of the messages of this chapter is that the same general mathematical concepts can apply on many different levels and to many different things. Nature respects this unity, and makes good use of it.




The organizing principle behind many such biological cycles is the mathematical concept of an oscillator-a unit whose natural dynamic causes it to repeat the same cycle of behavior over and over again. Biology hooks together huge "circuits" of oscillators, which interact with each other to create complex patterns of behavior. Such "coupled oscillator networks" are the unifying theme of this chapter.

Why do systems oscillate at all? The answer is that this is the simplest thing you can do if you don't want, or are not allowed, to remain still. Why does a caged tiger pace up and down? Its motion results from a combination of two constraints. First, it feels restless and does not wish to sit still. Second, it is confined within the cage and cannot simply disappear over the nearest hill. The simplest thing you can do when you have to move but can't escape altogether is to oscillate. Of course, there is nothing that forces the oscillation to repeat a regular rhythm; the tiger is free to follow an irregular path around the cage. But the simplest option-and therefore the one most likely to arise both in mathematics and in nature--is to find some series of motions that works, and repeat it over and over again. And that is what we mean by a periodic oscillation. In chapter 5, I described the vibration of a violin string. That, too, moves in a periodic oscillation, and it does so for the same reasons as the tiger. It can't remain still because it has been plucked, and it can't get away altogether because its ends are pinned down and its total energy cannot increase.

Many oscillations arise out of steady states. As conditions change, a system that has a steady state may lose it and begin to wobble periodically. In 1942, the German mathematician Eberhard Hopf found a general mathematical condition that guarantees such behavior: in his honor, this scenario is known as Hopf bifurcation. The idea is to approximate the dynamics of the original system in a particularly simple way, and to see whether a periodic wobble arises in this simplified system. Hopf proved that if the simplified system wobbles, then so does the original system. The great advantage of this method is that the mathematical calculations are carried out only for the simplified system, where they are relatively straightforward, whereas the result of those calculations tells us how the original system behaves. It is difficult to tackle the original system directly, and Hopf's approach sidesteps the difficulties in a very effective manner.

The word "bifurcation" is used because of a particular mental image of what is happening, in which the periodic oscillations "grow out from" the original steady state like a ripple on a pond growing out from its center. The physical interpretation of this mental picture is that the oscillations are very small to start with, and steadily become larger. The speed with which they grow is unimportant here.

For example, the sounds made by a clarinet depend on Hopf bifurcation. As the clarinetist blows air into the instrument, the reed-which was stationary-starts to vibrate. If the air flows gently, the vibration is small and produces a soft note. If the musician blows harder, the vibration grows and the note becomes louder. The important thing is that the musician does not have to blow in an oscillatory way (that is, in a rapid series of short puffs) to make the reed oscillate. This is typical of Hopf bifurcation: if the simplified system passes Hopf's mathematical test, then the real system will begin to oscillate of its own accord. In this case, the simplified system can be interpreted as a fictitious mathematical clarinet with a rather simple reed, although such an interpretation is not actually needed to carry out the calculations.

Hopf bifurcation can be seen as a special type of symmetry breaking. Unlike the examples of symmetry breaking described in the previous chapter, the symmetries that break relate not to space but to time. Time is a single variable, so mathematically it corresponds to a line-the time axis. There are only two types of line symmetry: translations and reflections. What does it mean for a system to be symmetric under time translation? It means that if you observe the motion of the system and then wait for some fixed interval and observe the motion of the system again, you will see exactly the same behavior. That is a description of periodic oscillations: if you wait for an interval equal to the period, you see exactly the same thing. So periodic oscillations have time-translation symmetry.

What about reflectional symmetries of time? Those correspond to reversing the direction in which time flows, a more subtle and philosophically difficult concept. Time reversal is peripheral to this chapter, but it is an extremely interesting question, which deserves to be discussed somewhere, so why not here? The law of motion is symmetric under time reversal.

If you make a film of any "legal" physical motion (one that obeys the laws), and run the movie backward, what you see is also a legal motion. However, the legal motions common in our world often look bizarre when run backward. Raindrops falling from the sky to create puddles are an everyday sight; puddles that spit raindrops skyward and vanish are not. The source of the difference lies in the initial conditions. Most initial conditions break time-reversal symmetry. For example, suppose we decide to start with raindrops falling downward. This is not a time-symmetric state: its time reversal would have raindrops falling upward. Even though the laws are time-reversible, the motion they produce need not be, because once the time-reversal symmetry has been broken by the choice of initial conditions, it remains broken.

Back to the oscillators. I've now explained that periodic oscillations possess time-translation symmetry, but I haven't yet told you what symmetry is broken to create that pattern. The answer is "all time translations." A state that is invariant under these symmetries must look exactly the same at all instants of time--not just intervals of one period. That is, it must be a steady state. So when a system whose state is steady begins to oscillate periodically, its time-translational symmetries decrease from all translations to only translations by a fixed interval. This all sounds rather theoretical. However, the realization that Hopf bifurcation is really a case of temporal symmetry breaking has led to an extensive theory of Hopf bifurcation in systems that have other symmetries as well-especially spatial ones. The mathematical machinery does not depend on particular interpretations and can easily work with several different kinds of symmetry at once. One of the success stories of this approach is a general classification of the patterns that typically set in when a symmetric network of oscillators undergoes a Hopf bifurcation, and one of the areas to which it has recently been applied is animal locomotion.

Two biologically distinct but mathematically similar types of oscillator are involved in locomotion. The most obvious oscillators are the animal's limbs, which can be thought of as mechanical systems-linked assemblies of bones, pivoting at the joints, pulled this way and that by contracting muscles. The main oscillators that concern us here, however, are to be found in the creature's nervous system, the neural circuitry that generates the rhythmic electrical signals that in tum stimulate and control the limbs' activity. Biologists call such a circuit a CPG, which stands for "central pattern generator." Correspondingly, a student of mine took to referring to a limb by the acronym LEG, allegedly for "locomotive excitation generator." Animals have two, four, six, eight, or more LEGs, but we know very little directly about the ePGs that control them, for reasons I shall shortly explain. A lot of what we do know has been arrived at by working backward-or forward, if you like-from mathematical models.

Some animals possess only one gait-only one rhythmic default pattern for moving their limbs. The elephant, for example, can only walk. When it wants to move faster, it ambles-but an amble is just a fast walk, and the patterns of leg movement are the same. Other animals possess many different gaits; take the horse, for example. At low speeds, horses walk; at higher speeds, they trot; and at top speed they gallop. Some insert yet another type of motion, a canter, between a trot and a gallop. The differences are fundamental: a trot isn't just a fast walk but a different kind of movement altogether. In 1965, the American zoologist Milton Hildebrand noticed that most gaits possess a degree of symmetry. That is, when an animal bounds, say, both front legs move together and both back legs move together; the bounding gait preserves the animal's bilateral symmetry. Other symmetries are more subtle: for example, the left half of a camel may follow the same sequence of movements as the right, but half a period out of phase-that is, after a time delay equal to half the period. So the pace gait has its own characteristic symmetry: "reflect left and right, and shift the phase by half a period." You use exactly this type of symmetry breaking to move yourself around: despite your bilateral symmetry, you don't move both legs simultaneously! There's an obvious advantage to bipeds in not doing so: if they move both legs slowly at the same time they fall over.

The seven most common quadrupedal gaits are the trot, pace, bound, walk, rotary gallop, transverse gallop, and can ter, In the trot, the legs are in effect linked in diagonal pairs. First the front left and back right hit the ground together, then the front right and back left. In the bound, the front legs hit the ground together, then the back legs, The pace links the movements fore and aft: the two left legs hit the ground, then the two right. The walk involves a more complex but equally rhythmic pattern: front left, back right, front right, back left, then repeat. In the rotary gallop, the front legs hit the ground almost together, but with the right (say) very slightly later than the left; then the back legs hit the ground almost together, but this time with the left very slightly later than the right. The transverse gallop is similar, but the sequence is reversed for the rear legs. The canter is even more curious: first front left, then back right, then the other two legs simultaneously, There is also a rarer gait, the pronk, in which all four legs move simultaneously.

The pronk is uncommon, outside of cartoons, but is sometimes seen in young deer. The pace is observed in camels, the bound in dogs; cheetahs use the rotary gallop to travel at top speed, Horses are among the more versatile quadrupeds, using the walk, trot, transverse gallop, and canter, depending on circumstances.

The ability to switch gaits comes from the dynamics of CPGs. The basic idea behind CPG models is that the rhythms and the phase relations of animal gaits are determined by the natural oscillation patterns of relatively simple neural circuits. What might such a circuit look like? Trying to locate a specific piece of neural circuitry in an animal's body is like searching for a particular grain of sand in a desert: to map out the nervous system of all but the simplest of animals is well beyond the capabilities even of today's science. So we have to sneak up on the problem of ePG design in a less direct manner.

One approach is to work out the simplest type of circuit that might produce all the distinct but related symmetry patterns of gaits. At first, this looks like a tall order, and we might be forgiven if we tried to concoct some elaborate structure with switches that effected the change from one gait to another, like a car gearbox. But the theory of Hopf bifurcation tells us that there is a simpler and more natural way. It turns out that the symmetry patterns observed in gaits are strongly reminiscent of those found in symmetric networks of oscillators. Such networks naturally possess an entire repertoire of symmetry-breaking oscillations, and can switch between them in a natural manner. You don't need a complicated gearbox. For example, a network representing the ePG of a biped requires only two identical oscillators, one for each leg. The mathematics shows that if two identical oscillators are coupled- connected so that the state of each affects that of the other-then there are precisely two typical oscillation patterns. One is the in-phase pattern, in which both oscillators behave identically. The other is the out-oJ-phase pattern, in which both oscillators behave identically except for a halfperiod phase difference. Suppose that this signal from the ePG is used to drive the muscles that control a biped's legs, by assigning one leg to each oscillator. The resulting gaits inherit the same two patterns. For the in-phase oscillation of the network, both legs move together: the animal performs a two-legged hopping motion, like a kangaroo. In contrast, the out-of-phase motion of the ePG produces a gait resembling the human walk. These two gaits are the ones most commonly observed in bipeds, (Bipeds can, of course, do other things; for example, they can hop along on one leg-but in that case they effectively turn themselves into one-legged animals.)

What about quadrupeds? The simplest model is now a system of four coupled oscillators-one for each leg. Now the mathematics predicts a greater variety of patterns, and nearly all of them correspond to observed gaits. The most symmetric gait, the pronk, corresponds to all four oscillators being synchronized- that is, to unbroken symmetry. The next most symmetric gaits-the bound, the pace, and the trot-correspond to grouping the oscillators as two out-of-phase pairs: front/back, left/right, or diagonally. The walk is a circulating figure-eight pattern and, again, occurs naturally in the mathematics. The two kinds of gallop are more subtle. The rotary gallop is a mixture of pace and bound, and the transverse gallop is a mixture of bound and trot. The canter is even more subtle and not as well understood.

The theory extends readily to six-legged creatures such as insects. For example, the typical gait of a cockroach-and, indeed, of most insects-is the tripod, in which the middle leg on one side moves in phase with the front and back legs on the other side, and then the other three legs move together, half a period out of phase with the first set. This is one of the natural patterns for six oscillators connected in a ring.

The symmetry-breaking theory also explains how animals can change gait without having a gearbox: a single network of oscillators can adopt different patterns under different conditions. The possible transitions between gaits are also organized by symmetry. The faster the animal moves, the less symmetry its gait has: more speed breaks more symmetry. But an explanation of why they change gait requires more detailed information on physiology. In 1981, D. F. Hoyt and R. C. Taylor discovered that when horses are permitted to select their own speeds, depending on terrain, they choose whichever gait minimizes their oxygen consumption.

I've gone into quite a lot of detail about the mathematics of gaits because it is an unusual application of modern mathematical techniques in an area that at first sight seems totally unrelated. To end this chapter, I want to show you another application of the same general ideas, except that in this case it is biologically important that symmetry not be broken.

One of the most spectacular displays in the whole of nature occurs in Southeast Asia, where huge swarms of fireflies flash in synchrony. In his 1935 article" Synchronous Flashing of Fireflies" in the journal Science, the American biologist Hugh Smith provides a compelling description of the phenomenon:

Imagine a tree thirty-five to forty feet high. apparently with a firefly on every leaf. and all the fireflies flashing in perfect unison at the rate of about three times in two seconds. the tree being in complete darkness between flashes. Imagine a tenth of a mile of river front with an unbroken line of mangrove trees with fireflies on every leaf flashing in synchronism, the insects on the trees at the ends of the line acting in perfect unison with those between. Then. if one's imagination is sufficiently vivid. he may form some conception of this amazing spectacle.

Why do the flashes synchronize? In 1990, Renato Mirollo and Steven Strogatz showed that synchrony is the rule for mathematical models in which every firefly interacts with every other. Again, the idea is to model the insects as a population of oscillators coupled together-this time by visual signals. The chemical cycle used by each firefly to create a flash of light is represented as an oscillator. The population of fireflies is represented by a network of such oscillators with fully symmetric coupling-that is, each oscillator affects all of the others in exactly the same manner. The most unusual feature of this model, which was introduced by the American biologist Charles Peskin in 1975, is that the oscillators are pulsecoupled. That is, an oscillator affects its neighbors only at the instant when it creates a flash of light.

The mathematical difficulty is to disentangle all these interactions, so that their combined effect stands out clearly. Mirollo and Strogatz proved that no matter what the initial conditions are, eventually all the oscillators become synchronized. The proof is based on the idea of absorption, which happens when two oscillators with different phases "lock together" and thereafter stay in phase with each other. Because the coupling is fully symmetric, once a group of oscillators has locked together, it cannot unlock. A geometric and analytic proof shows that a sequence of these absorptions must occur, which eventually locks all the oscillators together.

The big message in both locomotion and synchronization is that nature's rhythms are often linked to symmetry, and that the patterns that occur can be classified mathematically by invoking the general principles of symmetry breaking. The principles of symmetry breaking do not answer every question about the natural world, but they do provide a unifying framework, and often suggest interesting new questions. In particular, they both pose and answer the question, Why these patterns but not others?

The lesser message is that mathematics can illuminate many aspects of nature that we do not normally think of as being mathematical. This is a message that goes back to the Scottish zoologist D' Arcy Thompson, whose classic but maverick book On Growth and Form set out, in 1917, an enormous variety of more or less plausible evidence for the role of mathematics in the generation of biological form and behavior.

In an age when most biologists seem to think that the only interesting thing about an animal is its DNA sequence, it is a message that needs to be repeated, loudly and often.

Chapter 6 : Broken Symmetry
Chapter 7 : The Rhythm of Life
Chapter 8 : Do dice Play God 
Chapter 9 : Drops Dynamics and Daisies