Getting in Sync

...the behavior of a possibly very large number of individuals becomes synchronized to a rhythm shared by the ensemble. How does this happen without some kind of external organization? How does such a system naturally evolve into this kind of equilibrium? And how does mathematics help us to understand these phenomena? These are the questions we explore in this column. ...

Introduction

Toward the end of a concert I attended recently, the band left the stage, and the audience of several hundred began applauding urging the band back for an encore. The applause was at first disorganized with each audience member clapping in their own rhythm. Before long, however, one could detect a beat in the applause and soon everyone was clapping in unison, at the same time and tempo.

How did it happen? There was no leadership providing direction, and there was, initially at least, no shared goal to organize ourselves in this way.

Far from being rare, this type of self-organizing behavior is frequently observed in nature, most strikingly among firefly populations in southeast Asia. Hugh Smith offered this description in 1935:

Imagine a dozen such trees standing close together along the river's edge with synchronously flashing fireflies on every leaf. Imagine a tenth of a mile of river front with an unbroken line of Sonneratia 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.

 

Of course, the demands on one's imagination are less than they used to be as this video of fireflies in Thailand is easily found on YouTube:

One might be tempted to explain the synchronous flashing of fireflies as the product of some unrecognized intelligence. Yet we also observe similar phenomena in the inanimate world. The Dutch physicist Christian Huygens, whose interest in astronomy led him to investigate precise timekeeping and clock making, was bedridden for a time in 1665 while recovering from an illness. With nothing much to do, he noticed that the pendula of two clocks located near one another swung in unison, the clocks keeping perfect time with one another. When separated, however, the clocks were seen to differ by about five seconds a day.

To explain what he called "an odd kind of sympathy," Huygens speculated that the clocks were interacting through imperceptible movements in the wooden beams supporting the clocks. This video from MIT's McGovern Institute illustrates:

In all of these examples, the behavior of a possibly very large number of individuals becomes synchronized to a rhythm shared by the ensemble. How does this happen without some kind of external organization? How does such a system naturally evolve into this kind of equilibrium? And how does mathematics help us to understand these phenomena? These are the questions we explore in this column.

Pulse-coupled oscillators

We'll begin by studying the synchrony of fireflies flashing. Two observations are important. First, as any child who's ever caught a firefly in a jar on a summer's evening will tell us, an individual firefly will flash on its own in some periodic fashion. For us, anything that exhibits periodic behavior will be called an oscillator so we imagine the group of fireflies along a river bank as being composed of individual oscillators that evolve toward synchrony.

John and Elisabeth Buck made a second observation when they traveled to Thailand and captured some fireflies that they later released in their hotel room. After turning off the lights, they were able to observe the evolution of the fireflies' flashing. First, small groups of fireflies began flashing synchronously. These groups then grew and merged until the entire ensemble was blinking on and off in unison. This behavior is illustrated in the following demonstration.

The Bucks speculated that individual fireflies adjusted the rhythm of their flashing in response to the flashing of other fireflies. Indeed, laboratory experiments later provided confirmation; by flashing a light at a firefly, observers saw that an individual firefly responded to the flash by modifying the rhythm of its own flashing.

Because two fireflies only interact with one another at the moment when one of them flashes, we call this a system of pulse-coupled oscillators. The resulting discontinuity in the interaction between the fireflies makes this a difficult situation to analyze mathematically.

Charles Peskin made the first successful attempt in 1975 with his work to understand the synchronization of cardiac pacemaker cells. This pacemaker, a collection of roughly 10,000 cells in the sinoatrial node of the human heart, is responsible for regulating the heart's beating about once a second over an entire lifetime.

Kept in isolation, an individual pacemaker cell fires at a regular interval. In an ensemble, however, each cell adjusts its firing so that the collection fires synchronously, the same behavior we observe in fireflies. The evolutionary advantage should be clear: the responsibility for firing periodically is invested in a large group of cells so that the pacemaker is able to survive the failure of a single cell.

Peskin created a mathematical model of the electro-chemical mechanism by which the pacemaker cells fire and respond to the firing of other cells. There are two important parts to Peskin's analysis. First, he modeled the firing of an individual cell as a simple $RC$ circuit comprised of a resistor and capacitor in parallel. The voltage across the capacitor increases, though the rate of increase slows as it approaches a firing threshold. Once the voltage reaches that threshold, the cell fires, the voltage returns to zero, and the cycle begins anew. The cyclic behavior of the voltage in the circuit is shown below.

Second, Peskin modeled the pulse-coupling of the cells by assuming that the firing of one cell increases the voltage of all the other cells by some fixed amount $\epsilon$. Using this model, Peskin was able to prove that a system of two such cells would synchronize. Though his model becomes too unwieldy when there are more than two cells, he made the following conjectures:

1. A system with many more identical pulse-coupled oscillators will synchronize.

2. If the oscillators are allowed to fire at different frequencies, the system will synchronize.

In 1990, Renato Mirollo and Steven Strogatz simplified Peskin's model by identifying the essential nature of the concavity in the firing graph and proved that an arbitrary number of identical pulse-coupled oscillators will synchronize, Peskin's first conjecture. We will describe their idea in the case where we have two oscillators.

The location of an oscillator during one period of its firing is described by its phase $\phi$ where $0\leq \phi\leq 1$. The voltage is then a function $f(\phi)$ as shown with firing threshold $1$. As we will see, it is important that the graph of $f$ is concave down. Typically, an oscillator's phase $\phi$ is a linear function of time so that the phase increases at a constant rate.

Suppose that we have two oscillators $A$ and $B$, and we consider an initial time when the phase of $A$ is zero, meaning that $A$ has just fired. We denote the phase of $B$ by $\phi$. Mirollo and Strogatz studied the evolution of this system of two oscillators by comparing the phase of $B$ at this time to the phase of $B$ at the next time immediately after $A$ has fired. Let's track the system between these two times.

After some time, the phase of $B$ increases by $1-\phi$ to reach one, which means that $B$ is ready to fire. Since the phase of $A$ increases at the same rate, the phase of $A$ is now $1-\phi$ giving a voltage of $f(1-\phi)$.

When $B$ fires, its phase returns to zero. This firing adds $\epsilon$ to the voltage of $A$ giving the voltage $f(1-\phi)+\epsilon$. The phase of $A$ is now $$h(\phi)=\phi'= f^{-1}(f(1-\phi)+\epsilon).$$ Notice that $h(\phi)$ tells us the phase of $A$ after $B$, whose phase begins at $\phi$, fires. Mirollo and Strogatz call $h$ the firing map.

We wish to trace the system until $A$ fires so we allow time to continue until the phase of $A$ increases by $1-\phi'$ to $1$. The phase of $B$ is then $1-\phi'$.

After $A$ fires, its phase returns to zero while the phase of $B$ is $$\phi''= h(\phi')=f^{-1}(f(1-\phi')+\epsilon).$$ We have now been through one cycle in which $A$ and $B$ have both fired with the phase of $A$ beginning and ending at zero. The phase of $B$, which was initially $\phi$, is now $$ R(\phi) = \phi''= h(\phi') = h\left(h(\phi)\right). $$ We call $R$ the return map and note that it records the new phase of $B$ as we progress through one of these cycles.

In this way, we can plot the evolution of the phase of $B$ through many cycles beginning with an initial phase $\phi$.

If we remember that $\phi=1$ is equivalent to $\phi=0$, we see that, in both of these cases, the two oscillators synchronize.

 

In fact, the figure below shows how the system evolves given several initial conditions, all of which lead to synchronization.

If we imagine filling in the gaps on this plot, we suspect that there is one initial condition that will not lead to synchronization. For instance, suppose we begin with an initial phase $\phi$ so that the firing map gives $h(\phi)=\phi$. In this case, after $B$ fires, the phase of $A$ is the same as the initial phase of $B$, and we return to the initial state with $A$ and $B$ interchanged.

Then $R(\phi) = h(h(\phi)) = \phi$ so one cycle in which both $B$ and $A$ fire returns us to the state in which we begin. This state will, of course, persist indefinitely so we cannot expect the system to synchronize. It is straightforward to see that there is only one value of $\phi$ satisfying this property; that is, there is a unique fixed point $\phi^*$ such that $R(\phi^*) = \phi^*$.

In fact, the fixed point $\phi^*$ is unstable in the sense that an initial phase near $\phi^*$ will move away from $\phi^*$ and eventually to synchronization. This is where the concavity of the graph of $f$ becomes important. In the example we considered, the phase of $B$ lagged behind that of $A$.

When $B$ fires, $A$ is in the early part of its cycle. In this region, the graph of $f$ is fairly steep so that increasing the voltage of $A$ by $\epsilon$ leads to a relatively small increase in the phase of $A$.

However, when $A$ fires, $B$ is later in its cycle where the graph of $f$ is flatter. This means that increasing the voltage of $B$ by $\epsilon$ increases the phase of $B$ by a larger amount.

Over one cycle in which both $A$ and $B$ fire, the phase of $B$ is advanced more than that of $A$, so $B$, which lagged behind $A$, is catching up with $A$. In this way, the system is seen to be moving toward synchronization, which it reaches when the phase of $B$ is greater than $1-\epsilon$ at the time $A$ fires.

To extend this idea to more than two oscillators, Mirollo and Strogatz focus on an event they call an absorption, which occurs when the firing of one cell $A$ causes another cell $B$ to fire. Before the firing of $A$, the phase of $B$ must be greater than $1-\epsilon$. After the firing of $A$, the phase of both $A$ and $B$ are zero. Once this occurs, the phase of $A$ and $B$ will evolve in the same way so that their firing forever remains synchronized.

When two cells become synchronized, they fire at the same time, which increases the voltage of all the other cells by $2\epsilon$ giving greater impetus to the other cells to synchronize to $A$ and $B$. This explains the observation made by the Bucks, who noted that the flashing of fireflies tended to synchronize in groups that grew and merged until the entire group is in synchrony.

The argument given by Mirollo and Strogatz begins with some initial condition and considers the number of firings that occur before the first absorption. A somewhat elementary argument shows that the set of initial conditions that never lead to an absorption has measure zero and that this implies that the set of initial conditions that never synchonize has measure zero. In other words, for almost all initial conditions, the system of pulse-coupled oscillators will synchronize.

The demonstration below is same as the earlier one with the phases of the oscillators displayed on the right. The phases are initially randomly distributed, but absorptions, by which the phases of two or more oscillators coalesce, may soon be observed. After an absorption, the greater effect that a group of oscillators firing together has on the other oscillators, and the resulting push toward synchrony, is apparent.

Using these ideas, Mirollo and Strogatz proved the first of Peskin's conjectures, that a system of arbitrarily many identical pulse-coupled oscillators will synchronize. Recall that the second conjecture states that oscillators will synchronize even when their firing frequencies vary. We next look at a model that provides this kind of flexibility.

The Kuramoto model

While systems of pulse-coupled oscillators are certainly observed in nature, there are many other systems of coupled oscillators, such as Huygens' synchronizing clocks, that are characterized by continuous interactions between the oscillators. Norbert Wiener was among the first to explore such systems in the 1950s motivated by a desire to explain electrical activity in the brain and the presence of alpha rhythms.

Somewhat later, Arthur Winfree, while finishing an undergraduate degree at Cornell, began forming a model of a system of continuously interacting biological oscillators. In addition to analyzing these systems mathematically, Winfree constructed numerical simulations and even conducted experiments on a system of 71 electrically coupled neon-tube oscillators. It was Winfree who, in this way, first observed the phenomena we will soon describe.

Inspired by Winfree's work, Yoshiki Kuramoto introduced what is now called the Kuramoto model in a series of papers beginning in 1975. In addition to providing a framework for studying systems of continuously interacting oscillators, this line of investigation allows us to consider systems comprised of oscillators whose frequencies vary, the subject of Peskin's second conjecture.

In Kuramoto's model, we consider a system of $N$ oscillators with the phase $\theta_j$ of each varying from $0$ to $2\pi$ with frequency $\omega_j$. To allow for variation in the frequencies, we choose $\omega_j$ from a distribution $g(\omega)$, such as a familiar normal distribution with mean $\overline{\omega}$ as shown.

For future reference, let's assume that the distribution is symmetric about $\overline{\omega}$ so that $g(\overline{\omega}+\omega) = g(\overline{\omega}-\omega)$ which implies that $g'(\overline{\omega}) = 0$. The distribution also has a maximum at $\overline{\omega}$ so that $g''(\overline{\omega}) \lt 0$.

If the oscillators are uncoupled, the evolution of their phases is governed by the differential equations for the rate of change of $\theta_j$: $$ \theta_j' = \omega_j. $$ The evolution of this system is demonstrated below, where the phase $\theta_j$ of each oscillator is represented by a point on the unit circle. Since the oscillators are uncoupled, there is no reason to expect they will synchronize. Indeed, we see that the phase of each oscillator evolves according to its characteristic frequency.

We now allow the oscillators to interact with one another so that each oscillator exerts a kind of synchronizing pull on the others. We write $$ \theta_j' = \omega_j + \sum_{k\neq j} \Gamma_{jk}(\theta_k-\theta_j), $$ where $\Gamma_{jk}(\theta_k-\theta_j)$ represents the pull that oscillator $k$ exerts on $j$.

We have a few expectations of this interaction. First, when $\theta_k - \theta_j$ is small, the oscillators are nearly synchronized so the interaction should be small. If $\theta_k-\theta_j$ is positive, oscillator $j$ lags behind $k$ so we wish to speed $j$ up. Finally, $\Gamma_{jk}$ should be a periodic function of the difference in phases.

A simple model satisfying these expectations is $\Gamma_{jk} = \frac KN\sin(\theta_k-\theta_j)$, where $K$ is a positive constant, known as the coupling constant, and $N$ is the number of oscillators in our system. In this way, we have Kuramoto's equations $$ \theta_j' = \omega_j + \frac KN \sum_k \sin(\theta_k-\theta_j), $$

Before we look at the behavior of systems having different values of $K$, we note that Kuramoto introduced a quantity, called the order parameter, that measures the degree of synchronization in the system. If we imagine the phase $\theta_j$ of an oscillator as defining the point $e^{i\theta_j}$ on the unit circle, as in our demonstration, we compute the centroid $$ Re^{i\psi} = \frac 1N \sum_ke^{i\theta_k}. $$ We call $R$ the order parameter, and note $0\leq R \leq 1$. For instance, if the points are uniformly distributed around the unit circle, we have $R\approx 0$. Alternatively, if the system is completely synchronized with $\theta_j(t)=\theta_k(t)$ for all $j$ and $k$, then $R=1$.

The demonstrations below show systems for various values of the coupling constant $K$ as well as the point $Re^{i\psi}$.

Here are some numerical simulations demonstrating the behavior of the order parameter with $N=100$ oscillators.

In this way, we are able to observe some of what Winfree first noticed. In particular,

• When the coupling constant $K$ is small, the system is incoherent and $R\approx 0$.

• At some critical value of the coupling constant, which we'll call $K_c$, the system becomes partially synchronized and the order parameter $R$ fluctuates around some positive value.

• As $K$ is increased, we see a greater degree of synchronization with more locked oscillators and an increase in the order parameter $R$.

We would like to explore and explain this behavior. Let us first recast the expression for the centroid by multiplying by $e^{-i\theta_j}$: $$ \begin{array}{rcl} Re^{i\psi} & = & \frac 1N\sum_k e^{i\theta_k} \\ Re^{i(\psi-\theta_j)} & = & \frac1N\sum_k e^{i(\theta_k-\theta_j)}. \\ \end{array} $$ Consideration of the imaginary parts gives $$ R\sin(\psi-\theta_j) = \frac1N\sum_k\sin(\theta_k-\theta_j), $$ which leads to $$ \theta_j' = \omega_j+KR\sin(\psi-\theta_j). $$

This expression represents a convenient repackaging of the interactions between the oscillators. First, rather than considering the interactions between each pair of oscillators, we only need to consider the interaction between an oscillator and a "mean" oscillator whose phase is $\psi$. Second, we see that the interaction between an oscillator and the mean oscillator is proportional to the order parameter $R$. Therefore, as some oscillators become locked, the order parameter $R$ grows, which increases the collective tug on each individual oscillator thus forming a positive feedback loop.

The simulations we looked at above were created with a finite number of oscillators. Kuramoto simplified the problem by studying the behavior as the number of oscillators $N\to\infty$ and looking for solutions where the centroid $Re^{i\psi}$ maintains a constant distance from the origin and rotates at a constant velocity. In other words, we will look for solutions where the order parameter $R$ and the velocity $\psi'$ is constant.

Kuramoto imagined that the frequency of the mean oscillator is $\overline{\omega}$, the mean of the frequency distribution, so that $\psi=\overline{\omega}t$. He then constructed a rotating reference frame by replacing $\theta_j$ with $ \theta_j+\psi =\theta_j+\overline{\omega}t$. In this reference frame, the equations become $$ (\theta_j+\psi)' = \omega_j + KR\sin(\psi-(\theta_j+\psi)), $$ which simplify to $$ \theta_j' = \omega_j - KR\sin\theta_j, $$ where we interpret $\theta_j$ and $\omega_j$ as the phase and frequency relative to the mean oscillator.

This expression explains one behavior we observed in the earlier demonstrations. When $\left|\omega_j\right| \leq KR$, oscillator $j$ moves to the fixed point $\theta_j$ (in the rotating reference frame) defined by $$\omega_j=KR\sin\theta_j$$ where $\left|\theta_j\right|\leq \frac{\pi}{2}$. These are the oscillators we earlier observed to be "locked." Notice that, as $K$ increases, an increasing number of oscillators become locked.

Oscillators with $\left|\omega_j\right|\gt KR$ will drift. If $\omega_j \gt KR$, the oscillator runs faster than the mean oscillator and makes laps around it. If $\omega_j \lt -KR$, the situation is reversed with the mean oscillator lapping oscillator $j$. This also explains another of Winfree's observations: a larger coupling constant is needed to bring a system whose frequencies are more widely distributed into synchrony.

This may, however, seem counterintuitive: if some of the oscillators drift, how can we have a solution for which the order parameter $R$ is constant? To answer this question, Kuramoto assumed that the drifting oscillators form a stationary distribution around the circle so that even as individual oscillators drift around the circle, their centroid is unchanged. We'll take some time to investigate the implications of this assumption.

We first take note of a particular symmetry. We denote the distribution of drifting oscillators at frequency $\omega$ and phase $\theta$ by $\rho(\theta,\omega)$; that is, $\rho(\theta, \omega)d\theta$ represents the fraction of oscillators having frequency $\omega$ and phase between $\theta$ and $\theta+d\theta$.

If the drifting oscillators are to form a stationary distribution, then the fraction of oscillators passing through phase $\theta$ in some time interval $dt$ must be independent of $\theta$. In this time interval, the phase of a drifting oscillator having frequency $\omega$ has changed by $$ d\theta = \theta'dt = \left|\omega-KR\sin\theta\right|~dt. $$ The fraction of drifting oscillators moving this far is therefore $$ \rho(\theta,\omega)~d\theta = \rho(\theta,\omega)\left|\omega - KR\sin\theta\right| ~dt. $$

The upshot of this analysis is that $\rho(\theta,\omega)\left|\omega-KR\sin\theta\right|$ is independent of $\theta$. As a result, we find the symmetry condition $$ \rho(\theta+\pi,-\omega) = \rho(\theta, \omega), $$ which will become important momentarily.

Let's now return to the expression defining the centroid $Re^{i\psi}$ and decompose the average as a sum over the locked oscillators and a sum over the drifting oscillators. Using $E$ to denote the expected value, we have $$ E(e^{i\theta}) = R = E_{\text{locked}}(e^{i\theta}) + E_{\text{drifting}}(e^{i\theta}), $$ remembering that $\psi = 0$ in our rotating reference frame.

Let's first consider the average of the drifting oscillators: $$ E_{\text{drifting}}(e^{i\theta}) = \int_{-\pi}^\pi\int_{\left|\omega\right| \gt KR} e^{i\theta} \rho(\theta,\omega)g(\omega)~d\omega~d\theta. $$ Remember that $g(\omega)$ is the frequency distribution of the oscillators; in the rotating reference frame, the symmetry condition $g(\omega+\overline{\omega})=g(\omega-\overline{\omega})$ becomes $g(\omega) = g(-\omega)$. Together with the symmetry condition $\rho(\theta+\pi,-\omega) = \rho(\theta,\omega)$, we see that this integral vanishes so we have $$E_{\text{drifting}}(e^{i\theta}) = 0. $$

This leaves us with $$ \begin{array}{rcl} R & = & E_{\text{locked}}(e^{i\theta}) = \int_{-KR}^{KR} e^{i\theta(\omega)} g(\omega)~d\omega \\ & = & \int_{-KR}^{KR} \cos\theta ~g(\omega)~d\omega + i\int_{-KR}^{KR} \sin\theta ~g(\omega)~d\omega. \end{array} $$ Once again, due to the symmetry of the distribution, $g(-\omega) = g(\omega)$, and the fact that $\sin\theta = \omega/KR$ for locked oscillators, we have $\int_{-KR}^{KR}\sin\theta ~g(\omega)~d\omega = 0$, which leaves us with $$R = \int_{-KR}^{KR} \cos\theta(\omega) ~g(\omega)~d\omega. $$ Applying the change of variable $\omega = KR\sin\theta$, we have $$ R = \int_{-\pi/2}^{\pi/2}\cos\theta ~g(KR\sin\theta) KR\cos\theta~d\theta $$ or $$ R = KR\int_{-\pi/2}^{\pi/2}\cos^2\theta ~g(KR\sin\theta) ~d\theta. $$

One trivial solution is, of course, $R=0$, which represents an incoherent solution. This is present for any value of $K$. If we seek a nontrivial solution, however, we have $$1 = K\int_{-\pi/2}^{\pi/2}\cos^2\theta ~g(KR\sin\theta) ~d\theta. $$ Approximating $g$ by its quadratic Taylor polynomial, we have $$ g(\omega) \approx g(0) + g'(0)\omega + \frac12 g''(0)\omega^2 $$ or, remembering that the symmetry of the distribution implies that $g'(0) = 0$, $$g(\omega) \approx g(0) + \frac12 g''(0)\omega^2. $$ This gives us $$ \begin{array}{rcl} 1 & \approx & K\int_{-\pi/2}^{\pi/2}\cos^2\theta\left(g(0) + \frac12g''(0)K^2R^2\sin^2\theta\right) ~d\theta \\ 1 & \approx & \frac{\pi}{2}Kg(0) + \frac{\pi}{16}K^3R^2g''(0) \\ \end{array} $$

Because of the shape of the distribution, we have $g''(0) \lt 0$ meaning the second term on the right side is negative. Therefore, there is no nontrivial steady solution if $1 \gt \frac{\pi}{2}Kg(0)$ or if $K$ is less than the critical value $$K\lt K_c=\frac{2}{\pi g(0)}.$$

When $K\gt K_c$, we find, after rearranging the expression above, the nontrivial solution $$ R = A\sqrt{\frac{K-K_c}{K_c}} $$ where $A$ is some constant.

To summarize, when the coupling constant $K$ is less than the critical value $K_c$, we see that there is only the incoherent steady solution $R=0$. However, at $K=K_c$, a bifurcation occurs and a coherent steady solution $R=A\sqrt{\frac{K-K_c}{K_c}}$ appears, as shown below.

Kuramoto's argument beautifully explained the observations Winfree made through numerical simulations and, like all good work, raises more questions than it answers. In particular, how can we use these ideas to understand the situation when $N$ is finite? Here, it is not reasonable to expect solutions where $R$ is constant, and indeed our simulations above persistently show fluctuations in $R$. We would also like to understand the stability of the constant solutions that Kuramoto found. These issues are more fully described in Strogatz's Physica D paper.

In the model above, we assumed that the coupling between every pair of oscillators is the same. It's easy, however, to imagine situations in which oscillators interact only with other oscillators that are physically close. A simulation of one such example shows that we can see new and interesting phenomena. Note that the connected page gives a simulation similar to ours above but with a wider choice of parameters to explore.

Summary

Strogatz's wonderful book Sync gives a popular account of many of the ideas in this column and is easily accessible to a general audience. Besides bringing the personalities behind the science to life and revealing the serendipity of mathematical discovery, Strogatz makes a strong case for mathematicians to communicate their work more seriously to a wider audience.

For instance, Strogatz relates the story of Rep. Tom Petri's (R-Wisconsin) attempt to score political points by complaining about federal funds being spent on firefly research. Of course, it's easy to make this work sound like a ridiculous waste of taxpayer money in the absence of any deeper narrative of the work's significance. However, the ubiquity of synchronization in both the biological and physical world demands our serious study.

Indeed, we have described some biological appearances of synchrony as being advantageous. Cardiac pacemaker cells synchronize to create a robust timer. Male fireflies flash in unison for reproductive advantage. There are, however, some areas in which synchronization appears to deleterious effect. Since we have seen that many systems have a tendency to evolve toward synchrony, this is particularly concerning.

One famous example is the construction of London's Millennium Bridge. On its opening day, the bridge began swaying from side to side. The sway caused pedestrians to unconsciously synchronize their strides, which only reinforced and amplified the bridge's sway. Understanding how synchronization occurs can help us design ways to avoid it.

Internet routers, as they automatically relay periodic messages to one another, have fallen into synchronization leading to periodic spikes in Internet traffic. It should now be clear that a strategy, such as randomizing the interval at which messages are exchanged, could lead to a solution.

Part of the power of mathematics is its ability to detect common features of problems in seemingly disparate areas and to develop an understanding of all of them at once. This message needs to be continually reinforced for the public.

References

• Steven Strogatz. Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life. Hachette Books. New York. 2015.

  Strogatz's book is an excellent popular account of how synchronization occurs in a wide range of phenomena and the mathematics that underpins our understanding of these phenomena. Though there are no equations or what may be called "hard mathematics," Strogatz manages to convey both the significance of mathematical thought as well as the joy found in it.

• Hugh M. Smith. Synchronous Flashing of Fireflies. Science, New Series, Vol. 82, No. 2102. 1935. 151-152.

• John Bonner Buck. Synchronous Rhythmic Flashing of Fireflies. The Quarterly Review of Biology, Vol. 13, No. 3. 1938. 301-314.

• Charles S. Peskin Mathematical Aspects of Heart Physiology.

  Chapter 7 presents Peskin's analysis of pulse coupling and cardiac pacemaker cells.

• Renato Mirollo and Steven Strogatz. Synchronization of Pulse Coupled Oscillators. SIAM J. Appl. Math, 50(6), 1645-1662. 1990.

  This paper gives a proof of the first of Peskin's conjectures, that a system of many identical pulse-coupled oscillators will almost always synchronize.

• Arthur Winfree. The Geometry of Biological Time. Springer, New York. 1980.

• Yoshiki Kuramoto. Chemical Oscillations, Waves, and Turbulence. Springer, Berlin. 1984.

• Steven Strogatz. From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators. Physica D. 143(2000). 1-20.

• Steven Strogatz, Daniel Abrams, Allan McRobie, Bruno Eckhard, Edward Ott. Crowd synchrony on the Millennium Bridge. Nature, Vol. 438. November 2005. 43-44.

• Steven Strogatz and Ian Stewart. Coupled Oscillators and Biological Synchronization. Scientific American, Vol. 269, No. 6. 1993. 68-74.

• Renato Mirollo and Steven Strogatz. Stability of Incoherence in a Population of Coupled Oscillators. Journal of Statistical Physics. Vol. 63, Nos. 3/4. 1991.

  Mirollo and Strogatz analyze the stability of the steady solution $R=0$ in the Kuramoto model described in this article.

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