# 12.6 Many Oscillators and Biological Sync

## KURAMOTO

• The Kuramoto model mathematically captures the behavior of systems of many coupled oscillators.
• Unlike other models of this type, the Kuramoto model is solvable.

We've now seen one possible way to model the rather complicated process of two individual fireflies coming into sync with each other. The mechanism by which this happens is based on each firefly being aware of the other's cycle and making modifications in its own cycle to match it. Synchronization between these fireflies would not be possible were it not for this visual communication taking place.

It's interesting to think of this from the firefly's perspective. At some level, the firefly is aware of what its neighbor is doing and can, intentionally or not, adapt its own cycle to match. With only one neighbor, this may not seem like a big deal, but what about when there are two neighbors? How does our model change if there are more than just two oscillators? In reality, synchronous flashing has been observed in groups of many thousands of fireflies. If we want our model to be as accurate and useful as possible, we must find a way to generalize our model of coupled oscillators to account for synchronization within groups of many oscillators.

One such model was developed by Yoshiki Kuramoto at Kyoto University in the 1970s. In considering large groups of oscillators, it makes things significantly easier to assume that every oscillator affects each of the others equally. In the context of a group of biological oscillators, such as fireflies, one could reasonably expect that fireflies that are further away will actually have less influence than fireflies that are closer. This geographical/spatial factor is ignored in the Kuramoto model. This provides an example of how it is often necessary to make simplifying assumptions about a situation in order to create an understandable, workable model. Doing so provides a foothold from which we can then explore what happens as that model is modified.

What is remarkable about the Kuramoto model is that it is a potentially infinite set of nonlinear, coupled differential equations, and yet it can be solved exactly. The general model itself resembles our system of two equations from the previous section:

EQ:

, i=1...N

This form uses summation notation to compactly state a system of N differential equations, one for each oscillator. What it says is that the change in phase for a specific oscillator (the ith oscillator) depends on both its natural frequency, ωi, and the sum of the influences of the other oscillators. These influences are each related to the difference in phase between the ith oscillator and each other oscillator taken individually, which is why the sum is over j oscillators, even though the equation gives the behavior of the ith oscillator. Furthermore, the amount of influence that each other oscillator has on the ith one, K, is divided evenly by the total number of oscillators, N.

The Kuramoto model can be used to explain many different biological phenomena because of its simplicity and the fact that it can be solved. Systems of nonlinear, coupled differential equations can only rarely be solved exactly. Solutions to the Kuramoto model resemble somewhat our conclusions from the two-oscillator model, most notably the finding that spontaneous synchronization occurs depending on the relationships between differences in natural frequency and the strength of the interaction between oscillators.

In the realm of biology, there are many examples of situations in which the Kuramoto model is applicable. We've already seen how it applies to fireflies, and there are a couple of other fairly common yet fascinating examples from the biological world.

Crickets and frogs communicate with cyclic sound much as fireflies do with cyclic light. In some parts of the country, the night-time soundscape is full of the chirps of crickets and the chorus of frogs croaking. Sometimes these sounds can spontaneously synchronize within a species in a process that is similar to how fireflies synchronize their flashes.

## BEYOND BUGS

• Heart pacemaker cells exhibit spontaneous synchronization in their firing of electrical impulses.

Biological synchronization is by no means limited to insects and amphibians, however. The cells that make up the human heart's natural pacemaker, the rhythm keeper that controls the electrical signals that cause the heart to pump, display a propensity for spontaneous synchronization. Each cell can be thought of as an individual oscillator, in much the same way that a firefly can, but with a few key differences.

Recall that with the firefly, we modeled the cycle of its flashes as a smooth sinusoidally varying function. A heart cell's electrical firing is better modeled as a pulse. The voltage across a cell builds slowly until it reaches some threshold; at that point the cell discharges most of its voltage rapidly.

Each cell has a form of communication with its neighbors via the voltages that discharge. When one cell fires, it kicks up the voltages of its neighbors so that if they are close to their firing threshold, they fire. This has a synchronizing effect on all the nearby cells that were approaching their firing threshold when the first one fired. Cells that were not close to firing get knocked further out of sync with the others.

At first glance, it might seem that this would lead to disorganized behavior among some cells and organized behavior among others. What actually happens is that as certain cells near their firing threshold, voltage begins to leak out in small amounts, to be absorbed by the neighboring cells. This leakage would have little effect if there were only one or two cells, but in a group of thousands, the leakage has a homogenizing effect on the average voltage across each cell. In time, this leads to synchronization of the entire system, not just particular groups of cells.

Cells that build up charge and then discharge precipitously are not modeled well by the Kuramoto model. Math that involves sharp changes often gets tricky. These issues were successfully tackled, however, by Charlie Peskin at New York University in 1975. He was able to show mathematically how synchronization is possible for the entire cardiac firing system.

We have been talking mainly about cyclical synchronization up to this point, but there are other forms of spontaneous order that arise in nature, such as flocking and schooling. Believe it or not, even traffic congestion/flow often results in spontaneous order. The models for these phenomena are not as simple as the Kuramoto model, but the basic mechanism is the same. Spontaneous order emerges naturally in systems in which the individuals communicate with each other in some fashion and make small group-adaptive changes based on those signals. What's fascinating is that these individuals need not be organisms, and the signals exchanged can be much simpler than a cricket's chirp or the voltage spikes of the heart's pacemaker cells. Let us now turn our attention to synchronization of inanimate objects.