The mathematical explanation for “spontaneous synchronization”

Birds do it. Bugs do it. Even audiences at a play do it. The cells in your body are doing it right now, and it’s pretty amazing. 

What they all are doing is synchronizing. From lightning bugs flashing in rhythm in a summer field, to an audience’s thunderous applause somehow falling into a beat, life and the Universe offer multiple, remarkable examples of spontaneous synchronization across populations. While there are still deep mysteries about how this occurs, scientists have already captured the basic mechanism that not only explains spontaneous synchronization, but may offer some fundamental clues about life and its use of information.

Science of synchronization

Scientists have confronted the mystery of synchronization since the very birth of science. In 1665, Christiaan Huygens, who invented pendulum clocks, wrote of seeing a strange kind of sympathy shared by pendulums positioned next to each other. After each one started out of phase — in other words, swinging at its own rhythm — the two pendulums soon entered into a perfect dance. Being the brilliant physicist he was, Huygens deduced there must be some subtle and imperceptible movements of the material supporting both pendulums that drove them to synchronize.

The subject would later broaden beyond mechanical phenomena. In 1948, Norbert Weiner wrote a book called Cybernetics that focused on the twin problems of control and communication in systems. In his book, Weiner asked how large populations of crickets or neurons manage to synchronize their behavior so that their chirps or their neural firings end up moving in lock-step progression. 

So, if both the living and non-living worlds display spontaneous synchronization, what are the key elements needed to capture its essence?

Couplings and oscillators

The critical advance in the field came by recognizing that all cases of synchronization could be captured mathematically using two components. First, there is a population of oscillators — a fancy mathematical way of saying anything that repeats itself. A pendulum is a mechanical oscillator. A neuron repeatedly firing in a brain is a cellular oscillator. Lightning bugs flashing in a field are animal oscillators. 

The next step is to allow for some kind of coupling between all the individuals. The pendulums rest on a table. The neurons have connections to other neurons. The fireflies can see each other lighting up. These are all examples of couplings. 

With these two components, the entire problem can be cleanly captured in mathematics using what’s called dynamical systems, which is basically differential equations on steroids. That’s exactly what Yoshiki Kuramoto did in two papers, written in 1975 and 1982. The so-called Kuramoto Model has become the gold-standard foundation for studying spontaneous synchronization. The Kuramoto Model revealed the balance between the strength of the coupling between oscillators and the diversity of innate frequencies within each of them. 

What’s the frequency, Kuramoto?

If every cricket is chirping with its own pulse — a pulse completely random when compared with all the other crickets — then only a very strong coupling will lead to a beautiful synchronization of chirps. Here, “strong coupling” means the crickets are really paying attention to each other. Weak coupling would mean the crickets do hear each other, but they are not motivated to pay much attention. Only if all the crickets have innate chirping frequencies that are relatively close to each other can they fall into synchronization, and then they can do so even with weak coupling. 

A  wide range of innate frequencies needs strong couplings for synchronization. A small range of innate frequencies only needs weak couplings for synchronization.

The most important feature the Kuramoto Model revealed, however, was the distinct phase transition in these kinds of systems. A phase change is a relatively abrupt shift from one kind of behavior (no synchronization) to another (complete synchronization). Scientists found the Kuramoto Model displayed a clear onset of synchronization, which is the hallmark of a phase change. As the coupling strength between a population of oscillators increases, they will make the sudden transition from chaos into chorus. 

The Kuramoto Model is a beautiful example of a simple mathematical system that is able to capture complicated behavior in a complex system. That’s why my colleagues and I are using it as a first step in trying to develop a theory of semantic information. We were recently awarded a grant from the Templeton Foundation to understand how life uses information to create meaning — something normal information theory doesn’t really address. Because the Kuramoto Model is both simple and speaks to the kind of remarkable behavior life displays, we plan on seeing whether we can recast it into an information-theoretic framework. If it works, then we might just see a little deeper into how life and the Universe make meaning out of harmony.