Many of the new synchronization patterns arise in networks of oscillators, which have specific sets of connections, rather than all being coupled to one another, as assumed in the original Kuramoto model. Networks are better models of many real-world systems, like brains and the internet.

In a seminal paper in 2014, Louis Pecora of the United States Naval Research Laboratory and his coauthors put the pieces together about how to understand synchronization in networks. Building on previous work, they showed that networks break up into “clusters” of oscillators that synchronize. A special case of cluster sync is “remote synchronization,” in which oscillators that are not directly linked nonetheless sync up, forming a cluster, while the oscillators in between them behave differently, typically syncing up with another cluster. Remote synchronization jibes with findings about real-world networks, such as social networks. “Anecdotally it’s not your friend who influences your behavior so much as your friend’s friend,” D’Souza said.

In 2017, Motter’s group discovered that oscillators can remotely synchronize even when the oscillators between them are drifting incoherently. This scenario “breeds remote synchronization with chimera states,” he said. He and his colleagues hypothesize that this state could be relevant to neuronal information processing, since synchronous firing sometimes spans large distances in the brain. The state might also suggest new forms of secure communication and encryption.

Then there’s chaotic synchronization, where oscillators that are individually unpredictable nonetheless sync up and evolve together.

As theorists explore the math underpinning these exotic states, experimentalists have been devising new and better platforms for studying them. “Everyone prefers their own system,” said Matthew Matheny of the California Institute of Technology. In a paper in Science last month, Matheny, D’Souza, Michael Roukes, and 12 coauthors reported a menagerie of new synchronous states in a network of “nanoelectromechanical oscillators,” or NEMs — essentially miniature electric drumheads, in this case. The researchers studied a ring of eight NEMs, where each one’s vibrations send electrical impulses to its nearest neighbors in the ring. Despite the simplicity of this eight-oscillator system, “we started seeing a lot of crazy things,” Matheny said.

The researchers documented 16 synchronous states that the system fell into under different initial settings, though many more, rare states might be possible. In many cases, NEMs decoupled from their nearest neighbors and remotely synchronized, vibrating in phase with tiny drumheads elsewhere in the ring. For example, in one pattern, two nearest neighbors oscillated together, but the next pair adopted a different phase; the third pair synced up with the first and the fourth pair with the second. They also found chimeralike states (though it’s hard to prove that such a small system is a true chimera).

NEMs are more complicated than simple Kuramoto oscillators in that the frequency at which they oscillate affects their amplitude (roughly, their loudness). This inherent, self-referential “nonlinearity” of each NEM gives rise to complex mathematical relationships between them. For instance, the phase of one can affect the amplitude of its neighbor, which affects the phase of its next-nearest neighbor. The ring of NEMs serves as “a proxy for other things that are out in the wild,” said Strogatz. When you include a second variable, like amplitude variations, “that opens up a new zoo of phenomena.”

Roukes, who is a professor of physics, applied physics, and biological engineering at Caltech, is most interested in what the ring of NEMs suggests about huge networks like the brain. “This is very, very primordial compared to the complexity of the brain,” he said. “If we already see this explosion in complexity, then it seems feasible to me that a network of 200 billion nodes and 2,000 trillion [connections] would have enough complexity to sustain consciousness.”

Broken Symmetries

In the quest to understand and control the way things sync up, scientists are searching for the mathematical rules dictating when different synchronization patterns occur. That major research effort is unfinished, but it’s already clear that synchronization is a direct manifestation of symmetry — and the way it breaks.