Scientists have discovered new exotic forms of synchronization

In a world that seems filled with chaos, physicists have discovered new forms of synchronization, and are now learning to predict and control them.

Male fireflies of the Luciola cruciata species synchronize flashes on the banks of a river in Japan.

When the incoherent applause of the crowd suddenly turns into a single pulse, when everyone starts clapping in unison - who decided that would be so? Not you, and not someone else. Crickets make sounds synchronously; nearby metronomes swing at the same time; some fireflies flicker in the dark together. Across the United States, the grid operates at a frequency of 60 Hz, and all its innumerable AC inflows are synchronized on their own. Our life depends on synchronization. The neurons in the brain are activated by synchronous waves to control our body and mind, and the cells of the pacemaker are synchronized, creating a heartbeat.

Objects that have a rhythm synchronize naturally. However, no one described this phenomenon until 1665, when the Dutch physicist and inventor Christian Huygens spent several days in bed due to illness. A couple of hours with a pendulum hung on his wall next to him - he invented these devices. Huygens noticed that the pendulums swing exactly in unison, drawing closer, and then moving away from each other. Perhaps they are synchronized by air pressure? He conducted many experiments. For example, setting the table vertically between them did not affect synchronization. However, when he moved the clock away and at a right angle, they soon went out of sync. In the end, Huygens decided that the “sympathy” of the watch, as he called it, was due to the blows transmitted by the pendulums of the clock to each other through the wall.

When the left pendulum swings to the left, it passes a blow to the wall and moves the other pendulum to the right, and vice versa. The watches exchange strokes with each other until they come to the most stable and relaxed state with the wall. The most stable behavior for pendulums will be movement in opposite directions, when each of them pushes the other in the direction in which he moves - like how you swing a child on a swing. For a wall, this option is the easiest; it no longer moves, since the pendulums tell each other the same, but opposite in direction, kicks. The system no longer deviates from such a self-sustaining synchronous state. Many systems are synchronized for similar reasons, and shocks in them are replaced by other forms of interaction.


A sketch of a Huygens experiment with a couple of hours with a pendulum and his attempt to understand synchronization. “B again went through position BD when A is in AG, while the suspension A is pulled to the right, and therefore the vibration of the pendulum A is accelerated,” he wrote. “B is again in BK when A returns to position AF, while the suspension B pulls to the left, and therefore the vibration of the pendulum B slows down. Therefore, when the vibration of the pendulum B uniformly slows down and A accelerates, they must necessarily move in different phases. "

Another Dutchman, Engelbert Kempfer , traveled to Thailand in 1690 and watched there as local fireflies blink simultaneously "with the utmost regularity and accuracy." Two centuries later, English physicist John William Strett(better known as Lord Rayleigh), noted that if you put two organ pipes side by side, this leads to the fact that "the pipes begin to speak in absolute unison, despite the slight inevitable differences." Radio engineers in the 1920s found that connecting two electric generators with different frequencies makes them vibrate at a common frequency - this principle underlies radio transmission systems.

It was only in 1967 that the pulsating chirping of crickets inspired the American theoretical biologist Art Winfrey to create a mathematical model of synchronization. Winfrey's equation was too complicated to solve, but in 1974, Japanese physicist Yoshiki Kuramotofigured out how to simplify math. The Kuramoto model described a population of oscillators (objects that have a rhythm, such as a metronome or heart), and showed why the connected oscillators spontaneously synchronize.

Kuramoto, who was then 34 years old, did not have much experience in nonlinear dynamics - the study of feedback loops linking variables together. When he showed his model to experts in his field, they did not see its significance. Frustrated, he abandoned this work.

Five years later, Winfrey came across an abstractKuramoto’s speech about his model, and realized that it gives a new, revolutionary understanding of the subtle phenomenon that pervades the whole world. Kuramoto's mathematics turned out to be multifaceted and expandable enough to be responsible for the synchronization of clusters of neurons, fireflies, heart cells, starlings in a flock, reacting chemicals, alternating current and a huge number of other populations of interconnected “oscillators”.

“I could not imagine that my model would have such widespread use,” Kuramoto, now 78

, told us in an email. However, despite the universality of the Kuramoto model, all the illusions of physicists about understanding synchronization crashed in 2001. And again Kuramoto was at the center of what was happening.

Watches go differently

In the original Kuramoto model, the oscillator can be represented by an arrow rotating in a circle at a certain natural frequency. (If it is a firefly, it may flash each time the arrow points up). When two arrows are connected, the strength of their interaction depends on the sine of the angle between their directions. The larger the angle, the greater the sine, and the stronger the mutual influence. Only when the arrows are parallel, and rotate together, they stop influencing each other. Therefore, the arrows will move until they detect the synchronization state. Even oscillators with different natural frequencies, when combined, reach a compromise and oscillate in tandem.

However, this basic picture explains only a small part of the overall synchronization, in which the population of oscillators does the same thing. Although this synchronization is in its simplest form, “there are many examples of global synchronization; therefore, people pay so much attention to this, ”said Edilson Motter , a physicist at Northwestern University of Chicago and a leading sync specialist. “But in 2001, Kuramoto discovered something completely different. And from here begins the story of various conditions. ”


Yoshiki Kuramoto, professor of physics at Kyoto University

The first new type of synchronized behavior in a population of coupled oscillators simulated on a computer was noticed by Kuramoto's postdoc from Mongolia, Dorjsuren Battogtokh. Identical oscillators, equally connected with their neighbors, somehow broke into two groups: some oscillated synchronously, others incoherently.

Kuramoto presented the discovery made by him and Buttogtoch in 2001 in Bristol, but this result was not noticed by the community until Stephen Strogatz , a mathematician at Cornell University, came across it, studying conference materials two years later. “When I realized what I see on the charts, I did not believe this,” said Strogac.

“It was very strange that the Universe seemed the same in different places” of the system. And at the same time, the oscillators reacted differently to identical conditions, some of them stacked together, while others went their own way, as if they were not combined with anything. The symmetry of the system “broke,” said Strogac, “in an unprecedented way.”

Strogac and his graduate student Daniel Abrams , who now studies synchronization as a professor at Northwestern University, have reproduced this strange mixture of synchronism and asynchrony in their own computer simulations and studied the conditions for its appearance. Strogac called it a “chimeric state” in honor of a mythological fire-breathing monster made of incompatible parts. (A few months earlier, Strogac wrote a popular science bookSync on the prevalence of global synchronization).

Two independent teams, working with different physical systems, realized this chimeric state in the laboratory in 2012, and since then many more experiments have been carried out. Many researchers suspect that chimeric states appear naturally. The brain itself, apparently, is a complex type of chimera, in the sense that it simultaneously supports synchronous and asynchronous triggering of neurons. Researchers found qualitative similarities last yearbetween destabilization of chimeric states and epileptic seizures. “We believe that further research may discover new therapeutic methods for predicting and ending seizures,” said co-author Irina Omelchenko from Berlin University.

However, the chimeric state is not yet fully understood. Kuramoto designed all the mathematics, confirming that this state is consistent, and therefore possible, but this does not explain its appearance. Strogatz and Abrams worked the math even further, but other researchers would like to get a “more intuitive, physical explanation,” Strogatz said, and added: “I think we can say that we still have not completely understood” why the chimeric state arises.

Good fluctuations *

* Reference to the popular song The Beach Boys - Good Vibrations / approx. perev.

With the discovery of chimeras in the science of synchronization, a new era has begun, opening, presumably, the myriad of exotic forms that synchronization can take. Now theorists are working to formulate the rules and reasons for the appearance of various synchronization schemes. They have bold dreams of understanding how to predict and control synchronization in many real-world situations.

Motter and his team are looking for rules to stabilize the synchronization of power grids so that the integration of volatile power supplies, such as solar panels and windmills, into the power system is more stable. Other researchers are looking for ways to move systems from one state to another, which may be useful for correcting cardiac arrhythmias. New forms of synchronization may come in handy in encryption. Scientists argue that the work of the brain and even consciousness can perhaps be represented as a complex and delicate balance of synchronism and asynchrony.

“The topic of synchronization gets a big resonance,” said Raisa Dysusa, professor of computer science and engineering at the University of California, Davis. “We are creating new tools to study these exotic and intricate patterns that go beyond simply dividing into synchronized and random sections.”

Many of the new patterns of synchronization arise in networks of oscillators with special connections, and not just connected in pairs, as was assumed in the original Kuramoto model. Networks turn out to be better models of many real systems, such as the brain and the Internet.

In fruitful workfrom 2014, Luis Pecora of the US Navy Research Laboratory and his co-authors put together a model for synchronization within networks. Based on previous work, they showed that networks are divided into “clusters” of synchronized oscillators. A special case of cluster synchronization is “remote synchronization”, in which oscillators that are not directly connected to each other are synchronized anyway, forming a cluster, while the oscillators located between them behave differently, usually synchronizing with another cluster.

In 2017, Motter's group discoveredthat the oscillators can be synchronized remotely, even if the oscillators between them behave non-uniformly. This option "crosses remote synchronization with chimeric states," he said. He and colleagues suggested that this condition may be related to information processing by neurons, since synchronized triggering sometimes spreads to large areas in the brain. Also, this condition can lead to the creation of new forms of communication and encryption.

And there is also chaotic synchronization , in which the oscillators, being unpredictable separately, are still synchronized and developed together.

While theorists are studying the mathematics underlying these exotic states, experimenters are developing new, improved platforms for their study. “Everyone prefers their own system,” said Matthew Matheny of the California Institute of Technology. In a work in Science magazine from last month by Matheny, Dysus, Michael Rawkesand 12 of their co-authors talked about the whole zoo of new synchronous states in the network of "nanoelectromechanical oscillators", or NEM - in fact, miniature electric eardrums. Researchers studied a ring of eight NEMs, the vibrations of each of which sent electrical impulses to its nearest neighbors in the ring. Despite the simplicity of this system of eight oscillators, “we began to discover many crazy things,” Matheny said.

Researchers have documented 16 synchronous states that the system entered under different initial conditions, although there may be a much larger number of them and rarer states. In many cases, NEMs disconnected from their closest neighbors and synchronized remotely, vibrating in phase with tiny membranes located elsewhere in the ring. For example, in one case, the two closest neighbors oscillated together, but the next pair was in a different phase; the third pair synchronized with the first, and the fourth with the second. They also discovered conditions similar to chimeric ones (although it is hard to prove that such a small system is a true chimera).






In experiments with a ring of eight coupled oscillators, many synchronization sequences were found. In the “canted” state from above, the phases of each of the oscillators differ from the neighbors by a certain value. In the middle is a "wandering wave", and only opposite arrows remain in phase. Below is the state of the “chimera with noise recharge”. Two sets of arrows are always synchronized, and the arrows between them seem to randomly get into synchronization with their neighbors and exit it.

NEM is more complicated than simple Kuramoto oscillators, since the frequency of their oscillations affects their amplitude (roughly speaking, volume). This internal independent non-linearity of NEM leads to the appearance of complex mathematical relationships between them. For example, the phase of one can affect the amplitude of the neighbor, which in turn affects the phase of the next neighbor. The NEM ring serves as a “mediator for other unknown things,” said Strogac. When you turn on the second variable, for example, amplitude variations, "a new zoo of phenomena arises."

Rocks, a professor of physics, applied physics and bioengineering at Caltech, is more interested in what behaviors of large networks, such as the brain, stem from the properties of the NEM ring, “These are all very basic things compared to brain complexity,” he said. “If we are already witnessing an explosion of complexity, it is quite reasonable to assume that a network of 200 billion nodes and 2,000 trillion connections will have difficulties to support consciousness.”

Broken symmetry

In search of understanding and control over synchronization, scientists are trying to establish mathematical rules governing the appearance of various types of synchronization. This problem has not yet been solved, but it is already clear that synchronization is a direct manifestation of symmetry, as well as its violation.

The connection between synchronization and symmetry was first established by Pekora and his coauthors in their 2014 work on cluster synchronization. Scientists have linked various synchronized groups that can occur in a network of oscillators with network symmetry. In this context, symmetry means the possibility of replacing oscillators with places without changing the network, much like a square can be rotated 90 degrees or reflected horizontally, vertically or diagonally without changing its appearance.

Dysusa, Matheny and their colleagues applied the same powerful formalism in their latest NEM studies. Roughly speaking, a ring of eight NEM has octagon symmetry. But with the vibration of eight tiny membranes and the development of the system, some of these symmetries are spontaneously broken; NEM are divided into synchronous groups corresponding to subgroups in the symmetry group D8, which defines all the methods of rotation and reflection of the octagon, leaving it unchanged. For example, when NEMs are synchronized with their nearest neighbor, propagating patterns of oscillations along the ring in a checkerboard pattern, D8 is reduced to a subgroup D4. This means that the NEM network can be rotated by two positions or flipped relative to two axes without changing the pattern.

Even chimeras can be expressed in the language of clusters and subgroups of symmetry. “The synchronized part is one large synchronized cluster, and the desynchronized part is a bunch of individual clusters,” said Joe Hart, an experimenter at the Navy's Research Laboratory, working with Pecor and Motter.

Synchronization appears to arise from symmetry, and yet scientists also found that asymmetry helps stabilize synchronized states. “It's a little paradoxical,” Hart admitted. In February, Motter, Hart, Raj Roy from the University of Maryland and Yuanzhao Zhang from Northwestern University reported in a journalPhysical Review Letters, that the introduction of asymmetry in a cluster actually enhances its synchronization. For example, the organization of one-way communication of two oscillators, instead of two-way, not only does not violate the synchronization of the cluster, but makes it more resistant to noise and disturbances from the rest of the network.

These discoveries related to asymmetry are confirmed by experiments with artificial energy networks. At a meeting of the American physical community in Boston last month, Motter presented unpublished results suggesting that “it is easier for generators to oscillate with a frequency exactly the same if their parameters are specially configured differently in a special way,” he said. He believes that the tendency of nature to asymmetry will facilitate the task of stable synchronization of various energy sources.

“By creating the right combination of synchronism and asynchrony, you can solve a variety of problems,” said Kuramoto in the email. - Without a doubt, the processes of biological evolution are responsible for this extremely useful mechanism. I think that human-created systems will also become much more flexible if you introduce support for similar mechanisms in them. "