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.

Males of fireflies of the species Luciola cruciata synchronize flashes on the river bank in Japan



When the rambling applause of the crowd suddenly turns into a single pulse, when everyone starts to clap in unison - who decided it would be like that? Not you, and not someone else. Crickets make sounds synchronously; metronomes set next to swing simultaneously; some fireflies flicker in the dark together. Throughout the US, the grid operates at 60 Hz, and all of its innumerable ac supplies are synchronized by themselves. Our life depends on synchronization. The neurons in the brain are activated by synchronous waves to control our body and mind, and the pacemaker's cells are synchronized to create a heartbeat.



Objects with a rhythm are synchronized in a natural way. 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 pair of clocks with a pendulum hung on the wall next to it - he invented these devices himself. Huygens noticed that the pendulums swayed exactly in unison, drawing closer, and then moving away from each other. Perhaps their air pressure synchronizes? He conducted many experiments. For example, setting the table vertically between them had no effect on synchronization. However, when he moved the clock away and at a right angle, they soon became out of sync. As a result, Huygens decided that the “sympathy” of the clock, as he called it, was due to the strikes transmitted by the pendulums of the clock to each other through the wall.



When the left pendulum swings to the left, it transmits a blow to the wall and leads another pendulum to the right, and vice versa. The clock exchanges beats with each other until it comes to the most stable and relaxed state with the wall. The most stable behavior for the pendulums will be movement in opposite directions, when each of them pushes the other in the direction in which it moves itself - such as how you swing a child on a swing. For a wall, this is the easiest option; it no longer moves, since the pendulums communicate to 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 the blows in them are replaced by other forms of interaction.

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Sketch Huygens experiment with a pair of hours with a pendulum and his attempt to understand synchronization. “B again went through the BD position when A is in AG, while suspension A is attracted to the right and, therefore, the vibration of the pendulum A accelerates,” he wrote. “B is in BK again when A returns to the AF position, 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 evenly slows down and A accelerates, they must move in different phases. "



Another Dutchman, Engelbert Kempfer , traveled to Thailand in 1690 and watched there as the local fireflies blink at the same time “with the utmost regularity and accuracy”. Two centuries later, the English physicist John William Strett (better known as Lord Rayleigh) remarked 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 small inevitable differences." Radio engineers in the 1920s discovered that connecting two electric generators with different frequencies caused them to vibrate at a common frequency — this principle underlies the systems for transmitting radio messages.



And only in 1967, the pulsating chatter of crickets inspired the American theorist biologist Arta Winfrey to create a mathematical model of synchronization. Winfrey’s equation was too complicated to solve, but in 1974, the Japanese physicist Yoshiki Kuramoto understood how to simplify mathematics. The Kuramoto model described a population of oscillators (objects with a rhythm, such as a metronome or a heart), and showed why the combined oscillators synchronize spontaneously.



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



Five years later, Winfrey came across a summary of Kuramoto's speech about his model, and realized that it provides a new, revolutionary understanding of a subtle phenomenon that permeates the whole world. Kuramoto's mathematics was multifaceted and expandable enough to be responsible for synchronizing 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 a widespread use,” Kuramoto told us in his email, who is now 78.



However, despite the universality of the Kuramoto model, all the illusions of physicists about the understanding of synchronization broke in 2001. Once again, Kuramoto was at the center of what was happening.

Watches go differently

In the original Kuramoto model, the oscillator can be depicted as an arrow rotating in a circle at some natural frequency. (If it is a firefly, it may flash whenever the arrow points up). When two arrows connect, the strength of their interaction depends on the sine of the angle between their directions. The greater the angle, the greater the sine, and the stronger the mutual influence. Only when the arrows are parallel, and rotate together, do they cease to influence each other. Therefore, the arrows will shift until they detect a synchronization state. Even oscillators with different natural frequencies, when combined, reach a compromise and fluctuate in tandem.



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





Yoshiki Kuramoto, Professor of Physics from Kyoto University



The first kind of synchronous behavior in a population of coupled oscillators simulated on a computer was noticed by the postdoc Kuramoto from Mongolia, Dorsjuren Battogtokh [Dorjsuren Battogtokh]. Identical oscillators, equally connected with their neighbors, were somehow divided into two groups: some oscillated synchronously, others were incoherent.



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



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



Strogau and his graduate student Daniel Abrams , who is currently studying synchronization as a processor at Northwestern University, reproduced this strange mixture of synchronicity and asynchrony in their own computer simulations and studied the conditions for its appearance. Strohats called it a “chimeric state” in honor of a mythological fire-breathing monster made of incompatible parts. (A few months before, Strohats wrote the popular science book Sync on the prevalence of global synchronization).



Two independent teams, working with different physical systems, implemented 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, to all appearances, is a complex type of chimera, in the sense that it simultaneously supports the synchronous and asynchronous triggering of neurons. Last year, researchers discovered a qualitative similarity between destabilization of chimeric states and epileptic seizures. “We believe that further research may reveal new therapeutic methods for predicting and stopping seizures,” said co-author Irina Omelchenko of the University of Berlin.



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

Good vibrations *

* Link to the popular song The Beach Boys - Good Vibrations / approx. trans.



With the discovery of chimeras in the science of synchronization, a new era began, revealing, presumably, countless exotic forms that synchronization can take. Now the theorists are working to form the rules and reasons for the emergence of various synchronization schemes. They have wildest dreams to understand how to predict and control synchronization in many real-world situations.



Motter and the team are looking for rules to stabilize the synchronization of power grids so that the integration of non-permanent power sources, such as solar panels and wind turbines, into the power grid 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 can be useful in encryption. Scientists argue that the work of the brain and even consciousness, perhaps, can be imagined as a complex and delicate balance of synchrony and asynchrony.



“The topic of synchronization is getting a big resonance,” said Raissa Disuz , a professor of computer science and engineering from the University of California, Davis. "We are creating new tools to study these exotic and intricate patterns that go beyond the simple division 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 suggested in the original Kuramoto model. Networks turn out to be better models of many real-world systems, such as the brain and the Internet.



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



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



And there is also a 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 studying them. “Everyone prefers their own system,” said Matthew Matheny of the California Institute of Technology. In last month’s work in the journal Science, Matheny, Disuza, Michael Rowks and 12 of their co-authors spoke about the whole zoo of new synchronous states in the network of “nanoelectromechanical oscillators”, or NEM - in fact, miniature electric eardrums. The researchers studied a ring of eight NEMs, the vibrations of each of which sent electrical impulses to its closest neighbors in the ring. Despite the simplicity of this system of eight oscillators, “we began to discover many crazy things,” said Matheny.



The researchers documented 16 synchronous states, which the system entered under different initial conditions, although there may be a much larger number of them and more rare states. In many cases, NEMs were disconnected from their nearest 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 couple was in another phase; the third pair was synchronized with the first, and the fourth - with the second. They also discovered chimeric-like states (although it’s 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 discovered. In the "canted" state, the phases of each of the oscillators on top differ from their neighbors by a certain value. In the middle is the "wandering wave", and only opposite arrows remain in phase. Below - the state of "chimera with noise feed." Two sets of arrows are always synchronized, and the arrows between them seem to randomly synchronize with and out of their neighbors.



NEM is more complicated than simple Kuramoto oscillators, since the frequency of their oscillations affects their amplitude (roughly speaking, loudness). This internal independent nonlinearity of NEM leads to the emergence 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 Strohats. When the second variable is turned on, for example, amplitude variations, “a new zoo of phenomena arises”.



Rowx, a professor of physics, applied physics and bioengineering sciences 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 the complexity of the brain,” he said. “If we are already seeing an explosion of complexity, then it is quite reasonable to assume that the network of 200 billion nodes and 2,000 trillion connections will have difficulty in supporting consciousness.”

Broken symmetries

In search of understanding and control over synchronization, scientists are trying to establish mathematical rules governing the emergence 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 link between synchronization and symmetry was first established by Pecora and his co-authors in the cluster synchronization paper of 2014. Scientists have linked various synchronized groups that can occur in a network of oscillators with network symmetry. In this context, symmetry refers to the possibility of replacing oscillators in places without changing the network, much like a square can be rotated 90 degrees or reflected horizontally, vertically or diagonally without changing its appearance.



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



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



Synchronization appears to arise from symmetry, and yet scientists also found that asymmetry helps stabilize synchronized states. “This is a bit of a paradox,” Hart admitted. In February, Motter, Hart, Raj Roy from the University of Maryland and Yuanzhao Zhang from Northwestern University reported in the journal Physical Review Letters that introducing asymmetry into the cluster actually enhances its synchronization. For example, the organization of one-way communication between two oscillators instead of two-way does not only disrupt cluster synchronization, 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 power grids. At the meeting of the American physical community in Boston last month, Motter presented unpublished results, saying that "generators are easier to oscillate with a frequency that is exactly the same if they specifically adjust their parameters 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.



“Creating a suitable combination of synchrony and asynchrony, you can solve a variety of tasks,” Kuramoto noted in an email. - Without a doubt, biological evolution processes are responsible for this extremely useful mechanism. I think that man-made systems will also become much more flexible if we introduce support for similar mechanisms into them. ”