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Focus on Mathematics

Mathematics of Synchronization Phenomena: Hayato Chiba, Assistant Professor

1. History of synchronization phenomena

The history of synchronization phenomena dates back to 17th century Europe during the Age of Discovery. At that time, an error-free timepiece was essential for safe and accurate navigation, but sundials, which were still the most common device for keeping time, were not much use on board a ship since the position of the sun at any time of the day differed according to location. Thus, for scientists, the development of an accurate mechanical clock became a prime objective. Against this background, Christiaan Huygens (1629 – 1695, known for the Huygens principal of wave diffraction) invented the pendulum clock and devoted much effort to improving it. He also noticed that the pendulums of two pendulum clocks hanging on a wall would eventually move in unison. This was because the two pendulums would exert a force on each other through minute vibrations in the wall. In this way, Huygens is said to have discovered synchronization phenomena. We can perform a simple experiment on pendulum synchronization using metronomes. Please take a look at the following animation.

Initially moving separately……
but eventually moving in unison.


In general, synchronization phenomena refer to the ways in which a set of identical “things” eventually behave in unison through mutual interaction.




2. What kinds of synchronization phenomena exist in the world?

At present, a variety of synchronization phenomena have been discovered in the natural world especially in relation to living things. For example, if a large number of fireflies gather around a single tree, they will eventually start to flash at the same time creating a bright flashing light. (Please check out videos of this phenomenon on YouTube.) Similarly, there are types of birds and frogs that chirp or croak with the same period as other birds or frogs in the same area producing a very noisy effect.

The human body, as well, is a treasure house of synchronization phenomena. For example, our heartbeat originates in the pulsation of the myocardial cells making up the muscles of the heart. If each of these cells was to pulsate on its own, the heart would not be able to operate smoothly. Cells that vibrate in sync, however, give rise to a healthy heartbeat. In fact, poorly synchronized cells can result in a heart attack or other disorder. To give another example, we know that our limbs move on the basis of instructions received from the brain. These instructions are conveyed to the limbs through electrical signals, so we might ask, “How is it that the brain creates electricity?” Well, each cell (neuron) in the brain can produce a small current by using the potential difference between the inner side and outer side of the cell membrane. If many cells discharge this current in sync, however, a large electrical signal can be produced.




3. What makes synchronization phenomena interesting?

Synchronization phenomena have come to be researched in an interdisciplinary manner by various types of researchers. As shown by the above example, the research of synchronization phenomena can aid in obtaining a greater understanding of the human body, so it is not surprising that researchers like physicists and biologists are researching synchronization phenomena with much zeal. Here, using a few specialized terms, I would like to show you how synchronization phenomena can also be interesting as a target of research in the field of mathematics.

I think that anyone who has studied some physics even in high school will know that the motion of a charge or mass point can be described by Newton’s equations of motion. Generally speaking, a representation of natural phenomena using mathematical equations is called “modeling.” Now, let’s describe the positionx1(t) of a single pendulum by the following differential equation:

For readers not familiar with differential equations, just consider for now that solving the above equation will give x1(t), which is the function that describes the position of the pendulum (the pendulum’s angle of deflection) with respect to time. Next, let’s prepare another pendulum clock completely identical to the first one. Since this second clock is identical to the first one, its position x2(t) will satisfy the same equation as the one given above. Additionally, if we hang these two clocks next to each other on a wall, we can make them interact with each other. We must then add a term expressing this interaction to the differential equation for each of the clocks in the following way:

Here, G(x1, x2) expresses the magnitude of the force exerted by the first pendulum on the second one, and G(x2, x1) expresses the magnitude of the force exerted by the second pendulum on the first one. The field that studies how the solution of these equations behaves as time t progresses is called “dynamical systems theory.” If x1(t) = x2(t) for sufficiently large t, the two pendulums are said to be synchronized. What happens now if we have many more pendulums? If you have trouble imagining such a large number of pendulums, just think of cells instead. Now, N number of objects that mutually interact with each other should behave according to the following N-dimensional equation (N variables):

The term G(xi, xj) expresses the magnitude of the effect exerted by the jth object on the ith object. The difficulty of researching synchronization phenomena lies in the fact that N may be extremely large. Equations that have so far been studied in dynamical systems theory have consisted only of a very small number of variables. For example, N=2 when studying the joint movement of the earth and sun. In the case of cells, however, this number may be on the order of hundreds of millions or trillions, which means that existing theory is incapable of attacking such problems. At present, the theory of dynamical systems with many or infinite degrees of freedom in which the number of variables is overwhelmingly large is making steady progress by adding techniques from other fields like statistical mechanics, graph theory, and functional analysis to existing techniques in dynamical systems theory.




4. In what way can synchronization phenomena be useful?

Given that we understand the mechanism behind synchronization phenomena, what kinds of applications can we consider? What we need to keep in mind here is that objects or things that are not very powerful individually can join forces to collectively produce a large amount of power. This property, if skillfully applied, should be applicable to various types of scenarios. For example, in a circuit that connects many Josephson junctions (each of which produces a small amount of current based on quantum-mechanical principles), a large current can be made to flow if the junctions are connected in a way conducive to synchronization. This technique is already being used in medical devices. We can also envision the construction of a pacemaker for the heart by controlling the synchronization of myocardial cells.

The strong point of mathematics is its universality. Whether we are talking about cells or circuits, either subject can be studied by the same means as long as the underlying mathematical structure is the same. Although the research of mathematics progresses at a slow pace, the impact on neighboring fields when a certain problem is solved is immeasurable. Since the field of dynamical systems with many degrees of freedom is still developing, researchers in physics, engineering, and other fields are eagerly anticipating the emergence of powerful new mathematical tools.