Steven Strogatz:
"Sync" (Hyperion, 2003)

(Copyright © 2009 Piero Scaruffi | Legal restrictions - Termini d'uso )
There is a way to look at processes of self-organization that is alternative to the view based on phase transitions: it is based on synchronized oscillators. Self-organization of a system implies that every part of that system is somewhat "synchronized" in realizing some kind of order.

Clocks and cycles are pervasive in our universe, ranging from the biological clocks of the tiniest organisms to the cycles of human history, from the cycles of electrons around the nucleus to the cycles of planets around their stars. The order of these cycles emerges from the synchronized behavior of "coupled oscillators". The tendency for members of a population to work in unison, to get synchronized, is ubiquitous in nature, from flocks of birds to groups of neurons. It is a general property of our universe as much as gravity and electricity.

The USA mathematician Norbert Wiener (the inventor of Cybernetics) believed that the activity of the brain is synchronized by a clock that is implemented by a group of neurons. Each neuron is a terrible clock, and is highly vulnerable. A group of neurons, though, each of them a bad clock, that influences each other, tends to converge towards a synchronized state that constitutes a much more reliable clock. Wiener proved mathematically that a population of interacting neurons (that adjust their frequency based on what their neighbors are doing) yields over time a population of synchronized neurons. He speculated that "frequency pulling" might be a universal method of self-organization for any complex system.

The USA mathematician Charles Peskin ("Mathematical Aspects of Heart Physiology", 1975) reached a similar conclusion when studying the "clock" of the heart: its rhythm is set not by a specific cell but by the collective behavior of a group of cells. In both cases the advantage of having a population (rather than just one member) provide the clock is that the process becomes fault tolerant: if one fails, the whole corrects it; if one dies, the others are enough for the process to continue.

The USA biologists David Welsh and Steve Reppert ("Individual Neurons Dissociated From Rat Suprachiasmatic Nucleus Express Independently Phased Circadian Firing Rhythms", 1995) discovered that the suprachiasmatic nucleus of the mammalian hypothalamus contains a circadian clock. This "clock" is actually made of a large population of independent, single-cell oscillators that collectively provide an accurante time-keeping. This "master" clock is probably responsible for all the rhythms in the body of a mammal.

Populations of synchronized oscillators seem to be pervasive in nature. The Indian physicist Satyendra Bose discovered in 1924 that at low temperatures all bosons behave like one. Such a Bose-Einstein condensate was first achieved in a gas in 1995. This means that all their quantum waves are synchronized (or, better, "phase coherent").

The British physicist Brian Josephson ("Coupled Superconductors", 1963) discovered an odd quantum phenomenon called "Johnson junction", which is a consequence of quantum tunneling effect: an electric current arises between two weakly-coupled superconductors (whose waves overlap slightly but don't interfere with each other too much) that are separated by a very thin non-conducting barrier. Richard Feynman immediately realized that this was a special case of a universal phenomenon: the Josephson effect will occur for any pair of weakly-coupled phase-coherent systems. It was later realized that the equations for the electrical oscillations in a Josephson junction are identical to the equations of the motion of a pendulum.

The USA physicist Arthur Winfree studied the nonlinear equations of a population of coupled oscillators and verified that, if each oscillator can influence the others, the population as a whole has a tendency to get synchonized. The population does not need any leader in order to achieve this: it's the interaction among the various oscillators of the population that creates the order. Winfree realized that synchronization of a population of oscillators is a phenomenon similar to the phase transition of a substance (for example, water), in which all the molecules of that substance have to "cooperate" in order for the substance to change state (for example, to turn into ice). Organizing a population of oscillators is similar to organizing a population of molecules, except that the former gets organized in time and the latter gets organized in space.

The Japanese physicist Yoshiki Kuramoto proved that a system of such equations always has one obvious solution (the state of incoherence, in which the population is completely disorganized, a state which can actually be implemented in a very large number of ways) and sometimes also has another solution, one of complete synchronization. The latter occurs only when the initial states are not too chaotic. There is a threshold value for their chaos below which that solution of complete synchronization exists and above which it does not exist. The initial states have to be at least partially synchronized. Again, this principle evokes the threshold above or below which a phase transition occurs in a substance.

The USA mathematician Steven Strogatz then proved that the state of incoherence is a state of unstable equilibrium, which means that it will sooner or later collapse into one of the other possible states, the states that represent synchronous behavior. In other words, Strogatz proved that synchronicity (and therefore self-organization) "will" emerge at some point in any system that exhibits partial synchronicity above the threshold.

Strogatz also figured out that, under certain circumstances, a population of coupled Josephson junctions will behave like Kuramoto's biological oscillators: the junctions will suddenly synchronize. Any system made of many independent oscillators that are weakly coupled, and that are coupled with the same intensity to all the others, will exhibit spontaneous synchrony.

The USA physicist John Hopfield ("Neurons, Dynamics and Computation", 1994), the man who had rescued neural networks from oblivion, made the connection with self-organized systems: a system of synchronized oscillator is a self-organized system of the kind studied by Kauffman and Bak. Self-organization can be achieved in time or in space. Some interacting molecules, cells and atoms achieve self-organization in space through spontaneous reorganization, whereas some coupled oscillators achieve self-organization in time through spontaneous synchrony.

Synchronized oscillators exhibit mathematical properties that might explain natural phenomena. For example, chemical reactions can oscillate spontaneously. During such oscillations there is a point (the "phase singularity") that does not oscillate like the rest, in which the cycle amplitude collapses down to zero and its phase cannot be determined anymore. It turns out that this phase singularity generates a spiral wave that cannot be destroyed for as long as the phase singularity exists. The spiral is extremely resilient, almost invulnerable. These spiral waves emerge in chemical, biological and physical systems under the right conditions.

Winfree, Arthur: THE GEOMETRY OF BIOLOGICAL TIME (Springer-Verlag, 1980)
Kuramoto, Yoshiki: CHEMICAL OSCILLATIONS, WAVES, AND TURBULENCE (Springer-Verlag, 1984)

TM, ®, Copyright © 2009 Piero Scaruffi