Strogatz examines the underlying process of creating patterned behavior in situations where there is no obvious conscious control or even intention. These phenomena arise from “coupled oscillation”—that is, the tendency of phenomena at all levels of existence to synchronize their rhythmic features. The classic example: southeast Asian fireflies that flash in synchrony over miles of countryside.

Other natural examples are discussed:

• earthquakes
• forest fires
• mass extinctions
• synchronization of female menarche
• heart attacks

The Mathematics of Sync

The underlying requirement for coupled oscillation or sync to occur is for phenomena to operate in cycles and for the players in the phenomena to be able to influence each other mutually. In addition, a catalyst may sometimes be necessary. One of variables is pulsed communication vs. continuous interaction: continuous interaction creates more complex, subtle sync.

Some observed qualities/principles of sync:

• One-to-one coupling grows to many-to-many coupling.
• Catalysts tend to initially sync some players but desync others further.
• If one oscillator kicks another over a threshold, they will remain synchronized forever.
• Self-organized criticality and cascading behavior (described by Per Bak’s statistics of cascades).
• Frequency pulling (from Norbert Wiener): the tendency of oscillators to pull stray patterns into sync with the group is the universal mechanism of self-organization.
• Influence function (amplitude), sensitivity function (tendency to speed up or slow down), and level of connectivity shape the individual behaviors of players in the system and the time it takes them to self-organize (Winfree).
• Whenever the whole is different from the sum of the parts; whenever cooperation or competition is going on, the governing equations are nonlinear.
• A threshold of similarity is equivalent to a “phase transition.”
• Kuramoto’s rule: oscillators symmetrically adjust by making the minimum adjustment that allows them to communicate.

Disturbances to an equilibrium system grow as a function of the similarity of the individual players; if the players are nearly identical, the disturbances grow exponentially.

Link between biology and physics: “mutual syncronization is analogous to a phase transition, like the freezing of water into ice. The main difference is that when oscillators freeze into sync, they line up in time, not space.”

Frequency pulling tends to produce a pattern distribution that is unlike the familiar bell curve; instead it has a tall, narrow central peak and two weak peaks on either side—this is a possibly a description of a “standard” distribution to a synchronized or self-organized system.

“Virtually all major unsolved problems in science today have this intricate character…a complex, self-organizing system where everyone changes the state of everyone else.” Examples cited: biochemical cell reactions that lead to cancer; stock market booms and crashes; emergence of consciousness from firings of brain neurons; origin of life in the chemical reactions of the primordial soup.

Kuramoto’s rule in more detail: the amount of adjustment between pairs of oscillators is given by the sine function of the ange between them, multiplied by a number called the “coupling strength,” which determines the maximum possible adjustment. Breakthrough in this idea was the symmetrical relationship between oscillators, compared to Winfree’s concepts of frequency pull and sensitivity.

Kuramoto continued: all systems will migrate toward a state in which the order parameter and speed of the pack are constants. There are ultimately only two such states: an order parameter of 0, in which the system will never display synchrony; a “partially synchronized” state consisting of three groups: a synchronized pack of average speed, a slower desynchronized swarm of dawdlers, and a faster desynchronized swarm of sprinters. This latter case is possible only up to a certain threshold of diversity. You can predict how ordered the pack will be as a function of with width of the bell curve.

David Welsch & Steve Reppert (Mass Gen Hospital): “the brain contains a population of oscillators with distributed natural frequencies which pull one another into synchrony and make a more accurate oscillator en masse than individually. Wiener anticipated all that, but he missed an important detail: Instead of cycling 10 times per second, these cells cycle about a million times slower. These are the cells of the circadian pacemaker, the internal chronometer that keeps us in sync with the world around us.”

Strogatz’s breakthrough idea was to view oscillators as fluids.

Sync and cooperation:

“Reproductive sync has benefits for all if the females in the group are cooperative…It could be that women unconsciously strive to ovulate and conceive in step with their friends (to allow them to share child-rearing and breast-feeding duties) and to keep out of step with their enemies (to avoid competing with them for scarce resources)….Female rats in a synchronized group produce larger and healthier offspring than those reared by a solo mother.”

Cooperation in the context of oscillators means ability to sense one another’s rhythms and react to stay in step. [implications for growth of sensors?]

“When the system was self-synchronizing, Winfree found that no oscillator was indispensable. There was no boss. Any oscillator could be removed and the process would still work. Furthermore, the pack did not necessarily run at the speed of its fastest member. Depending on the choice of influence and sensitivity functions, the group could run at a pace nearer the average speed of those in the pack, or it could go faster or slower than any of its members. It was all wonderfully counterintuitive. Group synchronization was not hierarchical, but it wasn’t always purely democratic either.” (p. 52-53).

Human problems that sync can help explain, solve, interpret:

What causes fads, crowd behavior, and mob psychology?

While much of sync theory focuses on rhythmic phenomena, repeating the same cycles, human behavior is more complex. Thresholds are a focus here. Relevant research comes from:

• Thomas Schelling: discoverer of the “tipping point.”
• Mark Granovetter: rioting behavior (interesting that it is hard to distinguish a person who will readily join a riot from one who will join only after someone else starts rioting; i.e., normal sociological and psychological segmentations won’t catch the difference); also the threshold is quite low.
• Duncan Watts: innovation modeling by network analysis reveals two tipping points: 1) islands link together creating global cascades; 2) a dilution effect when cascading vanishes entirely because a node has too many neighbors. Also the concept of a “vulnerable cluster” which is equivalent to “early adopters.”

Traffic congestion

A basic chaos theory problem; key research comes from:

• Dirk Helbing and Bernardo Huberman: auto traffic synchronizes at a certain density, with large trucks setting the gating speed.
• Boris Kerner and Hubert Rehborn: traffic slows to crawl as on-ramp increases density, but stays in that sync for two hours after the heavy on-flow, “trapped” in a stable, but suboptimal sync; they have to be “defibrillated” in order to regain free flow.

Intentional collective action:

Examples of this are dance, singing, “waves” at football games, audience applause (in Europe); and on the dark side: totalitarianism: Nietszsche: “In individual, insanity is rare, but in groups, parties, nations, and epochs it is the rule.”

How the brain gives rise to the mind:

Acts of cognition are linked to brief surges of neural synchrony.

• Christoph von der Malsburg (USC) asked what physical process binds all the chaos of sensations to allow the perception of a single object. Hypothesis: separate banks of neurons throughout the brain all oscillate in sync for a fraction of second.
• Charles Gray and Wolf Singer: tests hypothesis with cats.
• Jurgen Fell (U. of Bonn) tests human subjects (epileptics with implanted electrodes), found link between sync activity on first viewing words to be memorized and ability to recall later.
• Francisco Varela: “mooney face” experiments reveal a flurry of “gamma oscillations” that mark the moment of perception, recognition occurs when discharges in multiple regions of the brain sync, followed by “active desynchronization.”
• Christof Koch and Francis Crick: Consciousness may be the subjective experience of these states of synchrony passing through our brains.