Synchronization in Nature
Module 2: Synchronization and Collective Behaviour
Synchronization is the adjustment of the rhythms of self-sustained oscillators due to a weak interaction between them (Pikovsky, Rosenblum, and Kurths 2001). Two ingredients matter. Each unit oscillates on its own, with its own natural frequency. The units interact, and the interaction changes their timing, not their existence. This module models both ingredients with phase variables.
Three classical examples
Fireflies
Males of some Southeast Asian firefly species, such as Pteroptyx malaccae, gather in trees and flash in unison. Buck (1988) reviews the field and laboratory evidence. A single firefly can also entrain to a periodic flashing light, which is the experiment behind the model in Phase Locking (G. B. Ermentrout and Rinzel 1984; B. Ermentrout 1991).
Metronomes on a moving base
Place two or more metronomes on a light board that rests on two cans. The board moves slightly with each swing, and this motion couples the pendulums. Pantaleone (2002) shows that the metronomes usually end up swinging in phase, with antiphase motion only under special conditions. He also shows that the system is a mechanical realization of the Kuramoto model you study in Session 11.
Cardiac pacemaker cells
The sinoatrial node sets the heartbeat. Michaels, Matyas, and Jalife (1987) simulated two-dimensional arrays of electrically coupled sinus node cells with different intrinsic periods. The activation maps looked like a wave spreading from a single leading pacemaker, yet the pattern came from the mutual entrainment of all the spontaneously beating cells.
Modelling choices
The examples share a structure, and the models in this module keep only that structure.
| Example | Oscillator | Coupling |
|---|---|---|
| Fireflies | Flash cycle of one insect | Light from the other insects |
| Metronomes | Pendulum with escapement | Motion of the common base |
| Pacemaker cells | Membrane potential cycle | Electrical current between cells |
| Millennium Bridge | Gait of one pedestrian | Lateral motion of the deck (Steven H. Strogatz et al. 2005) |
Winfree (1967) proposed describing each unit by a phase alone and studying large populations of such phase oscillators. Kuramoto (1975) simplified Winfree’s model into the solvable form that now carries his name (Steven H. Strogatz 2000). A different route treats the interaction as short pulses, as in the integrate-and-fire model of Mirollo and Strogatz (1990).
Pick the metronome example. Name the state variable of one oscillator, the physical quantity that carries the coupling, and one parameter that you expect to control whether synchronization happens.
The state of one metronome is the angle of its pendulum, which you can reduce to a phase on the circle. The coupling is carried by the horizontal displacement of the base. A natural control parameter is the mass of the base relative to the pendulums: a heavier base moves less and couples the metronomes more weakly. The spread of natural frequencies (the tempo settings) matters too. Pantaleone (2002) analyses both.
Where this module goes
- Flows on the Circle introduces the phase variable for one oscillator.
- Phase Locking couples one oscillator to a periodic stimulus.
- The Kuramoto Model couples many oscillators to each other.