E64 Synchronization and coordination

Topic

This collective synchronization within an insect colony—the very phenomenon modeled elsewhere in this set—manifests itself in the simultaneous flashing of a firefly (*Photinus pyralis*), the coordinated flight of a swarm, and the colony's overall activity cycles.

The Kuramoto model provides the canonical mathematical framework for describing this synchronization: each insect in the colony acts as an individual oscillator with its own natural frequency—varying across individuals according to the population's specific distribution—while being coupled to the other colony members with a specific coupling strength. As the coupling strength between a given oscillator and the other individuals increases, so too does the oscillator's tendency to synchronize with them.

This coupling between the colony's oscillators involves a critical threshold—analogous to the critical point defining the transition between disordered and ordered swarms in another model in this set: below this threshold, the colony remains asynchronous, whereas above it, partial or total synchronization emerges among the oscillators.

The degree of this partial or total synchronization is quantified by an order parameter—analogous to the one used elsewhere in this set to measure swarm spatial alignment—though here it quantifies temporal alignment (of flashes or movements) rather than spatial alignment between individuals. The specific communication delay between the oscillators in the colony—the time elapsed between an individual detecting a neighbor's flash and subsequently emitting its own—introduces a phase lag between them. If this phase lag between the two flashes increases, the resulting final synchrony among the oscillators may either stabilize or destabilize, depending on the sign of that lag.

Thus, in the model introduced via the example of firefly synchronization, the variability in the oscillators' natural frequencies acts as a form of noise that opposes synchronization, whereas the coupling between the oscillators—also mentioned at the start—reinforces it. If the coupling between the colony's oscillators significantly outweighs the variability of their natural frequencies, the resulting collective synchronization—of the firefly, the swarm, and the colony—becomes even more complete and stable.

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