Firefly Synchrony — Pulse-Coupled Oscillators (Mirollo-Strogatz)

Each firefly is an integrate-and-fire oscillator. When it fires (phase φ=1), all others jump by ε·f(φ). Mirollo & Strogatz (1990) proved that for excitatory coupling, total synchrony is reached in finite time from almost any initial condition.

Parameters

Synchrony

Kuramoto R —
Avg phase —
Flashes/cycle —
Time —
Mirollo-Strogatz (1990): Theorem — for any concave-up f, excitatory coupling (ε>0) leads to absorption events. Eventually all oscillators merge into one cluster and fire simultaneously.

Coupling: when firefly j fires, φᵢ → f⁻¹(f(φᵢ)+ε)
f(φ) = [exp(bφ)−1]/(exp(b)−1)