This is the Mirollo–Strogatz model of pulse-coupled biological oscillators (SIAM J. Appl. Math., 1990) — the accepted explanation for how Pteroptyx firefly swarms flash in unison with no leader or shared clock. Each firefly carries a phase θᵢ that rises linearly from 0 toward 1 at its own natural period Tᵢ. Its visible "charge" is a concave function of phase:
V(θ) = ln(1 + (e^b − 1)·θ) / b
When θᵢ reaches 1 the firefly flashes and resets to 0. Every other firefly's charge is instantly nudged up by ε: V ← min(1, V + ε), then mapped back to phase with the inverse of V(θ). Because V is concave, a nudge advances a nearly-charged firefly by more than a barely-charged one — an "absorption" effect. If the nudge itself pushes a neighbor over threshold, it fires too, cascading within the same instant.
- Order parameter r — the Kuramoto measure |mean(e^i·2πθ)| of how tightly the phases are bunched; r→1 means every firefly is charging in lockstep.
- ε (coupling) — how hard one flash pulls its neighbors forward. Mirollo & Strogatz proved that for any ε>0 and any concave charging curve, an all-to-all coupled population synchronizes from almost any starting phases — try ε=0 to see the swarm never lock.
- Period spread — natural variation between individual fireflies' free-running periods; more spread makes sync slower and less perfect.