Firefly Synchrony
Pulse-coupled integrate-and-fire oscillators (Mirollo & Strogatz 1990)
Mirollo-Strogatz Theorem (1990): N identical pulse-coupled integrate-and-fire oscillators with excitatory coupling converge to synchrony from almost all initial conditions.
Each firefly has an internal phase φ ∈ [0,1]. When φ=1, it fires and resets to 0, while all fireflies within range receive a pulse that advances their phase:
φ → f⁻¹(f(φ) + ε), where f(φ) = (1/b)ln(1+[e^b−1]φ) (concave PRC)
The concavity of the phase response curve (PRC) guarantees absorption — fired pulses can merge phases, creating synchrony clusters that grow until all oscillators fire together.