Firefly Synchronization: 2D Kuramoto Flash-Coupling Lab

The 3D original renders fireflies drifting through a forest scene with a soft flicker on each particle, but nothing in it actually couples the fireflies to each other — every flash is an independent, unrelated animation. This 2D companion replaces that with the real phenomenon "fireflies in a forest" is known for: synchronized flashing, modeled with the mean-field Kuramoto equations that explain how swarms of species like Pteroptyx malaccae lock their flashes into unison purely through local visual coupling. Each firefly carries its own phase and a slightly different natural frequency (drawn from a normal distribution you widen or narrow with the frequency-spread slider); every frame, each one nudges its phase toward the population's average phase, weighted by the coupling strength K. A live readout tracks the Kuramoto order parameter R — 0 when flashes are scattered randomly, climbing toward 1 as the swarm locks into a single pulse — alongside the mean phase and the swarm's aggregate flash rate, so the underlying synchronization dynamics are visible rather than merely implied by a pretty scene.