Barkley Excitable Medium

DEFINITION

∂u/∂t = ε⁻¹·u(1−u)(u − (v+b)/a) + ∇²u
∂v/∂t = u − v

NOTES

A medium where every point can fire once and then must rest: u is the excitation racing through, v the recovery chasing it, and the whole chemistry of an excitable reaction is reduced to one cubic and one line. That reduction was the paper's point — the qualitative life of excitable media, spiral waves whose tips drift and meander, needs nothing more expensive. A broken wavefront cannot die out or reconnect, so it curls, and the curl is the spiral: the same figure that organises heart arrhythmia, slime-mould aggregation and the Belousov–Zhabotinsky dish, here from the cheapest equations that produce it.

PROVENANCE

Origin

D. Barkley, "A model for fast computer simulation of waves in excitable media", Physica D 49(1–2), 1991, 61–70

Standing

Public domain — a pair of partial differential equations. No patent

Constants

a = 0.75, b = 0.06, ε = 0.02 sits in the meander region of the paper's own a–b parameter plane; the b slider walks from rigid rotation toward the excitability edge, and the ranges were measured live here rather than transcribed

Source

doi:10.1016/0167-2789(91)90194-E