Barkley's model of an excitable medium

Consider two-variable system of reaction - diffusion equations [1]
    ∂u/∂t = (1/ε) u(1 - u)(u - uth ) + Δu ,
    ∂v/∂t = u - v ,     uth = (v - b)/a .

The local dynamics (in the absence of diffusion) is illustrated in Fig.1. u = v = 0 is the stable fixed point. For u > uth the variable u (the red curve) switches quickly to 1. Due to the small parameter ε it is fast in comparison to the recovery variable v (the green curve) which grows exponentially. When uth = (v + b)/a exceeds u the excitation is quenched to 0 and v decays exponentially. In order to take large time steps a semi-implicit integration scheme is used [1,2].

On 2D plane the u values are displayed by the red color and the v values by the green one. Diffusion propagates excitation through the plane and we get nonlinear waves.

In this script a = 0.5, b = 0.04, ε = 0.02. As since 8 bit floating point values (in RGBA textures) are not enough, therefore floating numbers are stored broken in two cells (bytes).

[1] Dwight Barkley "A model for fast computer simulation of waves in excitable media" Physica D 49 (1991) 61-70
[2] M.Dowle, R.M.Mantel and D.Barkley "Fast simulations of waves in three-dimensional excitable media" Int. Journal of Bifurcation and Chaos, Vol. 7, No. 11 (1997) 2529-2545

Simulations on GPU     updated 14 July 2010