Rotating waves in simple scalar excitable media: approximations and numerical solutions

Rotating waves, in a two-dimensional annular region, are studied using a number of semi-analytical methods. For our kinetics, we use a simple one-variable phase model that arises as the normal form in the transition between excitability and oscillation at a saddle-node invariant circle (SNIC) bifurcation. After deriving asymptotic expressions for the scalar dispersion relationship, we use this to compare the approximations to a direct numerical simulation of the governing nonlinear partial differential equation. All of the approximation methods are based on writing the full solution [Formula: see text] as [Formula: see text] + "perturbation" terms where V is some function and [Formula: see text] is the radial phase shift. Finding an expression for the radial phase shift is the main aim of this paper. It is found that the total twist of the spiral is not a monotone function of the excitability. The largest twist occurs in the vicinity of the transition from excitable to oscillatory behavior.