Abstract
We study a sigmoidal version of the FitzHugh-Nagumo reaction-diffusion system based on an analytic description using piecewise linear approximations of the reaction kinetics. We completely describe the dynamics of wave fronts and discuss the properties of the speed equation. The speed diagrams show front bifurcations between branches with one, three, or five fronts that differ significantly from the classical FitzHugh-Nagumo model. We examine how the number of fronts and their speed vary with the model parameters. We also investigate numerically the stability of the front solutions in a case when five fronts exist.
The classic FitzHugh–Nagumo (FHN) equations have been widely studied as a model for wave propagation and pattern formation in excitable media such as neural and reaction-diffusion systems. In that model, the inhibitor or recovery variable, v, has linear kinetics. More sophisticated models employ a more realistic sigmoidal kinetics for v. Here, we study a sigmoidal FHN-type model modified by replacing the nonlinear terms in the kinetics of both variables by piecewise linear functions, which enables us to obtain analytic expressions for wavefronts and their speeds. We examine how the number of fronts and their dynamics vary with such model parameters as the excitation threshold of the activator variable and the ratios of kinetic time scales and diffusion coefficients between activator and inhibitor.