 Cubic fast-slow reaction-diffusion model

This nondimensional two-variable reaction–diffusion system combines a fast cubic local response with slow negative feedback. With $D>0$, $0<\alpha <1$, $\beta >0$ and $\epsilon \ll 1$, its spatially uniform rest state at $I=0$ is linearly stable: the Jacobian has trace $-\alpha -\epsilon \beta$ and determinant $\epsilon (1+\alpha \beta )$. For suitable parameters a finite perturbation can nevertheless cause a large excursion followed by recovery. Frozen recovery gives a bistable cubic reaction-diffusion equation; the full coupled travelling-wave boundary-value problem determines whether front pairs form a pulse. This is a generic excitable-medium formulation of the FitzHugh-Nagumo model, without assigning any biological variable or application.