Stability of front solutions in inhomogeneous media

Abstract

We use the FitzHugh–Nagumo (F–N) model to study front solutions in inhomogeneous media. Inhomogeneity is modeled by adding a space-dependent term, z( x), in the equations. Changes in z shift the cubic nullcline up and down in the phase-plane of the F–N model. This is used to mimic shifts in the cubic-shaped nullcline of a bidomain model of calcium waves in egg cells [Physica D, in press] caused by changes in the total calcium concentration. Conditions for the existence and stability of stationary front solutions for a general class of z functions are obtained analytically when the dynamics are piecewise linear. We show that spatial inhomogeneities cause pinning and oscillations of front solutions. This is best demonstrated when z( x) is a linear ramp. When the slope is large, a linear ramp breaks the translational invariance of the stationary front and stabilizes the front. The front becomes less stable as the slope decreases. At a critical slope, the front becomes unstable through a Hopf bifurcation beyond which oscillations in the front occur. This also occurs for nonlinear spatial inhomogeneities including a step increase with varying magnitude and steepness. This bifurcation has been found previously [Physica D 106 (1997) 270; Phys. Rev. E 55 (1997) 366] in a local analysis of the F–N model near a co-dimension two point (sometimes called a non-equilibrium Ising–Bloch (NIB) bifurcation) and with inhomogeneities of small magnitude. The analysis presented here applies globally in parameter space and is valid for inhomogeneities of arbitrary magnitude. We show that this bifurcation need not occur in conjunction with the NIB point. It is caused exclusively by spatial inhomogeneity. A rich variety of wave phenomena including front oscillation, front pinning, and front reflection are shown to occur in inhomogeneous media.