Wave-Block in Excitable Media Due to Regions of Depressed Excitability

We study propagation failure using the one-dimensional scalar bistable equation with a passive “gap” region. By applying comparison principles for this type of equation, the problem of finding conditions for block is reduced to finding conditions for the existence of steady state solutions. We present a geometrical method that allows one to easily compute the critical gap length above which a steady state solution, and thus block, first occurs. The method also helps to uncover the general bifurcation structure of the problem including the stability of the steady state solutions. In obtaining these results, we characterize the relationship between the properties of the system and propagation failure. The method can easily be extended to other gap dynamics. We use it to show that block associated with any local inhomogeneity must be associated with a limit point bifurcation.