Wave propagation and blocking in inhomogeneous media

Wave propagation governed by reaction-diffusion equations in homogeneous media has been studied extensively, and initiation and propagation are well understood in scalar equations such as Fisher's equation and the bistable equation. However, in many biological applications the medium is inhomogeneous, and in one space dimension a typical model is a series of cells, within each of which the dynamics obey a reaction-diffusion equation, and which are coupled by reaction-free gap junctions. If the cell and gap sizes scale correctly such systems can be homogenized and the lowest order equation is the equation for a homogeneous medium [11]. However this usually cannot be done, as evidenced by the fact that such averaged equations cannot predict a finite range of propagation in an excitable system; once a wave is fully developed it propagates indefinitely. However, recent experimental results on calcium waves in numerous systems show that waves propagate though a fixed number of cells and then stop. In this paper we show how this can be understood within the framework of a very simple model for excitable systems.