Reflected waves in an inhomogeneous excitable medium

Propagation can be encumbered in an excitable cable in which intrinsic properties change abruptly. A sudden increase in diameter or a decrease in conductivity or excitability can lead to propagation block or delay in propagation with or without reflection. We study such transient phenomena from a geometric point of view. A simple two-cell caricature, with one cell enlarged to mimic diameter increase, is developed and analyzed. Our analysis indicates that reflected waves may result from the existence of an unstable periodic orbit. As the inhomogeneity parameter is varied, this unstable cycle is nearer to and then farther from the initial state that mimics an incoming wave. This fact leads to a variety of complicated reflected waves. Correspondingly, we find numerically complex sequences of reflected-transmitted waves in biophysically more realistic cable analogues. The unstable periodic orbit in the cable appears to be related to a one-dimensional spiral wave described by Kopell and Howard [Stud. Appl. Math., 64 (1981), pp. 1-56]. Finally, we argue that reflection phenomena occur more robustly when excitability is due to saddle-type threshold behavior (type I excitability in the sense of Rinzel and Ermentrout [in Methods in Neuronal Modeling: FP om Synapses to Networks, C. Koch and I. Segev, eds., MIT Press, Cambridge, MA, 1989]).