Singular perturbation theory of traveling waves in excitable media (a review)

Abstract

Waves of chemical or electrical activity, traveling through space, have been observed in several contexts: chemical reaction mixtures in non-convecting liquid phase, cell suspensions, nerve axons, and neuromuscular tissues. Typically, wave-supporting preparations are excitable; that is, they respond and sensitivity to perturbations are rapidly damped out, but suprathreshold disturbances trigger an abrupt and substantial response. The abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media. Singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious, but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways. We compare and contrast the approaches taken by Greenberg, Zykov, Fife, Krinskii and his collaborators, and ourselves, with particular emphasis on the case of rotating spiral waves. Our intention is to bring some order to the important but difficult theory of propagating waves in two-dimensional excitable media. In conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three-dimensional space.