Abstract
Signals propagating along a nerve axon are modeled by traveling wave solutions of the Hodgkin–Huxley (HH) or FitzHugh–Nagumo(FHN) systems. The existence and the stability problems of the traveling wave solutions have been studied by many authors in various ways. Rinzel and Keller have shown numerically that there exist two branches of traveling train solutions of FHN system with different propagating velocities—namely, fast solutions and slow solutions. For the HH system, similar results have been obtained by Miller and Rinzel. The two branches connect each other at the minimal period. Traveling pulse solutions can be regarded as traveling train solutions with the infinite period. On the other hand, for the traveling train solutions, Maginu has shown the instability of the slow solutions and the fast solutions, whose periods are close to the minimal period, for FHN systems. This chapter describes the distribution of the spectrum of the linearized operator about the fast traveling train solutions of FHN system.
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