Analytical Solutions for Traveling Pulses and Wave Trains in Neural Models: Excitable and Oscillatory Regimes

Abstract

We consider a piecewise linear approximation of the diffusive Morris-Lecar model of neuronal activity, the Tonnelier-Gerstner model. Exact analytical solutions for one-dimensional excitation waves are derived. The dynamics of traveling waves is related to two basic regimes of wave propagation: excitable and oscillatory cases. In the first case we describe mathematically the structure of a solitary pulse and in the second case—the form of a periodic sequence of pulses (a periodic wave train).