Traveling waves in neural models

Since the fundamental work of Hodgkin and Huxley in 1952 on the propagation of action potentials in the giant axon of Loligo there has been an enormous amount of related research. It gave rise to a substantial literature on the modeling of neurons and systems of neurons, as well as investigations of the mathematical properties of the equations arising. Various simplified models have been introduced, including that of Fitzhugh and Nagumo, and a piecewise linear analog. For each single cell model there is a corresponding diffusive model for propagation of a voltage signal. The analysis of the Hodgkin and Huxley system of four differential equations done by Fitzhugh exhibited an important feature: the existence for a short time after excitation, of a saddle point in a reduced model. We investigate counterparts to the saddle in other models and find near singular behavior that sustains propagating waves. We compare various models, offer some of our own, and give numerical evidence to illuminate the robustness of stable propagation or the lack of propagating waves.