### Abstract or Additional Information

Two types of neural networks are considered. One is the usual FitzHugh-Nagumo pde model, except that more general functions f(u) are considered. The other type includes models such as that of Pinto and Ermentrout, where a differential-integral equation is coupled with an ode. Most of the talk will be about an existence proof for an integral equation type model recently studied by G. Faye. He proved the existence of a fast pulse, but he had to assume something about the speed of traveling fronts for the reduced case of only one equation. This assumption can only be verified by numerical integration. Our method does not require this assumption, and also proves the existence of a second, slower, pulse. We will also discuss why the method does not appear to apply to the Pinto-Ermentrout model, and how the type of assumption made by Faye also arises for FitzHugh-Nagumo type systems.