RESUMEN

We consider traveling front and pulse solutions to a system of integro-differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension. Our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions. This is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed. Our second goal is to establish the existence of traveling pulse solutions using more rigorous methods. Two techniques are presented. The first, a shooting argument, reduces the problem from finding a specific solution to an integro-differential equation system to finding any solution to an ODE system. The second, a singular perturbation argument, provides a construction of traveling pulse solutions under more general conditions.

RESUMEN

We use singular perturbation theory to analyze the dynamics of N weakly interacting pulses in a one-dimensional synaptically coupled neuronal network. The network is modeled in terms of a nonlocal integro-differential equation, in which the integral kernel represents the spatial distribution of synaptic weights, and the output activity of a neuron is taken to be a mean firing rate. We derive a set of N coupled ordinary differential equations (ODEs) for the dynamics of individual pulses, establishing a direct relationship between the explicit form of the pulse interactions and the structure of the long-range synaptic coupling. The system of ODEs is used to explore the existence and stability of stationary N-pulses and traveling wave trains.