Spatially Structured Activity in Synaptically Coupled Neuronal Networks: I. Traveling Fronts and Pulses

Abstract

Abstract. 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.

Pattern Formation in a Network of Excitatory and Inhibitory Cells with Adaptation

“…As we have shown, increasing τ A leads to a transition from the zero eigenvalue stationary-state case to the oscillatory Hopf case. Waves have been associated with epileptic behavior in previous models (see, e.g., [20]). …”

“…Pinto and Ermentrout [20] analyzed a model like this, without the inhibitory interactions, in order to study propagating waves.…”

Two-Dimensional Bumps in Piecewise Smooth Neural Fields with Synaptic Depression

“…It is then possible to establish existence of stationary and traveling pulse solutions by explicit construction, and to determine local stability in terms of an associated Evans function by linearizing the neural field equations about the pulse solution [2,7,30,31,32,43]. In the case of stationary pulses or bumps, local stability reduces to the problem of calculating the effects of perturbations at the bump boundary, where u(x) = θ.…”

“…The inclusion of negative feedback is motivated by the fact that the scalar model given by (1.1) cannot support traveling pulse solutions in the absence of synaptic inhibition, which is inconsistent with what is observed in disinhibited slice experiments [6,41,42]. Negative feedback is typically taken to be linear by analogy with the FitzHugh-Nagumo equation [13], leading to a neural field model of the form [30,31] …”

“…As in the case of nonlinear PDE models of diffusively coupled excitable systems [17], neural field models can exhibit a variety of coherent pulse-like structures, including both stationary and traveling solitary pulses. Traveling pulses tend to occur when synaptic connections are predominantly excitatory and there is some form of slow local adaptation or recovery [30], whereas stationary pulses (activity bumps) occur in the presence of lateral inhibition [2,31]. The formation of localized activity states can be used to model a number of neurobiological phenomena.…”