Activity Patterns in Lateral-Inhibition Type Neural Fields with Asymmetric Excitatory Distal Components

In this work, we generalize the dynamics governing activity patterns in a neural field model under an extension to the conventional lateral-inhibition kernel which incorporates homogeneous excitatory distal connections. We demonstrate that the integro-differential equation representing the activity of the field can be represented as a delay-differential equation whose order depends on the number of exponential components present in the kernel, and we determine the exact form of this equation for an arbitrary firing rate function. We prove the existence and provide a stability analysis for a traveling pulse solution in the case of a Heaviside firing rate function, and later prove the existence of the same pattern under a piecewise-linear gain. We derive the corresponding Evans function which is the primary tool in the stability analysis, and suggest methods for ultimately determining the stability of these last solutions.Ph.D., Mathematics -- Drexel University, 201