| We study spatially patterned stationary solutions of an integro-differential equation introduced by Wilson and Cowan and proposed by Amari as a model of neural activity on a single layer of interconnected neurons: 6ux,t 6t=-ux,t +Rw x-yfuy,t dy+h. In particular, we investigate the existence of N-bump stationary solutions, or solutions positive on a region that can be decomposed into a disjoint union of N finite intervals. N-bump solutions of this equation have been proposed to model the persistent localized activity observed in regions of the brain responsible for navigation, short-term memory, and biological motion recognition.; Previous research has established criteria for the existence of equal width two-bump stationary solutions of the Amari equation when considering a Heaviside firing rate function and a coupling function of a "Mexican hat" type. One such condition concerns the sign of the external stimulus which is quite difficult to establish under general assumptions. We address this problem early in the dissertation and determine necessary conditions for the existence of two-bump stationary solutions that assure the stimulus is non-positive. We generalize the two-bump case and study a system of equations whose solutions correspond to non-equal width stationary solution candidates.; Later in the dissertation we introduce a class of "Mexican hat" coupling functions different than the couplings seen in the current literature. Using this class of functions we are able to identify a region in the parameter space wherein two-bump solution candidates exist. Furthermore, we prove the candidates are genuine solutions to the original equation, a task bypassed by many authors. We then establish the linear stability of the one- and two-bump solutions admitted by this particular class of coupling functions.; In the final portion of the dissertation we move away from the "Mexican hat" coupling and investigate the same system with a piecewise linear firing rate and a two-parameter family of oscillating coupling functions. We identify regions in the parameter space in which solutions exhibit persistent localized activity. We then study the stability behavior of such solutions. |
