| In this dissertation, I study the class of nonlinear integro-differential equations 6u&parl0;x,t&parr0;6t =-u&parl0;x,t&parr0;+-infinityinfinity wx-yf&parl0;u&parl0; y,t&parr0;-q&parr0;dy +h. These equations arise in neuroscience for modeling short-term memory and were introduced by Shun-ichi Amari in 1977. Here u( x, t) is the average membrane potential of the neurons, w(x) accounts for the coupling between neurons and is of Mexican-hat type, and f(u) is the firing rate of a neuron with input u, taken to be a Heaviside step function. Finally, the parameter h denotes a constant external stimulus applied uniformly.; I will study the patterns exhibited by these equations. A region of neuronal excitation is a set R(u) = {lcub}x| u(x) > 0{rcub}. If R(u) is a finite, connected, open interval, the pattern is a "1-bump" solution. If R(u) consists of N > 1 disjoint, finite, connected, open intervals, then the solution is called an N-bump solution.; For 1-bump solutions, I review the Amari existence theorem and establish necessary and sufficient conditions for their existence. Then, I study the stability of 1-bump solutions. In previous work, perturbations of the endpoints of the intervals of the excited region were considered. This criterion is a necessary condition for stability. However, one must also account for shape changes. Hence, the questions arise here whether this elementary condition is also sufficient and whether there is some rigidity in these solutions that their properties (lengths, heights) are related. To this end, I carry out a classical stability analysis, and I show that this criterion is also sufficient to establish their linear stability.; For 2-bump solutions, I show that the necessary criteria for the existence of equal width 2-bump solutions, developed by William Troy and Carlo Laing, are sufficient as well, with one extra, natural condition. Further, I explore 1-bump solutions with a dimple and show that for some coupling functions it is possible for this type of solution to become a 2-bump solution. In addition, I generalize the 2-bump results to N-bump solutions, as well as to spatially periodic solutions, identifying their origins in classical Turing bifurcations.; One of the main analytical techniques I use is to approximate w(x) by a piecewise linear function that shapes the essential qualitative and quantitative properties of smooth w (x). This facilitates determining how solution properties depend on the parameters and characteristics of w( x).; In addition, I study the steady-state equation as an example of a broad class of nonlinear integral equations known as Hammerstein equations. We will use the calculus of variations to study this equation with the energy function constructed by Donald French. |
