This dissertation studies a one dimensional neural network rate model that supports localized self-sustained solutions. These solutions could be an analog for working memory in the brain. Working memory refers to the temporary storage of information necessary for performing different mental tasks. Cortical neurons that show persistent activity are observed in animals during working memory tasks. The physical process underlying this persistent activity could be due to self-sustained network activity of the neurons in the brain.The term `bump' has been coined to imply a spatially localized persistent activity state that is sustained internally by a network of neurons. Many researchers have analyzed the bump state using Firing rate models with either the Heaviside gain function or a saturating sigmoidal one. These gain functions imply that neurons begin to fire once their synaptic input reaches threshold, and the firing rate saturates to a maximal value almost immediately. However, cortical neurons that exhibit persistent activity usually are well below their maximal attainable rate. To resolve this paradox, I study a single population rate model using a biophysically relevant firing rate function.I consider the existence and the stability of standing single-pulse solutions of an integro-diferential neural network equation. In this network, the synaptic coupling has local excitatory coupling with distal lateral inhibition and the non-saturating gain function is piece-wise linear. A standing pulse solution of this network is a synaptic input pattern that supports a bump state. I show that the existence condition for single-pulses of the integro-differential equation can be reducedto the solution of an algebraic system. With this condition, I map out the shape of the pulsesfor different coupling weights and gains. By a similar approach, I also find the conditions for the existences of dimple-pulses and double-pulses. For a fixed gain and connectivity, there are at least two single-pulse solutions - a "large" one and a "small" one. However, more than two single-pulses can coexist depending on the parameter range. To have standing single-pulses, the gain function and synaptic coupling are both important.I also derive a stability criteria for the standing pulse solutions. I show that the large pulse is stable and the small pulse is unstable. If there are more than two pulse solutions coexisting, the first pulse is the small one and it is unstable. The second one is a large stable pulse. The third pulse is wider than the second one and it is unstable. More importantly, the second single-pulse (which could be a dimple pulse) is bistable with the "all-off" state. The stable pulse represents the memory. When the network is switched to the "all-off" state, the memory is erased.

In this thesis methods from nonlinear dynamical systems, pattern formation and bifurcation theory, combined with numerical simulations, are applied to three models in neuroscience. In Chapter 1 we analyze the Wilson-Cowan equations for a single self-excited population of cells with absolute refractory period. We construct the normal form for a Hopf bifurcation, and prove that by increasing the refractory period the network switches from a steady state to an oscillatory behavior. Numerical simulations indicate that for large values of refractoriness the oscillation converges to a relaxation-like pattern, the period of which we estimate. Chapter 2 brings new results for the rate model introduced by Hansel and Sompolinsky who study feature selectivity in local cortical circuits. We study their model with a more general, nonlinear sigmoid gain function, and prove that the system can exhibit different kind of patterns such as stationary states, traveling waves and standing waves. Standing waves can be obtained only if the threshold is sufficiently high and only for intermediate values of the strength of adaptation. A large adaptation strength destabilizes the pattern. Therefore the localized activity starts to propagate along the network, resulting in a traveling wave. We construct the normal form for Hopf and Takens-Bogdanov with O(2)-symmetry bifurcations and study the interactions between spatial and spatio-temporal patterns in the neural network. Numerical simulations are provided.Chapter 3 addresses several questions with regard to the traveling wave propagation in a leaky-integrate-and-fire model for a network with purely excitatory (exponentially decaying) synaptic coupling. We analyze the case when the neurons fire multiple spikes and derive a formula for the voltage. We compute in a certain parameter space, the curves that delineate the region where single-spike traveling wave solutions exist, and show that there is a region of parameter space where neurons can propagate a two-spike traveling wave.