Waves and Oscillations in Model Neuronal Networks

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