Stochastic Synchronization and Coherence Resonance Near a Hopf Bifurcation

Noise is an inherent part of neuronal dynamics. Experimental studies have revealed that noise plays an important role in neural dynamics. It has been shown, for example, that noise can have a constructive effect on the functioning of biological systems such as noise-induced synchronization, which is typically studied in the context of excitable neural networks. Neural excitability and the bifurcations which result in the transition from quiescence to oscillation or bursting/spike emission largely determine the neurophysiological properties of neurons. For example, excitable neurons in the vicinity of a Hopf bifurcation have been shown to respond preferably to excitation and can be easily synchronized by a stochastic stimulus. In this thesis we study the roles of stochastic synchronization and coherence resonance in neuro-physiological models near a Hopf bifurcation. We begin by considering a mathematical model for a neural network in the vicinity of a Hopf bifurcation, where bursting can be induced by a stochastic stimulus. We show that the coherence of the network is optimized by an optimal level of stochastic stimulus. Then, we study the general class of such models by considering the canonical model for a normal form near a Hopf bifurcation. We show that synchronization is optimized by an optimal level of stochastic stimulus and may be further tuned by adjusting the coupling of our model in a non-trivial way.