The work we present in this thesis is a series of studies of how heterogeneities in coupling affect the synchronization of coupled neural oscillators. We begin by examining how heterogeneity in coupling strength affects the equilibrium phase difference of a pair of coupled, spiking neurons when compared to the case of identical coupling. This study is performed using pairs of Hodgkin-Huxley and Wang-Buzsaki neurons. We find that heterogeneity in coupling strength breaks the symmetry of the bifurcation diagrams of equilibrium phase difference versus the synaptic rate constant for weakly coupled pairs of neurons. We observe important qualitative changes such as the loss of the ubiquitous in-phase and anti-phase solutions found when the coupling is identical and regions of parameter space where no phase locked solution exists.

Another type of heterogeneity can be found by having different types of coupling between oscillators. Synaptic coupling between neurons can either be exciting or inhibiting. We examine the synchronization dynamics when a pair of neurons is coupled with one excitatory and one inhibitory synapse. We also use coupled pairs of Hodgkin-Huxley neurons and Wang-Buzsaki neurons for this work. We then explore the existance of 1 : n coupled states for a coupled pair of theta neurons. We do this in order to reproduce an observed effect called quantal slowing. Quantal slowing is the phenomena where jumping between different 1 : n coupled states is observed instead of gradual changes in period as a parameter in the system is varied. All of these topics fall under the general heading of coupled, non-linear oscillators and specifically weakly coupled, neural oscillators.

The audience for this thesis is most likely going to be a mixed crowd as the research reported herein is interdisciplinary. Choosing the content for the introduction proved far more challenging than expected. It might be impossible to write a maximally useful introductory portion of a thesis when it could be read by a physicist, mathematician, engineer or biologist. Undoubtedly readers will find some portion of this introduction elementary. At the risk of boring some or all of my readers we decided it was best to proceed so that enough of the mathematical (biological) background is explained in the introduction so that a biologist (mathematician) is able to appreciate the motivations for the research and the results presented. We begin with an introduction in nonlinear dynamics explaining the mathematical tools we use to characterize the excitability of individual neurons, as well as oscillations and synchrony in neural networks. The next part of the introductory material is an overview of the biology of neurons. We then describe the neuron models used in this work and finally describe the techniques we employ to study coupled neurons.