| Excitatory-inhibitory networks arise in many neuronal systems. Examples include models for thalamic sleep rhythms and parkinsonian tremor. Such networks have been shown to exhibit a rich structure of firing patterns, including synchronous activity, irregular and chaotic dynamics and propagating wave-like behavior. Computational and analytical methods have been employed extensively to understand those patterns in networks with one spatial dimension. However there has been very little work devoted to the numerical or analytical investigation of higher dimensional networks. In this thesis, I consider two-dimensional sheets of synaptically coupled excitatory and inhibitory neurons and explore the types of additional patterns that emerge.;The models consist of large systems of nonlinear differential equations and represent the interactions between two neural populations: the subthalamic nucleus and the globus pallidus. The membrane potential in those models exhibits bursting patterns and thus reveals several time scales. Using the discrepancy in time scale of the variables involved I analyze the mechanisms underlying such bursts and then reduce this complex high-dimensional model to a simpler yet biophysically meaningful system. In the second part of this project, I use the reduced model to derive conditions on network parameters for the existence of various propagating patterns. I compute the functional dependence of the velocity on parameters controlling the inhibitory synaptic input and explain the failure of propagation that occurs for a certain parameter range. |
