Partial Synchronization in a Network of Neural Oscillators

A network with three types of model neurons is considered. These are designated as excitatory, fast inhibitory, and slow inhibitory cells. Each excitatory cell is modeled as a relaxation oscillator, while, for simplicity, the inhibitory cells are modeled by a single equation. The coupling between cells is modeled in a way that mimics chemical synapses. This network is introduced as a simplification of the model in Traub and Miles [Neuronal Networks of the Hippocampus, Cambridge University Press, New York, 1991] for oscillations in the CA3 region of the hippocampus. A geometric approach for analyzing the solution structure of the network is developed. The analysis clarifies the role that each type of cell and each parameter plays in obtaining partially synchronized rhythms, determines how the parameters should be chosen for this collective behavior, and indicates possible instabilities which arise as parameters are varied. A symbolic scheme is derived which accurately determines the firing patterns for a large class of networks. The symbolic scheme predicts which cells fire during each event of activity and demonstrates why simple networks may give rise to multiple stable firing patterns.