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Multistability in Spiking Neuron Models of Delayed Recurrent Inhibitory Loops

majianfu{at}mathstat.yorku.ca Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada

Jianhong Wu

wujh{at}mathstat.yorku.ca Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada

We consider the effect of the effective timing of a delayed feedback on the excitatory neuron in a recurrent inhibitory loop, when biological realities of firing and absolute refractory period are incorporated into a phenomenological spiking linear or quadratic integrate-and-fire neuron model. We show that such models are capable of generating a large number of asymptotically stable periodic solutions with predictable patterns of oscillations. We observe that the number of fixed points of the so-called phase resetting map coincides with the number of distinct periods of all stable periodic solutions rather than the number of stable patterns. We demonstrate how configurational information corresponding to these distinct periods can be explored to calculate and predict the number of stable patterns.