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Abstract
Coupled nonlinear differential equations are derived for the dynamics of spatially
localized populations containing both excitatory and inhibitory model neurons. Phase
plane methods and numerical solutions are then used to investigate population responses
to various types of stimuli. The results obtained show simple and multiple hysteresis
phenomena and limit cycle activity. The latter is particularly interesting since the
frequency of the limit cycle oscillation is found to be a monotonic function of stimulus
intensity. Finally, it is proved that the existence of limit cycle dynamics in response
to one class of stimuli implies the existence of multiple stable states and hysteresis
in response to a different class of stimuli. The relation between these findings and
a number of experiments is discussed.