When postsynaptic conductance varies slowly compared to the spike generation process, a straightforward averaging scheme can be used to reduce the system's complexity. Our model consists of a Hodgkin-Huxley-like membrane description for each cell; synaptic activation is described by first order kinetics, with slow rates, in which the equilibrium activation is a sigmoidal function of the presynaptic voltage. Our work concentrates on a two-cell network and it applies qualitatively to the activity patterns, including bistable behavior, recently observed in simple in vitro circuits with slow synapses (Kleinfeld et al. 1990). The fact that our averaged system is derived from a realistic biophysical model has important consequences. In particular, it can preserve certain hysteresis behavior near threshold that is not represented in a simple ad hoc sigmoidal input-output network. This behavior enables a coupled pair of cells, one excitatory and one inhibitory, to generate an alternating burst rhythm even though neither cell has fatiguing properties.

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