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We introduce and study an extension of the classical elapsed time equation inthe context of neuron populations that are described by the elapsed time sincethe last discharge, i.e., the refractory period. In this extension weincorporate the elapsed since the penultimate discharge and we obtain a morecomplex system of integro-differential equations. For this new system we proveconvergence to stationary state by means of Doeblin's theory in the case ofweak non-linearities in an appropriate functional setting, inspired by the caseof the classical elapsed time equation. Moreover, we present some numericalsimulations to observe how different firing rates can give different types ofbehaviors and to contrast them with theoretical results of both classical andextended models.

A multiple time renewal equation for neural assemblies with elapsed time model