On Spiking Models for Synaptic Activity and Impulsive Differential Equations

Abstract

We illustrate the problems that can arise in writing differential equations that include Dirac delta functions to model equations with state-dependent impulsive forcing. Specifically, difficulties arise in the interpretation of the products of distributions with discontinuous functions. We suggest several methods to resolve these ambiguities, such as using limiting sequences and asymptotic analysis, with applications of the results given for discrete maps. These suggestions are applied to a popular model describing synaptic connections in the brain.