On the interspike-intervals of periodically-driven integrate-and-fire models

TL;DR

This paper investigates the properties of interspike-intervals in periodically-driven integrate-and-fire models, providing new theoretical insights into their regularity, dependence on input functions, and distribution behavior, supported by computational examples.

Contribution

It generalizes existing results by analyzing locally integrable inputs and offers a comprehensive description of interspike-interval regularity and distribution in these models.

Findings

Proves continuous dependence of the firing map on input functions.

Provides a complete description of interspike-interval distribution behavior.

Explains numerically observed facts about interspike-interval distributions.

Abstract

We analyze properties of the firing map, which iterations give information about consecutive spikes, for periodically driven linear integrate-and-fire models. By considering locally integrable (thus in general not continuous) input functions, we generalize some results of other authors. In particular we prove theorems concerning continuous dependence of the firing map on the input in suitable function spaces. Using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism, we provide also a complete description of the regularity properties of the sequence of interspike-intervals and behaviour of the interspike-interval distribution. Our results allow to explain some facts concerning this distribution observed numerically by other authors. These theoretical findings are illustrated by carefully chosen computational examples.