A Spectral Analysis of the Sequence of Firing Phases in Stochastic Integrate-and-Fire Oscillators

TL;DR

This paper analyzes the spectral properties of stochastic integrate-and-fire neuron models, deriving asymptotic eigenvalue approximations for the phase transition operator, and connects these findings to neuronal firing behavior under small noise conditions.

Contribution

It provides the first rigorous spectral analysis of stochastic integrate-and-fire oscillators, including asymptotic eigenvalue approximations and law of large numbers results for first passage times.

Findings

Eigenvalues of the transition operator are approximated asymptotically.

Strong laws of large numbers and CLT are established for first passage times.

Connections to numerical studies of stochastic integrate-and-fire models are discussed.

Abstract

Integrate and fire oscillators are widely used to model the generation of action potentials in neurons. In this paper, we discuss small noise asymptotic results for a class of stochastic integrate and fire oscillators (SIFs) in which the buildup of membrane potential in the neuron is governed by a Gaussian diffusion process. To analyze this model, we study the asymptotic behavior of the spectrum of the firing phase transition operator. We begin by proving strong versions of a law of large numbers and central limit theorem for the first passage-time of the underlying diffusion process across a general time dependent boundary. Using these results, we obtain asymptotic approximations of the transition operator's eigenvalues. We also discuss connections between our results and earlier numerical investigations of SIFs.