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

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.