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

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