  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.
