The First Passage Time Problem for Gauss-Diffusion Processes: Algorithmic Approaches and Applications to LIF Neuronal Model

Motivated by some unsolved problems of biological interest, such as the description of firing probability densities for Leaky Integrate-and-Fire neuronal models, we consider the first-passage-time problem for Gauss-diffusion processes along the line of Mehr and McFadden (J R Stat Soc B 27:505-522, 1965). This is essentially based on a space-time transformation, originally due to Doob (Ann Math Stat 20:393-403, 1949), by which any Gauss-Markov process can expressed in terms of the standard Wiener process. Starting with an analysis that pinpoints certain properties of mean and autocovariance of a Gauss-Markov process, we are led to the formulation of some numerical and time-asymptotically analytical methods for evaluating first-passage-time probability density functions for Gauss-diffusion processes. Implementations for neuronal models under various parameter choices of biological significance confirm the expected excellent accuracy of our methods.