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

Motivated by some as yet unsolved problems of biological interest, such as the description of firing probability densities for Leaky-and-Integrate neuronal models, we consider the first-passage-time problem for Gauss-diffusion processes along the line of Mehr and McFadden (1965). This is essentially based on a space-time transformation, originally due to Doob (1949), by which any Gauss-Markov process can expressed in terms of the standardWiener process. Starting with an analysis that pinpoints certain properties of mean and autocovariance of a Gauss-Markov process, we are lead 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 parameters choices of biological significance confirms the expected excellent accuracy of our methods.