Closed-form solutions for the first-passage-time problem and neuronal modeling

The Gauss–Diffusion processes are here considered and some relations between their infinitesimal moments and mean and covariance functions are remarked. The corresponding linear stochastic differential equations are re-written specifying the coefficient functions and highlighting their meanings in theoretical and application contexts. We resort the Doob-transformation of a Gauss–Markov process as a transformed Wiener process and we represent some time-inhomogeneous processes as transformed Ornstein–Uhlenbeck process. The first passage time problem is considered in order to discuss some neuronal models based on Gauss–Diffusion processes. We recall some different approaches to solve the first passage time problem specifying when a closed-form result exists and numerical evaluations are required when the latter is not available. In the contest of neuronal modeling, relations between firing threshold, mean behavior of the neuronal membrane voltage and input currents are given for the existence of a closed-form result useful to describe the firing activity. Finally, we collect in an unified way some models and the corresponding Gauss–Diffusion processes already considered by us in some previous papers.