A Markov approximation and related bounds for stochastically drifted processes and applications to neuronal modelling.

In order to include a stochastic input in the Leaky Integrate-and-Fire (LIF) neuronal model, we consider a linear stochastic differential equation (SDE) having the stochastic process $Z(t)$ in the drift coefficient. We characterize the solution $X(t)$ of this SDE by specifying its expectation and covariance functions depending on whether the drift process $Z(t)$ is jointly distributed with the Brownian motion or not. We consider an ad hoc Gauss-Markov (GM) process for approximating the process $X(t)$ and we specify a bound for the $L^2$-distance between the processes. We also prove that the upper bound allows to obtain an estimation of the probability that the process $X(t)$ is close to a time-varying boundary (the firing threshold) by means of the distribution function of the first passage time of the GM process. The bounds are evaluated in the cases where $Z(t)$ is an Ornstein-Uhlenbeck (OU) process, a Poisson process, a compound Poisson process or a shot noise. Finally, these results are applied in the study of a LIF model stochastically driven by a shot noise process $Z(t)$.