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The leaky integrate-and-fire neuronal model proposed in Stevens and Zador (1998), in which time constant and resting potential are postulated to be time dependent, is revisited within a stochastic framework in which the membrane potential is mathematically described as a gauss-diffusion process. The first-passage-time probability density, miming in such a context the firing probability density, is evaluated by either the Volterra integral equation of Buonocore, Nobile, and Ricciardi ( 1987 ) or, when possible, by the asymptotics of Giorno, Nobile, and Ricciardi (1990). The model examined here represents an extension of the classic leaky integrate-and-fire one based on the Ornstein-Uhlenbeck process in that it is in principle compatible with the inclusion of some other physiological characteristics such as relative refractoriness. It also allows finer tuning possibilities in view of its accounting for certain qualitative as well as quantitative features, such as the behavior of the time course of the membrane potential prior to firings and the computation of experimentally measurable statistical descriptors of the firing time: mean, median, coefficient of variation, and skewness. Finally, implementations of this model are provided in connection with certain experimental evidence discussed in the literature.

Mean and variance of the first passage time through a constant boundary for the Ornstein-Uhlenbeck process are determined by a straight-forward differentiation of the Laplace transform of the first passage time probability density function. The results of some numerical computations are discussed to shed some light on the input-output behavior of a formal neuron whose dynamics is modeled by a diffusion process of Ornstein-Uhlenbeck type.

Making use of a Rice-like series expansion, for a class of stationary Gaussian processes the asymptotic behavior of the first passage time probability density function through certain time-varying boundaries, including periodic boundaries, is determined. Sufficient conditions are then given such that the density asymptotically exhibits an exponential behavior when the boundary is either asymptotically constant or asymptotically periodic. [ABSTRACT FROM AUTHOR]