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.

Mathematical models are an important tool for neuroscientists. During the last thirty years many papers have appeared on single neuron description and specifically on stochastic Integrate and Fire models. Analytical results have been proved and numerical and simulation methods have been developed for their study. Reviews appeared recently collect the main features of these models but do not focus on the methodologies employed to obtain them. Aim of this paper is to fill this gap by upgrading old reviews on this topic. The idea is to collect the existing methods and the available analytical results for the most common one dimensional stochastic Integrate and Fire models to make them available for studies on networks. An effort to unify the mathematical notations is also made. This review is divided in two parts:1. Derivation of the models with the list of the available closed forms expressions for their characterization; 2. Presentation of the existing mathematical and statistical methods for the study of these models.

A mathematical model for the clock phase and frequency deviation based on the theory of stochastic differential equations (SDEs) is discussed. In particular, we consider a model that includes what are called the 'white and random walk frequency noises' in time metrology, which give rise in a mathematical context to a Wiener and an integrated Wiener process on the clock phase. Due to the particular simple expression of the functions involved an exact solution exists, and we determine it by considering the model in a wider theoretical context, which is suitable for generalizations to more complex instances. Moreover, we determine an iterative form for the solution, useful for simulation and further processing of clock data, such as filtering and prediction. The Euler method, generally applied in literature to approximate the solution of SDEs, is examined and compared to the exact solution, and the magnitude of the approximation is evaluated. The possible extension of the model to other noise sources, such as the flicker and white phase noises, is also sketched.

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.