Diffusion approximation and first-passage-time problem for a model neuron. III A birth-and-death approach

A stochastic model for single neuron's activity is constructed as the continuous limit of a birth-and-death process in the presence of a reversal hyper-polarization potential. The resulting process is a one dimensional diffusion with linear drift and infinitesimal variance, somewhat different from that proposed by Lánský and Lánská in a previous paper. A detailed study is performed for both the discrete process and its continuous approximation. In particular, the neuronal firing time problem is discussed and the moments of the firing time are explicitly obtained. Use of a new computation method is then made to obtain the firing p.d.f. The behaviour of mean, variance and coefficient of variation of the firing time and of its p.d.f. is analysed to pinpoint the role played by the parameters of the model. A mathematical description of the return process for this neuronal diffusion model is finally provided to obtain closed form expressions for the asymptotic moments and steady state p.d.f. of the neuron's membrane potential.