Diffusion approximation and first-passage-time problem for a model neuron

Abstract

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 hyperpolarization 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.

Continuum Approximation of Invasion Probabilities

“…Let u \epsilon (x) be the Diffusion Approximation for the extinction probabilities, satisfying (4) of Definition 1.6. Then (14) u \epsilon (x) = 1 -\int x 0 e - \Psi (y)/\epsilon dy \int x\ast 0 e - \Psi (y)/\epsilon dy , where x \ast = min\{ x : \lambda (x) = \mu (x)\} .…”

“…Integrating to get u \epsilon (x) = \int x 0 h(y) dy and enforcing the boundary conditions u \epsilon (0) = 1 and u \epsilon (x \ast ) = 1 yields the solution (14).…”

“…The integrals that appear in (14) are in a form compatible with applying Watson's lemma [28]. In fact, Watson's lemma can be applied directly to the denominator.…”

“…(See section 3.1 for a discussion.) This technique has been used prominently to model epidemics [1], neuronal activity [14,10], and branching processes with logistic growth limits [20]. In the SDE setting, solutions to invasion probability and mean extinction time questions can be obtained by solving certain ODEs (or PDEs, depending on the dimension).…”

On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

“…Accordingly, in this paper we aim to discuss some properties of a bilateral birth-death process characterized by nonlinear rates, namely of sigmoidal form, whose transition probabilities are bimodal. We recall that the large interest on birth-death processes in biomathematics is mainly due to their wide applicability, not only in the classical area of population dynamics but also in the realm of stochastic neuronal modeling (see, for instance, [7,8]). …”