On Neuronal Firing Modeling via Specially Confined Diffusion Processes

First passage time problems for diffusion processes have been extensively investigated to' model neuronal firing activity or extinction times in population dynamics (see, for instance, [IO)). In this paper we study the asymptotic behavior of first passage times densities for a class of specially confined temporally homogeneous diffusion processes in the presence of an entrance or a reflecting boundary. The emphasis is on problems of a rather mathematical nature, concerning the behavior of the first passage time density and of its moments when the neuronal firing threshold is in the neighborhood of the reflecting boundary, and when it moves indefinitely away from it. Our asymptotic results are obtained without need to determine beforehand the transition probability density in the presence of entrance or reflecting boundariesj they depend, instead, only on drift, infinitesimal variance, threshold and on the entrance or the reflecting boundary of the processo Some evaluations of moments of first passage time, in particular, mean and variance, are performed by solving numerically, or analytically whenever possible, Siegert's recursion equations [12], and by comparing the results with those obtained through our approximate formulas. In the case where the transition probability density is known, the goodness of the obtained approximations can be verified. Such results appear to be useful for neuronal modeling in the presence of reversal potential especially to pinpoint the role of the involved parameters in various models, some of which are the object of a somewhat detailed analysis.