Abstract
An Ornstein-Uhlenbeck diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that the neuron is subject to a sequence of inhibitory and excitatory post-synaptic potentials that occur with time-dependent rates. The resulting process is characterized by time-dependent drift. For this model, we construct the return process describing the membrane potential. It is a non homogeneous Ornstein-Uhlenbeck process with jumps on which the effect of random refractoriness is introduced. An asymptotic analysis of the process modeling the number of firings and the distribution of interspike intervals is performed under the assumption of exponential distribution for the firing time. Some numerical evaluations are performed to provide quantitative information on the role of the parameters.
Highlights
Introduction and backgroundIn 1964, in [7] Gernstein and Mandelbrot proposed a model of neuronal activity based on the Wiener process
Several ways exist to derive this model, one of these consists of assuming that the neuron is subject to a sequence of inhibitory and excitatory postsynaptic potentials (PSP’s)
Note that in the Case 1. of Section 2 the firing threshold is asymptotically constant so the firing time mean is time independent. This situation has been widely studied in [6], so that in the following we will analyze the return process constructed on X (t) making use of the exponential approximation of the first passage time (FPT) pdf (19)
Summary
Introduction and backgroundIn 1964, in [7] Gernstein and Mandelbrot proposed a model of neuronal activity based on the Wiener process. This situation has been widely studied in [6], so that in the following we will analyze the return process constructed on X (t) making use of the exponential approximation of the FPT pdf (19).
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