Jacobi processes with jumps as neuronal models: a first passage time analysis

TL;DR

This paper introduces a Markovian extension of classical neuronal models using Jacobi processes with jumps, analyzing first-passage times to better understand neuron firing behavior.

Contribution

It presents a novel Markovian Jacobi process with jumps for neuronal modeling, including a new hypergeometric function generalization and a closed-form expectation for first-passage times.

Findings

Laplace transform of first-passage times characterized

Closed-form expression for expected first-passage time derived

Numerical analysis of firing rate for various parameters

Abstract

To overcome some limits of classical neuronal models, we propose a Markovian generalization of the classical model based on Jacobi processes by introducing downwards jumps to describe the activity of a single neuron. The statistical analysis of inter-spike intervals is performed by studying the first-passage times of the proposed Markovian Jacobi process with jumps through a constant boundary. In particular, we characterize its Laplace transform which is expressed in terms of some generalization of hypergeometric functions that we introduce, and deduce a closed-form expression for its expectation. Our approach, which is original in the context of first passage time problems, relies on intertwining relations between the semigroups of the classical Jacobi process and its generalization, which have been recently established in [11]. A numerical investigation of the firing rate of the considered neuron is performed for some choices of the involved parameters and of the jumps distributions.