Jacobi Processes with Jumps as Neuronal Models: A First Passage Time Analysis

.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 interspike 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 [P. Cheridito et al., J. Ec. Polytech. - Math., 8 (2021), pp. 331–378]. A numerical investigation of the firing rate of the considered neuron is performed for some choices of the involved parameters and of the jump distributions.Keywordsnon-local Wright–Fisher processfirst passage timeintertwining relationsMarkov semigroupsinfinitesimal generatorsneuronal modelingMSC codes37A3047D0647G2060J7560J70