  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.
