Joint distribution of first exit times of a two dimensional Wiener process with jumps with application to a pair of coupled neurons

Abstract

Motivated by a neuronal modeling problem, a bivariate Wiener process with two independent components is considered. Each component evolves independently until one of them reaches a threshold value. If the first component crosses the threshold value, it is reset while the dynamics of the other component remains unchanged. But, if this happens to the second component, the first one has a jump of constant amplitude; the second component is then reset to its starting value and its evolution restarts. Both processes evolve once again until one of them reaches again its boundary. In this work, the coupling of the first exit times of the two connected processes is studied.

Keywords: First passage time; Integrate and fire neuronal model; Jump diffusion process; Wiener process.