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

• Published in: Mathematical biosciences Vol. 245; no. 1; pp. 61 - 69
• Main Authors: Sacerdote, Laura, Zucca, Cristina
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.09.2013
• Subjects: Action Potentials - physiology; First passage time; Integrate and fire neuronal model; Jump diffusion process; Mathematical Concepts; Membrane Potentials - physiology; Models, Neurological; Nerve Net - physiology; Neurons - physiology; Wiener process; First passage time; Integrate and fire neuronal model; Wiener process; Jump diffusion process
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/j.mbs.2013.06.005

•We model the spiking activity of a pair of coupled neurons.•The model considers a bivariate Wiener process with jumps.•We study the joint distribution of the forward and inter-spike times.•Closed form expression and bounds of the joint distribution are determined. 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.