TY - JOUR

T1 - Weak convergence of marked point processes generated by crossings of multivariate jump processes

T2 - Applications to neural network modeling

AU - Tamborrino, Massimiliano

AU - Sacerdote, Laura

AU - Jacobsen, Martin

PY - 2014

Y1 - 2014

N2 - We consider the multivariate point process determined by the crossing times of the components of a multivariate jump process through a multivariate boundary, assuming to reset each component to an initial value after its boundary crossing. We prove that this point process converges weakly to the point process determined by the crossing times of the limit process. This holds for both diffusion and deterministic limit processes. The almost sure convergence of the first passage times under the almost sure convergence of the processes is also proved. The particular case of a multivariate Stein process converging to a multivariate Ornstein–Uhlenbeck process is discussed as a guideline for applying diffusion limits for jump processes. We apply our theoretical findings to neural network modeling. The proposed model gives a mathematical foundation to the generalization of the class of Leaky Integrate-and-Fire models for single neural dynamics to the case of a firing network of neurons. This will help future study of dependent spike trains.

AB - We consider the multivariate point process determined by the crossing times of the components of a multivariate jump process through a multivariate boundary, assuming to reset each component to an initial value after its boundary crossing. We prove that this point process converges weakly to the point process determined by the crossing times of the limit process. This holds for both diffusion and deterministic limit processes. The almost sure convergence of the first passage times under the almost sure convergence of the processes is also proved. The particular case of a multivariate Stein process converging to a multivariate Ornstein–Uhlenbeck process is discussed as a guideline for applying diffusion limits for jump processes. We apply our theoretical findings to neural network modeling. The proposed model gives a mathematical foundation to the generalization of the class of Leaky Integrate-and-Fire models for single neural dynamics to the case of a firing network of neurons. This will help future study of dependent spike trains.

U2 - 10.1016/j.physd.2014.08.003

DO - 10.1016/j.physd.2014.08.003

M3 - Journal article

VL - 288

SP - 45

EP - 52

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -