Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling

摘要

We consider the multivariate point process determined by the crossing timesof the components of a multivariate jump process through a multivariateboundary, assuming to reset each component to an initial value after itsboundary crossing. We prove that this point process converges weakly to thepoint process determined by the crossing times of the limit process. This holdsfor both diffusion and deterministic limit processes. The almost sureconvergence of the first passage times under the almost sure convergence of theprocesses is also proved. The particular case of a multivariate Stein processconverging to a multivariate Ornstein-Uhlenbeck process is discussed as aguideline for applying diffusion limits for jump processes. We apply ourtheoretical findings to neural network modeling. The proposed model gives amathematical foundation to the generalization of the class of LeakyIntegrate-and-Fire models for single neural dynamics to the case of a firingnetwork of neurons. This will help future study of dependent spike trains.