  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.
