2013/10/31 by Massimiliano Tamborrino, Laura Sacerdote, Martin Jacobsen · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Applied mathematics #Artificial intelligence #Artificial neural network #Boundary (topology) #Computer science #Convergence (economics) #Diffusion and Search Dynamics #Generalization #Jump #Jump process #Limit (mathematics) #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Multivariate statistics #Physics #Point process #Process (computing) #Statistical physics #Statistics #math.PR #stochastic dynamics and bifurcation
paper · pdf · doi:10.1016/j.physd.2014.08.003
published as Physica D: Nonlinear Phenomena, 288, 45--52, 2014 · 20 pages, 1 figure
arxiv created 2014/07/13 · openalex publication_date 2014/09/06 · arxiv updated 2015/02/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.