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

Massimiliano Tamborrino; Laura Sacerdote; Martin Jacobsen

doi:10.1016/j.physd.2014.08.003

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

Massimiliano Tamborrino, Laura Sacerdote, Martin Jacobsen

Department of Mathematical Sciences

8 Citations (Scopus)

Abstract

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.

Original language	English
Journal	Physica D: Nonlinear Phenomena
Volume	288
Pages (from-to)	45-52
ISSN	0167-2789
DOIs	https://doi.org/10.1016/j.physd.2014.08.003
Publication status	Published - 15 Nov 2014

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10.1016/j.physd.2014.08.003

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title = "Weak convergence of marked point processes generated by crossings of multivariate jump processes: Applications to neural network modeling",

abstract = "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.",

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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 - 2014/11/15

Y1 - 2014/11/15

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.

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DO - 10.1016/j.physd.2014.08.003

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JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

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