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

Institut for Matematiske Fag

8 Citationer (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.

Originalsprog	Engelsk
Tidsskrift	Physica D: Nonlinear Phenomena
Vol/bind	288
Sider (fra-til)	45-52
ISSN	0167-2789
DOI	https://doi.org/10.1016/j.physd.2014.08.003
Status	Udgivet - 15 nov. 2014

Adgang til dokumentet

10.1016/j.physd.2014.08.003

Citationsformater

@article{3df2bcef4dad420cabd5320abacee26f,

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.",

author = "Massimiliano Tamborrino and Laura Sacerdote and Martin Jacobsen",

year = "2014",

month = nov,

day = "15",

doi = "10.1016/j.physd.2014.08.003",

language = "English",

volume = "288",

pages = "45--52",

journal = "Physica D: Nonlinear Phenomena",

issn = "0167-2789",

publisher = "Elsevier BV * North-Holland",

}

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 - 2014/11/15

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.

U2 - 10.1016/j.physd.2014.08.003

DO - 10.1016/j.physd.2014.08.003

M3 - Journal article

SN - 0167-2789

VL - 288

SP - 45

EP - 52

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -