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Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective

Publication ,  Journal Article
Liu, J-G; Wang, Z; Xie, Y; Zhang, Y; Zhou, Z
Published in: Mathematical Neuroscience and Applications
November 30, 2021

In the mean field integrate-and-fire model, the dynamics of a typical neuron within a large network is modeled as a diffusion-jump stochastic process whose jump takes place once the voltage reaches a threshold. In this work, the main goal is to establish the convergence relationship between the regularized process and the original one where in the regularized process, the jump mechanism is replaced by a Poisson dynamic, and jump intensity within the classically forbidden domain goes to infinity as the regularization parameter vanishes. On the macroscopic level, the Fokker-Planck equation for the process with random discharges (i.e. Poisson jumps) are defined on the whole space, while the equation for the limit process is on the half space. However, with the iteration scheme, the difficulty due to the domain differences has been greatly mitigated and the convergence for the stochastic process and the firing rates can be established. Moreover, we find a polynomial-order convergence for the distribution by a re-normalization argument in probability theory. Finally, by numerical experiments, we quantitatively explore the rate and the asymptotic behavior of the convergence for both linear and nonlinear models.

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Published In

Mathematical Neuroscience and Applications

DOI

EISSN

2801-0159

Publication Date

November 30, 2021

Volume

Volume 1

Publisher

Centre pour la Communication Scientifique Directe (CCSD)
 

Citation

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Liu, J.-G., Wang, Z., Xie, Y., Zhang, Y., & Zhou, Z. (2021). Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective. Mathematical Neuroscience and Applications, Volume 1. https://doi.org/10.46298/mna.7203
Liu, Jian-Guo, Ziheng Wang, Yantong Xie, Yuan Zhang, and Zhennan Zhou. “Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective.” Mathematical Neuroscience and Applications Volume 1 (November 30, 2021). https://doi.org/10.46298/mna.7203.
Liu J-G, Wang Z, Xie Y, Zhang Y, Zhou Z. Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective. Mathematical Neuroscience and Applications. 2021 Nov 30;Volume 1.
Liu, Jian-Guo, et al. “Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective.” Mathematical Neuroscience and Applications, vol. Volume 1, Centre pour la Communication Scientifique Directe (CCSD), Nov. 2021. Crossref, doi:10.46298/mna.7203.
Liu J-G, Wang Z, Xie Y, Zhang Y, Zhou Z. Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective. Mathematical Neuroscience and Applications. Centre pour la Communication Scientifique Directe (CCSD); 2021 Nov 30;Volume 1.

Published In

Mathematical Neuroscience and Applications

DOI

EISSN

2801-0159

Publication Date

November 30, 2021

Volume

Volume 1

Publisher

Centre pour la Communication Scientifique Directe (CCSD)