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Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective
- Source :
- Mathematical Neuroscience and Applications. 1
- Publication Year :
- 2021
- Publisher :
- Centre pour la Communication Scientifique Directe (CCSD), 2021.
-
Abstract
- 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.<br />This is a new version of the existed paper and I should not submit a new one. Please see arXiv:2009.04679
- Subjects :
- Stochastic process
Probability (math.PR)
60B10, 60H30, 92B20, 92C20
Poisson distribution
symbols.namesake
Nonlinear system
Mathematics - Analysis of PDEs
Distribution (mathematics)
Probability theory
Mean field theory
FOS: Mathematics
symbols
Jump
Limit (mathematics)
Statistical physics
Mathematics - Probability
Analysis of PDEs (math.AP)
Mathematics
Subjects
Details
- ISSN :
- 28010159
- Volume :
- 1
- Database :
- OpenAIRE
- Journal :
- Mathematical Neuroscience and Applications
- Accession number :
- edsair.doi.dedup.....2606b9f19628df4db39e4ac6a1b63260