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Relaxation in time elapsed neuron network models in the weak connectivity regime

Mischler, Stéphane; Weng, Qilong (2018), Relaxation in time elapsed neuron network models in the weak connectivity regime, Acta Applicandae Mathematicae, 157, 1, p. 45-74. 10.1007/s10440-018-0163-4

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01148645
Date
2018
Journal name
Acta Applicandae Mathematicae
Volume
157
Number
1
Publisher
Springer
Pages
45-74
Publication identifier
10.1007/s10440-018-0163-4
Metadata
Show full item record
Author(s)
Mischler, Stéphane
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Weng, Qilong
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
In order to describe the firing activity of a homogenous assembly of neurons, we consider time elapsed models, which give mathematical descriptions of the probability density of neurons structured by the distribution of times elapsed since the last discharge. Under general assumption on the firing rate and the delay distribution, we prove the uniqueness of the steady state and its nonlinear exponential stability in the weak connectivity regime. The result generalizes some similar results obtained in [10] in the case without delay. Our approach uses the spectral analysis theory for semigroups in Banach spaces developed recently by the first author and collaborators.
Subjects / Keywords
Neuron networks; time elapsed dynamics; semigroup spectral analysis; weak connectivity; long time asymptotic

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