General time elapsed neuron network model: well-posedness and strong connectivity regime
Weng, Qilong (2016), General time elapsed neuron network model: well-posedness and strong connectivity regime. https://basepub.dauphine.fr/handle/123456789/17139
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-01243163
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
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Abstract (EN)For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models, indicating how the time elapsed since the last discharge construct the probability density of neurons. Through the spectral analysis theory for semigroups in Banach spaces developed recently in [6, 9], on the one hand, we prove the existence and the uniqueness of the weak solution in the whole connectivity regime as well as the parallel results on the long time behavior of solutions obtained in  under general assumptions on the ring rate and the delay distribution. On the other hand, we extend those similar results obtained in [11, 12] in the case without delay to the case taking delay into account and both in the weak and the strong connectivity regime with a particular step function ring rate. Our approach uses the spectral analysis theory for semigroups in Banach spaces developed recently by the first author and collaborators.
Subjects / Keywordssemigroup spectral analysis; Neuron networks; time elapsed dynamics; strong connectivity; exponential asymptotic stability
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