GLOBAL SOLVABILITY OF A NETWORKED INTEGRATE-AND-FIRE MODEL OF MCKEAN-VLASOV TYPE

• 成果类型: Article
• 署名作者: Delarue, Francois; Inglis, James; Rubenthaler, Sylvain; Tanre, Etienne
• 署名单位: Universite Cote d'Azur; Universite Cote d'Azur
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1044
• 发表日期: 2015
• 页码: 2096-2133
• 关键词: neurons; DYNAMICS

摘要

We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by a, is of great importance as the resulting system is known to blow-up for large values of a. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when a is small enough.