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Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

François Delarue 1 James Inglis 2 Sylvain Rubenthaler 1 Etienne Tanré 2 
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : 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 is of great importance as the resulting system is known to blow-up as this becomes large. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when the coefficient of proportionality is small enough.
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François Delarue, James Inglis, Sylvain Rubenthaler, Etienne Tanré. Global solvability of a networked integrate-and-fire model of McKean-Vlasov type. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2015, 25 (4), pp.2096--2133. ⟨10.1214/14-AAP1044⟩. ⟨hal-00747565v4⟩



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