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Pré-Publication, Document De Travail Année : 2018

Long time behavior of a mean-field model of interacting neurons

Résumé

We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean-Vlasov equation.
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Dates et versions

hal-01903857 , version 1 (24-10-2018)
hal-01903857 , version 2 (14-05-2019)
hal-01903857 , version 3 (14-08-2020)

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Quentin Cormier, Etienne Tanré, Romain Veltz. Long time behavior of a mean-field model of interacting neurons. 2018. ⟨hal-01903857v1⟩
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