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

Quentin Cormier 1 Etienne Tanré 1 Romain Veltz 2
1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
2 MATHNEURO - Mathématiques pour les Neurosciences
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : 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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Submitted on : Tuesday, May 14, 2019 - 11:53:32 AM
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Quentin Cormier, Etienne Tanré, Romain Veltz. Long time behavior of a mean-field model of interacting neurons. Stochastic Processes and their Applications, Elsevier, 2019, ⟨10.1016/j.spa.2019.07.010⟩. ⟨hal-01903857v2⟩

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