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

Quentin Cormier, Etienne Tanré and Romain Veltz

Stochastic Processes and their Applications, 2020, vol. 130, issue 5, 2553-2595

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 probability 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 probability 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.

Keywords: McKean–Vlasov SDE; Long time behavior; Mean-field interaction; Volterra integral equation; Piecewise deterministic Markov process (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2019.07.010

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