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

Abstract : We study the long time behavior of a McKean-Vlasov stochastic differential equation (SDE), driven by a Poisson measure. In neuroscience, this SDE models the dynamics of the membrane potential of a typical neuron in a large network. The model can be derived by considering a finite network of generalized Integrate-And-Fire neurons and by taking the limit where the number of neurons goes to infinity. Hence the McKean-Vlasov SDE is a mean-field model of spiking neurons.We study existence and uniqueness of the solution this McKean-Vlasov SDE and describe its invariant probability measures. For small enough interaction parameter J, we prove uniqueness and global stability of the invariant measure. For J arbitrary large however, the invariant measures may not be unique. We give a sufficient condition ensuring the local stability of such a given invariant probability measure. Our criterion involves the location of the zeros of an explicit holomorphic function associated to the considered stationary solution. When all the zeros have negative real part, we prove that stability holds. We then give sufficient general conditions ensuring the existence of periodic solutions through a Hopf bifurcation: at some critical interaction parameter J0, the invariant probability losses its stability and periodic solutions appear for J close to J0. To obtain these results, we combine probabilistic and deterministic methods. In particular, a key tool in this analysis is a nonlinear Volterra Integral equation satisfied by the synaptic current.Finally, we illustrate these results with examples which are tractable analytically. Additionally, we give numerical methods to approximate the solution of the mean-field equation and to predict numerically the bifurcations.
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https://hal.inria.fr/tel-03123384
Contributor : Quentin Cormier Connect in order to contact the contributor
Submitted on : Wednesday, January 27, 2021 - 6:44:13 PM
Last modification on : Monday, October 11, 2021 - 5:09:11 PM
Long-term archiving on: : Wednesday, April 28, 2021 - 7:22:02 PM

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  • HAL Id : tel-03123384, version 1

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Quentin Cormier. Long time behavior of a mean-field model of interacting spiking neurons. Probability [math.PR]. Université Côte d'Azur, 2021. English. ⟨NNT : 2021COAZ4008⟩. ⟨tel-03123384⟩

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