Title :

Wasserstein contraction for the stochastic Morris-Lecar neuron model

Author(s) :

Herda, Maxime [Auteur]
Reliable numerical approximations of dissipative systems [RAPSODI]
Monmarché, Pierre [Auteur]
Laboratoire de chimie théorique [LCT]
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Perthame, Benoît [Auteur]
Modelling and Analysis for Medical and Biological Applications [MAMBA]
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]

Publication date :

2023-07-25

English keyword(s) :

Voltage-conductance kinetic equation
Neural networks
Reflected SDEs
Fokker-Planck equation
Wasserstein distance
Couplings

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Mathématiques [math]/Probabilités [math.PR]

English abstract : [en]

Neuron models have attracted a lot of attention recently, both in mathematics and neuroscience. We are interested in studying long-time and largepopulation emerging properties in a simplified toy model. From a mathematical ...
Show more >Neuron models have attracted a lot of attention recently, both in mathematics and neuroscience. We are interested in studying long-time and largepopulation emerging properties in a simplified toy model. From a mathematical perspective, this amounts to study the long-time behaviour of a degenerate reflected diffusion process. Using coupling arguments, the flow is proven to be a contraction of the Wasserstein distance for long times, which implies the exponential relaxation toward a (non-explicit) unique globally attractive equilibrium distribution. The result is extended to a McKean-Vlasov type non-linear variation of the model, when the mean-field interaction is sufficiently small. The ergodicity of the process results from a combination of deterministic contraction properties and local diffusion, the noise being sufficient to drive the system away from non-contractive domains.Show less >

Language :

Anglais

ANR Project :

Diffusions modulées : métastabilité et algorithmes stochastiques
Enjeux mathématiques issus des neurosciences

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL