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H 1 Mean-field analysis of an excitatory neuronal network: application to systemic risk modeling?

Auteurs : Delarue, François (Auteur de la Conférence)
CIRM (Editeur )

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integrate and default model mean-field interaction blow-up phenomenon systemic risk nonlinear Fokker Planck equation change of regime in existence and uniqueness

Résumé : Inspired by modeling in neurosciences, we here discuss the well-posedness of a networked integrate-and-fire model describing an infinite population of companies which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the debt of a company increases when some of the others default: precisely, the loss it receives is proportional to the instantaneous proportion of companies that default at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by a, is of great importance as the resulting system is known to blow-up when a takes large values, a blow-up meaning that a macroscopic proportion of companies may default at the same time. In the current talk, we focus on the complementary regime and prove that existence and uniqueness hold in arbitrary time without any blow-up when the excitatory parameter is small enough.

Codes MSC :
35K60 - "Nonlinear boundary value problems for linear parabolic PDE; boundary value problems for nonlinear parabolic PDE"
82C31 - Stochastic methods in time-dependent statistical mechanics (Fokker-Planck, Langevin, etc.)
92B20 - Neural networks, artificial life and related topics

    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 11/09/14
    Date de captation : 09/09/14
    Collection : Research talks ; Partial Differential Equations ; Mathematics in Science and Technology
    Format : QuickTime (.mov) Durée : 00:34:23
    Domaine : PDE ; Mathematics in Science & Technology
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2014-09-09_Delarue.mp4

Informations sur la rencontre

Nom de la rencontre : Advances in stochastic analysis for risk modeling / Analyse stochastique pour la modélisation des risques
Organisateurs de la rencontre : Bouchard, Bruno ; Chassagneux, Jean-François ; Elie, Romuald ; Réveillac, Anthony ; Soner, H. Mete
Dates : 08/09/14 - 12/09/14
Année de la rencontre : 2014

Citation Data

DOI : 10.24350/CIRM.V.18560203
Cite this video as: Delarue, François (2014). Mean-field analysis of an excitatory neuronal network: application to systemic risk modeling?. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18560203
URI : http://dx.doi.org/10.24350/CIRM.V.18560203

Bibliographie

  • M. J. Cáceres, J. A. Carrillo, and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states, J. Math. Neurosci., 1 (2011), p. 7 - http://dx.doi.org/10.1186/2190-8567-1-7

  • F. Delarue, J. Inglis, S. Rubenthaler, and E. Tanré, Global solvability of a networked integrate-and-fire model of McKean-Vlasov type. arxiv:1211.0299v4, 2014 - http://arxiv.org/abs/1211.0299v4

  • F. Delarue, J. Inglis, S. Rubenthaler, and E. Tanré, Particle systems with a singular mean-field self-excitation. Application to neuronal networks. arxiv:1406.1151, 2014 - http://arxiv.org/abs/1406.1151v2



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