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Mean-field analysis of an excitatory neuronal network: application to systemic risk modeling?
Post-edited
Authors : Delarue, François (Author of the conference)
CIRM (Publisher )
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
CIRM (Publisher )
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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 |
Abstract :
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.
MSC Codes : 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
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 11/09/14 Conference Date : 09/09/14 Subseries : Research talks arXiv category : Quantitative Biology ; Analysis of PDEs ; Probability Mathematical Area(s) : PDE ; Mathematics in Science & Technology Format : QuickTime (.mov) Video Time : 00:34:23 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2014-09-09_Delarue.mp4 
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Information on the Event
Event Title : Advances in stochastic analysis for risk modeling / Analyse stochastique pour la modélisation des risquesEvent Organizers : Bouchard, Bruno ; Chassagneux, Jean-François ; Elie, Romuald ; Réveillac, Anthony ; Soner, H. Mete Dates : 08/09/14 - 12/09/14 Event Year : 2014
Citation Data
DOI : 10.24350/CIRM.V.18560203Cite 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 |
Bibliography
- 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
