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Particle systems with a singular mean-field self-excitation. Application to neuronal networks

Author

Listed:
  • Delarue, F.
  • Inglis, J.
  • Rubenthaler, S.
  • Tanré, E.

Abstract

We discuss the construction and approximation of solutions to a nonlinear McKean–Vlasov equation driven by a singular self-excitatory interaction of the mean-field type. Such an equation is intended to describe an infinite population of neurons which interact with one another. Each time a proportion of neurons ‘spike’, the whole network instantaneously receives an excitatory kick. The instantaneous nature of the excitation makes the system singular and prevents the application of standard results from the literature. Making use of the Skorohod M1 topology, we prove that, for the right notion of a ‘physical’ solution, the nonlinear equation can be approximated either by a finite particle system or by a delayed equation. As a by-product, we obtain the existence of ‘synchronized’ solutions, for which a macroscopic proportion of neurons may spike at the same time.

Suggested Citation

  • Delarue, F. & Inglis, J. & Rubenthaler, S. & Tanré, E., 2015. "Particle systems with a singular mean-field self-excitation. Application to neuronal networks," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2451-2492.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:6:p:2451-2492
    DOI: 10.1016/j.spa.2015.01.007
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    References listed on IDEAS

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    1. Ward Whitt, 2001. "The Reflection Map with Discontinuities," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 447-484, August.
    2. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers, 2011. "Default clustering in large portfolios: Typical events," Papers 1104.1773, arXiv.org, revised Feb 2013.
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    Cited by:

    1. Cormier, Quentin & Tanré, Etienne & Veltz, Romain, 2020. "Long time behavior of a mean-field model of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2553-2595.
    2. Erhan Bayraktar & Gaoyue Guo & Wenpin Tang & Yuming Paul Zhang, 2022. "Systemic robustness: a mean-field particle system approach," Papers 2212.08518, arXiv.org, revised Aug 2023.
    3. Sirignano, Justin & Spiliopoulos, Konstantinos, 2020. "Mean field analysis of neural networks: A central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1820-1852.
    4. Kumar, Santosh & Singh, Paramjeet, 2019. "High order WENO finite volume approximation for population density neuron model," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 173-189.
    5. Ben Hambly & Andreas Søjmark, 2019. "An SPDE model for systemic risk with endogenous contagion," Finance and Stochastics, Springer, vol. 23(3), pages 535-594, July.
    6. Sean Ledger & Andreas Sojmark, 2018. "Uniqueness for contagious McKean--Vlasov systems in the weak feedback regime," Papers 1811.12356, arXiv.org, revised Oct 2019.
    7. Sean Ledger & Andreas Sojmark, 2018. "At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem," Papers 1807.05126, arXiv.org, revised Mar 2024.
    8. Chevallier, Julien, 2017. "Mean-field limit of generalized Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3870-3912.
    9. Erhan Bayraktar & Gaoyue Guo & Wenpin Tang & Yuming Zhang, 2020. "McKean-Vlasov equations involving hitting times: blow-ups and global solvability," Papers 2010.14646, arXiv.org, revised Jul 2023.
    10. Ben Hambly & Andreas Sojmark, 2018. "An SPDE Model for Systemic Risk with Endogenous Contagion," Papers 1801.10088, arXiv.org, revised Sep 2018.
    11. Roman Gayduk & Sergey Nadtochiy, 2020. "Control-Stopping Games for Market Microstructure and Beyond," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1289-1317, November.

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    More about this item

    Keywords

    McKean nonlinear diffusion process; Counting process; Propagation of chaos; Integrate-and-fire network; Skorohod M1 topology; Neuroscience;
    All these keywords.

    JEL classification:

    • M1 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Administration

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