IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v125y2015i6p2451-2492.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915000186
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2015.01.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. Chevallier, Julien, 2017. "Mean-field limit of generalized Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3870-3912.
    4. Sean Ledger & Andreas Sojmark, 2018. "Uniqueness for contagious McKean--Vlasov systems in the weak feedback regime," Papers 1811.12356, arXiv.org, revised Oct 2019.
    5. 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.
    6. 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.
    7. Ben Hambly & Andreas Sojmark, 2018. "An SPDE Model for Systemic Risk with Endogenous Contagion," Papers 1801.10088, arXiv.org, revised Sep 2018.
    8. 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.
    9. 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.
    10. 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.
    11. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Agostino Capponi & Xu Sun & David D. Yao, 2020. "A Dynamic Network Model of Interbank Lending—Systemic Risk and Liquidity Provisioning," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 1127-1152, August.
    2. Jean-David Fermanian, 2020. "On the Dependence between Default Risk and Recovery Rates in Structural Models," Annals of Economics and Statistics, GENES, issue 140, pages 45-82.
    3. Egami, M. & Kevkhishvili, R., 2017. "An analysis of simultaneous company defaults using a shot noise process," Journal of Banking & Finance, Elsevier, vol. 80(C), pages 135-161.
    4. Muralidharan, Ajith & Pedarsani, Ramtin & Varaiya, Pravin, 2015. "Analysis of fixed-time control," Transportation Research Part B: Methodological, Elsevier, vol. 73(C), pages 81-90.
    5. Konstantinos Spiliopoulos & Jia Yang, 2018. "Network effects in default clustering for large systems," Papers 1812.07645, arXiv.org, revised Feb 2020.
    6. Burzoni, Matteo & Campi, Luciano, 2023. "Mean field games with absorption and common noise with a model of bank run," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 206-241.
    7. Ben Hambly & Nikolaos Kolliopoulos, 2020. "Fast mean-reversion asymptotics for large portfolios of stochastic volatility models," Finance and Stochastics, Springer, vol. 24(3), pages 757-794, July.
    8. Frikha, Noufel & Li, Libo, 2021. "Well-posedness and approximation of some one-dimensional Lévy-driven non-linear SDEs," Stochastic Processes and their Applications, Elsevier, vol. 132(C), pages 76-107.
    9. Ben Hambly & Andreas Sojmark, 2018. "An SPDE Model for Systemic Risk with Endogenous Contagion," Papers 1801.10088, arXiv.org, revised Sep 2018.
    10. Fei Fang & Yiwei Sun & Konstantinos Spiliopoulos, 2016. "The effect of heterogeneity on flocking behavior and systemic risk," Papers 1607.08287, arXiv.org, revised Jun 2017.
    11. Anastasia Borovykh & Andrea Pascucci & Stefano La Rovere, 2018. "Systemic risk in a mean-field model of interbank lending with self-exciting shocks," IISE Transactions, Taylor & Francis Journals, vol. 50(9), pages 806-819, September.
    12. Amarjit Budhiraja & Michael Conroy, 2022. "Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions," Journal of Theoretical Probability, Springer, vol. 35(1), pages 295-349, March.
    13. Fang Fei & Sun Yiwei & Spiliopoulos Konstantinos, 2017. "On the effect of heterogeneity on flocking behavior and systemic risk," Statistics & Risk Modeling, De Gruyter, vol. 34(3-4), pages 141-155, September.
    14. Tang, Qihe & Tang, Zhaofeng & Yang, Yang, 2019. "Sharp asymptotics for large portfolio losses under extreme risks," European Journal of Operational Research, Elsevier, vol. 276(2), pages 710-722.
    15. Josselin Garnier & George Papanicolaou & Tzu-Wei Yang, 2015. "A risk analysis for a system stabilized by a central agent," Papers 1507.08333, arXiv.org, revised Aug 2015.
    16. 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.
    17. Justin Sirignano & Kay Giesecke, 2019. "Risk Analysis for Large Pools of Loans," Management Science, INFORMS, vol. 65(1), pages 107-121, January.
    18. Ahmad, F. & Hambly, B.M. & Ledger, S., 2018. "A stochastic partial differential equation model for the pricing of mortgage-backed securities," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3778-3806.
    19. 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.
    20. Tomoyuki Ichiba & Michael Ludkovski & Andrey Sarantsev, 2019. "Dynamic contagion in a banking system with births and defaults," Annals of Finance, Springer, vol. 15(4), pages 489-538, December.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:125:y:2015:i:6:p:2451-2492. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.