IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v148y2022icp180-226.html
   My bibliography  Save this article

Interacting Hawkes processes with multiplicative inhibition

Author

Listed:
  • Duval, Céline
  • Luçon, Eric
  • Pouzat, Christophe

Abstract

In the present work, we introduce a general class of mean-field interacting nonlinear Hawkes processes modeling the reciprocal interactions between two neuronal populations, one excitatory and one inhibitory. The model incorporates two features: inhibition, which acts as a multiplicative factor onto the intensity of the excitatory population and additive retroaction from the excitatory neurons onto the inhibitory ones. We give first a detailed analysis of the well-posedness of this interacting system as well as its dynamics in large population. The second aim of the paper is to give a rigorous analysis of the longtime behavior of the mean-field limit process. We provide also numerical evidence that inhibition and retroaction may be responsible for the emergence of limit cycles in such system.

Suggested Citation

  • Duval, Céline & Luçon, Eric & Pouzat, Christophe, 2022. "Interacting Hawkes processes with multiplicative inhibition," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 180-226.
    • Handle: RePEc:eee:spapps:v:148:y:2022:i:c:p:180-226
    DOI: 10.1016/j.spa.2022.02.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414922000448
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2022.02.008?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. Bonnet, Anna & Martinez Herrera, Miguel & Sangnier, Maxime, 2021. "Maximum likelihood estimation for Hawkes processes with self-excitation or inhibition," Statistics & Probability Letters, Elsevier, vol. 179(C).
    2. Ditlevsen, Susanne & Löcherbach, Eva, 2017. "Multi-class oscillating systems of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1840-1869.
    3. Chevallier, J. & Duarte, A. & Löcherbach, E. & Ost, G., 2019. "Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 1-27.
    4. 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.
    5. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
    6. Chevallier, Julien, 2017. "Mean-field limit of generalized Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3870-3912.
    7. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Post-Print hal-01313994, HAL.
    8. Heesen, Sophie & Stannat, Wilhelm, 2021. "Fluctuation limits for mean-field interacting nonlinear Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 280-297.
    Full references (including those not matched with items on IDEAS)

    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. Agathe-Nerine, Zoé, 2022. "Multivariate Hawkes processes on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 86-148.
    2. Pfaffelhuber, P. & Rotter, S. & Stiefel, J., 2022. "Mean-field limits for non-linear Hawkes processes with excitation and inhibition," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 57-78.
    3. Li, Bo & Pang, Guodong, 2022. "Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 285-339.
    4. Charlotte Dion & Sarah Lemler, 2020. "Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 489-515, October.
    5. Ulrich Horst & Wei Xu, 2019. "Functional Limit Theorems for Marked Hawkes Point Measures ," Working Papers hal-02443841, HAL.
    6. Patrick Chang & Roger Bukuru & Tim Gebbie, 2019. "Revisiting the Epps effect using volume time averaging: An exercise in R," Papers 1912.02416, arXiv.org, revised Feb 2020.
    7. Thibault Jaisson & Mathieu Rosenbaum, 2015. "Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes," Papers 1504.03100, arXiv.org.
    8. Paul Jusselin & Mathieu Rosenbaum, 2020. "No‐arbitrage implies power‐law market impact and rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1309-1336, October.
    9. Omar Euch & Masaaki Fukasawa & Mathieu Rosenbaum, 2018. "The microstructural foundations of leverage effect and rough volatility," Finance and Stochastics, Springer, vol. 22(2), pages 241-280, April.
    10. Aur'elien Alfonsi & Pierre Blanc, 2014. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Papers 1404.0648, arXiv.org, revised Jun 2015.
    11. Cattiaux, Patrick & Colombani, Laetitia & Costa, Manon, 2022. "Limit theorems for Hawkes processes including inhibition," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 404-426.
    12. Hillairet, Caroline & Réveillac, Anthony & Rosenbaum, Mathieu, 2023. "An expansion formula for Hawkes processes and application to cyber-insurance derivatives," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 89-119.
    13. Heesen, Sophie & Stannat, Wilhelm, 2021. "Fluctuation limits for mean-field interacting nonlinear Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 280-297.
    14. Takeuchi, Atsushi, 2019. "Integration by parts formulas for marked Hawkes processes," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 229-237.
    15. Emmanuel Bacry & Jean-Fran�ois Muzy, 2014. "Hawkes model for price and trades high-frequency dynamics," Quantitative Finance, Taylor & Francis Journals, vol. 14(7), pages 1147-1166, July.
    16. Aurélien Alfonsi & Pierre Blanc, 2016. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Post-Print hal-00971369, HAL.
    17. Aditi Dandapani & Paul Jusselin & Mathieu Rosenbaum, 2019. "From quadratic Hawkes processes to super-Heston rough volatility models with Zumbach effect," Papers 1907.06151, arXiv.org, revised Jan 2021.
    18. Fuentes, Fernanda & Herrera, Rodrigo & Clements, Adam, 2018. "Modeling extreme risks in commodities and commodity currencies," Pacific-Basin Finance Journal, Elsevier, vol. 51(C), pages 108-120.
    19. Roueff, Francois & von Sachs, Rainer & Sansonnet, Laure, 2015. "Time-frequency analysis of locally stationary Hawkes processes," LIDAM Discussion Papers ISBA 2015011, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

    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:148:y:2022:i:c:p:180-226. 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.