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Multi-class oscillating systems of interacting neurons

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  • Ditlevsen, Susanne
  • Löcherbach, Eva

Abstract

We consider multi-class systems of interacting nonlinear Hawkes processes modeling several large families of neurons and study their mean field limits. As the total number of neurons goes to infinity we prove that the evolution within each class can be described by a nonlinear limit differential equation driven by a Poisson random measure, and state associated central limit theorems. We study situations in which the limit system exhibits oscillatory behavior, and relate the results to certain piecewise deterministic Markov processes and their diffusion approximations.

Suggested Citation

  • Ditlevsen, Susanne & Löcherbach, Eva, 2017. "Multi-class oscillating systems of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1840-1869.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:6:p:1840-1869
    DOI: 10.1016/j.spa.2016.09.013
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    References listed on IDEAS

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    1. Scheutzow, Michael, 1985. "Noise can create periodic behavior and stabilize nonlinear diffusions," Stochastic Processes and their Applications, Elsevier, vol. 20(2), pages 323-331, September.
    2. Benaim, Michel & Hirsch, Morris W., 1999. "Mixed Equilibria and Dynamical Systems Arising from Fictitious Play in Perturbed Games," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 36-72, October.
    3. Douc, Randal & Fort, Gersende & Guillin, Arnaud, 2009. "Subgeometric rates of convergence of f-ergodic strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 897-923, March.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. Anna Melnykova, 2020. "Parametric inference for hypoelliptic ergodic diffusions with full observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 595-635, October.
    4. 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.
    5. Agathe-Nerine, Zoé, 2022. "Multivariate Hawkes processes on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 86-148.
    6. Simon Clinet, 2022. "Quasi-likelihood analysis for marked point processes and application to marked Hawkes processes," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 189-225, July.
    7. Quentin Clairon & Adeline Samson, 2022. "Optimal control for parameter estimation in partially observed hypoelliptic stochastic differential equations," Computational Statistics, Springer, vol. 37(5), pages 2471-2491, November.
    8. Holbach, Simon, 2020. "Positive Harris recurrence for degenerate diffusions with internal variables and randomly perturbed time-periodic input," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6965-7003.
    9. Gao, Fuqing & Zhu, Lingjiong, 2018. "Some asymptotic results for nonlinear Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4051-4077.
    10. Chevallier, Julien & Ost, Guilherme, 2020. "Fluctuations for spatially extended Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5510-5542.
    11. Collet, Francesca & Kraaij, Richard C., 2020. "Path-space moderate deviations for a class of Curie–Weiss models with dissipation," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4028-4061.
    12. Susanne Ditlevsen & Adeline Samson, 2019. "Hypoelliptic diffusions: filtering and inference from complete and partial observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 361-384, April.
    13. Paolo Dai Pra & Elena Sartori & Marco Tolotti, 2019. "Climb on the Bandwagon: Consensus and Periodicity in a Lifetime Utility Model with Strategic Interactions," Dynamic Games and Applications, Springer, vol. 9(4), pages 1061-1075, December.
    14. Schmutz, Valentin, 2022. "Mean-field limit of age and leaky memory dependent Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 39-59.
    15. Pigato, Paolo, 2022. "Density estimates and short-time asymptotics for a hypoelliptic diffusion process," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 117-142.
    16. 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.
    17. 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.

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