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Stability of wandering bumps for Hawkes processes interacting on the circle

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  • Agathe-Nerine, Zoé

Abstract

We consider a population of Hawkes processes modeling the activity of N interacting neurons. The neurons are regularly positioned on the circle [−π,π], and the connectivity between neurons is given by a cosine kernel. The firing rate function is a sigmoid. The large population limit admits a locally stable manifold of stationary solutions. The main result of the paper concerns the long-time proximity of the synaptic voltage of the population to this manifold in polynomial times in N. We show in particular that the phase of the voltage along this manifold converges towards a Brownian motion on a time scale of order N.

Suggested Citation

  • Agathe-Nerine, Zoé, 2025. "Stability of wandering bumps for Hawkes processes interacting on the circle," Stochastic Processes and their Applications, Elsevier, vol. 182(C).
    • Handle: RePEc:eee:spapps:v:182:y:2025:i:c:s0304414925000183
    DOI: 10.1016/j.spa.2025.104577
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    References listed on IDEAS

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    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. 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.
    3. 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.
    4. Chevallier, Julien, 2017. "Mean-field limit of generalized Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3870-3912.
    5. Chevallier, Julien & Ost, Guilherme, 2020. "Fluctuations for spatially extended Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5510-5542.
    6. 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).
    7. Ditlevsen, Susanne & Löcherbach, Eva, 2017. "Multi-class oscillating systems of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1840-1869.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Prodhomme, Adrien, 2023. "Strong Gaussian approximation of metastable density-dependent Markov chains on large time scales," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 218-264.
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