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Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels

Author

Listed:
  • Chevallier, J.
  • Duarte, A.
  • Löcherbach, E.
  • Ost, G.

Abstract

We consider spatially extended systems of interacting nonlinear Hawkes processes modeling large systems of neurons placed in Rd and study the associated mean field limits. As the total number of neurons tends to infinity, we prove that the evolution of a typical neuron, attached to a given spatial position, can be described by a nonlinear limit differential equation driven by a Poisson random measure. The limit process is described by a neural field equation. As a consequence, we provide a rigorous derivation of the neural field equation based on a thorough mean field analysis.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:1:p:1-27
    DOI: 10.1016/j.spa.2018.02.007
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    References listed on IDEAS

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    1. Chevallier, Julien, 2017. "Mean-field limit of generalized Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3870-3912.
    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.
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    Cited by:

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    2. Luçon, Eric, 2020. "Quenched asymptotics for interacting diffusions on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6783-6842.
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    4. 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.
    5. 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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