Stability of wandering bumps for Hawkes processes interacting on the circle

Zoé Agathe-Nerine

Stochastic Processes and their Applications, 2025, vol. 182, issue C

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.

Keywords: Multivariate nonlinear Hawkes processes; Mean-field systems; Neural field equation; Spatially extended system; Stationary bumps (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2025.104577

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