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Pré-Publication, Document De Travail Année : 2023

Stability of wandering bumps for Hawkes processes interacting on the circle

Zoé Agathe-Nerine
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Résumé

We consider a population of Hawkes processes modeling the activity of $N$ interacting neurons. The neurons are regularly positioned on the circle $[-\pi, \pi]$, 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 $N$.
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Dates et versions

hal-04157323 , version 1 (10-07-2023)

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Zoé Agathe-Nerine. Stability of wandering bumps for Hawkes processes interacting on the circle. 2023. ⟨hal-04157323⟩
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