  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 $N$.
