  We consider a stochastic system of spiking neurons which was previously
proven to present a metastable behavior for a suitable choice of the parameter,
in the sense that the time of extinction is asymptotically memory-less when the
number of components in the system goes to $\infty$. In the present article we
complete this work by showing that, previous to extinction, the system tends to
stabilize in the sense that temporal means taken on an appropriate time scale
converge in probability to some fixed value. This property is sometime called
thermalization.
