  Neuronal networks can generate burst events. It remains unclear how to
analyse interburst periods and their statistics. We study here the phase-space
of a mean-field model, based on synaptic short-term changes, that exhibit burst
and interburst dynamics and we identify that interburst corresponds to the
escape from a basin of attraction. Using stochastic simulations, we report here
that the distribution of the these durations do not match with the time to
reach the boundary. We further analyse this phenomenon by studying a generic
class of two-dimensional dynamical systems perturbed by small noise that
exhibits two peculiar behaviors: 1- the maximum associated to the probability
density function is not located at the point attractor, which came as a
surprise. The distance between the maximum and the attractor increases with the
noise amplitude $\sigma$, as we show using WKB approximation and numerical
simulations. 2- For such systems, exiting from the basin of attraction is not
sufficient to characterize the entire escape time, due to trajectories that can
return several times inside the basin of attraction after crossing the
boundary, before eventually escaping far away. To conclude, long-interburst
durations are inherent properties of the dynamics and sould be expected in
empirical time series.
