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

Exit versus escape in a stochastic dynamical system of neuronal networks explains heterogenous bursting intervals