Modeling bursting in neuronal networks using facilitation-depression and afterhyperpolarization

In the absence of inhibition, excitatory neuronal networks can alternate between bursts and interburst intervals (IBI), with heterogeneous length distributions. As this dynamic remains unclear, especially the durations of each epoch, we develop here a bursting model based on synaptic depression and facilitation that also accounts for afterhyperpolarization (AHP), which is a key component of IBI. The framework is a novel stochastic three dimensional dynamical system perturbed by noise: numerical simulations can reproduce a succession of bursts and interbursts. Each phase corresponds to an exploration of a fraction of the phase-space, which contains three critical points (one attractor and two saddles) separated by a two-dimensional stable manifold $\Sigma$. We show here that bursting is defined by long deterministic excursions away from the attractor, while IBI corresponds to escape induced by random fluctuations. We show that the variability in the burst durations, depends on the distribution of exit points located on $\Sigma$ that we compute using WKB and the method of characteristics. Finally, to better characterize the role of several parameters such as the network connectivity or the AHP time scale, we compute analytically the mean burst and AHP durations in a linear approximation. To conclude the distribution of bursting and IBI could result from synaptic dynamics modulated by AHP.