Bursting in a next generation neural mass model with synaptic dynamics: a slow-fast approach

We report a detailed analysis on the emergence of bursting in a recently developed neural mass model that includes short-term synaptic plasticity. Neural mass models can mimic the collective dynamics of large-scale neuronal populations in terms of a few macroscopic variables like mean membrane potential and firing rate. The present one is particularly important, as it represents an exact meanfield limit of synaptically coupled quadratic integrate and fire (QIF) neurons. Without synaptic dynamics, a periodic external current with slow frequency epsilon can lead to burst-like dynamics. The firing patterns can be understood using singular perturbation theory, specifically slow-fast dissection. With synaptic dynamics, timescale separation leads to a variety of slow-fast phenomena and their role for bursting becomes inordinately more intricate. Canards are crucial to understand the route to bursting. They describe trajectories evolving nearby repelling locally invariant sets of the system and exist at the transition between subthreshold dynamics and bursting. Near the singular limit epsilon = 0, we report peculiar jump-on canards, which block a continuous transition to bursting. In the biologically more plausible eregime, this transition becomes continuous and bursts emerge via consecutive spike-adding transitions. The onset of bursting is complex and involves mixed-type-like torus canards, which form the very first spikes of the burst and follow fast-subsystem repelling limit cycles. We numerically evidence the same mechanisms to be responsible for bursting emergence in the QIF network with plastic synapses. The main conclusions apply for the network, owing to the exactness of the meanfield limit.