Averaging, Folded Singularities, and Torus Canards: Explaining Transitions between Bursting and Spiking in a Coupled Neuron Model

In this work, we identify generic bifurcation scenarios corresponding to transitions between bursting and tonic spiking solutions in a model for a coupled pair of burst-capable neurons, and we elucidate the central role of folded singularities in these scenarios. The folded singularities in our work arise in the context of fast-slow averaging, and hence our results link with the study of torus canards, a recently identified class of ordinary differential equation (ODE) solutions featuring oscillatory excursions along repelling structures in phase space [J. Burke et al., J. Math. Neurosci., 2 (2012), pp. 1-30]; in particular, our work extends this study to systems featuring two slow variables and symmetry and goes significantly beyond the analysis of activity transitions presented by Best et al. [SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 1107-1139].