Averaging, folded singularities, and torus canards in bursting/spiking transitions

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.