Modeling neuronal bursting: singularity theory meets neurophysiology∗

Exploiting the specific structure of neuron conductance-based models, the paper investigates the mathematical modeling of neuronal bursting. The proposed approach combines singularity theory and geometric singular perturbations to unfold the geometry of multiple time-scales attractors in the neighborhood of high-codimension singularities. We detect a three-time scale bursting attractor in the universal unfolding of the winged cusp singularity and discuss the physiological relevance of the bifurcation and unfolding parameters in determining a physiological route into bursting, that is, the transition between tonic spiking and bursting. The results suggest generality and simplicity in the organizing role of the winged cusp singularity for the global dynamics of conductance based models.