  Exploiting the specific structure of neuron conductance-based models, the
paper investigates the mathematical modeling of neuronal bursting modulation.
The proposed approach combines singularity theory and geometric singular
perturbations to capture 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 modulation of bursting. The results
suggest generality and simplicity in the organizing role of the winged cusp
singularity for the global dynamics of conductance based models.
