Voltage interval mappings for activity transitions in neuron models for elliptic bursters

We performed a thorough bifurcation analysis of a mathematical elliptic bursting model, using a computer-assisted reduction to equationless, one-dimensional Poincaré mappings for a voltage interval. Using the interval mappings, we were able to examine in detail the bifurcations that underlie the complex activity transitions between: tonic spiking and bursting, bursting and mixed-mode oscillations, and finally mixed-mode oscillations and quiescence in the FitzHugh–Nagumo–Rinzel model. We illustrate the wealth of information, qualitative and quantitative, that was derived from the Poincaré mappings, for the neuronal models and for similar (electro)chemical systems.

► Elliptic burster models are studied via reduction to equationless Poincaré mappings.
► The approach is applicable to elliptic bursters and other electrochemical systems.
► Transitions between tonic spiking, mixed mode oscillations and bursting are examined.
► Emergence and bifurcations of torus are reported in such elliptic slow–fast models.