A neuron is said to have bursting dynamics when its activity alternates between a quiescent state (equilibrium) and spiking state (limit cycle). Most models of bursting have singularly perturbed form

x’ = f(x,y)

y’ = \mu g(x,y)

Slow changes in y cause x’ = f(x,y) to bifurcate from equilibrium to limit cycle attractor and back. We review relevant bifurcation theory and use it to classify bursting dynamics. We also review classical methods of nonlinear analysis, such as averaging, singular perturbations, etc., and discuss challenges and pitfalls when one applies those methods to study bursting dynamics.

REFERENCE:

Izhikevich (2000) Neural Excitability, Spiking, and Bursting,

International Journal of Bifurcation and Chaos, 10:1171--1266.

http://www.nsi.edu/users/izhikevich/publications/nesb.htm