Minimal models of bursting neurons: How multiple currents, conductances, and timescales affect bifurcation diagrams

After reviewing the Hodgkin-Huxley ionic current formulation, we introduce a three-variable generic model of a single-compartment neuron comprising a two-dimensional fast subsystem and a very slow recovery variable. We study the effects of fast and slow currents on the existence and stability of equilibria and periodic orbits for the fast subsystem, presenting a classification of currents and developing graphical tools that aid in the analysis and construction of models with specified properties. We draw on these to propose a minimal model of a bursting neuron, identifying biophysical parameters that can shape and regulate key characteristics of the membrane voltage pattern: bursting frequency, duty cycle, spike rate, and the number of action potentials per burst. We present additional examples from the literature for comparison and illustration, and in a companion paper [SIAM J. Appl. Dyn. Syst., 3 ( 2004), pp. 671-700], we construct a model of an insect central pattern generator using these methods.