MULTIRHYTHMIC BURSTING

A complex modeled bursting neuron [C. C. Canavier, J. W. Clark, and J. H. Byrne, J. Neurophysiol. 66, 2107-2124 (1991)] has been shown to po ssess seven coexisting limit cycle solutions at a given parameter set [Canavier et at, J. Neurophysiol 69, 2252-2259 (1993); 72, 872-882 (19 94)]. These solutions are unique in that the limit cycles are concentr ic in the space of the slow variables. We examine the origin of these solutions using a minimal 4-variable bursting cell model. Poincare map s are constructed using a saddle-node bifurcation of a fast subsystem such as our Poincare section. This bifurcation defines a threshold bet ween the active and silent phases of the burst cycle in the space of t he slow variables. The maps identify parameter spaces with single limi t cycles, multiple limit cycles, and two types of chaotic bursting. To investigate the dynamical features which underlie the unique shape of the maps, the maps are further decomposed into two submaps which desc ribe the solution trajectories during the active and silent phases of a single burst. From these findings we postulate several necessary cri teria for a bursting model to possess multiple stable concentric limit cycles. These criteria are demonstrated in a generalized 3-variable m odel. Finally, using a less direct numerical procedure, similar return maps are calculated for the original complex model [C. C. Canavier, J . W. Clark, and J. H. Byrne, J. Neurophysiol. 66, 2107-2124 (1991)], w ith the resulting mappings appearing qualitatively similar to those of our 4-variable model. These multistable concentric bursting solutions cannot occur in a bursting model with one slow variable. This type of multistability arises when a bursting system has two or more slow var iables and is viewed as an essentially second-order system which recei ves discrete perturbations in a state-dependent manner.