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A complex modeled bursting neuron (Canavier, 1991) has been shown to possess seven coexisting limit cycle solutions at a given parameter set (Canavier 1993, 1994). These solutions are unique in that the limit cycles are concentric in the space of the slow variables. We examine the origin of these solutions using a minimal 4 variable bursting cell model. Poincare maps are constructed using a saddle-node bifurcation of a fast subsystem as our Poincare section. This bifurcation defines a threshold between the active and silent phases of the burst cycle in the space of the slow variables. The maps identifies parameter spaces with single limit 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 describe the solution trajectories during the active and silent phases of a single burst. From these findings we postulate several criteria which a bursting model must satisfy to possibly possess multiple stable limit cycles. Multistability may not occur in a bursting system with one slow variable. Multistability may occur when a bursting system has two (or more) slow variables and is viewed as a second-order system which receives discrete perturbations in a state-dependent manner. These criteria are demonstrated in an abstract 3 variable model. Finally, using a less direct numerical procedure, similar maps are calculated for the original complex model (Canavier 1991), with the resulting mappings appearing qualitatively similar to those of our 4 variable model.
Research Interests
Publications
BACK to Rob Butera's Hompage
BACK to the Mathematical Research Branch homepage