Multirhythmic bursting

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Abstract
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 possess seven coexisting limit cycle solutions at a given parameter set [Canavier et al., J. Neurophysiol 69, 2252–2259 (1993); 72, 872–882 (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. Poincaré maps are constructed using a saddle-node bifurcation of a fast subsystem such as our Poincaré 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 identify 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 necessary criteria for a bursting model to possess multiple stable concentric limit cycles. These criteria are demonstrated in a generalized 3-variable model. 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)], with 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 variables and is viewed as an essentially second-order system which receives discrete perturbations in a state-dependent manner.