| Neurons have a multitude of distinct oscillatory behaviors because of previous conditions, intrinsic cellular properties, and network connections. The qualitative theory of differential equations offers tools that can be used to describe solutions for the differential equations that model neurons. The primary tool for investigation of global stability is the Poincare return map. The maps capitalize on the recurrent nature of oscillations and are able to analyze changes in dynamics even at bifurcation points where most other methods fail.;Elliptic bursting models are found in numerous biological systems, including the external Globus Pallidus section of the brain; the focus for studies of epileptic seizures and Parkinson's disease. However the bifurcation structure for changes in dynamics remains incomplete. This dissertation develops computer assisted Poincar maps for mathematical and biologically relevant elliptic bursting neuron models and central pattern generators, CPG's. The method employed for individual neurons offers the advantage of an entire family of computationally smooth and complete mappings which can explain all dynamical transitions of the system. A complete bifurcation analysis was performed detailing the mechanisms for the transitions from tonic spiking to quiescence in elliptic bursters. A previously unknown unstable torus bifurcation was found to give rise to small amplitude oscillations.;The focus shifts from individual neuron models to small networks of neuron models, in particular 3-cell CPG's. A CPG is a small network which is able to produce a specific phasic relationships between the individual cells. The output rhythms may represent a number of biologically observable actions, such as walking or running gaits. A 2-dimensional map is derived from the phase-lags of the CPG. The cells are endogenously bursting neuron models mutually coupled using the fast threshold synaptic paradigm. Internal parameters, which change the burst duration of the individual cells , as well as type and strength of synaptic coupling; inhibitory and excitatory. The mappings generate clear explanations for rhythmic outcomes as well as basins of attraction for specific rhythms and possible mechanisms for switching between rhythms. A complete description of bifurcations and rhythmic patterns for a 3-cell network is given.;INDEX WORDS: Central pattern generator, Bifurcation, Return mappings, Polyrhythmic, Bursting, Duty cycle, Elliptic bursting, Motifs, Inhibitory, Multistability, Poincare map, Interneuron, Phaselag, Network, Synaptic, Fast threshold modulation, Saddlenode, Multifunctional. |
