Abstract
Degeneracy is a ubiquitous feature of many biological systems, especially neural circuits. Degenerate systems exhibit multifunctionality and robustness in the face of environmental perturbations. Currently no rigorous measures for quantifying the degeneracy of dynamical systems exist. Here, I introduce a measure of degeneracy for biophysical neuron models based on the theory of time delay embeddings that is also applicable in experimental settings. Additionally, I construct a new two-compartment model of the conditional burster AB of the crustacean pyloric network. This model was initially conceived to rememdy shortcomings in single compartment models of AB based on the work of Liu et al. (1998); these models fail to exhibit the transition from a tonic spiking state to a bursting state in the presence of a neuromodulatory current, I_MI. I describe these deficiencies in this thesis. In order to compare the new two-compartment models to the single compartment ones, my colleagues and I began to analyze their respective phase response properties. Here, I report the emergence of fine structural features in the phase response curves of these models. These fine structures can in part be accounted for by the dynamical phenomena of spike addition and spike deletion during a burst. Moreover, I report the discovery of multi-stability in the single compartment bursting models.