Phase Response in Networks of Bursting Neurons: Modeling Central Pattern Generators
Sherwood, William Erik
Central pattern generators (CPGs) are localized, autonomous neuronal networks that coordinate the multilayered, rhythmic neuromuscular activity underlying essential behaviors such as respiration, digestion, circulation, and locomotion. A key step in deciphering CPGs' production and modulation of broad repertoires of patterned rhythmic output is understanding how phase relationships are established and maintained in networks of rhythmically active, or bursting, neurons. This thesis presents mathematical and computational investigations of several problems involving phase response, bursting neural models, and phasing in neuronal networks, studied in the context of modeling CPGs. We first consider the problem of modeling the locomotor central pattern generator responsible for coordinating hindlimb movement in the rodent (RSHL CPG). We propose the first mathematical model for the RSHL CPG, and from the comprehensive, full model we derive several reduced models to test specific hypotheses concerning RSHL CPG architecture. We establish through computational experiments that our models are capable of reproducing the fundamental locomotor rhythm of the RSHL CPG. Our investigations also uncover surprising phase sensitivities and transient behaviors, phenomena unexpected from intuitions based on studies of networks of phase oscillators or tonically spiking neurons. We pursue the origins of the models' phase sensitivity by studying phase response in single, endogenously bursting neurons. We examine the validity of several assumptions commonly made by modelers and experimentalists regarding the phase resetting behavior of endogenously bursting neurons in response to single spike perturbations. Our empirical study of burst phase response for a large combination of neuronal models and perturbation types demonstrates that in many circumstances, these assumptions are incorrect. Furthermore, we find that the phase response curves of endogenous bursters differ significantly from those of non-bursting neural oscillators in characteristic ways. We use fast-slow dissection, phase plane analysis, and isochron portraits to analyze the distinctive shape of burst phase response curves. Our analysis explains the dynamics of burst phase response in regimes of both weak and strong coupling, highlighting the role of fast subsystem structures and bifurcations in determining phase response. We also explain the mechanisms of spike number change due to strong perturbations. Finally, we apply insights from our study of burst phase response for single neurons to develop a set of discrete maps describing the changes in bursting neurons' phase and spike number in response to single spike perturbations. We also develop a set of map-coupling algorithms that can represent burst activity in arbitrary network architectures, thus reducing the interaction of bursting neurons to the properly sequenced iteration of low-dimensional maps. Our method produces good agreement with the transient and asymptotic phasing behavior of a simple network of biophysically realistic bursters. With further refinement, our algorithmically coupled burst maps may serve as a useful tool for exploring phasing behavior in networks of bursting neurons, especially models of CPGs.
dynamical systems; central pattern generators; neural models; bursting; locomotion; fast-slow system; phase response; isochron; spinal cord; discrete maps
dissertation or thesis