UC Santa Barbara

For more than half a century, phase reduction techniques have been a remarkably powerful tool to aid in the understanding of oscillatory nonlinear dynamical systems, with a wide array of examples spanning the biological, physical, and chemical sciences. In the first part of this dissertation, we draw on the utility of phase reduction to investigate three problems with neuroscientific applications. Chapter 2 details a strategy to efficiently desynchronize a population of pathologically synchronized limit cycle oscillators which could represent a low energy treatment for Parkinson's disease. Chapter 3 gives theoretical details and experimental validation of a strategy to entrain a noisy and heterogeneous population of limit cycle oscillators with potential applications to the treatment of pathological conditions such as hearing loss and circadian dysfunction. Also, Chapter 4 details a general methodology which could make the implementation of control strategies from Chapters 2 and 3 more feasible in an in vivo setting.

While phase reduction has been exceedingly useful for systems with limit cycle solutions, analogous methodologies are not similarly developed for systems with excitable dynamics. The second part of this dissertation (Chapters 5 and 6) remedies this situation by developing a reduction methodology for excitable systems which approach a stable stationary solution. This reduction method relies on the notion of isostables, which are defined as sets of points in phase space that approach a stationary solution together, in a well-defined sense. An adjoint method is derived for calculating infinitesimal isostable response curves and this strategy is used to devise and implement sophisticated control strategies in complex dynamical systems of neurological and cardiological behavior. Emphasis is given to the problem of eliminating cardiac alternans which has been implicated as a precursor to fibrillation, a leading cause of death in the industrialized world. It is envisioned that this new reduction strategy could be as useful as a phase reduction has been in the past decades as it shows tremendous promise as a tool for the understanding and control of nonlinear systems.