UC Santa Barbara

Nonlinear oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. This dissertation investigates the dynamics of such oscillators arising in biology, and develops several control algorithms to modify their collective behavior. We demonstrate that these control algorithms have potential in devising treatments for Parkinson's disease, cardiac alternans, and jet lag.

Phase reduction, a classical reduction technique, has been instrumental in understanding such biological oscillators. In this dissertation, we investigate a new reduction technique called augmented phase reduction, and calculate its associated analytical expressions for six dynamically different planar systems: This helps us to understand the dynamical regimes for which the use of augmented phase reduction is advantageous over the standard phase reduction.

We further this study by developing a novel optimal control algorithm based on the augmented phase reduction to change the phase of a single oscillator using a minimum energy input. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier of the oscillator is close to unity; in such cases, the previously proposed control algorithm based on the standard phase reduction fails.

We then devise a novel framework to control a population of biological oscillators as a whole, and change their collective behavior. Our first two control algorithms are Lyapunov-based, and our third is an optimal control algorithm which minimizes the control energy consumption while achieving the desired collective behavior of an oscillator population. We show that the developed control algorithms can synchronize, desynchronize, cluster, and phase shift the population.

We continue this investigation by developing two novel machine learning control algorithms, which have a simple and intelligent structure that makes them effective even with a sparse data set. We show that these algorithms are powerful enough to control a wide variety of dynamical systems and not just biological oscillators. We conclude this study by understanding how the developed machine learning algorithms work in terms of phase reduction.

In this dissertation, we have developed all these algorithms with the goal of ease of experimental implementation, for which the model parameters/training data can be measured experimentally. We close the loop on this dissertation by carrying out robustness analysis for the developed algorithms; demonstrating their resilience to noise, and thus their suitability for controlling living biological tissue. They truly hold great potential in devising treatments for Parkinson's disease, cardiac alternans, and jet lag.