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.

Many models of disease and rumor spreading phenomena average the behavior of individuals in a population in order to obtain a coarse description of expected system behavior. For these types of models, we determine how close the coarse population-level approximation is to its corresponding agent-based system and discuss the accuracy of the population-level approximation. We apply these theoretical results to real social network data to see how well they describe the contagious nature of social phenomena. Specifically, we consider hashtag adoption data collected from the Twitter social network. To assimilate the Twitter data to a simple contagion model, we developed and implemented statistical learning methods to construct an adaptive state estimator for systems described by nonlinear stochastic differential equations.

We found that the static network structure alone is not sufficient for explaining hashtag adoption among users in the Twitter social network, and our result suggest that a user-centric model would be more appropriate for this task. We propose a model for individual social media users, termed a \textit{genotype}, which is a \textit{per-topic} summary of a user's interest, activity and susceptibility to adopt new information. We show that the genotype framework is capable of accurately quantifying the adoption behavior of individual users with respect to hashtag topics.

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.

A key consideration for nonlinear oscillators is how they respond to parametric perturbations; such changes in parameters can often lead to effects on both the amplitude and phase of these oscillators. When the underlying process is sensitive to these changes, there may be large effects on the system leading to unwanted conditions. Additionally, models for nonlinear oscillators are often high-dimensional and intractable, making analysis in terms of the original coordinates nonintuitive. Utilizing an isostable and isochron coordinate frame, the transient dynamics of nonlinear oscillators to parametric perturbations can be analyzed in a convenient way by making use of the phase and impulse response curves from augmented phase reduction. Although isochron and isostable coordinates are convenient to use in analysis, typically one is interested in understanding what happens in terms of the original variables, and thus, a coordinate recovery framework is devised to take the results from the isostable and isochron frame back to the original coordinates. This approach shows great promise for understanding the transient behavior of nonlinear oscillators due to parametric perturbations.

Vibrational energy harvesting is a subject that has received much attention as a possible replacement for remote battery-operated sensor networks. We describe a vibrational energy harvester with an asymmetric buckled beam which is constructed out of commercially available components, and has demonstrated a significantly increased bandwidth compared to a device exhibiting linear resonance. This particular beam could not be mathematically modeled by existing analytic techniques, so a method is developed to produce a reduced order analytic model based on a finite element representation of the system. Moreover, we present an argument for why common single-mode Galerkin projection models are incapable of accurately reproducing snap-through behavior at higher than infinitesimal buckling levels. The model developed here demonstrates good agreement with the behavior exhibited by the experimental system around the parameter region of high power output, as evidenced by similar phase portraits and frequency response plots. Further, an argument is presented for why current trends towards testing non-linear systems with constant acceleration frequency sweeps are misleading, and an alternative comparison platform is suggested. The model is analyzed from a dynamical systems perspective, and it is shown that the transitions between high and low power output can be associated with a period doubling cascade or a boundary crisis where a chaotic attractor stabilizes through an intermittency transition. Chaotic behavior is observed to be closely related to the high power output region, but it is possible to have appreciable power output with a periodic response. Potential future work involves analyzing alternate beam configurations in search of an optimal solution to the high power output bandwidth problem.

Oscillators are ubiquitous in nature. As such, a significant body of literature has been devoted to studying their dynamics and how to control those dynamics. Many systems, however, do not maintain the core assumptions guiding the development of this literature-- systems that are either too complex or too poorly understood to allow for simple mathematical representations. Machine learning can serve as a powerful tool to supplement our understanding of dynamical systems in situations where traditional methods fail. In this dissertation, we develop first a control strategy for oscillators using standard techniques and assumptions about our dynamical system, then explore the ways in which machine learning can replace some of the strictest requirements on developing control strategies. We demonstrate how machine learning can extract meaningful information from complex systems in neuroscience and use that information as the basis for control. Lastly, we discuss some emerging strategies for further marrying the disciplines of dynamics and control and machine learning.

When confronted with an ever-changing and often perilous environment, how an organism behaves in response to uncertain and incomplete sensory information can be a matter of life and death. Besides the need to assess individual sensory signals accurately, sensory systems must also be able to integrate signals from multiple sensory modalities (e.g. visual, auditory, haptic), some of which may produce conflicting information. Through studying the insect brain of the Drosophila larva, we sought to unwrap the mathematical principles behind how animals process sensory signals to guide their behavior, with a focus on olfaction.

In my dissertation, we employ computational models to investigate how the Drosophila larva transduces odors through its olfactory sensory neurons and combines these cues with other sensory modalities. We obtain three important clues towards understanding the neural implementation of sensory systems: 1. Drosophila larvae are capable of computing and combining the variance of sensory inputs to organize orientation behavior, suggesting that even relatively simple nervous systems can achieve probabilistic inference. 2. Upon prolonged increasing excitation, olfactory sensory neurons can counterintuitively transition from a spiking state to a silent state called depolarization block, which preserves sparsity in the neural code. 3. The bifurcation of spiking and silent states in olfactory sensory neurons driven by depolarization block allows Drosophila larvae to encode and discriminate different odors.