CAREER: A Nonlinear Model Reduction Framework for Oscillatory Systems and Associated Data-Driven Inference Strategies

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).This Faculty Early Career Development Program (CAREER) grant will fund research that enables improved understanding and control of collective neurological rhythms, for example pathological synchronization of neurons contributing to the motor symptoms of Parkinson’s disease, thereby promoting the progress of science, and advancing the national health. Deep brain stimulation injects electrical pulses into the brains of patients suffering from Parkinson’s to alleviate muscle tremors and rigidity. Because of the large inputs required, standard theoretical tools for predicting and analyzing the neuronal response are inadequate, since they assume small deviations from the synchronized behavior. This project will overcome such limitations by developing a new theoretical approach, suitable for complex and high-dimensional systems with oscillatory dynamics even for large deviations dominated by system nonlinearities. By combining this approach with machine-learning techniques, it will be possible to derive relevant dynamical models entirely from measurements, with potential applications to treating recovery from jet lag or controlling the air flow around vehicles and aircraft. Through close integration of research and education, this project will contribute to an engineering and science curriculum of inquiry-based, hands-on learning experiences for high school students attending outreach activities or specially designed, multi-day immersive programs at Lone Oaks Farm, a STEM education center in west Tennessee that serves large populations of underrepresented students from under-resourced local communities. Completion of this project will also yield a series of tutorial sessions, a set of online learning modules, and a computational toolbox, each promoting the use of powerful mathematical techniques for dynamical systems analysis to members of the larger research community.This research aims to make fundamental contributions to a theory of model reduction techniques for oscillatory high-dimensional systems whose dynamics are dominated by system nonlinearities, with particular emphasis on accuracy, analytical tractability, and suitability for control design. It achieves this aim by augmenting traditional phase-based reduction methods with a description of transversal dynamics in terms of isostable coordinates, which characterize the slowest decaying modes of the system Koopman operator. Adaptive updates to model parameters are then introduced to bound the time evolution of the isostable coordinates and ensure convergence of asymptotic expansions used in the model reduction. Generalizations to non-periodic dynamics and, importantly, to data-driven model identification in the absence of known underlying dynamical equations will be explored in theoretical models and in applications to circadian cycles, neural brain rhythms, and fluid flow systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.