Graduation Semester and Year

Spring 2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Dr. Pedro D. Maia

Second Advisor

Dr. Jianzhong Su

Third Advisor

Dr. Hristo Kojouharov

Fourth Advisor

Dr. Keaton Hamm

Abstract

This study utilizes local field potential (LFP) and electroencephalography (EEG) signals to rigorously analyze neural dynamics, enhancing our understanding of brain function. A significant aspect of this research is the application of state-space modeling and identification systems to investigate neural activity across various experimental conditions, including euthanasia, electroconvulsive therapy (ECT), and cardiac arrest. The Eigensystem Realization Algorithm (ERA) is employed to extract key neural patterns, facilitating dimensionality reduction while preserving essential features of brain activity.

We introduce an innovative strategy for determining the number of stacks (NS), which refines the approach to system identification. Our findings indicate that NS reflects the complexity of neural signals, showing an increase following an acute pain-like injection, which suggests notable alterations in the neural network. Furthermore, combining ERA with feature analysis provides deeper insights into neural state transitions across differing conditions. In the context of cardiac arrest, our study uncovers a sequential decline in neural connectivity, thereby challenging the prevailing notion of a uniform deterioration in brain activity. We effectively map these transitions using state-space modeling, presenting a more nuanced understanding of terminal brain activity.

Additionally, this research introduces a novel method that integrates the Bayesian Estimator of Abrupt Seasonality and Trend (BEAST) with ERA to enhance the robustness of neural signal segmentation and reconstruction. Traditional time series analysis techniques often fail to detect abrupt neural transitions; however, the integration of BEAST with ERA enables the identification of critical change points (CPs) and the characterization of underlying brain dynamics. This approach is further augmented by the Nystrom method, which improves spectral approximations and signal reconstruction. The Nystrom-BEAST framework optimally selects key data points, addressing the limitations of conventional random sampling methods and advancing the analysis of complex brain states. This research paves the way for more effective diagnostic and therapeutic applications in neuroscience.

Keywords

Bayesian Estimator of Abrupt Change, Seasonality, and Trend (BEAST), Brain, Eigensystem Realization Algorithm (ERA), Euthanasia, Local Field Potential (LFP), Model Reduction, Pain, Nociceptive Mechanisms, Electroconvulsive Therapy (ECT), Functional Connectivity, EEG/ECG signal processing, Cardiac arrest (CA), Terminal brain states, Consciousness at end-of-life, Dynamic network reorganization, low-rank approximation, column selection (CS), Model Reduction, landmarks, change points (CPs)

Disciplines

Analysis | Anesthesia and Analgesia | Applied Statistics | Behavior and Behavior Mechanisms | Clinical Trials | Data Science | Dynamical Systems | Dynamic Systems | Medical Neurobiology | Medical Physiology | Neurology | Neurosciences | Numerical Analysis and Computation | Ordinary Differential Equations and Applied Dynamics | Other Applied Mathematics | Other Psychiatry and Psychology | Statistical Methodology | Statistical Models | Vital and Health Statistics

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Available for download on Wednesday, April 28, 2027

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