Access Type

Open Access Dissertation

Date of Award

January 2011

Degree Type

Dissertation

Degree Name

Ph.D.

Department

Electrical and Computer Engineering

First Advisor

Hao Ying

Abstract

The topic of this dissertation is the study of noise in electrical engineering, neuroscience, biomedical engineering, and operations research through mathematical models that describe, explain, predict and control dynamic phenomena. Noise is modeled through Brownian Motion and the research problems are mathematically addressed by different versions of a generalized Langevin equation. Our mathematical models utilize stochastic differential equations (SDEs) and stochastic optimal control, both of which were born in the soil of electrical engineering. Central to this dissertation is a brain-physics based model of cerebrospinal fluid (CSF) dynamics, whose structure is fundamentally determined by an electrical circuit analogy. Our general Langevin framework encompasses many of the existing equations used in electrical engineering, neuroscience, biomedical engineering and operations research.

The generalized SDE for CSF dynamics extends a fundamental model in the field to discover new clinical insights and tools, provides the basis for a nonlinear controller, and suggests a new way to resolve an ongoing controversy regarding CSF dynamics in neuroscience. The natural generalization of the SDE for CSF dynamics is a SDE with polynomial drift. We develop a new analytical algorithm to solve SDEs with polynomial drift, thereby contributing to the electrical engineering literature on signal processing models, many of which are special cases of SDEs with polynomial drift. We make new contributions to the operations research literature on marketing communication models by unifying different types of dynamically optimal trajectories of spending in the framework of a classic model of market response, in which these different temporal patterns arise as a consequence of different boundary conditions.

The methodologies developed in this dissertation provide an analytical foundation for the solution of fundamental problems in gas discharge lamp dynamics in power engineering, degradation dynamics of ultra-thin metal oxides in MOS capacitors, and molecular motors in nanotechnology, thereby establishing a rich agenda for future research.

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