Thesis

Mathematical Analysis and Numerical Computation of the Brain Response to Single-Event-Related Stimuli

The goal of this study is to analyze mathematically and numerically the hemodynamic mathematical model that describes changes in blood flow and blood oxygenation during brain activation. At the mathematical level, we have established results pertaining to the existence, uniqueness, and asymptotic behavior of the state vector as well as the bloodoxygen-level dependent (BOLD) signal. At the numerical level, the goal is to propose a numerical strategy for retrieving accurately and efficiently the biophysiological parameters as well as the external stimulus characteristics of the considered hemodynamic model. The proposed method employs the RNA-CKF method developed in [21], but in a prediction/correction framework. Furthermore, numerical experiments have been conducted using synthetic functional Magnetic Resonance Imaging (fMRI) measurements, tainted with varying noise, to highlight the performance characteristics of this computational methodology. Finally, the algorithm was implemented to calibrate the model using real data obtained from a finger-tapping fMRI experiment conducted at the Nationwide Children's Hospital in Columbus, Ohio.

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