This dissertation deals with a range of mathematical research problems that have arisen out of the study of a neurophysiological phenomenon known as cortical spreading depression (CSD). CSD is a slow-moving wave of depolarization and subsequent electrical depression that occurs whenever the brain is in one or more of several different distressed states. CSD is of particular interest in migraine research, as it is thought to be responsible in migraine with aura, and in stroke research, where the number of CSD events is a strong determinant of patient outcome.
This dissertation presents a mathematical model for CSD that predicts the feed-forward and feed-back effects of vasoconstriction. The model predicts that the main constraint on recovery from CSD is metabolic - the facts that recovery from CSD is energetically expensive and oxygen flux is finite play a larger role than vasoconstriction in inhibiting CSD recovery. Yet, vasoconstriction and vasodilation both affect CSD.
In the laboratory, CSD is often studied using optical imaging techniques, where a need for extracting information out of images is needed. This dissertation first considers the more-general problem of object identification in images (segmentation), where a-priori shape-information is known in the form of an ensemble of templates. Using kernel-density-estimation, these templates are constructed into a prior probability distribution over the space of desired shapes. A nonlinear energy functional is defined where the desired segmentation is the minimum-energy state. This functional is minimized using a majorization-minimization algorithm where each iteration is solved quickly using graph cuts.
The image segmentation technique is then extended to the tracking of monotonically advancing boundaries like those observed in CSD. Here, a statistical model of boundary motion provides a set of templates that is used to regularize boundary segmentation. The boundary is modeled using Gaussian random fields, where boundary speed is expected to vary smoothly in space, and is expected to be locally correlated in space.
Finally, when studying CSD and other wave phenomenon, one often encounters inverse problems with partial differential equations. For example, one encounters inverse problems of the eikonal equation when studying CSD wave geometry. This dissertation provides a general framework for solving inverse problems involving partial differential equations. By connecting the method of Tikhonov-regularization to the method of Bayesian statistical regularization using a-priori Gaussian random fields, we develop a probabilistic model that is useful for both solving inverse problems, and for determining uncertainty. The mathematical framework is based on methods developed by the physics community to study the statistical properties of particles.
