I am an NSF postdoctoral fellow at the Mathematical Biosciences Institute in Columbus Ohio. I am generally interested in using stochastic processes, partial differential equations, and adapting/extending methodology used in theoretical physics with the aim of explainings physical processes occuring in biological systems.
I have previously worked on inverse problems using methods of statistical physics with applications to imaging and partial differential equations. On the biological end, I have paid particular attention to ion and oxygen homeostasis in the brain, with the goal of gaining an understanding of a phenomenon known as cortical spreading depression.From 2008-2012, I was a graduate student in the UCLA Department of Biomathematics (Department of Mathematical and Computational Biology), an applied mathematics program with emphasis on biological applications. My graduate advisors were Tom Chou, KC Brennan, and Van Savage. Additionally, I previously had completed an MS in Statistics under the tutelage of Rick Schoenberg, and Jan de Leeuw.
Abstract: We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (innite dimensional) coefficient functions from ordinary or partial dierential equations (ODE, PDE), a problem which is typically ill-posed. Regularization of these problems using $\ell^2$ function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem namely whether the subjective choice of regularization is compatible with prior knowledge. The path-integral approach, while offering an alternative to standard computational methods of inversion which typically involve the use of Markov-Chain-Monte-Carlo (MCMC) methods, also gives way naturally to Monte-Carlo methods that can be used to refine perturbative approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theory involving the Poisson equation.
Spreading depression is a wave of depolarization and subsequent depression that travels in gray matter in the brain. It involves drastic changes in ionic balances and brain metabolism. Previously, the connection between these two aspects of this phenomenon were not clear. Using a lumped vascular model that is controlled by extracellular ion concentrations, and a model for oxygen use and delivery, we examined the connection.
Usually, when seeking to identify objects in images, one knows the general shapes of the objects that one is looking for. When this knowledge is available as a set of possible templates, a natural technique for representation of said knowledge is through kernel density estimation. With the prior knowledge, one can frame segmentation as Bayesian inference, resulting in an energy minimization problem that is nonlinear. This nonlinearity can be ameliorated through the use of majorization minimization to find an algorithm such that image segmentations can be found quickly using graph cuts.
This project extended particle filtering, a stochastic variant of the Kalman filter, to the tracking of moving interfaces in image sequences. The fast marching method is used as a model for the evolution of the interface, and Gaussian random fields are used to estimate and predict the future speed and position of the interface for use in regularization.
Spreading depression is a large metabolic event, where the dicharge of many ions into the extracellular space requires the large use of ATP and hence oxygen for recovery. Furthermore, blood flow in the brain is disrupted as the vascular system reacts to local increases in extracellular potassium. This study investigated these changes using second-derivative optical spectroscopy. Second-derivative spectroscopy shows changes in blood oxygenation level because of spectroscopic signatures of oxy- and deoxy- hemoglobin are different. Furthermore, both spectra have high curvature, so second derivative spectroscopy is able to control for background absorbers while maintaining the blood signal. In this study, we also found a prolonged second phase of imbalance lasting approximately an hour and starting approximately five minutes after the initial spreading depression event. This phase of the phenomenon is poorly understood and may be highly relevant to brain disorder.
Modeling ambulance dispatch events as a spatial-temporal point-process, one is able to identify clustering in ambulance response time. To some degree, this clustering is expected since ambulances are a finite resource. If a large amount of EMS calls are concentrated in a single area, it adversely affects the ability of the system to respond to future calls, at least while the current ambulances are still busy servicing previous calls. Point-process modeling allows one to identify areas that are performing well with adversely high loads, and areas that are more vulnerable to extraordinary emergencies.