Dynamical systems, Oscillations, Synchronization and Parkinson's Disease - Leonid Rubchinsky
Parkinson's disease is marked by synchronized oscillatory dynamics of neural activity. It is believed that different kinds of oscillations are a) responsible for major motor symptoms of the disorder and b) known symptomatic therapies actually suppress synchronized oscillatory activity. It is of both fundamental and practical importance to understand the nature and dynamics of these oscillations, as well as to consider new means of their suppression. Moreover, understanding the neurodynamics of the Parkinson's disease will also promote our understanding of the healthy functioning of the brain parts, impacted in the disease. Mathematical and computational approaches are essential in this regard. Applied dynamical systems, time-series analysis, numerical ODE solution and other parts of applied mathematics are used in our group to understand the dynamics of the brain in parkinsonian state.
There are several related lines of research, REU students can participate in, ranging from the analysis of synchronization in the data, recorded in Parkinsonian patients during deep brain stimulation surgeries, to the data-constrained modeling of the brain circuits impacted in Parkinson's disease, to the modeling of Parkinsonian tremor genesis, to the exploration of new algorithms for adaptive deep brain stimulation.
Students ideally should have some familiarity with ODEs and their numerical solutions, with some basic time-series analysis techniques and with MATLAB, some knowledge of neurobiology is a plus, however we are mostly applied mathematicians and will be happy to teach students everything needed. Interested students will have a chance to interact with our biomedical collaborators (neurosurgery and neurology).
Mathematical Modeling of Ocular Blood Flow and its Relation to Glaucoma - Giovanna Guidoboni
Glaucoma is a disease in which the optic nerve is damaged, leading to progressive, irreversible loss of vision. Glaucoma is the second leading cause of blindness worldwide, and yet the mechanisms underlying its occurrence remain elusive. The proposed research projects focus on open angle glaucoma (OAG) which progresses at a slow rate and the loss of vision may not be noticed by the patient until the disease is significantly advanced. OAG is often associated with increased intraocular pressure (IOP), which is the pressure of the aqueous humor in the eye. Elevated IOP remains the current focus of therapy, but unfortunately many glaucoma patients continue to experience disease progression despite lowered IOP, even to target levels. Clinical observations show that alterations in ocular blood flow play a very important role in the progression of glaucoma. Significant correlations have been found between impaired vascular function and optic nerve damage, but the mechanisms giving rise to these correlations are still unknown. The goal of this project is to investigate the bio-mechanical connections between vascular function and optic nerve damage, in order to gain a better understanding of the risk factors that may be responsible for glaucoma onset and progression. To reach this goal, a variety of mathematical models will be used to describe the different ocular anatomical components, including humors, retina, choroid and sclera.
Students ideally should have some familiarity with ODEs and their numerical solutions, with some basic fluid and solid mechanics and with MATLAB. Interested students will have a chance to interact with our collaborators in the department of ophthalmology.