Visualizing the Hessian
I learned to code in Matlab, my preferred programming language for smaller projects, through this research with Dr. Alan Dorval in the Biomedical Engineering Department at the University of Utah. I started this project while taking Calculus III and Physics, and this work allowed me to better understand vector and matrix visualizations.
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Considering a 3-dimensional space in the brain in which constant current is delivered, we can observe the voltage at every point and the corresponding electric field vectors. Currently, visualization tools in the field allow neuro-engineers to view this rather naively. I worked to create a more detailed model that accurately represents the electric field in a 3D space, such that the eigenvectors and values of the Hessian matrix of the voltage function at each point exactly corresponded to the size and shape of the model. This model would help neuroscientists, especially those studying neurological disorders such as epilepsy, better understand the optimal placements of contacts during deep brain stimulation treatments.
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It would be nice to go back and make this code much more efficient and sophisticated, and plot my figures over a CT scan.
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I toyed with plotting many figures and rotating each of them individually. The axes with which each figure aligns would eventually relate to the current sources. These placements would provide information about the polarization of axons, dependent on the location of the source of current.

The glyph. This was created with the intention of overlaying many glyphs over a medical scan of the brain. The length of the primary axis corresponds with the largest eigenvalue of the Hessian matrix. If an axon lay in this area, it would be hyperpolarized. The radius of the torus corresponds with the secondary eigenvector and eigenvalue. If an axon lay here, it would be depolarized. I recall the (not-quite) torus shape to be difficult to perfect. We needed the inner part to be exactly linear and the outer edge to be round, in order to leave a space that showed where hypothetical axons wouldn't be polarized.

The electric field. I visualized the less sophisticated model we intended to improve upon. Here, the source of current is at (0,0,0).
