Collaborative Research: Quantifying Human Retinotopic Mapping by Conformal Geometry

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Retinotopy is the mapping of visual inputs from the retina to neurons in the brain. A retinotopic map is a visual of a particular occurrence of neuron activity taking place on a specific location in the brain. By analyzing the stimulus-referred functional magnetic resonance imaging (fMRI) response, retinotopic maps of the human visual cortex are generated. It has been hypothesized that human retinotopic maps are conformal mappings, but to date no theoretical models have been developed to quantify these maps. This project uses conformal geometry and fMRI data to study retinotopic maps in an attempt to model visual cortical organizations in the brain. This project will: (i) compute the intrinsic geometrical features that will determine and/or validate the conformality in the human retinotopy; (ii) model the relationships between the retinotopic maps in extrastriate visual areas and those in the primary visual cortex; (iii) develop methods to quantify retinotopic maps of individual subjects. The mathematical models applied in this project combine tools from topology, conformal geometry, complex analysis, optimization, and quasiconformal Teichmüller theory. Visual processing areas have been estimated to occupy more than half of the total surface of the primate neocortex. However, an understanding of visual cortical organizations still remains elusive due to their biological complexity. Retinotopic mapping of the human visual cortex, therefore, can be an important tool for studying the brain's circuitry. This project will produce theoretically sound and practically efficient methods for quantifying retinotopic maps. These methods will lead to non-invasive biomarkers of visual functions and may lead to cures for visual deficits. The computational theories and algorithms developed in this project will have applications in other fields, including computer vision, computer graphics, sensor networks, and geometric modeling. This project contributes to the BRAIN Initiative by increasing our knowledge of visual cortical organizations in the human brain and by developing laboratory infrastructure for knowledge discovery from brain imaging data. This project will create an interdisciplinary environment for graduate and undergraduate students and will develop interdisciplinary courses at the interface of mathematics and neuroscience.