The cortical surface of the human brain is a highly folded, convoluted structure that is 3-5 mm thick and topologically equivalent to a sheet. Most of the functional processing of the brain occurs on this sheet. However, the folding patterns vary considerably between individuals in terms of the shape, depth, length and location of the folds. As a result, neuroscientists are interested in 2D analysis methods and tools for comparing function and anatomy across subjects which can take into account some of this individual variability.
I am using methods from complex analysis, computational geometry, topology and image processing to "unfold" and "flatten" the cortical surface of the brain. While it is impossible to create areal or length preserving maps of a highly convoluted surface such as the brain, the Riemann Mapping Theorem describes the existence of conformal maps. I will discuss methods and theory from the area of "circle packings" which I am using to create approximations to discrete conformal maps of the brain. I will discuss some of the computational and topological issues that arise and how I am imposing coordinate systems on these maps. I will show some of the brain maps that I have created and discuss collaboration results with neuroscientists who are interested in depression, schizophrenia and Alzheimer's diseases.