Applied Mathematics Seminar

The human brain is composed of many folds and fissures which vary considerably in their size and extent between individuals. It is also known that most of the functional processing occurs on the surface of the brain in the grey matter, of which 60-70% is hidden from view as it is buried in the folds. This individual variability makes it very difficult to compare different brains across subjects. As a result, there is great interest among neuroscientists to unfold and flatten the cortical surface, with the aim of comparing brain maps across individuals. It is impossible to preserve linear or areal information when flattening a surface with intrinsic curvature such as the brain. However, it is possible to preserve conformal information. I will discuss a novel computer realization of the Riemann Mapping Theorem which uses circle packings to compute a discrete approximation of the conformal map of a surface embedded in 3-space. A circle packing is a collection of tangent circles representing a planar piecewise linear triangulated surface. These maps exhibit conformal behavior in that angular distortion is controlled and they can be created in the Euclidean and hyperbolic planes and on a sphere. I will present results of some of the maps I have created of the brain and discuss some of the topological and computational problems that arise.