The variability in the shape, location and size of the gyri and sulci of the human brain make it very difficult to compare functional and anatomical brain data across subjects. The belief that most of the cortical processing occurs in the thin layer of grey matter has lead to the idea of surface-based coordinate systems. "Unfolding" and "flattening" a cortical surface may make it easier to impose coordinate systems; comparing "flat" maps across subjects may lead to improved localization of functional foci.
In this presentation I will discuss a mathematical technique called Circle Packing that I am using to compute discrete conformal (angle-preserving) maps of the brain. A circle packing is a collection of circles with a specified pattern of tangencies and yields an approximation to a conformal map. We are interested in conformal mappings because the Riemann Mapping Theorem states that conformal mappings exist and are mathematically unique. In contrast, maps that attempt to preserve area or length will always have distortion. I will also discuss some of the topological issues that arise in computing maps of the brain and describe some of the conformal metrics we are exploring such as conformal modulus.