Cortical Mapping using Conformal Geometry

Monica Hurdal
Florida State University

The locations and patterns of functional brain activity in humans are difficult to compare across subjects because of individual differences in
cortical folding and the fact that functional foci are often buried within cortical sulci. While it is mathematically impossible to flatten curved
surfaces in 3-space without introducing metric and areal distortion, several algorithms can minimize such distortion; consequently, metric
flattening has been central in brain mapping efforts. On the other hand, while it has been known for 150 years that it is mathematically possible to flatten surfaces without any angular distortion, until the last decade there has been no algorithm for approximating these conformal flat maps. Conformal maps are particularly versatile, offering a variety of visual
presentations and manipulations backed by a uniquely rich mathematical theory. Using a circle packing algorithm we obtain discrete conformal
mappings which exploit this versatility. In this lecture, we present an introduction to the mathematics of conformal mappings and describe an approach based on circle packings, the first practical realization of the 150-year-old Riemann Mapping Theorem. We discuss the notion of a conformal structure on a surface, describe key
features of the three geometries of constant curvature where our maps reside: the sphere, plane and hyperbolic disc, and their classical
conformal automorphisms. We will also discuss conformal invariants, surface topology, cortical surface reconstruction and applications of
conformal maps in neuroscientific studies.

Audio (MP3 File, Podcast Ready)

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