Conformal Mapping of Brain Surfaces: Circle Packing and the Riemann Mapping Theorem

Ken Stephenson
University of Tennessee,Knoxville

The talk concerns mathematical efforts to "flat map" the human cortex, the highly convoluted but essentially 2D surface of neuronal tissue covering the brain. Flat maps identify the cortex
or cortical fragments with regions in the standard geometries: the (round) sphere, the plane, and the hyperbolic plane. The richest
mathematical structures on these surfaces are the "conformal" structures and the famous Riemann Mapping Theorem (1851) guarantees that these can be preserved during flat mapping --- there exist "conformal flat maps". However, it is only recently that these maps have become computable in practice. The talk will briefly describe preliminary surface extraction and meshing, then
will illustrate various flat map options and associated manipulations available using circle packing methods. The main goal is not visualization but rather development of tools which can bring conformal geometry into the analysis/comparison of
cortical structure and function. (Joint work with Monica Hurdal.)

Presentation (PDF File)

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