Conformal Mapping of Brain Surfaces: Circle Packing and the Riemann Mapping Theorem
Ken Stephenson
University of Tennessee,Knoxville
Mathematics
This talk concerns mathematical efforts to “flat map” the human cortex, the highly convoluted yet essentially two-dimensional surface of neuronal tissue covering the brain. Flat maps identify the cortex or cortical fragments with regions in standard geometries: the (round) sphere, the plane, and the hyperbolic plane.
The richest mathematical structures on these surfaces are conformal structures, and the Riemann Mapping Theorem (1851) guarantees that these structures can be preserved during flat mapping—that is, conformal flat maps exist. However, only recently have such maps become computable in practice.
The talk briefly describes preliminary surface extraction and meshing procedures and then illustrates various flat map options and associated manipulations available through circle packing methods. The primary goal is not visualization, but rather the development of tools that incorporate conformal geometry into the analysis and comparison of cortical structure and function.
This work is joint with Monica Hurdal.
