This poster was presented as Poster #194 at the Fifth International Conference on Functional Mapping of the Human Mapping (HBM'99) held in Düsseldorf, Germany, June 22-26, 1999.

The abstract for this poster was published in NeuroImage, vol. 9, part 2, pg. S194, 1999.



1Monica K. Hurdal, 1De Witt L. Sumners, 2Kelly Rehm, 3Kirt Schaper,
1Philip L. Bowers, 4Ken Stephenson, 2,3David A. Rottenberg
Hyperbolic Flat Map of the Cerebellum Displaying Colored Anatomical 
1Department of Mathematics, Florida State University, Tallahassee, FL, U.S.A.
2Department of Radiology, University of Minnesota, Minneapolis, MN, U.S.A.
3PET Imaging Center, VA Medical Center, Minneapolis, MN, U.S.A.
4Department of Mathematics, University of Tennessee, Knoxville, TN, U.S.A.
1,2,3,4Members of the International Neuroimaging Consortium
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  • mapping the cerebellar cortex is essential for the description and analysis of spatial and functional relationships within and between cortical regions
  • flat maps may assist in the analysis of activated foci buried in interfolial fissures
We present cortical flat maps of the cerebellum using our novel method of circle packing. These maps are conformal in character as angular distortion is controlled.
Our Approach to Flat Mapping
  • use math theory to create mathematically unique flat maps of the brain
  • it is impossible to flatten a curved surface in 3D space without linear and areal distortion
  • the Riemann Mapping Theorem (1854) says it is possible to preserve conformal (angular) information
  • our approach attempts to preserve the conformal structure between the original cortical surface and the flattened surface
Flat Mapping the Cerebellum
  • interested in describing activated cerebellar foci in functional neuroimaging
  • high resolution T1-weighted MRI volume obtained [1]
  • cerebellum extracted from volume and a topologically correct triangulated surface representing the cerebellum produced
  • functional data obtained from a target interception experiment (see Figure 1)
  • various lobes and fissures color coded for identification purposes (see Figures 2 & 3)
  • use circle packing [2] to create quasi-conformal flat maps of the cerebellum in the conventional Euclidean plane, in the hyperbolic plane and on a sphere (see Figures 4, 5, 6 & 7)
Axial Slice of Cerebellum with Functional Data Coronal Slice of Cerebellum with Functional Data Sagittal Slice of Cerebellum with Functional Data
Figure 1: Axial (left), coronal (middle) and sagittal (right) images showing functional activation of the cerebellum from a target interception task.
Color Coded Cerebellum in Sagittal MRI Slice

Figure 2: The human cerebellum is colored according to the following cortical regions: forest green = lobulus seminlunaris, lobulus semilunaris inferior, lobulus biventer; red = tonsils, flocculus; yellow = lingula, lobulus centralis, lobulus quadrangularis; blue = lobulus simplex, lobulus semilunaris superior; grey = white matter; bright green = fissura prima; cyan = fissure secunda; magenta = fissura horizontalis.
Front View of Color Coded Rendered Cerebellum Back View of Color Coded Rendered Cerebellum

Figure 3: The surface representing the cerebellum is exctracted from the MR images and rendered to show a view from the front and back. Colors correspond to regions shown in Figure 2 and purple corresponds to the boundary used for flat maps.
Euclidean Flat Map with Anatomical Regions Euclidean Flat Map with Functional Regions

Figure 4: Flattened map of the cerebellum (in the Euclidean plane). The colors in the left figure correspond to anatomical regions shown in Figure 2. The colors in the right figure correspond to regions of fucntional activation shown in Figure 1.
Hyperbolic Flat 
Map with Anatomical Regions
Hyperbolic Flat 
Map with Alternate Focus
Figure 5: Cerebellum mapped to a disk (in the hyperbolic plane). The origin (map focus) is marked in black in the center of the maps. The second map has been transformed to a different map focus to display regions in the lower portion of the map. The black circle located near the bottom of the first map is used as the map origin in the second map. Similarly, the black circle located near the top of the second map corresponds to the map focus of the first map.
Hyperbolic Flat 
Map with Functional Regions
Hyperbolic Flat 
Map with Alternate Focus
Figure 6: Same maps as Figure 5 but regions of functional activation are shown from a target interception task. The map foci are marked in black. Fissures are indicated in dark grey and lobes are indicated in light grey.
Spherical Map 
with Anatomical Regions
Rotated View 
of Spherical Flat Map with Anatomical Regions
Map with Functional Regions
View of Spherical Flat Map with Functional Regions
Figure 7: Cerebellum mapped to a sphere. Two different views are shown. Anatomical information is displayed on the upper maps and functional information is displayed on the lower maps.
Advantages and Benefits
  • conformal mappings presereve angle proportion, are canonical and hence mathematically unique
  • easy to impose a coordinate system when given two anatomical landmarks, which allows comparison of different maps
  • easy to transform and change locations of map distortion
  • no additional surface cuts are required
  • clinical tool for analyzing anatomical and functional differences
For more information...

[1] Holmes, C. J., Hoge, R., Collins, L., Evans, A. C., Neuroimage, 1996, 3:S28.
[2] Dubejko, T. Stephenson, K., Experimental Mathematics, 1995, 4:307-348.

Updated July 1999.
Copyright 1999 by Monica K. Hurdal. All rights reserved.