170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 239 10. 4 EXTENSIONS OF THE MODEL 239 Figure 10. 19 Relationshipbetweenspatialfrequencyandorientationmaps. Grayregionsdenote low spatial frequency. Note that iso-orientation contours tend to cross the border of spatial fre-quency domains at right angles, and that the pinwheel centers or singularities of orientation preference are often located at the centers of either low or high spatial frequency domains (not highlighted). Source: Reproduced with permission of Nature Publishing Group. ( See insert for colourrepresentationofthisfigure. ) 10. 4. 1 Spatial frequency preferences A first step in this direction is contained in Bressloff and Cowan (2003b). This is basedonapaperby Hübener etal. (1997)containingtheresultsofanopticalimag-ing study of the Cat V1, in which spatial frequency preference patches are plotted relative to the contours of orientation preference patches. Figure 10. 19 reproduces one of their illustrations. The observation that there may be an orthogonal relation-ship between orientation and spatial frequency is interesting but not unexpected, giventhatorientationandspatialfrequencyare,infact,the(orthogonal)components of two-dimensional spatial frequency. This suggested to us the need to include the spatialfrequencyorwidthofastripeorbar,inadditiontoitsorientationinthevisual field, asanother featuretobeincorporated intothe functional geometryof V1. The formulation we introduced is shown in figure 10. 20. Itwillbeseenthatinsteadofeigenfunctionscos2 𝜙whichliveonacircle,wenow need eigenfunctions which also live on 𝜃, the polar angle specifying the latitude of a point(𝜃,𝜙)of a sphere, where 𝜙is the azimuthal angle specifying the longitude. We therefore choose 𝜃=log(p∕pmin) log(pmax∕pmin)(10. 14)