170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 240 240 CH10 GEOMETRIC VISUAL HALLUCINATIONS Orientation ϕ pmaxpmin Spatial frequency p Figure 10. 20 Sphericalnetworktopology. Orientationandspatialfrequencylabelsaredenoted by(𝜙,p)with0 ≤𝜙<𝜋andpmin≤p≤pmax. so that𝜃varies linearly with log p, [This is consistent with observations which show that log pvaries linearly with cortical separation (Issa et al. 2000)], and the eigenfunctions we need now live on the sphere 𝕊2, instead of on the circle 𝕊. Thus the space our extended model of V1 lives in is now ℝ2×𝕊2, and the symmetry group𝕆(2)now generalizes to 𝕊𝕆(3)the group of rotations of the sphere, and the eigenfunctionsorharmonicsarenownolongersimplesinusoids,butinsteadarethe sphericalharmonics Ym n(𝜃,𝜙)∝Pm n(cos𝜃)e2im𝜙 where Pm n(cos𝜃)are associated Legendre polynomials. Thefirstfewoftheseare P0 0(cos𝜃)=1,P1 0(cos𝜃)=cos𝜃,P1 1(cos𝜃)=-sin𝜃,so thatsphericalharmonicsaremadeupofsumsofproductsofbothsinesandcosines of both the angles 𝜃and𝜙. It follows that whereas two points on a circle have an angular separation 𝛼such that cos𝛼=cos(𝜙-𝜙′), two points on asphere havean angular separation such that cos𝛼=cos𝜃cos𝜃′+sin𝜃sin𝜃cos2(𝜙-𝜙′). These angular separations are invariant under the actions, respectively, of the rota-tion groups 𝕆(2)and𝕊𝕆(3).