170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 234 234 CH10 GEOMETRIC VISUAL HALLUCINATIONS 10. 3. 1 Doubly periodic planforms Howeverthereisstillaproblemthatneedstobeaddressed. Thesymmetrywehave introducediscontinuous. Thisimpliesthatthereexistinfinitelymanysolutions. But the neocortex is not really a continuous plane. In fact, the way we have interpreted the data provided by optical imaging and cell staining suggests that the neocortex, and particularly the visual cortex, is approximately a doubly periodic array whose symmetriesarethoseofthesymmetrygroups 𝔻n,withn=2,3,4and6,androtation angles that are multiples of 𝜋∕n. It follows that the relevant planforms are doubly periodic. Figures 10. 14 and 10. 15 show some doubly periodic even planforms. 10. 3. 2 What do such planforms look like in the visual ﬁeld? We now turn to the question of what such platforms look like in visual field coordinates (r,𝜃). In order to compute their appearance we need to apply an inverse cortico-retinal transformation. It should be clear from the derivation of the retino-corticalmapthatthiswillinvolvetheorientationpreferencevariable 𝜙. Thus the retino-cortical map needs to be extended to deal with such a preference, that is, it needs to be extended to cover the mapping of local contours in the visual field. Let𝜙Rbe the orientation of such a contour in the visual field, and 𝜙its image in V1. Wenowask:Whatisthemap 𝜙R𝜙thatmustbeaddedtotheretino-cortical mapz Rzdescribedearlier?Notefirstthatalinein V1ofconstantslopegivenby tan𝜙is alevelcurve of the equation f(x,y)=ycos𝜙-xsin𝜙 (a) (b) Figure 10. 14 (a) Non-contoured even axial planform on the square lattice. (b) Non-contoured even axial planform on thehexagonal lattice.