170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 249 10. 5 SUMMARY AND CONCLUDING REMARKS 249 plane, and therefore it has the symmetry of the Euclidean Group in the plane, 𝔼(2). It followed from this mathematical formulation that we could immediately predict thatthenormalmodesofvibrationoftheneocortexwillbe planewaves oftheform cos(k⋅x) where kis the wave number or spatial frequency of the waves, and x=(x,y). W e then used the fact that the primary visual cortex, V1, is connected via the LGN directly to the retina, in such a fashion that the retino-cortical map is topographic. Fromsuchamap Ermentroutand Iwereabletocomputetheappearanceoftheplane waves in the visual field, using the inverse cortico-retinal map, and found that they generated the non-contoured form constants described by Klüver. It took another decade or so before Blasdel's pioneering work on imaging the details of the distribution of iso-orientation preference patches in V1 appeared in print. From this and related work combining such images with the staining of indi-vidualneuronalconnectivity,Iwasabletoworkouttheshift-twistsymmetryofthe connections between such patches. This led to the next model in which the orien-tation preference of a patch, 𝜙, was taken to be another coordinate of the space in which V1 lives. Thus the coordinate system became (x,y,𝜙)and the space ℝ2×𝕊, where𝕊isthecircle. Thepredictionfromthiswasthatthenormalmodesofvibration of V1 should now be of the form cos2𝜙⋅cos(k⋅x) that is, plane waves whose amplitude is modulated by the 𝜋-periodic function cos2𝜙. After suitable modification of the retino-cortical map to incorporate its actiononthe 𝜙variable,wewereabletocomputetheappearanceofsuchmodified plane waves in the visual field. We were also able to make use of another piece of mathematics, that is, the equivariant branching lemma worked out by Golubitsky et al., which enabled us to predict that our model of V1's functional architecture couldgenerateexactlyfourcombinationsoftheabovenormalmodes,andthatunder the action of the extended cortico-retinal map, these combinations corresponded exactlytothefour Klüverformconstants. Thusgeometry,symmetry,andnonlinear dynamics led us to an exact theory of the generation of the Klüver form constants, both contoured, andnoncontoured. The remaining papers described in this chapter contain various elaborations and extensions of these results. Thus Bressloff and I extended the theory to incorpo-rateanothervisualfeaturepreferenceencodedin V1,thatof spatialfrequency,first reported by Hübener et al. The main assumption we made is that orientation and spatialfrequencypreferencesareencodedin V1insuchawaythattheycanberep-resentedmathematicallyasthetwoangularcomponentsofthesurfaceofa sphere,𝜙, and𝜃,sothatthespace V1livesinisnow ℝ2×𝕊2. Itfollowsthatthenormalmodes of vibration of V1 should now be plane waves modulated by sphericalharmonics,