170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 241 10. 4 EXTENSIONS OF THE MODEL 241 An important consequence of these properties is that if we choose weighting functions that are invariant under such group actions, then, as we have described earlier, the connectivity pattern shows the shift-twist property for orientation pref-erence𝜙, but now in the case of pairs of angles (𝜃,𝜙), the spherical representation hassingularities at the two poles of the sphere at which the orientation preferences disappear. We can expect contoured patterns to be produced only in regions well away from the poles, where the horizontal connectivity pattern is isotropic. This is consistentwiththeobservationsof Livingstoneand Hubel(1984). Wecanconclude fromthisanalysisthatourextensionoftheearliertheorywillagaingenerateallthe form constants, but the mechanism is not the same as in the earlier study. In this modelitistheanatomyandnotthedynamicswhichgeneratesthefullrepertoireof all the imagery. 10. 4. 2 Pinning There is however a problem that requires further work. The V1 planforms we have shownarethosegeneratedbytheequationsofthemodelswehaveintroduced. How-ever when we attempt to simulate the generation of these patterns by computer simulation,wefindthattheplanformsarequiteirregular. Onewaytosolvethisprob-lemistoincorporatemorefullyintothe V1modelitscrystallinestructure(Hubeland Wiesel, 1974a). A first version of this was published by Bressloff (2002), and later morefullydevelopedin Bressloff(2003),and Bressloffand Cowan(2003a). Essen-tially,thebasicideaofthesepaperswastousethephysicist'sapproachtoanalyzethe properties of crystalline structures using Blochwaves. This methodology was later usedby Bakerand Cowan(2009)tosolvetheaboveirregularityproblem. Consider, forexample,theirregularstripesshowninfigure10. 21,producedbysimulatingthe Ermentrout-Cowanequations. Howcanwestraightenoutsuchanirregularpattern? One answer is that the irregular pattern is pinnedto a planar lattice generated by thehorizontalpatchyconnectionsthatcoupleeachvertexofthelattice(i. e. ahyper-column) to its neighbors [See figure 10. 12. ] The mechanism of pinning is the spa-tial analogue of the mechanism of synchronization of coupled nonlinear oscillators (Pikovsky,2003). Itcanbeexplainedinsimpletermsusingalittle Fourieranalysis. Consider figure 10. 22 showing (a) the connectivity of the lattice and of the Mexi-can Hatweightingfunctionofthe Turingmechanismusedinthe Ermentrout-Cowan model, and their associated Fourier spectra. Itwillbeseenthatthe Mexicanhatfitsintothelatticeina commensurate fashion, inthatthepeaksandvalleysofthetwodistributionsarealignedinspatial-frequency space. This is the prerequisite for pinning. The effects of such a pinning are shown in Figure 10. 21b. Of course V1 is not, in fact, a perfect crystal, but as Figure 10. 10 shows, it is quite irregular so that the stripes that form as a consequence of pinning should still be somewhat irregular, as is indicated in Figure 10. 8.