170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 248 248 CH10 GEOMETRIC VISUAL HALLUCINATIONS Transition Stage two Stage one Stage three Figure 10. 28 Stages in the development of complex hallucinations. Source: Reproduced with permissionof HTO/Wikimedia Commons/Public Domain. visual areas and beyond. Such areas are known to carry out computations on data provided by V1, and so on, generating ever more complex representations of the imagestherein. Thewholeprocessultimatelytravelstoprefrontalcorticalareassuch as area 46, and beyond; and of course there are feedbacks from such areas to lower levels. It is striking that there is an almost direct path through V4 directly to area 46 (Young, 1992) and back. There is some data that supports the proposed trigger location in V1, provided by Ganis etal. (2004) in a functional magnetic resonance imaging study. However it should be noted that the BOLD signal found in V1 and closely related regions is small compared with signals recorded in other regions during the hallucinatory experience. See for example ffytche etal. (1998); ffytche (2008), and Allen etal. (2008). But all the data is consistent with the idea that the initialtriggeringeventislocatedin V1andproceedsfromtheretohigherlocations. 10. 5 Summary and concluding remarks 10. 5. 1 Summary It should now be clear that all the results reviewed in this chapter have a common origin based on the idea that in confronting the immense complexity of neural cir-cuitry,somesimplificationisnecessary. Wethereforestartedwithtwofundamental ideas. First, despite the complexity, the neocortex is approximately homogeneous, thatis,anyonelocalregionlooks,moreorless,thesameasanyother,sothatifthe neocortex were to be removed from the skull and flattened out, it would look like a thinsheetorslabofhomogeneousneuraltissue. Theseobservationshavemathemat-icalconsequences. Theyindicatethatthe spacethecortex'livesin'canbespecified bythe Cartesiancoordinates (x,y,z)ofsuchaslab. Buttheslabdepth(3mm)isvery small compared with the lateral extent (1m) of each side, so that the zcoordinate can be neglected. We can therefore consider the space the cortex lives in to be the 2-dimensionalrealnumberplane ℝ×ℝ=ℝ2. Second,becauseofthehomogeneity, wecanmoveanywhereintheplane,andthecircuitrywillstilllookthesame. Math-ematicallywesaythattheneocorticalplaneisinvarianttorigidbodymotionsofthe