170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 223 10. 1 INTRODUCTION 223 But what are the spatial dimensions of the human neocortex? If it were to be extracted from the skull and flattened out it would be about 1m square and about 3mm thick, and contain about 5 ×1010neurons. To a first approximation it is a planewith Euclidean (or Cartesian) coordinates xandy. Thus our picture of the neocortex is very simple, any one local region or patch looks just like any other patch,andtheunderlyingsymmetryofthispictureisthatofthegroupofrigidbody motions in the plane. Usingthisinformationwecancalculatewhatshouldbethenormalmodesofvibra-tion of such a neocortex. The answer is planewaves of the form cos(k⋅x)=cos(kxx+kyy) wherekxandkyarespatialwavenumbersinthe xandydirections. Thecentralthesis ofalltheresultsthatwillbedescribedinthischapteristhattheneocortexproduces spatiallyorganizedpatternsthatarecombinationsoftheseplanewaves,whenitloses the dynamical stability of its (postulated) spatially homogeneous resting state. 10. 1. 2 The retino-cortical map Sowhydoweseetheformconstants?Alargepartoftheansweriscontainedinthe factthatthereexistsa topographicmap fromtheeyestotheprimaryvisualcortexor V1, (and beyond with a coarser topology). Every retinal ganglion cell has an axon that connects topographically to the lateral geniculate nucleus LGN, and thence by optic radiation to layer IV of V1. [Note that topographic means that neighbouring locationsinthevisualfieldstimulateneighbouringlocationsin V1. ]However,there are important differences between the neuroanatomy of the retina and that of V1. First,theretinaisathinlayerofphotoreceptorsandneuronsontheinnersurfaceof theeyeball,andrepresentspointsinthevisualfield(approximately)in polarcoordi-nates(r,𝜃)wheretheradialcoordinate rrepresentsdegreesofvisualanglefromthe location of the fixation point, and the angular coordinate 𝜃represents the elevation oftheobjectfromthehorizon. But V1hasa Cartesiancoordinatesystemx,y. Sothe topographicmaphastobe {1∶1}oftheform (r,𝜃)(x,y). Thesituationismade more complex by the fact that the packing density of retinal ganglion cells is not uniform,butinfoveatedprimatesfallswithan(approximate)inversesquarelawof the form (Drasdo, 1977): 𝜌G(r)=4 𝜋k 𝜋(𝑤0+𝜖r)2(10. 1) where𝑤0=0. 087istheaveragediameteroffovealretinalganglioncells, 𝜖=0. 051 is the rate of increase with rof𝑤G(r), the receptive field width of retinal ganglion cells,andk=7isthenumberofothercell'sreceptivefieldsoverlappingonerecep-tive field. This is to be compared with the packing density of cells in V1, 𝜌Cwhich is uniform.