170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 224 224 CH10 GEOMETRIC VISUAL HALLUCINATIONS We can now calculate the retino-cortical map by assuming that the number of retinal ganglion cell axons in a single elementary retinal area d AR=𝜌Grdrd𝜃maps ina{1∶1}fashionontothesurfaceofasingleelementaryareaof V1 d Ccontaining 𝜌Cdxdycortical cells, that is, 𝜌Grdrd𝜃=𝜌Cdxdy (10. 2) We solve this equation using equation 10. 1, by letting √𝜌Cdx=√𝜌Gdr;√𝜌Cdy=√𝜌Grd𝜃 whence x=√ 4k 𝜋𝜌C1 𝜖ln[ 1+𝜖 𝑤0r] ;y=√ 4k 𝜋𝜌Cr𝜃 𝑤0+𝜖r(10. 3) We note that near the fovea x√ 4k 𝜋𝜌Cr 𝑤0;y√ 4k 𝜋𝜌Cr𝜃 𝑤0. whereas away from the fovea 𝑤√ 4k 𝜋𝜌C1 𝜖ln[ 𝜖r 𝑤0] ;y√ 4k 𝜋𝜌C𝜃 𝜖. Note also that 𝑤G(r)=𝑤0+𝜖rso that its rate of increase is 𝜖. This was mea-sured both in the retina and in V1 by Hubel and Wiesel (1974b) and is called the retino-corticalmagnificationfactor. Thus, away from the fovea there is an (approximately) logarithmic map from the retinal surface onto the surface of V1, that is, x∼lnr,y∼𝜃. Letz=x+iybe a complex number. This in turn leads to the expression z=lnr+i𝜃ez=elnr+i𝜃=elnr⋅ei𝜃=rei𝜃 Such a complex map is called the complex logarithm and was first used to model the retino-cortical map by Schwartz (1977). Figure 10. 7 shows details of the map. Evidently it expresses the effects of the retino-cortical magnification factor in that, away from the fovea, the width of cortical receptive fields Δ𝑤C=Δx∼1 𝜖Δr r=1 sinceΔr r=Δ𝑤G r=𝜖. Thuscorticalreceptivefieldshave constantwidthsincorticalcoordinates,whichis a requirement for the spatial homogeneity of V1. [See also Cowan (1977). ]