Parkinson's disease. Current Opinion in Neurology, 14, 499-504. doi: 10. 1097/00019052-200208000-00011 Young, R. S. L., Cole, R. E., Gamble, M., & Rayner, D. M. (1975). Subjective patterns elicited by light flicker. Vision Research, 15, 1291-1293. Retrieved from http://ncbi. nlm. nih. gov/pubmed/1198944Zubek, J. P. (1969). Sensory deprivation: Fifteen years of research. New York, NY: Appleton-Century-Crofts. Zuckerman, M., & Cohen, N. (1964). Sources of reports of visual and auditory sensations in perceptual-isolation experiments. Psychological Bulletin, 62, 1-20. doi:10. 1037/h0048599 Appendix The Nonlinear Transform From Retinal to Cortical Coordinates Geometry is a magic that works. —Rene ´ Thom (1975) It has long been known that resolution of stimuli grows pro-gressively worse with the distance on retina away from the fovea(fixation point). Several factors contribute to this: Receptive fieldcenters in foveal retina are smaller than in the periphery, fovealoptics are better than peripheral optics, retinal ganglion cell densityis higher near fovea, and visual cortex devotes more neurons tofoveal inputs to peripheral ones. These factors fit neatly together;the cortex devotes its resources in accordance to the availability ofretinal information. Thus, the cortex magnifies the foveal repre-sentation and progressively minifies the peripheral representation. Originally, cortical magnification C Mwas thought of as a ratio of neural image size on cortex to optical image size on retina. An application of this is M-scaling: the physical magnification that must be applied to a stimulus at a peripheral location in the retinafor a particular perceptual task to be completed that could becompleted with an unmagnified stimulus presented to the fovea. For example, contrast sensitivity is relatively constant if spatialfrequency is expressed in cycles/millimeter on visual cortex. Thecortical magnification factor can also be inferred from the scalingof fortification serrations in migraine (Gru ¨sser, 1995). In general, cortical magnification C Mat a distance r(in degrees of visual angle) from the fovea along a constant retinal direction /H9258, is fit by a simple monotonically decreasing function CM/H11005k//H20849r/H11001a/H20850. (A1) Because CMcan be defined as a ratio of small changes of cortical and retinal extent, it resembles a differential. If we naively inte-grate a function of this form—as Fechner did for Weber's law (see Billock & Tsou, 2011, on Fechner's law)—a logarithmic functionis obtained. However, since both the retina and cortex are two-dimensional sheets and isotropy is not guaranteed, the situation isa little more complex (Schwartz, 1994). In 1977, two independentinvestigators noted that the cortical magnification factor can beintegrated in such a way as to yield a sensible mapping from thepolar ( r,/H9258) retina to the more rectilinear ( x,y) cortex (Cowan, 1977; Schwartz, 1977). x/H11005/H9252ln/H20851/H20849√/H20849εr/H20850/H11001/H20849w o2/H11001εr2/H208501/ 2/H20850/2wo/H20852;y/H11005/H9252r/H9258//H20849wo2/H11001εr2/H208501/ 2, (A2)where ris radial distance (in degrees of visual angle) on retina from fovea, /H9258is the angular direction from the fovea, wois the mean diameter of foveal receptive fields, εis the rate of increase of that diameter with r, dis the density of receptive field overlap, and/H9252/H11005 (4d//H9266ε)1/2. Equation A2 is the version that was employed by Ermentrout and Cowan (1979) and Ermentrout (1984); varia-tions can be found in other studies, but for our purposes, thedifferences are of no consequence. For distances greater than onedegree of visual angle from the fovea (e. g., for all but a tiny portionof the ca. 135° vertical /H11003150° horizontal degree visual field), this function is well approximated as x/H11005/H9252ln(r√ε/w o);y/H11005/H9252 /H9258. This mapping is sometimes called a complex logarithmic function (be-cause the same form can be used to represent the logarithm of acomplex number like ln[ r,i/H9258]; Cowan, 1977; Schwartz, 1977). This conceptual advance did not get the attention it deserved until Tootell, Silverman, Switkes, and De Valois (1982) found a directand astonishingly convincing way to image the mapping in pri-mates. They created a polar grid visual stimulus consisting ofradial lines intersecting logarithmically spaced concentric circles. Markings on the grid continually reversed in contrast, assuring thatthey would activate both on-and off-cells and both sustained-andtransient-cells for maximum neural response. The stimuli were shownfor 30 min to one eye of a macaque who had been dosed withradioactive 14C-labeled 2-deoxy-D-glucose. Increased neural activity led to increased uptake of the radioactive glucose; after metabolism, the radioactive tag accumulated in the active neurons. The animal wasthen sacrificed, and its occipital cortex was extracted and pressedagainst a photographic plate, which was exposed by emitted radiation(see Figure A1). The delineation between stimulated and inactivecortex is remarkably crisp. Although modern imaging techniquesallow the same mapping to be studied in intact humans (see Figure 5),this remains the most convincing single direct demonstration of anyneural inference that we are aware of. A1 Although the nonlinear retinocortical mapping has important implications for how the cortex may be organized to processspatial information (Schwartz, 1977, 1980), for our purposes itsmost important implication is that three very different kinds of A1Schwartz (1994) recounted that when Tootell first showed Figure A1 at a conference (the Association for Research in Vision and Ophthalmology annual meeting, ca. 1980), the audience spontaneously stood and applauded. (Appendix continues )772 BILLOCK AND TSOU