On the cortical mapping function - Visual space, cortical space, and crowding

The retino-cortical visual pathway is retinotopically organized: Neighbourhood relationships on the retina are preserved in the mapping. Size relationships in that mapping are also highly regular: The size of a patch in the visual field that maps onto a cortical patch of fixed size follows, along any radius and over a wide range, simply a linear function with retinal eccentricity. As a consequence, the mapping of retinal to cortical locations follows a logarithmic function along that radius. While this has already been shown by Fischer (1973, Vision Research, 13, 2113-2120), the link between the linear function - which describes the local behaviour by the cortical magnification factor M - and the logarithmic location function for the global behaviour, has never been made explicit. The present paper provides such a link as a set of ready-to-use equations using Levi and Klein's E₂ nomenclature, and examples for their validity and applicability in the mapping literature are discussed. The equations allow estimating M in the retinotopic centre; values thus derived from the literature show enormous, hitherto unnoticed, variability. A new structural parameter, d₂, is proposed to characterize the cortical map, as a counterpart to E₂; it shows much more stability. One pitfall is discussed and spelt out, namely the common myth that a pure logarithmic function, without constant term, will give an adequate map. The correct equations are finally extended to describe the cortical map of Bouma's law on visual crowding. The result contradicts recent suggestions that critical crowding distance corresponds to constant cortical distance.