The V1-V2-V3 complex: quasiconformal dipole maps in primate striate and extra-striate cortex

The mapping function w = k log(z + a) is a widely accepted approximation to the topographic structure of primate V1 foveal and parafoveal regions. A better model, at the cost of an additional parameter, captures the full field topographic map in terms of the dipole map function w = k log[(z + a)/(z + b)]. However, neither model describes topographic shear since they are both explicitly complex-analytic or conformal. In this paper, we adopt a simple ansatz for topographic shear in V1, V2, and V3 that assumes that cortical topographic shear is rotational, i.e. a compression along iso-eccentricity contours. We model the constant rotational shear with a quasiconformal mapping, the wedge mapping. Composing this wedge mapping with the dipole mapping provides an approximation to V1, V2, and V3 topographic structure, effectively unifying all three areas into a single V1-V2-V3 complex using five independent parameters. This work represents the first full-field, multi-area, quasiconformal model of striate and extra-striate topographic map structure.