Neural Differential Calculus and Functional Architectures

The paper starts with a basic presentation of Neurogeometry as geometrizing the connectivity of primary visual areas by means of concepts of modern differential geometry. The retinotopic parameterization of orientation (OR) hypercolumns in area V1 by positions in the retina can be interpreted as a fibre bundle, namely the fibration of 1-jets of planar curves. The functional architecture defined on V1 by the cortico-cortical connections between OR hypercolumns can then be interpreted as the canonical contact structure of the 1-jets fibre bundle. The paper presents next the pinwheel structure of OR maps with their singularities. The existence of a characteristic mesh of the network of pinwheels corresponds to the fact that the OR field is a solution of the Helmholtz equation. The concepts of a phase field and of a Gaussian field can consequently be applied to the statistics of pinwheels. Lastly, the paper focuses on the relations between the pinwheels of OR maps and the fractures of direction (DR) maps. It presents a neurogeometrical model of beautiful empirical results from Nicholas Swindale. In this model, one takes into account not only preferred ORs and DRs but also the tuning curves of which they are peaks. The spatial layout of DRs with their singularities (fractures ending at pinwheels) is then modeled by a universal unfolding (in Thom's sense) of tuning curves.