Functional architecture I

the pinwheels of V1

pp. 113-273

in: Jean Petitot, Elements of neurogeometry, Berlin, Springer, 2017

Abstract

This chapter presents a first set of experimental data on the functional architecture of area V1 and, in particular, on what is referred to as its "pinwheel' structure. Most of the so-called simple neurons in V1 detect positions and orientations in the visual field, with those detecting the various orientations for each position being grouped together into functional micromodules that can be defined anatomically and are called orientation hypercolumns or pinwheels. In this sense, V1 implements a 2D discrete approximation of the 3D fibre bundle (pi :mathbb {V}=R imes mathbb {P} ^{1} ightarrow R), with the retinal plane R as its base space and the projective line (mathbb {P}^{1}) of orientations in the plane as its fibre. V1 then appears to be a field of orientations in a plane (called an "orientation map' by neurophysiologists), a field whose singularities are the centres of the pinwheels . This chapter studies these singularities, gives their normal forms , and specifies the distortions and defects of their networks. One way to model such orientation maps is to treat them as phase fields, analogous to those encountered in optics, whose singularities have been thoroughly analyzed by specialists such as Michael Berry. These fields are superpositions of solutions to the Helmholtz equation, whose wave number depends in a precise manner on the mesh of the pinwheel lattice. They enable the construction of very interesting models, such as those proposed by Fred Wolf and Theo Geisel. These are explained here. However, in these models of phase fields , orientation selectivity must vanish at singularities. Yet many experimental results show that this is not the case. The chapter thus presents another model based on the geometric notion of blow-up. It also explains how the fibration that models the orientation variable interferes with other fibrations (other visual "maps') that model other variables such as direction , ocular dominance , phase, spatial frequency, or colour. For spatial frequency, it presents the dipole model proposed by Daniel Bennequin. V1 therefore implements fibrations of rather high dimension in two-dimensional layers. This leads to the problem of knowing how to express the independence of these different variables. A plausible hypothesis relies on a transversality principle. The chapter ends with data on two other aspects of neurobiology: (i) the relation between the cerebral hemispheres through callosal connections and (ii) the primary processing of colour in the "blobs' of V1.