Introduction to neurogeometry of vision

The term "neurogeometry" had been proposed by J. Petitot for the branch of neuroscience, which investigates different models of brain structures, mostly related with vision, in the language of differential geometry and differential equations. The structures are considered as continuum media with internal structure, described by properties of neurons. The approach is based on the principle of locality of visual neurons, whose excitation depends on energy density of light, coming to a small domain $D$ of retina ("receptive field"). Many visual neurons work as linear filters (generalized functions with support $D$) — their excitation is described by the integral of the intensity function $I$ over $D$ taking with some weight ("receptive profile").
In the talk, we shortly describe the structure and function of early visual systems — eye, retina, LGN. We discuss the basic structures of visual cortex VI — pinwheel field and hypercolumns , discovered by D. Hubel and T. Wiesel (Nobel prise 1981).
We give a short survey of geometric model of VI cortex (Petitot contact model, symplectic model by Petitot-Citti-Sarti, Bressloff-Cowan spherical model of a hypercolumn, Faugeras hyperbolic platform, evolution model by Geisel-Wolf)).
We consider a synthesis of the models by Petitot-Citti-Satri and Bressloff-Cowan and discuss its application to the solution the stability problem - problem of invariancy of perception with respect to fixation eye movements, discovered by A. Yarbus.