Spatio-temporal models of the functional architecture of the visual cortex

In this work I tried to explore many aspects of cognitive visual science, each one based on different academic fields, proposing mathematical models capable to reproduce both neuro-physiological and phenomenological results that were described in the recent literature. The structure of my thesis is mainly composed of three chapters, corresponding to the three main areas of research on which I focused my work. The results of each work put the basis for the following, and their ensemble form an homogeneous and large-scale survey on the spatio-temporal properties of the architecture of the visual cortex of mammals.

Neurogeometry of stereo vision

This work aims to develop a neurogeometric model of stereo vision, based on cortical architectures involved in the problem of 3D perception and neural mechanisms generated by retinal disparities. First, we provide a sub-Riemannian geometry for stereo vision, inspired by the work on the stereo problem by Zucker (2006), and using sub-Riemannian tools introduced by Citti-Sarti (2006) for monocular vision. We present a mathematical interpretation of the neural mechanisms underlying the behavior of binocular cells, that integrate monocular inputs. The natural compatibility between stereo geometry and neurophysiological models shows that these binocular cells are sensitive to position and orientation. Therefore, we model their action in the space R3xS2 equipped with a sub-Riemannian metric. Integral curves of the sub-Riemannian structure model neural connectivity and can be related to the 3D analog of the psychophysical association fields for the 3D process of regular contour formation. Then, we identify 3D perceptual units in the visual scene: they emerge as a consequence of the random cortico-cortical connection of binocular cells. Considering an opportune stochastic version of the integral curves, we generate a family of kernels. These kernels represent the probability of interaction between binocular cells, and they are implemented as facilitation patterns to define the evolution in time of neural population activity at a point. This activity is usually modeled through a mean field equation: steady stable solutions lead to consider the associated eigenvalue problem. We show that three-dimensional perceptual units naturally arise from the discrete version of the eigenvalue problem associated to the integro-differential equation of the population activity.