A metric model of the visual cortex

The purpose of this thesis is the development of a model for the geometry of the connectivity of the primary visual cortex (V1), by means of functional analysis tools on metric measure spaces. The metric structure proposed to describe the internal connections of V1 implements a notion of correlation between neurons, based on their feature selectivity: this is expressed through a connectivity kernel that is directly induced by the local feature analysis performed by the cells. Such kernel carries a geometrical structure consistent with the well-known properties of long-range horizontal connections in V1, and it is compatible with the perceptual rules synthesized by the concept of association field. Moreover, its construction can be applied to banks of filters not necessarily obtained through a group representation, and possibly only numerically known. This model is then applied to insert biologically inspired connections in deep learning algorithms, to enhance their ability to perform pattern completion in image classification tasks. The main novelty in our approach lies in its ability to recover global geometric properties of the functional architecture of V1 without imposing any parameterization or invariance, but rather by exploiting the local information naturally encoded in the behavior of single V1 neurons in presence of a visual stimulus.