Metodi variazionali nell'encoding di dati fMRI per la modellizzazione delle architetture corticali

This work aims to apply calculus of variations tools to estimate brain activity in the primary visual cortex (V1) to model the functional architecture of this area. The first chapter is devoted to a presentation of the wavelet theory and harmonic analysis tools necessary to model the receptive profiles (RPs) of simple cells in V1. The set of these RPs is shown to be accurately modeled by a family of 2D Gabor wavelets, each obtained as the product of an elliptical Gaussian and a complex plane wave. The receptive profiles modeled by these functions are shown to act, at least to a first approximation, as linear filters on an image. A proper countable subset of this family is also demonstrated to be a frame, and therefore to give a complete representation of any optic signal as a linear combination of its projections on these filters. Moreover, Gabor wavelets form the general class of functions which achieve the theoretical lower limit for uncertainty in the space and frequency domains. The fact that the set of receptive profiles of simple cells seems to be best modeled by a collection of 2D Gabor functions is therefore a crucial result, as it suggests that visual neurons act in an optimal way. In the second chapter we deal with some techniques of brain encoding, i.e. methods for estimating parameters for a suitable receptive field model, based on recorded brain activity. We first need to specify what kind of data we are referring to as "recorded brain activity": to this end, a first section is dedicated to functional magnetic resonance imaging (fMRI), a neuroimaging procedure that measures neuronal activation in an indirect way, that is by detecting changes associated with blood flow in the brain. The brain is reproduced as a volume specified by voxel indices in a 3D matrix, and values representing fMRI data are associated to each voxel. Specialized imaging softwares allow to extract surfaces from these volume data, displaying the values (as colours) relative to the cortex and making it easier to interpret them. Afterwards, we show how to combine the information obtained through fMRI with the mathematical instruments introduced earlier, to obtain a model able to reproduce the brain activity generated by previously unseen images, in the form of fMRI-like data. This goal is achieved through minimization of a properly chosen functional. The first addend to be introduced is a so-called fiducial term, namely the squared norm of the difference between the recorded fMRI response and the simulated one, depending on coefficients (weights) that are to be determined as minimum points of this functional. One can then add further terms for regularization. In particular, we examine the lasso term, that is the sum of the absolute values of the weights (which we aim to minimize to obtain a sparse representation, namely one involving as few coefficients as possible). The third chapter introduces some geometric information about cortical representation of optic signals. In particular we examine the retino-cortical mapping, which is shown to be well approximated, close to the center of the field of view, by a complex logarithmic function, depending on parameters whose role is to specify the fit. In this work, we develop a technique to determine the parameters of this map, guided by encoded fMRI data, and we suggest an approach to the still open problem of inverse mapping.