Symmetry Breaking and Pattern Selection in Models of Visual Development

Response properties of neurons in the visual cortex form spatial representations called maps of several stimulus features such as orientation preference (OP) or ocular dominance (OD). This thesis studies macroscopic models for the development of such visual cortical maps described by the time evolution of order parameter fields. It is plausible that map development is a process of optimization of the cortical connectivity and thus the map dynamics is proposed to be derived from an energy functional. As experimental evidence suggests that cortical maps are not independent of each other we study the joint optimization of various maps. Using generalized Swift-Hohenberg equations all terms that enter the dynamics are justified by symmetry considerations. This field dynamics is analyzed using two approaches: First, through a weakly nonlinear analysis which leads to a description of map development close to the pattern forming threshold given by amplitude equations. Second, through direct numerical simulation using a fully implicit integration scheme.We first study the joint development of OP and OD maps using various types of inter-map coupling energy functionals. Our results identify the commonality and specific fingerprints of particular optimization principles on the spatial layout of the maps. Moreover, we show that the influence of the OD map offers a solution to the problem of the stability of so-called pinwheels, topological defects in OP maps. In this model the uncoupled OP map disintegrates into pinwheel-free stripe patterns. However, for strong inter-map coupling and in case of the over-representation of one eye over the other pinwheel rich solutions are stable, where pinwheels are arranged in a crystal array. Next, we introduce non-local interactions terms to the dynamics to translate the long-range connectivity of visual cortical neurons. We show that this can lead to spatially irregular and pinwheel-rich patterns. We further present a potential explanation of the species differences in the OD map layout by considering a detuning of the typical wavelengths of OP and OD maps. The uncoupled OP dynamics has numerous stationary solutions. We present a comprehensive characterization of this large set of stationary solutions, study their stability properties, and in particular study how the stability of unrealistic solutions can be excluded. One class of stationary solutions is dominant i.e. it contains all energetic ground states of the model. These ground states are moreover found to be energetically degenerate. We identify two mechanisms that can lift this degeneracy and thus lead to pattern selection: Higher order corrections to the amplitude equations and the breaking of a symmetry of the field dynamics, namely permutation symmetry. For both cases we identify local minima and energetic ground states and study the impact of pattern selection on the quantitative properties of OP maps such as their pinwheel density.This thesis presents models for the development of visual cortical maps and demonstrates that the presence or absence of certain symmetries is crucial for the emergence of realistic patterns. We have identified models which lead to maps with a realistic spatial layout that moreover also quantitatively resemble maps obtained from experiment.