Course 12 Symmetry breaking and pattern selection in visual cortical development

Abstract

The chapter discusses the description of the development of orientation preference columns in terms of a dynamics of abstract order parameter fields and connects this description to the theory of Gaussian random fields, and examines how the theory of Gaussian random fields is used to obtain quantitative information on the generation and motion of pinwheels, in the two dimensional pattern of visual cortical orientation columns. The chapter also discusses the symmetry based approach used to derive this prediction to study also the kind of patterns to which the map will asymptotically converge and the interactions essential for the stabilization of different kinds of solutions. An exposition of an appropriate perturbation method called weakly nonlinear analysis for the problem of orientation column formation is provided. Using this method, a class of generalized Swift–Hohenberg models for the formation of patterns of contour detecting neurons during visual cortical development is constructed. In this model class, permutation symmetry of the model equations satisfies the requirement that the visual cortex develops selectivity for all contour orientations. Long-range interactions are found to be essential for the stability of realistic solutions.