Coordinated Optimization of Visual Cortical Maps (I) Symmetry-based Analysis

In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), and spatial frequency. It is a long-standing question in theoretical neuroscience whether the observed maps should be interpreted as optima of a specific energy functional that summarizes the design principles of cortical functional architecture. A rigorous evaluation of this optimization hypothesis is particularly demanded by recent evidence that the functional architecture of orientation columns precisely follows species invariant quantitative laws. Because it would be desirable to infer the form of such an optimization principle from the biological data, the optimization approach to explain cortical functional architecture raises the following questions: i) What are the genuine ground states of candidate energy functionals and how can they be calculated with precision and rigor? ii) How do differences in candidate optimization principles impact on the predicted map structure and conversely what can be learned about a hypothetical underlying optimization principle from observations on map structure? iii) Is there a way to analyze the coordinated organization of cortical maps predicted by optimization principles in general? To answer these questions we developed a general dynamical systems approach to the combined optimization of visual cortical maps of OP and another scalar feature such as OD or spatial frequency preference. From basic symmetry assumptions we obtain a comprehensive phenomenological classification of possible inter-map coupling energies and examine representative examples. We show that each individual coupling energy leads to a different class of OP solutions with different correlations among the maps such that inferences about the optimization principle from map layout appear viable. We systematically assess whether quantitative laws resembling experimental observations can result from the coordinated optimization of orientation columns with other feature maps. Author Summary: Neurons in the visual cortex form spatial representations or maps of several stimulus features. How are different spatial representations of visual information coordinated in the brain? In this paper, we study the hypothesis that the coordinated organization of several visual cortical maps can be explained by joint optimization. Previous attempts to explain the spatial layout of functional maps in the visual cortex proposed specific optimization principles ad hoc. Here, we systematically analyze how optimization principles in a general class of models impact on the spatial layout of visual cortical maps. For each considered optimization principle we identify the corresponding optima and analyze their spatial layout. This directly demonstrates that by studying map layout and geometric inter-map correlations one can substantially constrain the underlying optimization principle. In particular, we study whether such optimization principles can lead to spatially complex patterns and to geometric correlations among cortical maps as observed in imaging experiments.