Abstract
In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing. Now, existing models for the geometry and development of orientation preference maps in higher mammals make a crucial use of symmetry considerations. In this paper, we consider probabilistic models for V1 maps from the point of view of group theory; we focus on Gaussian random fields with symmetry properties and review the probabilistic arguments that allow one to estimate pinwheel densities and predict the observed value of π. Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry. We show that through the Plancherel decomposition of the space of complex-valued maps on the Euclidean plane, each infinite-dimensional irreducible unitary representation of the special Euclidean group yields a unique V1-like map, and we use representation theory as a symmetry-based toolbox to build orientation maps adapted to the most famous non-Euclidean geometries, viz. spherical and hyperbolic geometry. We find that most of the dominant traits of V1 maps are preserved in these; we also study the link between symmetry and the statistics of singularities in orientation maps, and show what the striking quantitative characteristics observed in animals become in our curved models.
Highlights
In the primary visual cortex, neurons are sensitive to selected features of the visual input: each cell analyzes the properties of a small window in the visual field, its response depends on the local orientations and spatial frequencies in the visual scene [1, 2], on velocities or time frequencies [3, 4], it is subject to ocular dominance [1], etc
We shall use the paragraph to give the necessary details on the geometry of the unit disk; the description of all unitary representations of SU(1, 1), we shall skip over8 in order to focus on the Plancherel decomposition of L2(G/K)
We started from a reformulation of existing work by Wolf, Geisel and colleagues, with the aim to understand the crucial symmetry arguments used in models with the help of noncommutative harmonic analysis, which is often a very wellsuited tool for using symmetry arguments in analysis and probability
Summary
In the primary visual cortex, neurons are sensitive to selected features of the visual input: each cell analyzes the properties of a small window in the visual field, its response depends on the local orientations and spatial frequencies in the visual scene [1, 2], on velocities or time frequencies [3, 4], it is subject to ocular dominance [1], etc These receptive profiles are distributed among the neurons of area V1, and in many species they are distributed in a remarkably orderly way [5,6,7]. Its beautiful geometrical properties (see Fig. 1) have prompted many experimental and theoretical studies (see [5, 7, 11,12,13]); the orientation map seems to be closely tied to the horizontal wiring (the layout of connectivities between microcolumns) of V1 [7, 14, 15], its geometry is correlated to that of all the other feature maps [16, 17], and while the geometrical properties of other feature maps vary much across species, those of orientation maps are remarkably similar [11].
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