Orientation Maps in V1 and Non-Euclidean Geometry

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.

Fig. 1

(Modified from Bosking et al. [7].) An Orientation Preference Map observed in the visual cortex of a tree shrew. The experimental procedure leading to this map is recalled in the main text. See also Swindale [18]. On the upper right corner, details at singular points (pinwheels) or regular points are shown

Fig. 2

Computer-generated map, sampled from a monochromatic field. This figure shows an orientation map which we have drawn from a simulated invariant Gaussian random field with circular power spectrum. This figure was generated using a superposition of 30 plane waves with frequency vectors at the vertices of a random polygon inscribed in a circle, and random Gaussian weights (see the Appendix); what is plotted is the argument. In the unit of length displayed on the x- and y-axes, the wavelength is $1/3$ here

Fig. 3

The “composite distance” to a point of the boundary. Definition of the quantity $\langle x,b\rangle$ if x is a point of $\mathbb{D}$ and b a point of its boundary: $\xi (b,x)$ is the horocycle through x which is tangent to the boundary at b, and $\mathrm{\Delta }(b,x)$ is the segment joining the origin O to the point on $\xi (b,x)$ which is diametrically opposite b; the number $\langle x,b\rangle$ is, up to a sign, the hyperbolic length of this segment

Fig. 4

Plot of the real part of a Helgason wave, with the exponential growth factor deleted: in the darkest regions the real part of the scaled wave vanishes, and in the brightest regions it is equal to one. Given the formula for $e_{\omega ,b}$, this plot also gives an idea of the argument as a function of z; notice that the argument is periodic when restricted to any geodesic whose closure in $\mathbb{C}$ contains the point of −1 of B

Fig. 5

Plot of a monochromatic “orientation map” on the hyperbolic plane. We used the spectral parameter $\omega =18$ in units of the disk’s radius. Because of the growth factor in the modulus of the $e_{\omega ,b}$, drawing a picture in which discretization effects do not appear calls for using more propagation directions than it did in the Euclidean case: 200 directions were used to generate the drawn picture

Fig. 6

An orientation map on the sphere sampled from the argument of a monochromatic $\mathit{SO}(3)$-invariant Gaussian random field on the sphere with spin 7. We plotted the restriction to a hemisphere; we used a superposition of spherical harmonics with spin seven and random, reduced independent Gaussian weights

Fig. 7

An orientation map on the sphere sampled from a random vector field which has $\mathit{SO}(3)$-shift-twist symmetry. We plotted the restriction to a hemisphere of the random map exploring ${\mathcal{H}}_{10}^{\mathrm{exact}}$; beware that the color coding has a different meaning than in Figs. 1, 2, 5, and 6. Here, the sample map is a vector field on the sphere, and there is no complex number; to visualize the direction of the emerging vector at each point, we apply the orthogonal projection from the drawn hemisphere to the “paper” plane, thus getting a vector field on the unit ball of the Euclidean plane, and plot the resulting orientation map using the same color code as in Figs. 1, 2, 5, and 6