Hyperbolic planforms in relation to visual edges and textures perception

Abstract

We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g., optical imaging, and opens the door to the design of experiments to test these hypotheses. We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific second-order tensor, the structure tensor, perhaps within a hypercolumn. We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. These patterns can be described by what we call hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of the planforms that were used in previous work to account for some visual hallucinations. If these patterns could be observed through brain imaging techniques they would reveal the built-in or acquired invariance of the neural organization to the action of the corresponding subgroups.

Figure 1. Geometric interpretation of the distance between two tensors.

The two structure tensors and are represented by the elliptic blobs shown in the lefthand side of the figure. After the change of coordinates defined by the matrix , is represented by the unit disk and the principal axes of are equal to the eigenvalues and that appear in (6), see text.

Figure 2. The orbits in the Poincaré disk of the three groups , and .

Figure 3. Simple transformations in the image plane.

The coordinate system which is used to estimate the image derivatives and some of its transformations under the action of some elements of (see text).

Figure 4. Horocyclic coordinates.

The horocyclic coordinates of the point of are the real values and such that . The horocycle through is the circle tangent to at and going through . is equal to the (hyperbolic) signed distance between the origin and the point which is equal to and is negative if is inside the circle of diameter and positive otherwise.

Figure 5. A periodic H-planform.

A representation of the periodic H-planform . The color represents the value of the magnitude of for z varying in . The periodicity is to be understood in terms of the hyperbolic distance . The hyperbolic distance between two consecutive points of intersection of the, say yellow, circles with the horizontal axis is the same. It does not look so to our “Euclidean” eyes and the distances look shorter when these points get closer to the point on the right and to the point of on the left. These points are actually at an infinite distance from the center of .

Figure 6. Color representation of the complex valued function .

Real (blue) and imaginary (red) parts of defined in equation (24) for H-planforms, , see text. We chose , , and in equation (3.

Figure 7. Color representation of the complex valued function .

Real (blue) and imaginary (red) parts of defined in equation (24) for H-planforms, , see text. We chose , , and and in equation (3.

Figure 8. An example of an H-planform which is invariant with respect to the octagonal Fuchsian group.

We have superimposed two fundamental domains: in the center the “main” one containing the origin, to its right another fundamental domain that shows the Euclidean distorsion due to the increase in the hyperbolic distance. In effect these two octagons can be exactly superimposed through the action of a hyperbolic isometry. The color encodes the value of the H-planform, blue indicates negative values, red indicate positive values, green indicates values close to 0.

Figure 9. Zoom on the first “octant” of the Poincaré disk.

It is at a higher spatial resolution than figure 8. In particular for the second octagon, the one to the right of the “main” one, it shows better the relationship between the intensity patterns within the two octagons.