The Operatorial Structure of the Cortex
We will next reinterpret the neurogeometric structure introduced in the first part of the paper from a probabilistic point of view, replacing vector field generators of the Lie algebra by the corresponding operators. This operation is called in physics second quantization.
FIGURE 7.5 A Kanitza triangle with curved boundaries (left) and its completion with geodesics in the group (right). The geodesics are not rectilinear, because they minimize the distance in the group (Equation 7.11) that contains the curvature k.
The Cortex as a Phase Space
In Sarti, Citti, and Petitot (2008), we identified the space of simple cells with the phase space of the retinal plane. A simple cell with the receptive field centered at a point (x, y) can be identified with an operator that selects the direction of the boundary at that point. It is then identified with the form
Note that this is a representation of a general unitary element of the con-tangent space at the fixed point or, equivalently, of the phase space. Its kernel is a bidimensional plane generated by the first-order operators X1 and X2, with the operatorial version of the vector fields Xj, generators of the group and defined in Equation (7.5):
These operators describe the propagation along the direction of the vector fields Xj, X2.
These operators, being identified with generators of a vector space, are defined up to a constant. This constant is generally chosen in such a way that the operators are self-adjoint, with respect to the standard scalar product defined on R2 # S1, which is defined as follows:
where h is the complex conjugate of h.
Recall that an operator X is self-adjoint if for every couple of functions b, h defined on R2 x S2,
Accordingly, we will choose as generators of the kernel of m the operators
where i is the imaginary unit.
The Generators of Rotations and Translations on the Retinal Plane The two operators X:, X2, introduced in Equation (7.10) satisfy the noncommutation relation (Equation [7.9]), which defines univocally the Lie algebra of rotation and translation. They act directly on the 3D phase space. However, through the action defined in Equation (7.2), we can obtain their projection on the 2D plane.
We saw that the group acts on the Hilbert space of functions f of two variables in the following way:
Since, for every point (^,f/) fixed, the action Axy6 can be considered a map from the phase space of variables (x, y, 6) to the 2D retinal plane, and then its differential map sends the operators Xx, X2 to operators Y1, Y2 on the 2D space:
simple derivatives show that
Hence, we obtain
The operators Y1, Y2 can be interpreted as the generators of translation in the direction | and rotation around the origin on the 2D retinal plane, respectively. The operators will also be interpreted as position and angular momentum operators, respectively. They satisfy the same commutation rules as the operators X1, X2, as well as the vector fields X1,X2 :
is the generator of translation in the direction orthogonal to Yp exactly as the corresponding vector X3 is the generator of translations in the direction orthogonal to X1.
The operators Y1 and Y2 represent the propagation along the curves of the bidimensional association fields (see Figure 7.4).
Note that they are self-adjoint with respect to the scalar product in R2:
The Uncertainty Principle
Receptive profiles have been interpreted as the minimal of the Heisenberg uncertainty principle by Daugman (1985). His crucial remark is the fact that simple cells are able to detect at the same time position and orientation, but these two quantities do not commute. Hence, the classical uncertainty principle first introduced by Heisenberg in quantum mechanics applies. Roughly speaking, it states that it is not possible to detect exactly both position and momentum and that the variance of their measurements cannot go below a fixed degree of uncertainty. However, there exist functions, the “coherent states,” that are able to minimize the degree of uncertainty in both quantities. These minima are Gabor (1946) filters, and Daugman (1985) proved that they are a good model of the receptive profiles of simple cells.
It was proven in Folland (1989) that the uncertainty principle does not apply only to the generators of the Heisenberg group. It applies to any couple of noncommuting operators. Hence, in our setting, it seems more natural to minimize the uncertainty principle for the generators of the Lie algebra of rotations and translations. Because we are looking for functions defined on the 2D space, we will use the representation of the vector fields in terms of the 2D variables, namely, the vector fields Y1 and Y2 introduced in Equation 7.11.
Proposition 7.1 The uncertainty principle in terms of these vector fields reads as follows:
where || || is the L2 norm. Proof. The proof is as follows:
(because the operators are self-adjoint)
where Im denotes the imaginary part. We conclude the proof, using the Cauchy-Schwartz inequality:
From this inequality and (7.14), the thesis immediately follows.
A slightly more general principle can be obtained if we substitute the norm of Yj with variance || Y – a.. ||, where a. is the mean value of Y.:
The products of variances of the position and angular momentum operators have a lower bound, so it is not possible to measure both quantities in an optimal way.
Coherent States and Receptive Profiles
The coherent states, minima of the uncertainty principle, are the functions that minimize the variance in the measure of position and momentum at the same time.
Proposition 7.2 The minimizers of inequality (7.13) satisfy the following equation:
Proof. The proof can be found, for example, in Theorem 1.34 in Folland (1989). It follows from the fact that minimizers must satisfy the equality in the uncertainty principle:
so that also the Schwartz inequality (Equation [7.15]) has to be an equality:
If the scalar product is equal to the product of the norms, then the vectors Y2u and Y1u have to be parallel:
A direct verification proves that the Gabor filter ^0 defined in Equation (7.1) satisfies this equation. As we said before, all the other filters were obtained from this via rotations and translations, so that they are coherent states in the sense of Perelomov (1986). Lee (1996) has shown these states are in agreement with the experimental data of the receptive profile (Figure 7.6).
Let us note that these minimizers are the coherent states of the reducible representation of the group. In Barbieri et al. (2010), we used the irreducible representation of the operators Y, so that we searched minimizers in the set of functions with fixed frequency in the Fourier domain, and we obtained the classical pinwheels structure, observed experimentaly by Bosking et al. (1997).
Output of Simple Cells and Bargmann Transform
We recall that the coherent states span the whole Hilbert space of functions defined on R2. They are not a basis of the space, but they form a frame. This amounts to saying that any function f on R2 can be represented as a linear combination of coherent states:
but the coefficients C(x, y, 6) are not unique (Folland, 1989).
FIGURE 7.6 Receptive profiles of simple cells measured by De Angelis (Reproduced with permission from De Angelis et al., 1995), and the corresponding Gabor functions (right), minimizers of the uncertainty principle (Equation [7.13]).
The output of single simple cells, defined in Equation (7.3), can be represented as a scalar product:
and it has the natural meaning of the projection of I in the direction of the specific element ¥ of the frame. Because the functions ¥ are coherent states, the whole output will be identified with a Bargmann transform, which is defined exactly as the scalar product with the entire bank of filters (Antoine, 2000):
This transform maps the image I(|, n) defined on the 2D space, to a function of three variables defined on the phase space. It inherits regularity properties from the properties of the filters. It is simple to verify that
Due to Proposition 7.2, this implies that the output satisfies
Hence, the Bargmann transform is an entire function in the Cauchy-Riemann (CR) structure generated by X1 and X2 (see Folland, 1989, for the definition of CR structure). In this context, the classical complex derivative
is replaced by the derivative with respect to the operators Xj. Hence, the function BI output of simple cells is holomorphic in this quasi-complex structure because
We quote Kritikos and Cho (1997) to outline the importance of holo-morphic functions when studying completion.