Geometric Visual Hallucinations, Euclidean Symmetry, and the Functional Architecture of Striate Cortex
Phil. Trans. Royal Soc. London B 356 (2001) 299-330.
In the first part of the paper we show that form constants, when viewed in
V1 coordinates, correspond essentially to combinations of plane waves, the
wavelengths of which are integral multiples of the width of a human
Hubel--Wiesel hypercolumn, about 2.00mm. We next introduce a
mathematical description of the
large--scale dynamics of V1 in terms of the continuum limit of a lattice
of interconnected
hypercolumns, each of which itself comprises a number of interconnected
iso--orientation columns.
We then show that the patterns of interconnection in V1 exhibit a very
interesting symmetry,
i.e., they are invariant under the action of the planar Euclidean group
E(2)---the group of rigid motions in
the plane---rotations, reflections and translations.
What is novel is that the lateral connectivity of V1 is such
that a new group action is needed to represent its properties: by virtue of its anisotropy it is invariant with
respect to certain shifts and twists of the plane.
It is this shift--twist invariance that generates
new representations of E(2). Assuming that the strength
of lateral connections is weak compared with that
of local connections, we next calculate the eigenvalues and eigenfunctions of the cortical dynamics, using
Rayleigh--Schrodinger perturbation theory.
The result is that in the absence of lateral connections, the
eigenfunctions are degenerate, comprising both even
and odd combinations of sinusoids in phi, the
cortical label for orientation preference, and plane waves in
In the second part of the paper we study the nature of various even and odd combinations of eigenfunctions or {\em planforms\/}, whose symmetries are such that they remain invariant under the particular action of E(2) we have imposed. These symmetries correspond to certain subgroups of E(2), the so--called axial subgroups. Axial subgroups are important in that the equivariant branching lemma indicates that when a symmetric dynamical system goes unstable, new solutions emerge which have symmetries corresponding to the axial subgroups of the underlying symmetry group. This is precisely the case studied in this paper. Thus we study the various planforms that emerge when our model V1 dynamics goes unstable under the presumed action of hallucinogens or flickering lights. We show that the planforms correspond to the axial subgroups of E(2), under the shift--twist action. We then compute what such planforms would look like in the visual field, given an extension of the retino--cortical map to include its action on local edges and contours. What is most interesting is that given our interpretation of the correspondence between V1 planforms and perceived patterns, the set of planforms generates representatives of all the form constants. It is also noteworthy that the planforms derived from our continuum model naturally divide V1 into regions in which the pattern has a near constant orientation, reminiscent of the iso--orientation patches constructed via optical imaging.
To complete the study we then investigate the stability of the planforms, using methods of nonlinear stability analysis, including Liapunov--Schmidt reduction and Poincare--Lindstedt perturbation theory. We find a close correspondence between stable planforms and form constants. The results are sensitive to the detailed specification of the lateral connectivity and suggest an interesting possibility, that the cortical mechanisms by which geometric visual hallucinations are generated, if sited mainly in V1, are closely related to those involved in the processing of edges and contours.