This paper is concerned with a striking visual experience: that of seeing geometric visual hallucinations. Hallucinatory images were classified by Klüver into four groups called form constants comprising (i) gratings, lattices, fretworks, filigrees, honeycombs and chequer–boards, (ii) cobwebs, (iii) tunnels, funnels, alleys, cones and vessels, and (iv) spirals. This paper describes a mathematical investigation of their origin based on the assumption that the patterns of connection between retina and striate cortex (henceforth referred to as V1)—the retinocortical map—and of neuronal circuits in V1, both local and lateral, determine their geometry.
In the first part of the paper we show that form constants, when viewed in V1 coordinates, essentially correspond to combinations of plane waves, the wavelengths of which are integral multiples of the width of a human Hubel–Wiesel hypercolumn, ca. 1.33–2 mm. 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–Schrödinger 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 ϕ, the cortical label for orientation preference, and plane waves in r, the cortical position coordinate. 'switching–on' the lateral interactions breaks the degeneracy and either even or else odd eigenfunctions are selected. These results can be shown to follow directly from the Euclidean symmetry we have imposed.
In the second part of the paper we study the nature of various even and odd combinations of eigenfunctions or planforms, the symmetries of which 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 symmetrical dynamical system becomes 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 become 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 retinocortical 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 what are called linear regions, in which the pattern has a near constant orientation, reminiscent of the iso–orientation patches constructed via optical imaging. The boundaries of such regions form fractures whose points of intersection correspond to the well–known ‘pinwheels’.
To complete the study we then investigate the stability of the planforms, using methods of nonlinear stability analysis, including Liapunov–Schmidt reduction and Poincaré–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.