Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex

This paper is concerned with a striking visual experience: that of seeing g eometric visual hallucinations. Hallucinatory images were classified by Klu ver into four groups called form constants comprising (i) gratings, lattice s, 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 assumpt ion that the patterns of connection between retina and striate cortex (henc eforth 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-W iesel hypercolumn, ca. 1.33-2 mm. We next introduce a mathematical descript ion 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 nu mber of interconnected iso-orientation columns. We then show that the patte rns of interconnection in V1 exhibit a very interesting symmetry i.e. they are invariant rotations, reflections and translations. What is novel is tha t 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 wi th respect to certain shifts and twists of the plane. It is this shift-twis t invariance that generates new representations of E(2). Assuming that the strength of lateral connections is weak compared with that of local connect ions, 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 r, the cortical posit ion coordinate. 'Switching-on' the lateral interactions breaks the degenera cy and either even or else odd eigenfunctions are selected. These results c an 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 s uch 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-c alled axial subgroups. Axial subgroups are important in that the equivarian t branching lemma indicates that when a symmetrical dynamical system become s unstable, new solutions emerge which have symmetries corresponding to the axial subgroups of the underlying symmetry group. This is precisely the ca se studied in this paper. Thus we study the various planforms that emerge w hen our model V1 dynamics become unstable under the presumed action of hall ucinogens or flickering lights. We show that the planforms correspond to th e axial subgroups of E(2), under the shift-twist action. We then compute wh at such planforms would look like in the visual field, given an extension o f the retinocortical map to include its action on local edges and contours. What is most interesting is that, given our interpretation of the correspo ndence between V1 planforms and perceived patterns, the set of planforms ge nerates representatives of all the form constants. It is also noteworthy th at the planforms derived from our continuum model naturally divide V1 into what are called linear regions, in which the pattern has a near constant or ientation, reminiscent of the iso-orientation patches constructed via optic al imaging. The boundaries of such regions form fractures whose points of i ntersection correspond to the well-known 'pinwheels'. To complete the study we then investigate the stability of the planforms, u sing methods of nonlinear stability analysis, including Liapunov-Schmidt re duction and Poincare-Lindstedt perturbation theory. We find a dose correspo ndence between stable planforms and form constants. The results are sensiti ve to the detailed specification of the lateral connectivity and suggest an interesting possibility, that the cortical mechanisms by which geometric v isual hallucinations are generated, if sited mainly in V1, are closely rela ted to those involved in the processing of edges and contours.