Geometric visual hallucinations and the architecture of the visual cortex Jack Cowan Department of Mathematics Department of Neurology Committee on Computational Neuroscience University of Chicago

Geometric visual hallucinations The hallucination is … not a static process but a dynamic process, the instability of which reflects an instability in its conditions of origin. R. MOURGUE (1932)

• CAUSES

OF HALLUCINATIONS

Flickering Light (Purkinje, Helmholtz) Anaesthesia Hypnagogic or Hypnopompic Near Death Entoptic Psychotropic drugs such as LSD, cannabis, mescaline, psilocybin

Entoptic forms

(a) Phosphene produced by deep binocular pressure on the eyeballs (Tyler 1978)

(b) Honeycomb image generated by marihuana (Siegel 1977)

Funnel and Spiral images generated by LSD (Oster 1970)

Funnel and Spiral images generated by LSD (Siegel 1977)

Cobweb petroglyph 1000 AD (Patterson 1992)

Cocaine induced imagery (Siegel 1977)

(a)

(b)

Fortification patterns seen during migraine. (a) Siegel (1977), (b) Richards (1971).

Namibian Rock Art

Blombos Cave 77000 BP

Piet Alberts Kopjes 25000 BP

European cave art

Chauvet 31000 BP

Pech-Merle 25000 BP

More Chauvet Cave Art

South African Cave Art

Solomonslagte 2000 BP

The first three stages of altered consciousness

Clottes & Lewis-Williams (1998)

• Klüver (1928) organized the images into four classes called form constants: (I) tunnels and funnels (II) spirals (III) lattices, honeycombs and checkerboards (IV) cobwebs

The retino-cortical map (Tootell et. al. 1982)

Entoptic images in visual cortex coordinates

(a) Funnel

(c) Spiral

(b) Funnel image in V1 coordinates

(d) Spiral image in V1 coordinates

• Many species have coat markings in the form of stripes

or spots. Turing (1952) provided a plausible theory for the emergence of such patterns. In what follows we develop a similar theory for the development of cortical activity patterns.

The Turing mechanism

u˙ = v˙ =

bu + r2 u + f (u, v) v + Dr2 v + g(u, v) Turing (1952)

u˙ =

bu +

Z

w(|r

r0 |) [u(r0 )]dr0 + I

Ermentrout and Cowan (1979)

*

Activity patterns in V1

Their Images in the Visual Field

Limitations of Ermentrout-Cowan model It cannot account for all of the basic hallucinations - those that consist of oriented edges such as honeycombs and cobwebs. •

The model ignores the fact that under normal conditions the visual cortex analyzes an image by breaking it up into various local features such as edges and contours.

Feature maps V1 cells are selective for various stimulus features such as orientation and spatial frequency. Cells tend to be grouped into columns with similar functional properties (Hubel and Wiesel (1962).

• The distribution of orientation preferences is roughly

periodic: about every 1.33 mm (in humans) there is an isoorientation patch of a given preference.

Pseudo-colored optical image of orientation map in Macaque V1 (Blasdel 1992).

There are at least two length-scales in the visual cortex: (a) local: cells less than about 1.33 mm apart tend to make connections with most neighboring cells in a roughly isotropic fashion. (b) long range: horizontal connections link cells signaling similar orientation preferences in different patches. Such connections are directional.

Connections made by an inhibitory interneuron in V1 (Eysel 1999)

Directional horizontal connections in a tree shrew. (Bosking et al. 1995)

The overall connectivity of the visual cortex

(Bressloff et.al. 2001)

• This visual cortex model exhibits a very interesting symmetry. Any shift in position from one hypercolumn to another, or any rotation of the direction of the lateral connections leaves it unchanged, provided the orientation label is also changed. We call this a shift-twist symmetry. Thus the model is invariant with respect to translations and shift-twist rotations. • The geometric operations of translation and shift-twist rotation provide a new way to generate the symmetry group of rigid body motions in twodimensions --the Euclidean group in the plane E(2). • By virtue of its pattern of connectivity, the visual cortex (and the equations which represent its dynamics) are invariant under the action of E(2).

∂a(r, φ,t) = a(r, φ,t) + µ ∂t

Z

0

πZ

0 0 dr dφ w(r, φ|r0, φ0)σ[a(r0, φ0,t)] + I(r, φ,t) 2 π R

where the pattern of connections is given w(r, φ|r0, φ0) = wloc(φ

φ0)δ(r

r0) + β wlat (r

r0, φ)δ(φ

φ0)

w is invariant under the action of the Euclidean group E(2) with a shift-twist symmetry:

Any shift in position from one hypercolumn to another, or any rotation of the direction of lateral connection leaves it unchanged, provided the orientation label is also changed.

• Every symmetry group can be decomposed into subgroups, each of which has less symmetry. Thus E(2) can be decomposed into the subgroup T(2) of simple translations (lateral movements) in the plane, together with a subgroup of various rotations and label permutations SO(2).

• In particular E(2) can be decomposed into a set of axial subgroups. An axial subgroup is one which leaves only one pattern or planform invariant.

• We can now utilize the equivariant

branching lemma (Golubitsky,

Stewart, & Schaeffer 1988)

• This says that when the homogeneous (no pattern) state of a dynamical system with symmetry becomes unstable, the new patterns which form have restricted symmetries corresponding to the axial subgroups of the system’s symmetry group.

• The new patterns are said to have broken symmetry, and the whole process is known as spontaneous symmetry breaking. It plays a key role in many natural process in which new patterns replace old ones. So in addition to animal coat markings and hallucinatory images a current example is the Higgs mechanism whereby elementary particles with mass are created, and another example is the mechanism underlying the Big Bang and the creation of the cosmos. All these examples make use of the Turing mechanism.

• Patterns that the visual cortex can generate when it

goes into an altered state.

Their corresponding entoptic images

(a)

Superimposed maps (a) white: ocular dominance contours separating regions driven by left or right eyes, (b) black: iso-orientation preference contours, (c) colored patches: spatial frequency tuning-red (high), yellow, blue (intermediate), purple (low).

(Issa et.al. 2000)

Spherical representation of orientation and spatial frequency preferences

Towards hallucinations of color

• Other hallucinatory images that don’t

correspond directly to axial planforms

Mean-field vs Fluctuation-driven pattern formation

Thus mean-field pattern formation, which is what we have been considering, is not robust (regions II and V) compared with quasi-(fluctuation-dependent) pattern formation (region IV), in a network in which long-range patchy connections to inhibitory Basket cells are just as numerous as long-range patchy connections to Pyramidal cells. However if in the visual cortex, such long -range connections to Basket cells are sparse, then the opposite is true. This implies that the most robust pattern formation would be mean-field. The best anatomical evidence to date is consistent with this conclusion. If this were not the case most people would see geometric visual hallucinations most of the time, and normal vision would be not be possible.

Conclusion The images seen in various hallucinatory episodes correspond exactly to those patterns that can be generated in the visual cortex when it becomes unstable. Since all humans have essentially the same visual cortex architecture, it follows that such images are universal archetypes, since one is, in effect, seeing one’s own visual cortex architecture.

Publications G.B. Ermentrout & J.D. Cowan: A Mathematical Theory of Visual Hallucination Patterns, Biological Cybernetics, 34, 137-150, (1979) P. Bressloff, J.D. Cowan, M. Golubitsky, P Thomas, & M. Wiener: Geometric Hallucinations, Euclidean Symmetry, and the Functional Architecture of Striate Cortex, Phil.Trans.Roy. Soc.(Lond.) B, 356, 299-330, (2001) P. Bressloff & J.D. Cowan: A spherical model for orientation and spatial frequency tuning in a cortical hypercolumn, Phil.Trans.Roy. Soc.(Lond.) B, 367, 1643-1667 (2002) P. Bressloff & J.D. Cowan: The visual cortex as a Crystal, Physica D, 173, 226-258 (2003) T.I. Baker & J.D. Cowan: Spontaneous pattern formation and pinning in Primary Visual Cortex, J. Physiology (Paris), 103, 52-68 (2009) T.C. Butler, M. Benayoun, E. Wallace, W. van Drongelen, N. Goldenfeld & J.D. Cowan, Evolutionary constraints on visual cortex architecture from the dynamics of hallucinations: Proc. U.S. Nat. Acad. Sci., 109, 2, 606-609, (2012)