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 ( b ) Honeycomb image generated by pressure on the eyeballs ( Tyler 1978 ) 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 Cocaine induced imagery ( Siegel 1977 ) ( Patterson 1992 )
( 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 ( b ) Funnel image in V1 coordinates ( c ) Spiral ( 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 = � bu + r 2 u + f ( u, v ) ˙ v = � v + D r 2 v + g ( u, v ) ˙ Turing ( 1952 ) Z w ( | r − r 0 | ) σ [ u ( r 0 )] dr 0 + I u = − bu + ˙ 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 fs equency. 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 iso - orientation 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 di ff erent 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. W e 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 two - dimensions -- 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 ) .
Z π ∂ a ( r , φ , t ) R 2 w ( r , φ | r 0 , φ 0 ) σ [ a ( r 0 , φ 0 , t )] d r 0 d φ 0 Z = � a ( r , φ , t )+ µ + I ( r , φ , t ) ∂ t π 0 where the pattern of connections is given w ( r , φ | r 0 , φ 0 ) = w loc ( φ � φ 0 ) δ ( r � r 0 )+ β w lat ( r � r 0 , φ ) δ ( φ � φ 0 ) 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.
• W e 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 e ff ect, 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 . Bresslo ff , 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 . Bresslo ff & J.D. Cowan: A spherical model for orientation and spatial frequency tuning in a cortical hypercolumn, Phil.T rans.Roy. Soc. ( Lond. ) B, 367, 1643 - 1667 ( 2002 ) P . Bresslo ff & 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 )
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