8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
1/50
Symmetry and Visual Hallucinations
Martin GolubitskyMathematical Biosciences Institute
Ohio State University
Paul Bressloff Jack CowanUtah Chicago
Peter Thomas Matthew Wiener
Case Western Neuropsychology, NIHLie June Shiau Andrew Trk
Houston Clear Lake Houston
Klver: We wish to stress . . . one point, namely, that under
diverse conditions the visual system responds in terms of a
limited number of form constants.
. 1/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
2/50
Why Study Patterns I
1. Science behind patterns
. 2/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
3/50
Columnar Joints on Staffa near Mull
. 3/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
4/50
Columns along Snake River
. 4/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
5/50
Giants Causeway - Irish Coast
. 5/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
6/50
Experiment on Corn Starch
Goehring and Morris, 2005
. 6/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
7/50
Why Study Patterns II
1. Science behind patterns
2. Change in patterns provide tests for models
. 7/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
8/50
A Brief History of Navier-Stokes
Navier-Stokes equations for an incompressible fluid
ut = 2u (u )u 1p0 =
u
u = velocity vector = mass densityp = pressure = kinematic viscosity
Navier (1821); Stokes (1856); Taylor (1923)
. 8/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
9/50
The Couette Taylor Experiment
o i
i = speed of innercylinder
o = speed of outercylinder
Andereck, Liu, and Swinney (1986)
Couette flow Taylor vortices
. 9/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
10/50
.I. Taylor: Theory & Experiment (1923
. 10/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
11/50
Why Study Patterns III
1. Science behind patterns2. Change in patterns provide tests for models
3. Model independence
Mathematics provides menu of patterns
. 11/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
12/50
Planar Symmetry-Breaking
Euclidean symmetry: translations, rotations, reflections
Symmetry-breaking from translation invariant state inplanar systems with Euclidean symmetry leads to
Stripes:States invariant under translation in one direction
Spots:
States centered at lattice points
. 12/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
13/50
Sand Dunes in Namibian Desert
. 13/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
14/50
Zebra Stripes
. 14/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
15/50
Mud Plains
. 15/
L d S
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
16/50
Leopard Spots
. 16/
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
17/50
Outline
1. Geometric Visual Hallucinations
2. Structure of Visual Cortex
Hubel and Wiesel hypercolumns; local and lateralconnections; isotropy versus anisotropy
3. Pattern Formation in V1Symmetry; Three models
4. Interpretation of Patterns in Retinal Coordinates
. 17/
Vi l H ll i i
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
18/50
Visual Hallucinations
Drug uniformly forces activation of cortical cells
Leads to spontaneous pattern formation on cortex
Map from V1 to retina;translates pattern on cortex to visual image
. 18/
Vi l H ll i ti
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
19/50
Visual Hallucinations
Drug uniformly forces activation of cortical cells
Leads to spontaneous pattern formation on cortex
Map from V1 to retina;translates pattern on cortex to visual image
Patterns fall into four form constants(Klver, 1928)
tunnels and funnels
spirals
lattices includes honeycombs and phosphenes cobwebs
. 18/
F l d S i l
http://scriptfunnels/http://scriptspirals/http://scripthoneycombs/http://scriptphosphenes/http://scriptcobwebs/http://scriptcobwebs/http://scriptphosphenes/http://scripthoneycombs/http://scriptspirals/http://scriptfunnels/8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
20/50
Funnels and Spirals
. 19/
Lattices: Hone combs & Phosphenes
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
21/50
Lattices: Honeycombs & Phosphenes
. 20/
C b b
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
22/50
Cobwebs
. 21/
Orientation Sensitivity of Cells in V1
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
23/50
Orientation Sensitivity of Cells in V1
Most V1 cells sensitive to orientation of contrast edge
Distribution of orientation preferences in Macaque V1 (Blasdel)
Hubel and Wiesel, 1974Each millimeter there is a hypercolumnconsisting of
orientation sensitive cells in every direction preference
. 22/
Structure of Primary Visual Cortex (V1)
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
24/50
Structure of Primary Visual Cortex (V1)
Optical imaging exhibits pattern of connection
V1 lateral connections: Macaque (left, Blasdel) and Tree Shrew (right, Fitzpatrick)
Two kinds of coupling: local and lateral
(a) local: cells < 1mm connect with most neighbors
(b) lateral: cells make contact each mm along axons;
connections in direction of cells preference
. 23/
Anisotropy in Lateral Coupling
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
25/50
Anisotropy in Lateral Coupling
Macaque: most anisotropydue to stretching indirection orthogonal toocular dominance
columns. Anisotropy isweak.
. 24/
Anisotropy in Lateral Coupling
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
26/50
Anisotropy in Lateral Coupling
Macaque: most anisotropydue to stretching indirection orthogonal toocular dominance
columns. Anisotropy isweak.
Tree shrew: anisotropypronounced
. 24/
Action of Euclidean Group: Anisotropy
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
27/50
Action of Euclidean Group: Anisotropy
hypercolumn
lateral connections
local connections
. 25/
Action of Euclidean Group: Anisotropy
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
28/50
Action of Euclidean Group: Anisotropy
hypercolumn
lateral connections
local connections
Abstract physical space ofV1 is R2 S1 not R2Hypercolumn becomes
circle of orientations
Euclidean group on R2:translations, rotations,
reflections
Euclidean groups acts
on R2
S1
byTy(x, ) = (Tyx, )
R(x, ) = (Rx, + )
(x, ) = (x,) . 25/
Isotropic Lateral Connections
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
29/50
Isotropic Lateral Connections
hypercolumn
lateral connections
local connections
. 26/
Isotropic Lateral Connections
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
30/50
Isotropic Lateral Connections
hypercolumn
lateral connections
local connections
New O(2) symmetry
(x, ) = (x, + )
Weak anisotropy isforced symmetrybreaking of
E(2)+O(2) E(2)
. 26/
Three Models
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
31/50
Three Models
E(2) acting on R2
(Ermentrout-Cowan)neurons located at each point xActivity variable: a(x) = voltage potential of neuronPattern given by threshold a(x) > v0
. 27/
Three Models
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
32/50
Three Models
E(2) acting on R2
(Ermentrout-Cowan)neurons located at each point xActivity variable: a(x) = voltage potential of neuronPattern given by threshold a(x) > v0
Shift-twist action of E(2) on R2 S1 (Bressloff-Cowan)hypercolumns located at x; neurons tuned to
strongly anisotropic lateral connectionsActivity variable: a(x, )Pattern given by winner-take-all
. 27/
Three Models
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
33/50
Three Models
E(2) acting on R2
(Ermentrout-Cowan)neurons located at each point xActivity variable: a(x) = voltage potential of neuronPattern given by threshold a(x) > v0
Shift-twist action of E(2) on R2 S1 (Bressloff-Cowan)hypercolumns located at x; neurons tuned to
strongly anisotropic lateral connectionsActivity variable: a(x, )Pattern given by winner-take-all
Symmetry breaking: E(2)+O(2) E(2)weakly anisotropic lateral couplingActivity variable: a(x, )
Pattern given by winner-take-all
. 27/
Planforms For Ermentrout-Cowan
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
34/50
Planforms For Ermentrout Cowan
Threshold Patterns
3 2 1 0 1 2 33
2
1
0
1
2
3
3 2 1 0 1 2 33
2
1
0
1
2
3
Square lattice: stripes and squares
3 2 1 0 1 2 33
2
1
0
1
2
3
2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.52
1.5
1
0.5
0
0.5
1
1.5
2
Hexagonal lattice: stripes and hexagons
. 28/
Winner-Take-All Strategy
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
35/50
a S a gy
Creation of Line Fields
Given: Activity a(x, ) of neuron in hypercolumn at x
sensitive to direction
Assumption: Most active neuron in hypercolumnsuppresses other neurons in hypercolumn
Consequence: For all x find direction x where activityis maximum
Planform: Line segment at each x oriented at angle x
. 29/
Planforms For Bressloff-Cowan
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
36/50
Planforms For Bressloff Cowan
1.5 1 0.5 0 0.5 1 1.5
1
0.5
0
0.5
1
x
y
O(2) Z2
Erolls
1.5 1 0.5 0 0.5 1 1.5
1
0.5
0
0.5
1
x
y
O(2) Z4
Orolls
1 0.5 0 0.5 1
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
x
y
D4
Esquares
1 0.5 0 0.5 1
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
x
y
D4
Osquares
. 30/
Cortex to Retina
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
37/50
Cortex to Retina
Neurons on cortex are uniformly distributed
Neurons in retina fall off by 1/r2 from fovea
Unique angle preserving map takes uniform density
square to 1/r2 density disk: complex exponential
Straight lines on cortex circles, logarithmic spirals, and rays in retina
. 31/
Visual Hallucinations
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
38/50
1
1 1
1 1 1 1 8
(I) funnel and (II) spiral images LSD [Siegel & Jarvik, 1975], (III) honeycomb marihuana
[Clottes & Lewis-Williams (1998)], (IV) cobweb petroglyph [Patterson, 1992]
. 32/
Planforms in the Visual Field
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
39/50
( a ) ( b )
( c )
( d )
V i s u a l f i e l d p l a n f o r m s
. 33/
Weakly Anisotropic Coupling
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
40/50
y p p g
In addition to equilibria found in Bressloff-Cowanmodel there exist periodic solutions that emanatefrom steady-state bifurcation
1. Rotating Spirals
2. Tunneling Blobs Tunneling Spiraling Blobs
3. Pulsating Blobs
. 34/
Pattern Formation Outline
http://scriptrotating/http://scripttunneling/http://scripttunnelingsp/http://scriptpulsating/http://scriptpulsating/http://scripttunnelingsp/http://scripttunneling/http://scriptrotating/8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
41/50
1. Bifurcation Theory with Symmetry
Equivariant Branching Lemma
Model independent analysis
2. Translations lead to plane waves
3. Planforms: Computation of eigenfunctions
. 35/
Primer on Steady-State Bifurcation
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
42/50
Solve x = f(x, ) = 0 where f : Rn R Rn
Local theory: Assume f(0, 0) = 0 & find solns near (0, 0)
If L = (dxf)0,0 nonsingular, IFT implies unique soln x()
. 36/
Primer on Steady-State Bifurcation
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
43/50
Solve x = f(x, ) = 0 where f : Rn R Rn
Local theory: Assume f(0, 0) = 0 & find solns near (0, 0)
If L = (dxf)0,0 nonsingular, IFT implies unique soln x()
Bifurcation of steady states ker L = {0}
Reduction theory implies that steady-states are foundby solving (y, ) = 0 where
: ker L R ker L
. 36/
Equivariant Steady-State Bifurcation
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
44/50
Let : Rn Rn be linear
is a symmetry iff (soln)=soln iff f(x,) = f(x, )
Chain rule = L = L = ker L is -invariant
Theorem: Fix symmetry group . Genericallyker L is an absolutely irreducible representation of
Reduction implies that there is a unique steady-statebifurcation theory for each absolutely irreducible rep
. 37/
Equivariant Bifurcation Theory
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
45/50
Let be a subgroup
Fix() =
{x
ker L : x = x
} is axial if dim Fix() = 1
Equivariant Branching Lemma:
Generically, there exists a branch of solutions with symmetryfor every axial subgroup
MODEL INDEPENDENT
Solution types do not depend on the equation onlyon the symmetry group and its representation on ker L
. 38/
Translations
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
46/50
Let Wk = {u()eikx + c.c.} k R2 = wave vector
Translations act on Wk by
Ty(u()eikx) = u()eik(x+y) =
eikyu()
eikx
L : Wk WkEigenfunctions of L have plane wavefactors
. 39/
Axials in Ermentrout-Cowan Model
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
47/50
Name Planform Eigenfunction
stripes cos xsquares cos x + cos y
hexagons cos(k0 x) + cos(k1 x) + cos(k2 x)
k0 = (1, 0) k1 =12(1,3) k2 = 12(1,3)
. 40/
Axials in Bressloff-Cowan Model
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
48/50
Name Planform Eigenfunction u
squares u()cos x+ u
2
cos y even
stripes u()cos x evenhexagons
2
j=0 u (j/3) cos(kj x) evensquare u()cos x
u
2 cos y odd
stripes u()cos x odd
hexagons
2
j=0 u (j/3) cos(kj x) odd
triangles2
j=0 u (j/3) sin(kj x) oddrectangles u
3
cos(k1 x) u
+
3
cos(k2 x) odd
. 41/
How to Find Amplitude Function u()
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
49/50
Isotropic connections imply EXTRA O(2) symmetry
O(2) decomposes Wk into sum of irreducible subspaces
Wk,p = {zepieikx + c.c. : z C} = R2
Eigenfunctions lie in Wk,p for some p
W+k,p = {cos(p)eikx} even case
Wk,p = {sin(p)eikx} odd case
With weak anisotropy
u() cos(p) or u() sin(p)
. 42/
Rotating waves
8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations
50/50
Suppose Fix() is two-dimensional
Suppose N() = SO(2)
Then generically solutions are rotating waves of apattern with symmetry
. 43/