martin golubitsky- symmetry and visual hallucinations

8/3/2019 Martin Golubitsky- Symmetry and Visual Hallucinations

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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.

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Why Study Patterns I

1. Science behind patterns

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Columnar Joints on Staffa near Mull

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Columns along Snake River

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Giants Causeway - Irish Coast

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Experiment on Corn Starch

Goehring and Morris, 2005

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Why Study Patterns II

1. Science behind patterns

2. Change in patterns provide tests for models

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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)

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The Couette Taylor Experiment

o i

i = speed of innercylinder

o = speed of outercylinder

Andereck, Liu, and Swinney (1986)

Couette flow Taylor vortices

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.I. Taylor: Theory & Experiment (1923

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Why Study Patterns III

1. Science behind patterns2. Change in patterns provide tests for models

3. Model independence

Mathematics provides menu of patterns

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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

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Sand Dunes in Namibian Desert

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Zebra Stripes

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Mud Plains

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L d S


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Leopard Spots

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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

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Vi l H ll i i


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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

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Vi l H ll i ti


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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

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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/


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Funnels and Spirals

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Lattices: Hone combs & Phosphenes


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Lattices: Honeycombs & Phosphenes

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C b b


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Cobwebs

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Orientation Sensitivity of Cells in V1


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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

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Structure of Primary Visual Cortex (V1)


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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

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Anisotropy in Lateral Coupling


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Macaque: most anisotropydue to stretching indirection orthogonal toocular dominance

columns. Anisotropy isweak.

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Macaque: most anisotropydue to stretching indirection orthogonal toocular dominance

columns. Anisotropy isweak.

Tree shrew: anisotropypronounced

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Action of Euclidean Group: Anisotropy


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hypercolumn

lateral connections

local connections

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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


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hypercolumn

lateral connections

local connections

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hypercolumn

lateral connections

local connections

New O(2) symmetry

(x, ) = (x, + )

Weak anisotropy isforced symmetrybreaking of

E(2)+O(2) E(2)

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Three Models


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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

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Three Models


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Three Models

E(2) acting on R2


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

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Three Models


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Three Models

E(2) acting on R2


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

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Planforms For Ermentrout-Cowan


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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

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Winner-Take-All Strategy


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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

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Planforms For Bressloff-Cowan


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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

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Cortex to Retina


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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

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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]

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Planforms in the Visual Field


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( a ) ( b )

( c )

( d )

V i s u a l f i e l d p l a n f o r m s

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Weakly Anisotropic Coupling


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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

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Pattern Formation Outline
http://scriptrotating/http://scripttunneling/http://scripttunnelingsp/http://scriptpulsating/http://scriptpulsating/http://scripttunnelingsp/http://scripttunneling/http://scriptrotating/


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1. Bifurcation Theory with Symmetry

Equivariant Branching Lemma

Model independent analysis

2. Translations lead to plane waves

3. Planforms: Computation of eigenfunctions

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Primer on Steady-State Bifurcation


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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()

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Primer on Steady-State Bifurcation


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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

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Equivariant Steady-State Bifurcation


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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

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Equivariant Bifurcation Theory


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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

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Translations


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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

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Axials in Ermentrout-Cowan Model


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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)

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Axials in Bressloff-Cowan Model


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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

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How to Find Amplitude Function u()


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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)

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Rotating waves


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Suppose Fix() is two-dimensional

Suppose N() = SO(2)

Then generically solutions are rotating waves of apattern with symmetry

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martin golubitsky- symmetry and visual hallucinations

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