Symmetry and Visual Hallucinations

Transcription

1 p. 1/32 Symmetry and Visual Hallucinations Martin Golubitsky Mathematical Biosciences Institute Ohio State University Paul Bressloff Utah Peter Thomas Case Western Lie June Shiau Houston Clear Lake Jack Cowan Chicago Matthew Wiener Neuropsychology, NIH Andrew Török Houston Klüver: We wish to stress... one point, namely, that under diverse conditions the visual system responds in terms of a limited number of form constants.

2 p. 2/32 Planar Symmetry-Breaking Euclidean symmetry: translations, rotations, reflections Symmetry-breaking from translation invariant state in planar systems with Euclidean symmetry leads to Stripes: States invariant under translation in one direction Spots: States centered at lattice points

3 Sand Dunes in Namibian Desert p. 3/32

4 Mud Plains p. 4/32

5 p. 5/32 Outline 1. Geometric Visual Hallucinations 2. Structure of Visual Cortex Hubel and Wiesel hypercolumns; local and lateral connections; isotropy versus anisotropy 3. Pattern Formation in V1 Symmetry; Three models 4. Interpretation of Patterns in Retinal Coordinates

6 p. 6/32 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

7 p. 6/32 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 (Klüver, 1928) tunnels and funnels spirals lattices includes honeycombs and phosphenes cobwebs

8 Funnels and Spirals p. 7/32

9 Lattices: Honeycombs & Phosphenes p. 8/32

10 Cobwebs p. 9/32

11 p. 10/32 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, 1974 Each millimeter there is a hypercolumn consisting of orientation sensitive cells in every direction preference

12 p. 11/32 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 cell s preference

13 p. 12/32 Anisotropy in Lateral Coupling Macaque: most anisotropy due to stretching in direction orthogonal to ocular dominance columns. Anisotropy is weak.

14 p. 12/32 Anisotropy in Lateral Coupling Macaque: most anisotropy due to stretching in direction orthogonal to ocular dominance columns. Anisotropy is weak. Tree shrew: anisotropy pronounced

15 p. 13/32 Action of Euclidean Group: Anisotropy local connections lateral connections hypercolumn

16 p. 13/32 Action of Euclidean Group: Anisotropy local connections Abstract physical space of V1 is R 2 S 1 not R 2 Hypercolumn becomes circle of orientations lateral connections Euclidean group on R 2 : translations, rotations, reflections hypercolumn Euclidean groups acts on R 2 S 1 by T y (x,ϕ) = (T y x,ϕ) R θ (x,ϕ) = (R θ x,ϕ + θ) κ(x, ϕ) = (κx, ϕ)

17 p. 14/32 Isotropic Lateral Connections local connections lateral connections hypercolumn

18 p. 14/32 Isotropic Lateral Connections New O(2) symmetry ˆφ(x,ϕ) = (x,ϕ + ˆφ) local connections hypercolumn lateral connections Weak anisotropy is forced symmetry breaking of E(2) +O(2) E(2)

19 p. 15/32 Three Models E(2) acting on R 2 (Ermentrout-Cowan) neurons located at each point x Activity variable: a(x) = voltage potential of neuron Pattern given by threshold a(x) > v 0

20 p. 15/32 Three Models E(2) acting on R 2 (Ermentrout-Cowan) neurons located at each point x Activity variable: a(x) = voltage potential of neuron Pattern given by threshold a(x) > v 0 Shift-twist action of E(2) on R 2 S 1 (Bressloff-Cowan) hypercolumns located at x; neurons tuned to ϕ strongly anisotropic lateral connections Activity variable: a(x, ϕ) Pattern given by winner-take-all

21 p. 15/32 Three Models E(2) acting on R 2 (Ermentrout-Cowan) neurons located at each point x Activity variable: a(x) = voltage potential of neuron Pattern given by threshold a(x) > v 0 Shift-twist action of E(2) on R 2 S 1 (Bressloff-Cowan) hypercolumns located at x; neurons tuned to ϕ strongly anisotropic lateral connections Activity variable: a(x, ϕ) Pattern given by winner-take-all Symmetry breaking: E(2) +O(2) E(2) weakly anisotropic lateral coupling Activity variable: a(x, ϕ) Pattern given by winner-take-all

22 p. 16/32 Planforms For Ermentrout-Cowan Threshold Patterns Square lattice: stripes and squares Hexagonal lattice: stripes and hexagons

23 p. 17/32 Winner-Take-All Strategy Creation of Line Fields Given: Activity a(x,ϕ) of neuron in hypercolumn at x sensitive to direction ϕ Assumption: Most active neuron in hypercolumn suppresses other neurons in hypercolumn Consequence: For all x find direction ϕ x where activity is maximum Planform: Line segment at each x oriented at angle ϕ x

24 p. 18/32 Planforms For Bressloff-Cowan O(2) Z 2 Erolls O(2) Z 4 Orolls y 0 y x x D 4 Esquares D 4 Osquares y 0 y x x

25 p. 19/32 Cortex to Retina Neurons on cortex are uniformly distributed Neurons in retina fall off by 1/r 2 from fovea Unique angle preserving map takes uniform density square to 1/r 2 density disk: complex exponential Straight lines on cortex circles, logarithmic spirals, and rays in retina

26 p. 20/32 Visual Hallucinations (I) (II) (III) (IV) (I) funnel and (II) spiral images LSD [Siegel & Jarvik, 1975], (III) honeycomb marihuana [Clottes & Lewis-Williams (1998)], (IV) cobweb petroglyph [Patterson, 1992]

27 p. 21/32 Planforms in the Visual Field (a) (b) (c) (d) Visual field planforms

28 p. 22/32 Weakly Anisotropic Coupling In addition to equilibria found in Bressloff-Cowan model there exist periodic solutions that emanate from steady-state bifurcation 1. Rotating Spirals 2. Tunneling Blobs Tunneling Spiraling Blobs 3. Pulsating Blobs

29 p. 23/32 Pattern Formation Outline 1. Bifurcation Theory with Symmetry Equivariant Branching Lemma Model independent analysis 2. Translations lead to plane waves 3. Planforms: Computation of eigenfunctions

30 p. 24/32 Primer on Steady-State Bifurcation Solve ẋ = f(x,λ) = 0 where f : R n R R n Local theory: Assume f(0, 0) = 0 & find solns near (0, 0) If L = (d x f) 0,0 nonsingular, IFT implies unique soln x(λ)

31 p. 24/32 Primer on Steady-State Bifurcation Solve ẋ = f(x,λ) = 0 where f : R n R R n Local theory: Assume f(0, 0) = 0 & find solns near (0, 0) If L = (d x f) 0,0 nonsingular, IFT implies unique soln x(λ) Bifurcation of steady states kerl {0} Reduction theory implies that steady-states are found by solving ϕ(y,λ) = 0 where ϕ : kerl R ker L

32 p. 25/32 Equivariant Steady-State Bifurcation Let γ : R n R n be linear γ is a symmetry iff γ(soln)=soln iff f(γx,λ) = γf(x,λ) Chain rule = Lγ = γl = kerl is γ-invariant Theorem: Fix symmetry group Γ. Generically ker L is an absolutely irreducible representation of Γ Reduction implies that there is a unique steady-state bifurcation theory for each absolutely irreducible rep

33 p. 26/32 Equivariant Bifurcation Theory Let Σ Γ be a subgroup Fix(Σ) = {x kerl : σx = x σ Σ} Σ is axial if dim Fix(Σ) = 1 Equivariant Branching Lemma: Generically, there exists a branch of solutions with Σ symmetry for every axial subgroup Σ MODEL INDEPENDENT Solution types do not depend on the equation only on the symmetry group and its representation on kerl

34 p. 27/32 Translations Let W k = {u(ϕ)e ik x + c.c.} k R 2 = wave vector Translations act on W k by T y (u(ϕ)e ik x ) = u(ϕ)e ik (x+y) = [ ] e ik y u(ϕ) e ik x L : W k W k Eigenfunctions of L have plane wave factors

35 p. 28/32 Reflections Choose REFLECTION ρ so that ρk = k ( ρ u(ϕ)e ik x) = ρ(u(ϕ))e ik x So ρ : W k W k

36 p. 28/32 Reflections Choose REFLECTION ρ so that ρk = k ( ρ u(ϕ)e ik x) = ρ(u(ϕ))e ik x So ρ : W k W k ρ 2 = 1 implies W k = W + k W k where ρ acts as +1 on W + k and 1 on W k

37 p. 28/32 Reflections Choose REFLECTION ρ so that ρk = k ( ρ u(ϕ)e ik x) = ρ(u(ϕ))e ik x So ρ : W k W k ρ 2 = 1 implies W k = W + k W k where ρ acts as +1 on W + k and 1 on W k Eigenfunctions are even or odd. When k = (1, 0) u( ϕ) = u(ϕ) u W + k u( ϕ) = u(ϕ) u W k

38 p. 29/32 Rotations Rotations act on spaces W k R θ ( u(ϕ)e ik x) = R θ (u(ϕ))e ir θ(k) x Therefore R θ (W k ) = W Rθ (k) Therefore ker L is -dimensional

39 p. 29/32 Rotations Rotations act on spaces W k R θ ( u(ϕ)e ik x) = R θ (u(ϕ))e ir θ(k) x Therefore Therefore R θ (W k ) = W Rθ (k) ker L is -dimensional Double-periodicity: Look for solutions on planar lattice F L = {f F : f(x + l) = f(x) l L}

40 p. 29/32 Rotations Rotations act on spaces W k R θ ( u(ϕ)e ik x) = R θ (u(ϕ))e ir θ(k) x Therefore Therefore R θ (W k ) = W Rθ (k) ker L is -dimensional Double-periodicity: Look for solutions on planar lattice F L = {f F : f(x + l) = f(x) l L} Finite number of rotations: ker L is finite-dimensional

41 p. 29/32 Rotations Rotations act on spaces W k R θ ( u(ϕ)e ik x) = R θ (u(ϕ))e ir θ(k) x Therefore Therefore R θ (W k ) = W Rθ (k) ker L is -dimensional Double-periodicity: Look for solutions on planar lattice F L = {f F : f(x + l) = f(x) l L} Finite number of rotations: ker L is finite-dimensional Choose lattice size so shortest dual vectors are critical

42 p. 30/32 Axials in Ermentrout-Cowan Model Name Planform Eigenfunction stripes cos x squares cosx + cos y hexagons cos(k 0 x) + cos(k 1 x) + cos(k 2 x) k 0 = (1, 0) k 1 = 2 1( 1, 3) k 2 = 1 2 ( 1, 3)

43 p. 31/32 Axials in Bressloff-Cowan Model Name Planform Eigenfunction u squares u(ϕ)cos x + u ( ϕ π 2) cos y even stripes u(ϕ) cos x even hexagons 2 j=0 u(ϕ jπ/3) cos(k j x) even square u(ϕ)cos x u ( ϕ π 2) cos y odd stripes u(ϕ) cos x odd hexagons 2 j=0 u(ϕ jπ/3) cos(k j x) odd triangles 2 j=0 u(ϕ jπ/3) sin(k j x) odd rectangles u ( ϕ π 3) cos(k1 x) u ( ϕ + π 3) cos(k2 x) odd

44 p. 32/32 How to Find Amplitude Function u(ϕ) Isotropic connections imply EXTRA O(2) symmetry O(2) decomposes W k into sum of irreducible subspaces W k,p = {ze pϕi e ik x + c.c. : z C} = R 2 Eigenfunctions lie in W k,p for some p W + k,p = {cos(pϕ)eik x } W k,p = {sin(pϕ)eik x } even case odd case With weak anisotropy u(ϕ) cos(pϕ) or u(ϕ) sin(pϕ)

45 p. 33/32 Rotating waves Suppose Fix(Σ) is two-dimensional Suppose N Γ (Σ) = Σ SO(2) Then generically solutions are rotating waves of a pattern with Σ symmetry Leads to rotating spirals and tunnels Suppose N Γ (Σ) = Σ D 4 Leads to pulsating solutions