Orientation maps on non-Euclidean spaces defined with group representations Alexandre Afgoustidis Institut de mathématiques de Jussieu - Paris Rive Gauche Euclidean symmetry in gaussian field models for V1 orientation maps I In this poster, we discuss a symmetry-based model for the layout of orientation preferences (in V1) in species with continuous orientation maps, and introduce generalizations to homogeneous non-Euclidean geometries. I Niebur and Worgötter, Wolf and Geisel, Barbieri-Citti-Sanguinetti-Sarti and many others : three ingredients are enough for getting realistic V1-like orientation maps (ask me for defintions !), From Bosking et al. (1997). V1 orientation map obtained by optical imaging in a tree shrew. Details in the upper right corner show pinwheels, points in the map where all orientations are represented. Gaussian random field on the plane + Euclidean invariance of the probability distribution + Dominance of a single wavelength in the correlation spectrum From Niebur and Worgötter (1994). Correlation spectrum in an experimental V1 map. From Schnabel’s thesis (2008). Correlation spectrum of a real tree shrew map. To simulate a monochromatic SE(2)-invariant field ΦΛ, I drew this map from a superposition of 30 plane waves with random propagation directions, gaussian random complex amplitudes and the same wavelength Λ. Pinwheel density and variance of hypercolumn size in invariant Gaussian fields The spectral thinness condition must correspond to having a well-defined hypercolumn size in the map. I Can we prove it ? Kaschube, Schnabel et al. (2010) : in many species, the correlation spectrum of experimental V1 maps does concentrate on a typical wavelength Λ, and Fig. below shows how, in SE(2)-invariant gaussian fields, the variance of typical spacing between iso-orientation domains varies with the spectral width (numerical evaluation from an exact formula). 2 Mean number of pinwheels in an area Λ = 3.14 ± 2% I In a centered gaussian field with SE(2)-invariant probability distribution, E (Nb. of points on a line segment of length ` with a given orientation preference) ∝ `. I Let I us define the quasiperiod Λth as the only number ` for which it is equal to one. For any invariant gaussian field on the Euclidean plane, with thin spectrum or not, E Number of of pinwheels in an area 2 Λth = π (for details, ask me !) Group representation theory : a symmetry-based toolbox Representation theory is a symmetry-based generalization of harmonic analysis to non-Euclidean spaces with a group action. I Weyl and Wigner proved that it can be used as an engineering tool in physics, to guess at models from little more than symmetry assumptions. I The set of maps which can be sampled from Φλ carries an irreducible unitary representation of SE(2). I I Yaglom : zero-mean Gaussian invariant random fields on a homogeneous space G/K correspond (via their correlation functions) to K -bi-invariant (deterministic) functions G → C. I Élie Cartan, Harish-Chandra, Helgason and others : for some pairs (G, K ), studying K -bi-invariant functions on G is the same as studying some unitary irreducible representations of G which can be realized as spaces of smooth functions on G/K . Using group representations to define orientation maps on non-Euclidean spaces I If G is a group and X a G-homogeneous riemannian space, we can look for gaussian fields defined on X whose probability distribution is G-invariant, and whose samples probe an irreducible representation of G contained in L2(X ). I We call these gaussian fields monochromatic ; when X is one of Élie Cartan’s riemannian symmetric spaces, they can be described explicitly. I There are generalizations to fields having shift-twist symmetry, but they are technically more difficult to study. Negative curvature Positive curvature When X is a riemannian symmetric space of the noncompact type, we can use Helgason waves to mimic the Euclidean constructions. Here the analogies with Euclidean space are strongest. When X is a compact symmetric space, the irreducible constituents of L2(X ) are finitedimensional and generated by (generalized) spherical harmonics ; there is but a countable set of monochromatic fields. Example with G = SL2(R) : Example with G = SO(3) : Pinwheel densities in non-Euclidean maps I I In riemannian symmetric spaces, let us first define the quasiperiod in an invariant gaussian field as the only real number Λ such that, on a geodesic segment of length Λ, the expectation for the number of points with a given orientation is equal to one. For invariant gaussian fields defined on symmetric spaces, Λ can be computed from an eigenvalue of the Laplace-Beltrami operator. Many thanks to : I My advisor Daniel Bennequin, for many suggestions and constant support ; I Alessandro Sarti, Michel Duflo, Alain Berthoz, for helpful discussion ; I Christine and Sophie Cachot, for help with computer software. We can then use recent generalizations of the Kac-Rice formula to evaluate the pinwheel density δ (mean number of zeroes in a region of invariant volume Λdim X ) of a monochromatic invariant field : δ = π dim(X )/2. Further reading (ask me for detailed references !) I Kaschube et al. (2010) Science Vol. 330 no. 6007, pp. 1113-1116 I F. Wolf and T. Geisel (2003) Journal of Physiology - Paris 97:253-264. I A. M. Yaglom (1961), 2d Order Homogeneous Random fields. I R. Adler, J. Taylor (2005). The geometry of Random Fields (Springer). I S. Helgason (1984) Groups and Geometric Analysis