# American Institute of Mathematical Sciences

March  2013, 8(1): 211-260. doi: 10.3934/nhm.2013.8.211

## A spatialized model of visual texture perception using the structure tensor formalism

 1 School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN 55455, United States 2 J-A Dieudonné Laboratory, CNRS and University of Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France

Received  January 2012 Revised  December 2012 Published  April 2013

The primary visual cortex (V1) can be partitioned into fundamental domains or hypercolumns consisting of one set of orientation columns arranged around a singularity or pinwheel'' in the orientation preference map. A recent study on the specific problem of visual textures perception suggested that textures may be represented at the population level in the cortex as a second-order tensor, the structure tensor, within a hypercolumn. In this paper, we present a mathematical analysis of such interacting hypercolumns that takes into account the functional geometry of local and lateral connections. The geometry of the hypercolumn is identified with that of the Poincaré disk $\mathbb{D}$. Using the symmetry properties of the connections, we investigate the spontaneous formation of cortical activity patterns. These states are characterized by tuned responses in the feature space, which are doubly-periodically distributed across the cortex.
Citation: Grégory Faye, Pascal Chossat. A spatialized model of visual texture perception using the structure tensor formalism. Networks & Heterogeneous Media, 2013, 8 (1) : 211-260. doi: 10.3934/nhm.2013.8.211
##### References:
 [1] R. Aurich and F. Steiner, Periodic-orbit sum rules for the hadamard-gutzwiller model,, Physica D, 39 (1989), 169. doi: 10.1016/0167-2789(89)90003-1. Google Scholar [2] T. I. Baker and J. D. Cowan, Spontaneous pattern formation and pinning in the primary visual cortex,, Journal of Physiology-Paris, 103 (2009), 52. doi: 10.1016/j.jphysparis.2009.05.011. Google Scholar [3] N. L. Balazs and A. Voros, Chaos on the pseudosphere,, Physics Reports, 143 (1986), 109. doi: 10.1016/0370-1573(86)90159-6. Google Scholar [4] R. Ben-Yishai, RL Bar-Or and H. Sompolinsky, Theory of orientation tuning in visual cortex,, Proceedings of the National Academy of Sciences, 92 (1995), 3844. doi: 10.1073/pnas.92.9.3844. Google Scholar [5] J. Bigun and G. Granlund, Optimal orientation detection of linear symmetry,, in, (1987), 433. Google Scholar [6] B. Blumenfeld, D. Bibitchkov and M. Tsodyks, Neural network model of the primary visual cortex: From functional architecture to lateral connectivity and back,, Journal of Computational Neuroscience, 20 (2006), 219. doi: 10.1007/s10827-006-6307-y. Google Scholar [7] I. Bosch Vivancos, P. Chossat and I. Melbourne, New planforms in systems of partial differential equations with Euclidean symmetry,, Archive for Rational Mechanics and Analysis, 131 (1995), 199. doi: 10.1007/BF00382886. Google Scholar [8] W. H. Bosking, Y. Zhang, B. Schofield and D. Fitzpatrick, Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex,, The Journal of Neuroscience, 17 (1997), 2112. Google Scholar [9] P. C. Bressloff and J. D.Cowan, The functional geometry of local and horizontal connections in a model of v1,, Journal of Physiology, 97 (2003), 221. doi: 10.1016/j.jphysparis.2003.09.017. Google Scholar [10] P. C. Bressloff and J. D. Cowan, A spherical model for orientation and spatial frequency tuning in a cortical hypercolumn,, Philosophical Transactions of the Royal Society B, (2003). doi: 10.1098/rstb.2002.1109. Google Scholar [11] P. C. Bressloff, Spatially periodic modulation of cortical patterns by long-range horizontal connections,, Physica D: Nonlinear Phenomena, 185 (2003), 131. doi: 10.1016/S0167-2789(03)00238-0. Google Scholar [12] P. C. Bressloff and J. D. Cowan, The visual cortex as a crystal,, Physica D: Nonlinear Phenomena, 173 (2002), 226. doi: 10.1016/S0167-2789(02)00677-2. Google Scholar [13] P. C. Bressloff, J. D. Cowan, M. Golubitsky and P. J. Thomas, Scalar and pseudoscalar bifurcations motivated by pattern formation on the visual cortex,, Nonlinearity, 14 (2001). doi: 10.1088/0951-7715/14/4/305. Google Scholar [14] P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas and M. C. Wiener, Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex,, Phil. Trans. R. Soc. Lond. B, 306 (2001), 299. doi: 10.1098/rstb.2000.0769. Google Scholar [15] P. C. Bressloff and A. M. Oster, Theory for the alignment of cortical feature maps during development,, Physical Review E, 82 (2010). doi: 10.1103/PhysRevE.82.021920. Google Scholar [16] I. Chavel, "Eigenvalues in Riemannian Geometry,", 115. Academic Press, 115 (1984). Google Scholar [17] S. Chemla and F. Chavane, Voltage-sensitive dye imaging: Technique review and models,, Journal of Physiology-Paris, 104 (2010), 40. doi: 10.1016/j.jphysparis.2009.11.009. Google Scholar [18] P. Chossat, G. Faye and O. Faugeras, Bifurcations of hyperbolic planforms,, Journal of Nonlinear Science, (2011). doi: 10.1007/s00332-010-9089-3. Google Scholar [19] P. Chossat and R. Lauterbach, "Methods in Equivariant Bifurcations and Dynamical Systems,", World Scientific Publishing Company, (2000). Google Scholar [20] P. Chossat and O. Faugeras, Hyperbolic planforms in relation to visual edges and textures perception,, Plos. Comput. Biol., 5 (2009). doi: 10.1371/journal.pcbi.1000625. Google Scholar [21] P. G. Ciarlet and J. L. Lions, "Handbook of Numerical Analysis,", II Finite Element Methods (part1). North-Holland, II (1991). Google Scholar [22] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space,, J. Math. Imaging Vis., (2006), 307. doi: 10.1007/s10851-005-3630-2. Google Scholar [23] D. P. Edwards, K. P. Purpura and E. Kaplan, Contrast sensitivity and spatial frequency response of primate cortical neurons in and around the cytochrome oxidase blobs,, Vision Research, 35 (1995), 1501. Google Scholar [24] I. Erdélyi, "Higher Transcendental Functions,", 1 Robert E. Krieger Publishing Company, 1 (1985). Google Scholar [25] G. B. Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns,, Biological Cybernetics, 34 (1979), 137. doi: 10.1007/BF00336965. Google Scholar [26] G. Faye and P. Chossat, Bifurcation diagrams and heteroclinic networks of octagonal h-planforms,, Journal of Nonlinear Science, 22 (2012), 277. doi: 10.1007/s00332-011-9118-x. Google Scholar [27] G. Faye, P. Chossat and O. Faugeras, Analysis of a hyperbolic geometric model for visual texture perception,, The Journal of Mathematical Neuroscience, 1 (2011). doi: 10.1186/2190-8567-1-4. Google Scholar [28] M. Golubitsky, L. J. Shiau and A. Török, Bifurcation on the visual cortex with weakly anisotropic lateral coupling,, SIAM Journal on Applied Dynamical Systems, 2 (2003), 97. doi: 10.1137/S1111111102409882. Google Scholar [29] M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory,", volume II. Springer, (1988). doi: 10.1007/978-1-4612-4574-2. Google Scholar [30] D. Hansel and H. Sompolinsky, Modeling feature selectivity in local cortical circuits,, Methods of Neuronal Modeling, (1997), 499. Google Scholar [31] M. Haragus and G. Iooss, "Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Systems,", EDP Sci. Springer Verlag UTX Series, (2010). doi: 10.1007/978-0-85729-112-7. Google Scholar [32] S. Helgason, "Groups and Geometric Analysis,", 83 of Mathematical Surveys and Monographs. American Mathematical Society, 83 (2000). Google Scholar [33] R. B. Hoyle, "Pattern Formation: an Introduction to Methods,", Cambridge Univ Pr, (2006). doi: 10.1017/CBO9780511616051. Google Scholar [34] D. H. Hubel and T. N. Wiesel, Receptive fields and functional architecture in two nonstriate visual areas (18 and 19) of the cat,, Journal of Neurophysiology, 28 (1965), 229. Google Scholar [35] D. H. Hubel and T. N. Wiesel, Receptive fields and functional architecture of monkey striate cortex,, The Journal of Physiology, 195 (1968). Google Scholar [36] D. H. Hubel and T. N. Wiesel, Functional architecture of macaque monkey,, Proceedings of the Royal Society, (1977), 1. Google Scholar [37] H. Iwaniec, "Spectral Methods of Automorphic Forms,", 53 of AMS Graduate Series in Mathematics, 53 (2002). Google Scholar [38] M. Kaschube, M. Schnabel, S. Löwel, D. M. Coppola, L. E. White and F. Wolf, Universality in the evolution of orientation columns in the visual cortex,, Science, 330 (2010). doi: 10.1126/science.1194869. Google Scholar [39] M. Kaschube, M. Schnabel and F. Wolf, Self-organization and the selection of pinwheel density in visual cortical development,, New Journal of Physics, 10 (2008). doi: 10.1088/1367-2630/10/1/015009. Google Scholar [40] S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics. The University of Chicago Press, (1992). Google Scholar [41] H. Kluver, "Mescal, and Mechanisms of Hallucinations,", University of Chicago Press Chicago, (1966). Google Scholar [42] H. Knutsson, Representing local structure using tensors,, Scandinavian Conference on Image Analysis, (1989), 244. doi: 10.1007/978-3-642-21227-7_51. Google Scholar [43] N. N. Lebedev, "Special Functions and Their Applications,", (edited by R. A. Silverman), (1972). Google Scholar [44] P. S. Leon, I. Vanzetta, G. S. Masson and L. U. Perrinet, Motion Clouds: Model-based stimulus synthesis of natural-like random textures for the study of motion perception,, Journal of Neurophysiology, 107 (2012), 3217. doi: 10.1152/jn.00737.2011. Google Scholar [45] M. S. Livingstone and D. H. Hubel, Anatomy and physiology of a color system in the primate visual cortex,, Journal of Neuroscience, 4 (1984), 309. Google Scholar [46] J. S. Lund, A. Angelucci and P. C. Bressloff, Anatomical substrates for functional columns in macaque monkey primary visual cortex,, Cerebral Cortex, 12 (2003), 15. doi: 10.1093/cercor/13.1.15. Google Scholar [47] I. Melbourne, A singularity theory analysis of bifurcation problems with octahedral symmetry,, Dynamics and Stability of Systems, 1 (1986). doi: 10.1080/02681118608806020. Google Scholar [48] W. Miller, "Symmetry Groups and Their Applications,", Academic Press, (1972). Google Scholar [49] M. Moakher, A differential geometric approach to the geometric mean of symmetric positie-definite matrices,, SIAM J. Matrix Anal. Appl., 26 (2005), 735. doi: 10.1137/S0895479803436937. Google Scholar [50] J. D. Murray, "Mathematical Biology II, Spatial Models and Biomedical Applications,", Springer, (2003). Google Scholar [51] G. A. Orban, H. Kennedy and J. Bullier, Velocity sensitivity and direction selectivity of neurons in areas V1 and V2 of the monkey: Influence of eccentricity,, Journal of Neurophysiology, 56 (1986), 462. Google Scholar [52] G. Oster, Phosphenes,, Scientific American, 222 (1970). doi: 10.1038/scientificamerican0270-82. Google Scholar [53] A. M. Oster and P. C. Bressloff, A developmental model of ocular dominance column formation on a growing cortex,, Bulletin of Mathematical Biology, (2006). doi: 10.1007/s11538-005-9055-7. Google Scholar [54] J. Petitot, The neurogeometry of pinwheels as a sub-Riemannian contact structure,, Journal of Physiology-Paris, 97 (2003), 265. Google Scholar [55] J. Petitot, "Neurogéométrie de la Vision,", Les Éditions de l'École Polytechnique, (2009). Google Scholar [56] G. Sanguinetti, A. Sarti and G. Citti, Implementation of a Model for Perceptual Completion in $\R^2\times S^1$,, Computer Vision and Computer Graphics, 24 (2009), 188. Google Scholar [57] A. Sarti and G. Citti, Non-commutative field theory in the visual cortex,, Computer Vision: from Surfaces to 3D Objects, (2010). Google Scholar [58] A. Sarti, G. Citti and J. Petitot, The symplectic structure of the primary visual cortex,, Biological Cybernetics, 98 (2008), 33. doi: 10.1007/s00422-007-0194-9. Google Scholar [59] J. Schummers, J. Mariño and M. Sur, Synaptic integration by v1 neurons depends on location within the orientation map,, Neuron, 36 (2002), 969. doi: 10.1016/S0896-6273(02)01012-7. Google Scholar [60] J. P. Serre, "Représentations Linéaires des Groupes Finis,", Hermann, (1978). Google Scholar [61] L. C. Sincich and J. C. Horton, Divided by cytochrome oxidase: A map of the projections from V1 to V2 in macaques,, Science, 295 (2002), 1734. Google Scholar [62] A. Terras, "Harmonic Analysis on Symmetric Spaces and Applications,", Springer-Verlag, 2 (1988). doi: 10.1007/978-1-4612-3820-1. Google Scholar [63] R. B. H. Tootell, S. L. Hamilton, M. S. Silverman, E. Switkes and R. L. De Valois, Functional anatomy of macaque striate cortex. V. Spatial Frequency,, Journal of Neuroscience, 8 (1988), 1610. Google Scholar [64] R. Veltz and O. Faugeras, Local/global analysis of the stationary solutions of some neural field equations,, SIAM Journal on Applied Dynamical Systems, 9 (2010), 954. doi: 10.1137/090773611. Google Scholar [65] R. Veltz and O. Faugeras, Illusions in the ring model of visual orientation selectivity,, Technical Report, (2010). doi: 10.1137/090773611. Google Scholar [66] G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1995). Google Scholar [67] H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons,, Biophys. Journal, 12 (1972), 1. doi: 10.1016/S0006-3495(72)86068-5. Google Scholar [68] H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue,, Biological Cybernetics, 13 (1973), 55. doi: 10.1007/BF00288786. Google Scholar [69] F. Wolf and T. Geisel, Spontaneous pinwheel annihilation during visual development,, Nature, 395 (1998), 73. Google Scholar [70] A. Zettl, "Sturm-Liouville Theory,", 121, 121 (2005). Google Scholar

show all references

##### References:
 [1] R. Aurich and F. Steiner, Periodic-orbit sum rules for the hadamard-gutzwiller model,, Physica D, 39 (1989), 169. doi: 10.1016/0167-2789(89)90003-1. Google Scholar [2] T. I. Baker and J. D. Cowan, Spontaneous pattern formation and pinning in the primary visual cortex,, Journal of Physiology-Paris, 103 (2009), 52. doi: 10.1016/j.jphysparis.2009.05.011. Google Scholar [3] N. L. Balazs and A. Voros, Chaos on the pseudosphere,, Physics Reports, 143 (1986), 109. doi: 10.1016/0370-1573(86)90159-6. Google Scholar [4] R. Ben-Yishai, RL Bar-Or and H. Sompolinsky, Theory of orientation tuning in visual cortex,, Proceedings of the National Academy of Sciences, 92 (1995), 3844. doi: 10.1073/pnas.92.9.3844. Google Scholar [5] J. Bigun and G. Granlund, Optimal orientation detection of linear symmetry,, in, (1987), 433. Google Scholar [6] B. Blumenfeld, D. Bibitchkov and M. Tsodyks, Neural network model of the primary visual cortex: From functional architecture to lateral connectivity and back,, Journal of Computational Neuroscience, 20 (2006), 219. doi: 10.1007/s10827-006-6307-y. Google Scholar [7] I. Bosch Vivancos, P. Chossat and I. Melbourne, New planforms in systems of partial differential equations with Euclidean symmetry,, Archive for Rational Mechanics and Analysis, 131 (1995), 199. doi: 10.1007/BF00382886. Google Scholar [8] W. H. Bosking, Y. Zhang, B. Schofield and D. Fitzpatrick, Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex,, The Journal of Neuroscience, 17 (1997), 2112. Google Scholar [9] P. C. Bressloff and J. D.Cowan, The functional geometry of local and horizontal connections in a model of v1,, Journal of Physiology, 97 (2003), 221. doi: 10.1016/j.jphysparis.2003.09.017. Google Scholar [10] P. C. Bressloff and J. D. Cowan, A spherical model for orientation and spatial frequency tuning in a cortical hypercolumn,, Philosophical Transactions of the Royal Society B, (2003). doi: 10.1098/rstb.2002.1109. Google Scholar [11] P. C. Bressloff, Spatially periodic modulation of cortical patterns by long-range horizontal connections,, Physica D: Nonlinear Phenomena, 185 (2003), 131. doi: 10.1016/S0167-2789(03)00238-0. Google Scholar [12] P. C. Bressloff and J. D. Cowan, The visual cortex as a crystal,, Physica D: Nonlinear Phenomena, 173 (2002), 226. doi: 10.1016/S0167-2789(02)00677-2. Google Scholar [13] P. C. Bressloff, J. D. Cowan, M. Golubitsky and P. J. Thomas, Scalar and pseudoscalar bifurcations motivated by pattern formation on the visual cortex,, Nonlinearity, 14 (2001). doi: 10.1088/0951-7715/14/4/305. Google Scholar [14] P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas and M. C. Wiener, Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex,, Phil. Trans. R. Soc. Lond. B, 306 (2001), 299. doi: 10.1098/rstb.2000.0769. Google Scholar [15] P. C. Bressloff and A. M. Oster, Theory for the alignment of cortical feature maps during development,, Physical Review E, 82 (2010). doi: 10.1103/PhysRevE.82.021920. Google Scholar [16] I. Chavel, "Eigenvalues in Riemannian Geometry,", 115. Academic Press, 115 (1984). Google Scholar [17] S. Chemla and F. Chavane, Voltage-sensitive dye imaging: Technique review and models,, Journal of Physiology-Paris, 104 (2010), 40. doi: 10.1016/j.jphysparis.2009.11.009. Google Scholar [18] P. Chossat, G. Faye and O. Faugeras, Bifurcations of hyperbolic planforms,, Journal of Nonlinear Science, (2011). doi: 10.1007/s00332-010-9089-3. Google Scholar [19] P. Chossat and R. Lauterbach, "Methods in Equivariant Bifurcations and Dynamical Systems,", World Scientific Publishing Company, (2000). Google Scholar [20] P. Chossat and O. Faugeras, Hyperbolic planforms in relation to visual edges and textures perception,, Plos. Comput. Biol., 5 (2009). doi: 10.1371/journal.pcbi.1000625. Google Scholar [21] P. G. Ciarlet and J. L. Lions, "Handbook of Numerical Analysis,", II Finite Element Methods (part1). North-Holland, II (1991). Google Scholar [22] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space,, J. Math. Imaging Vis., (2006), 307. doi: 10.1007/s10851-005-3630-2. Google Scholar [23] D. P. Edwards, K. P. Purpura and E. Kaplan, Contrast sensitivity and spatial frequency response of primate cortical neurons in and around the cytochrome oxidase blobs,, Vision Research, 35 (1995), 1501. Google Scholar [24] I. Erdélyi, "Higher Transcendental Functions,", 1 Robert E. Krieger Publishing Company, 1 (1985). Google Scholar [25] G. B. Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns,, Biological Cybernetics, 34 (1979), 137. doi: 10.1007/BF00336965. Google Scholar [26] G. Faye and P. Chossat, Bifurcation diagrams and heteroclinic networks of octagonal h-planforms,, Journal of Nonlinear Science, 22 (2012), 277. doi: 10.1007/s00332-011-9118-x. Google Scholar [27] G. Faye, P. Chossat and O. Faugeras, Analysis of a hyperbolic geometric model for visual texture perception,, The Journal of Mathematical Neuroscience, 1 (2011). doi: 10.1186/2190-8567-1-4. Google Scholar [28] M. Golubitsky, L. J. Shiau and A. Török, Bifurcation on the visual cortex with weakly anisotropic lateral coupling,, SIAM Journal on Applied Dynamical Systems, 2 (2003), 97. doi: 10.1137/S1111111102409882. Google Scholar [29] M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory,", volume II. Springer, (1988). doi: 10.1007/978-1-4612-4574-2. Google Scholar [30] D. Hansel and H. Sompolinsky, Modeling feature selectivity in local cortical circuits,, Methods of Neuronal Modeling, (1997), 499. Google Scholar [31] M. Haragus and G. Iooss, "Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Systems,", EDP Sci. Springer Verlag UTX Series, (2010). doi: 10.1007/978-0-85729-112-7. Google Scholar [32] S. Helgason, "Groups and Geometric Analysis,", 83 of Mathematical Surveys and Monographs. American Mathematical Society, 83 (2000). Google Scholar [33] R. B. Hoyle, "Pattern Formation: an Introduction to Methods,", Cambridge Univ Pr, (2006). doi: 10.1017/CBO9780511616051. Google Scholar [34] D. H. Hubel and T. N. Wiesel, Receptive fields and functional architecture in two nonstriate visual areas (18 and 19) of the cat,, Journal of Neurophysiology, 28 (1965), 229. Google Scholar [35] D. H. Hubel and T. N. Wiesel, Receptive fields and functional architecture of monkey striate cortex,, The Journal of Physiology, 195 (1968). Google Scholar [36] D. H. Hubel and T. N. Wiesel, Functional architecture of macaque monkey,, Proceedings of the Royal Society, (1977), 1. Google Scholar [37] H. Iwaniec, "Spectral Methods of Automorphic Forms,", 53 of AMS Graduate Series in Mathematics, 53 (2002). Google Scholar [38] M. Kaschube, M. Schnabel, S. Löwel, D. M. Coppola, L. E. White and F. Wolf, Universality in the evolution of orientation columns in the visual cortex,, Science, 330 (2010). doi: 10.1126/science.1194869. Google Scholar [39] M. Kaschube, M. Schnabel and F. Wolf, Self-organization and the selection of pinwheel density in visual cortical development,, New Journal of Physics, 10 (2008). doi: 10.1088/1367-2630/10/1/015009. Google Scholar [40] S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics. The University of Chicago Press, (1992). Google Scholar [41] H. Kluver, "Mescal, and Mechanisms of Hallucinations,", University of Chicago Press Chicago, (1966). Google Scholar [42] H. Knutsson, Representing local structure using tensors,, Scandinavian Conference on Image Analysis, (1989), 244. doi: 10.1007/978-3-642-21227-7_51. Google Scholar [43] N. N. Lebedev, "Special Functions and Their Applications,", (edited by R. A. Silverman), (1972). Google Scholar [44] P. S. Leon, I. Vanzetta, G. S. Masson and L. U. Perrinet, Motion Clouds: Model-based stimulus synthesis of natural-like random textures for the study of motion perception,, Journal of Neurophysiology, 107 (2012), 3217. doi: 10.1152/jn.00737.2011. Google Scholar [45] M. S. Livingstone and D. H. Hubel, Anatomy and physiology of a color system in the primate visual cortex,, Journal of Neuroscience, 4 (1984), 309. Google Scholar [46] J. S. Lund, A. Angelucci and P. C. Bressloff, Anatomical substrates for functional columns in macaque monkey primary visual cortex,, Cerebral Cortex, 12 (2003), 15. doi: 10.1093/cercor/13.1.15. Google Scholar [47] I. Melbourne, A singularity theory analysis of bifurcation problems with octahedral symmetry,, Dynamics and Stability of Systems, 1 (1986). doi: 10.1080/02681118608806020. Google Scholar [48] W. Miller, "Symmetry Groups and Their Applications,", Academic Press, (1972). Google Scholar [49] M. Moakher, A differential geometric approach to the geometric mean of symmetric positie-definite matrices,, SIAM J. Matrix Anal. Appl., 26 (2005), 735. doi: 10.1137/S0895479803436937. Google Scholar [50] J. D. Murray, "Mathematical Biology II, Spatial Models and Biomedical Applications,", Springer, (2003). Google Scholar [51] G. A. Orban, H. Kennedy and J. Bullier, Velocity sensitivity and direction selectivity of neurons in areas V1 and V2 of the monkey: Influence of eccentricity,, Journal of Neurophysiology, 56 (1986), 462. Google Scholar [52] G. Oster, Phosphenes,, Scientific American, 222 (1970). doi: 10.1038/scientificamerican0270-82. Google Scholar [53] A. M. Oster and P. C. Bressloff, A developmental model of ocular dominance column formation on a growing cortex,, Bulletin of Mathematical Biology, (2006). doi: 10.1007/s11538-005-9055-7. Google Scholar [54] J. Petitot, The neurogeometry of pinwheels as a sub-Riemannian contact structure,, Journal of Physiology-Paris, 97 (2003), 265. Google Scholar [55] J. Petitot, "Neurogéométrie de la Vision,", Les Éditions de l'École Polytechnique, (2009). Google Scholar [56] G. Sanguinetti, A. Sarti and G. Citti, Implementation of a Model for Perceptual Completion in $\R^2\times S^1$,, Computer Vision and Computer Graphics, 24 (2009), 188. Google Scholar [57] A. Sarti and G. Citti, Non-commutative field theory in the visual cortex,, Computer Vision: from Surfaces to 3D Objects, (2010). Google Scholar [58] A. Sarti, G. Citti and J. Petitot, The symplectic structure of the primary visual cortex,, Biological Cybernetics, 98 (2008), 33. doi: 10.1007/s00422-007-0194-9. Google Scholar [59] J. Schummers, J. Mariño and M. 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