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.

[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.

[3]

N. L. Balazs and A. Voros, Chaos on the pseudosphere,, Physics Reports, 143 (1986), 109. doi: 10.1016/0370-1573(86)90159-6.

[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.

[5]

J. Bigun and G. Granlund, Optimal orientation detection of linear symmetry,, in, (1987), 433.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

I. Chavel, "Eigenvalues in Riemannian Geometry,", 115. Academic Press, 115 (1984).

[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.

[18]

P. Chossat, G. Faye and O. Faugeras, Bifurcations of hyperbolic planforms,, Journal of Nonlinear Science, (2011). doi: 10.1007/s00332-010-9089-3.

[19]

P. Chossat and R. Lauterbach, "Methods in Equivariant Bifurcations and Dynamical Systems,", World Scientific Publishing Company, (2000).

[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.

[21]

P. G. Ciarlet and J. L. Lions, "Handbook of Numerical Analysis,", II Finite Element Methods (part1). North-Holland, II (1991).

[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.

[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.

[24]

I. Erdélyi, "Higher Transcendental Functions,", 1 Robert E. Krieger Publishing Company, 1 (1985).

[25]

G. B. Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns,, Biological Cybernetics, 34 (1979), 137. doi: 10.1007/BF00336965.

[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.

[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.

[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.

[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.

[30]

D. Hansel and H. Sompolinsky, Modeling feature selectivity in local cortical circuits,, Methods of Neuronal Modeling, (1997), 499.

[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.

[32]

S. Helgason, "Groups and Geometric Analysis,", 83 of Mathematical Surveys and Monographs. American Mathematical Society, 83 (2000).

[33]

R. B. Hoyle, "Pattern Formation: an Introduction to Methods,", Cambridge Univ Pr, (2006). doi: 10.1017/CBO9780511616051.

[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.

[35]

D. H. Hubel and T. N. Wiesel, Receptive fields and functional architecture of monkey striate cortex,, The Journal of Physiology, 195 (1968).

[36]

D. H. Hubel and T. N. Wiesel, Functional architecture of macaque monkey,, Proceedings of the Royal Society, (1977), 1.

[37]

H. Iwaniec, "Spectral Methods of Automorphic Forms,", 53 of AMS Graduate Series in Mathematics, 53 (2002).

[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.

[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.

[40]

S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics. The University of Chicago Press, (1992).

[41]

H. Kluver, "Mescal, and Mechanisms of Hallucinations,", University of Chicago Press Chicago, (1966).

[42]

H. Knutsson, Representing local structure using tensors,, Scandinavian Conference on Image Analysis, (1989), 244. doi: 10.1007/978-3-642-21227-7_51.

[43]

N. N. Lebedev, "Special Functions and Their Applications,", (edited by R. A. Silverman), (1972).

[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.

[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.

[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.

[47]

I. Melbourne, A singularity theory analysis of bifurcation problems with octahedral symmetry,, Dynamics and Stability of Systems, 1 (1986). doi: 10.1080/02681118608806020.

[48]

W. Miller, "Symmetry Groups and Their Applications,", Academic Press, (1972).

[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.

[50]

J. D. Murray, "Mathematical Biology II, Spatial Models and Biomedical Applications,", Springer, (2003).

[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.

[52]

G. Oster, Phosphenes,, Scientific American, 222 (1970). doi: 10.1038/scientificamerican0270-82.

[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.

[54]

J. Petitot, The neurogeometry of pinwheels as a sub-Riemannian contact structure,, Journal of Physiology-Paris, 97 (2003), 265.

[55]

J. Petitot, "Neurogéométrie de la Vision,", Les Éditions de l'École Polytechnique, (2009).

[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.

[57]

A. Sarti and G. Citti, Non-commutative field theory in the visual cortex,, Computer Vision: from Surfaces to 3D Objects, (2010).

[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.

[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.

[60]

J. P. Serre, "Représentations Linéaires des Groupes Finis,", Hermann, (1978).

[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.

[62]

A. Terras, "Harmonic Analysis on Symmetric Spaces and Applications,", Springer-Verlag, 2 (1988). doi: 10.1007/978-1-4612-3820-1.

[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.

[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.

[65]

R. Veltz and O. Faugeras, Illusions in the ring model of visual orientation selectivity,, Technical Report, (2010). doi: 10.1137/090773611.

[66]

G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1995).

[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.

[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.

[69]

F. Wolf and T. Geisel, Spontaneous pinwheel annihilation during visual development,, Nature, 395 (1998), 73.

[70]

A. Zettl, "Sturm-Liouville Theory,", 121, 121 (2005).

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.

[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.

[3]

N. L. Balazs and A. Voros, Chaos on the pseudosphere,, Physics Reports, 143 (1986), 109. doi: 10.1016/0370-1573(86)90159-6.

[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.

[5]

J. Bigun and G. Granlund, Optimal orientation detection of linear symmetry,, in, (1987), 433.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

I. Chavel, "Eigenvalues in Riemannian Geometry,", 115. Academic Press, 115 (1984).

[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.

[18]

P. Chossat, G. Faye and O. Faugeras, Bifurcations of hyperbolic planforms,, Journal of Nonlinear Science, (2011). doi: 10.1007/s00332-010-9089-3.

[19]

P. Chossat and R. Lauterbach, "Methods in Equivariant Bifurcations and Dynamical Systems,", World Scientific Publishing Company, (2000).

[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.

[21]

P. G. Ciarlet and J. L. Lions, "Handbook of Numerical Analysis,", II Finite Element Methods (part1). North-Holland, II (1991).

[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.

[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.

[24]

I. Erdélyi, "Higher Transcendental Functions,", 1 Robert E. Krieger Publishing Company, 1 (1985).

[25]

G. B. Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns,, Biological Cybernetics, 34 (1979), 137. doi: 10.1007/BF00336965.

[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.

[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.

[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.

[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.

[30]

D. Hansel and H. Sompolinsky, Modeling feature selectivity in local cortical circuits,, Methods of Neuronal Modeling, (1997), 499.

[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.

[32]

S. Helgason, "Groups and Geometric Analysis,", 83 of Mathematical Surveys and Monographs. American Mathematical Society, 83 (2000).

[33]

R. B. Hoyle, "Pattern Formation: an Introduction to Methods,", Cambridge Univ Pr, (2006). doi: 10.1017/CBO9780511616051.

[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.

[35]

D. H. Hubel and T. N. Wiesel, Receptive fields and functional architecture of monkey striate cortex,, The Journal of Physiology, 195 (1968).

[36]

D. H. Hubel and T. N. Wiesel, Functional architecture of macaque monkey,, Proceedings of the Royal Society, (1977), 1.

[37]

H. Iwaniec, "Spectral Methods of Automorphic Forms,", 53 of AMS Graduate Series in Mathematics, 53 (2002).

[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.

[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.

[40]

S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics. The University of Chicago Press, (1992).

[41]

H. Kluver, "Mescal, and Mechanisms of Hallucinations,", University of Chicago Press Chicago, (1966).

[42]

H. Knutsson, Representing local structure using tensors,, Scandinavian Conference on Image Analysis, (1989), 244. doi: 10.1007/978-3-642-21227-7_51.

[43]

N. N. Lebedev, "Special Functions and Their Applications,", (edited by R. A. Silverman), (1972).

[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.

[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.

[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.

[47]

I. Melbourne, A singularity theory analysis of bifurcation problems with octahedral symmetry,, Dynamics and Stability of Systems, 1 (1986). doi: 10.1080/02681118608806020.

[48]

W. Miller, "Symmetry Groups and Their Applications,", Academic Press, (1972).

[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.

[50]

J. D. Murray, "Mathematical Biology II, Spatial Models and Biomedical Applications,", Springer, (2003).

[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.

[52]

G. Oster, Phosphenes,, Scientific American, 222 (1970). doi: 10.1038/scientificamerican0270-82.

[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.

[54]

J. Petitot, The neurogeometry of pinwheels as a sub-Riemannian contact structure,, Journal of Physiology-Paris, 97 (2003), 265.

[55]

J. Petitot, "Neurogéométrie de la Vision,", Les Éditions de l'École Polytechnique, (2009).

[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.

[57]

A. Sarti and G. Citti, Non-commutative field theory in the visual cortex,, Computer Vision: from Surfaces to 3D Objects, (2010).

[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.

[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.

[60]

J. P. Serre, "Représentations Linéaires des Groupes Finis,", Hermann, (1978).

[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.

[62]

A. Terras, "Harmonic Analysis on Symmetric Spaces and Applications,", Springer-Verlag, 2 (1988). doi: 10.1007/978-1-4612-3820-1.

[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.

[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.

[65]

R. Veltz and O. Faugeras, Illusions in the ring model of visual orientation selectivity,, Technical Report, (2010). doi: 10.1137/090773611.

[66]

G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1995).

[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.

[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.

[69]

F. Wolf and T. Geisel, Spontaneous pinwheel annihilation during visual development,, Nature, 395 (1998), 73.

[70]

A. Zettl, "Sturm-Liouville Theory,", 121, 121 (2005).

[1]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

[2]

Yasuhito Miyamoto. Global bifurcation and stable two-phase separation for a phase field model in a disk. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 791-806. doi: 10.3934/dcds.2011.30.791

[3]

Hyung Ju Hwang, Thomas P. Witelski. Short-time pattern formation in thin film equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 867-885. doi: 10.3934/dcds.2009.23.867

[4]

Jong-Shenq Guo, Hirokazu Ninomiya, Chin-Chin Wu. Existence of a rotating wave pattern in a disk for a wave front interaction model. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1049-1063. doi: 10.3934/cpaa.2013.12.1049

[5]

Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357

[6]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395

[7]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

[8]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555

[9]

Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217

[10]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809

[11]

Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609

[12]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339

[13]

H. Malchow, F.M. Hilker, S.V. Petrovskii. Noise and productivity dependence of spatiotemporal pattern formation in a prey-predator system. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 705-711. doi: 10.3934/dcdsb.2004.4.705

[14]

Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111

[15]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597

[16]

Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056

[17]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505

[18]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions II: A Shovel-Bifurcation pattern. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 941-973. doi: 10.3934/dcds.2011.31.941

[19]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71

[20]

Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283

2016 Impact Factor: 1.2

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]