Comptes Rendus
Analyse numérique, Systèmes dynamiques
Spatial and color hallucinations in a mathematical model of primary visual cortex
Olivier D. Faugeras1  ; Anna Song2, 3 ; Romain Veltz1 
1 Université Côte d’Azur, Inria, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France
2 Department of Mathematics, Imperial College London, London, UK
3 Haematopoietic Stem Cell Laboratory, The Francis Crick Institute, London, UK
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 59-87.

We study a simplified model of the representation of colors in the primate primary cortical visual area V1. The model is described by an initial value problem related to a Hammerstein equation. The solutions to this problem represent the variation of the activity of populations of neurons in V1 as a function of space and color. The two space variables describe the spatial extent of the cortex while the two color variables describe the hue and the saturation represented at every location in the cortex. We prove the well-posedness of the initial value problem. We focus on its stationary, i.e. independent of time, and periodic in space solutions. We show that the model equation is equivariant with respect to the direct product 𝒢 of the group of the Euclidean transformations of the planar lattice determined by the spatial periodicity and the group of color transformations, isomorphic to O(2), and study the equivariant bifurcations of its stationary solutions when some parameters in the model vary. Their variations may be caused by the consumption of drugs and the bifurcated solutions may represent visual hallucinations in space and color. Some of the bifurcated solutions can be determined by applying the Equivariant Branching Lemma (EBL) by determining the axial subgroups of 𝒢. These define bifurcated solutions which are invariant under the action of the corresponding axial subgroup. We compute analytically these solutions and illustrate them as color images. Using advanced methods of numerical bifurcation analysis we then explore the persistence and stability of these solutions when varying some parameters in the model. We conjecture that we can rely on the EBL to predict the existence of patterns that survive in large parameter domains but not to predict their stability. On our way we discover the existence of spatially localized stable patterns through the phenomenon of “snaking”.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.289
Classification : 37L10, 35B06, 35B32, 35G25, 37M20, 45B05, 45K05, 46E35, 47G20, 47H30, 65J15, 65R0, 9208, 9210, 92B20
Olivier D. Faugeras 1 ; Anna Song 2, 3 ; Romain Veltz 1

1 Université Côte d’Azur, Inria, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France
2 Department of Mathematics, Imperial College London, London, UK
3 Haematopoietic Stem Cell Laboratory, The Francis Crick Institute, London, UK
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Olivier D. Faugeras and Anna Song and Romain Veltz},
     title = {Spatial and color hallucinations in a mathematical model of primary visual cortex},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {59--87},
     publisher = {Acad\'emie des sciences, Paris},
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     language = {en},
}
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Olivier D. Faugeras; Anna Song; Romain Veltz. Spatial and color hallucinations in a mathematical model of primary visual cortex. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 59-87. doi : 10.5802/crmath.289. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.289/

[1] Robert A. Adams Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press Inc., 1975 | MR

[2] Robert A. Adams; John J. F. Fournier Sobolev spaces, Pure and Applied Mathematics, 140, Elsevier, 2003

[3] Shun-Ichi Amari Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern., Volume 27 (1977) no. 2, pp. 77-87 | DOI | MR | Zbl

[4] Daniele Avitabile; David J. B. Lloyd; John Burke; Edgar Knobloch; Björn Sandstede To snake or not to snake in the planar Swift–Hohenberg equation, SIAM J. Appl. Dyn. Syst., Volume 9 (2010), pp. 704-733 | DOI | MR | Zbl

[5] Richard Barrett; Michael Berry; Tony F. Chan; James Demmel; June Donato; Jack Dongarra; Victor Eijkhout; Roldan Pozo; Charles Romine; Henk van der Vorst Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Society for Industrial and Applied Mathematics, 1993

[6] Michel Berthier Geometry of color perception. II: Perceived colors from real quantum states and Hering’s rebit., J. Math. Neurosci., Volume 10 (2020), 14, 25 pages | DOI | Zbl

[7] Tim Besard; Valentin Churavy; Alan Edelman; Bjorn De Sutter Rapid software prototyping for heterogeneous and distributed platforms, Adv. Eng. Softw., Volume 132 (2019), pp. 29-46 | DOI

[8] Tim Besard; Christophe Foket; Bjorn De Sutter Effective Extensible Programming: Unleashing Julia on GPUs, IEEE Trans. Parallel Distrib. Syst., Volume 30 (2019) no. 4, pp. 827-841 | DOI

[9] Paul C. Bressloff Spatiotemporal dynamics of continuum neural fields, J. Phys. A, Math. Theor., Volume 45 (2012) no. 3, 033001, 109 pages | DOI | MR | Zbl

[10] Paul C. Bressloff; Jack D. Cowan; Martin Golubitsky; Peter J. Thomas; Matthew C. Wiener Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex, Philos. Trans. R. Soc. Lond., Ser. B, Volume 306 (2001) no. 1407, pp. 299-330 | DOI

[11] Friedrich H. Busse Patterns of convection in spherical shells, J. Fluid Mech., Volume 72 (1975) no. 1, pp. 67-85 | DOI | Zbl

[12] Stephen J. Chapman; Gregory Kozyreff Exponential asymptotics of localised patterns and snaking bifurcation diagrams, Physica D, Volume 238 (2009) no. 3, pp. 319-354 | DOI | MR | Zbl

[13] Soumya Chatterjee; Kenichi Ohki; R. Clay Reid Chromatic micromaps in primary visual cortex, Nat. Commun., Volume 12 (2021) no. 1, 2315 | DOI

[14] Pascal Chossat Solutions avec symétrie diédrale dans les problèmes de bifurcation invariants par symétrie sphérique, C. R. Math. Acad. Sci. Paris, Volume 297 (1983) no. 12, pp. 639-642 | MR | Zbl

[15] Pascal Chossat; Reiner Lauterbach Methods in equivariant bifurcations and dynamical systems, Advanced Series in Nonlinear Dynamics, 15, World Scientific, 2000 | DOI

[16] Pascal Chossat; Reiner Lauterbach; Ian Melbourne Steady-state bifurcation with O(3)-symmetry, Arch. Ration. Mech. Anal., Volume 113 (1990) no. 4, pp. 313-376 | DOI | MR | Zbl

[17] Giampaolo Cicogna Symmetry breakdown from bifurcation, Lettere al Nuovo Cimento (1971-1985), Volume 31 (1981) no. 17, pp. 600-602 | DOI | MR

[18] Carol L. Colby; Jean-René R. Duhamel; Michael E. Goldberg Oculocentric spatial representation in parietal cortex, Cereb. Cortex, Volume 5 (1995) no. 5, pp. 470-481 | DOI

[19] Stephen Coombes; Peter beim Graben; Roland Potthast Tutorial on Neural Field Theory, Neural Fields: Theory and Applications (Stephen Coombes; Peter beim Graben; Roland Potthast; James Wright, eds.), Springer, 2014, pp. 1-43 | DOI | Zbl

[20] Sophie Denève; Christian K. Machens Efficient codes and balanced networks, Nature Neurosci., Volume 19 (2016) no. 3, pp. 375-382 | DOI

[21] Benoit Dionne; Martin Golubitsky Planforms in two and three dimensions, Z. Angew. Math. Phys., Volume 43 (1992) no. 1, pp. 36-62 | DOI | MR | Zbl

[22] Benoit Dionne; Martin Golubitsky; Mary Silber; Ian Stewart Time-periodic spatially periodic Planforms in Euclidean equivariant partial differential equations, Philos. Trans. R. Soc. Lond., Ser. A, Volume 352 (1995) no. 1698, pp. 125-168 | MR | Zbl

[23] Benoit Dionne; Mary Silber; Anne C. Skeldon Stability results for steady, spatially periodic planforms, Nonlinearity, Volume 10 (1997), pp. 321-353 | DOI | MR | Zbl

[24] Bard Ermentrout Neural networks as spatio-temporal pattern-forming systems, Rep. Prog. Phys., Volume 61 (1998), pp. 353-430 | DOI

[25] George B. Ermentrout; Jack D. Cowan Temporal oscillations in neuronal nets, J. Math. Biol., Volume 7 (1979) no. 3, pp. 265-280 | DOI | MR | Zbl

[26] Patrick E. Farrell; Ásgeir Birkisson; Simon W. Funke Deflation Techniques for Finding Distinct Solutions of Nonlinear Partial Differential Equations, SIAM J. Sci. Comput., Volume 37 (2015) no. 4, p. A2026-A2045 | DOI | MR | Zbl

[27] Olivier Faugeras; Romain Veltz; Francois Grimbert Persistent neural states: stationary localized activity patterns in nonlinear continuous n-population, q-dimensional neural networks, Neural Comput., Volume 21 (2009) no. 1, pp. 147-187 | DOI | MR | Zbl

[28] Shintaro Funahashi; Charles J. Bruce; Patricia S. Goldman-Rakic Mnemonic coding of visual space in the monkey’s dorsolateral prefrontal cortex, J. Neurophys., Volume 61 (1989), pp. 331-349 | DOI

[29] Martin Golubitsky; David G. Schaeffer Singularities and Groups in Bifurcation Theory. Vol. I., Applied Mathematical Sciences, 51, Springer, 1985 | DOI

[30] Martin Golubitsky; Ian Stewart The Symmetry Perspective. From equilibrium to chaos in phase space and physical space, Progress in Mathematics, 200, Birkhäuser, 2002

[31] Martin Golubitsky; Ian Stewart; David G. Schaeffer Singularities and Groups in Bifurcation Theory. Vol. II, Applied Mathematical Sciences, 69, Springer, 1988 | DOI

[32] Ewald Hering Outlines of a theory of the light sense, Harvard University Press, 1964

[33] Rebecca B. Hoyle Pattern formation: an introduction to methods, Cambridge University Press, 2006 | DOI

[34] David H. Hubel Blobs and color vision, Bioscience at the Physical Science Frontier, Springer, 1986, pp. 91-102 | DOI

[35] David H. Hubel Eye, Brain, and Vision, 22, Scientific American Library, 1995

[36] Edwin C. Ihrig; Martin Golubitsky Pattern selection with O(3) symmetry, Physica D, Volume 13 (1984) no. 1-2, pp. 1-33 | DOI | MR

[37] George H. Joblove; Donald Greenberg Color Spaces for Computer Graphics, ACM SIGGRAPH Comput. Graph., Volume 12 (1978) no. 3, pp. 20-25 | DOI

[38] Principles of neuroscience (Eric R. Kandel; Thomas M. Jessell; James H. Schwartz; Steven A. Siegelbaum; A. J. Hudspeth, eds.), McGraw-Hill, 2013

[39] E. Knobloch Spatial Localization in Dissipative Systems, Ann. Rev. Cond. Matter Phys., Volume 6 (2015) no. 1, pp. 325-359 | DOI

[40] Jan J. Koenderink Color for the Sciences, MIT Press, 2010

[41] Jennifer S. Lund; Alessandra Angelucci; Paul C. Bressloff Anatomical Substrates for Functional Columns in Macaque Monkey Primary Visual Cortex, Cereb. Cortex, Volume 13 (2003) no. 1, pp. 15-24 | DOI

[42] Earl K. Miller; Cynthia A. Erickson; Robert Desimone Neural mechanisms of visual working memory in prefrontal cortex of the Macaque, J. Neurosci., Volume 16 (1996), pp. 5154-5167 | DOI

[43] Patrick Monnier Standard definitions of chromatic induction fail to describe induction with S-cone patterned backgrounds, Vision Res., Volume 48 (2008), pp. 2708-2714 | DOI

[44] Patrick Monnier; Steven K. Shevell Chromatic induction from S-cone patterns, Vision Res., Volume 44 (2004), pp. 849-856 | DOI

[45] Jean B. Nganou How rare are subgroups of index 2?, Math. Mag., Volume 85 (2012) no. 3, pp. 215-220 | DOI | MR | Zbl

[46] Marc Olive Effective computation of SO(3) and O(3) linear representation symmetry classes, Math. Mech. Compl. Sys., Volume 7 (2019) no. 3, pp. 203-237 | DOI | MR

[47] Gerald Oster Phosphenes, Sci. Am., Volume 222 (1970) no. 2, pp. 82-87 | DOI

[48] Jean Petitot Elements of neurogeometry, Lecture Notes in Morphogenesis, Springer, 2017 | DOI | MR

[49] Roland Potthast; Peter Beim Graben Existence and properties of solutions for neural field equations, Math. Methods Appl. Sci., Volume 33 (2010) no. 8, pp. 935-949 | DOI | MR | Zbl

[50] Edoardo Provenzi Geometry of color perception. I: Structures and metrics of a homogeneous color space., J. Math. Neurosci., Volume 10 (2020), 7, 19 pages | DOI | MR | Zbl

[51] Paul H. Rabinowitz Some global results for nonlinear eigenvalue problems, J. Funct. Anal., Volume 7 (1971), pp. 487-513 | DOI | MR | Zbl

[52] Youcef Saad Numerical methods for large eigenvalue problems, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, 2011 no. 66 | DOI

[53] E. L. Schwartz Spatial mapping in the primate sensory projection: Analytic structure and relevance to perception, Biol. Cybern., Volume 25 (1977) no. 4, pp. 181-194 | DOI

[54] R. K. Siegel Hallucinations, Sci. Am., Volume 237 (1977) no. 4, pp. 132-140 | DOI

[55] Mary Silber; Edgar Knobloch Pattern selection in steady binary-fluid convection, Phys. Rev. A, Volume 38 (1988) no. 3, pp. 1468-1477 | DOI | MR

[56] Mary Silber; Edgar Knobloch Hopf bifurcation on a square lattice, Nonlinearity, Volume 4 (1991) no. 4, pp. 1063-1107 | DOI | MR | Zbl

[57] Anna Song; Olivier Faugeras; Romain Veltz A neural field model for color perception unifying assimilation and contrast, PLoS Comput. Biol., Volume 15 (2019) no. 6, e1007050 | DOI

[58] Daniel Y. Ts’o; Mark D. Zarella; Guy Burkitt Whither the hypercolumn?, J. Physiol., Volume 587 (2009) no. 12, pp. 2791-2805 | DOI

[59] Hannes Uecker; Daniel Wetzel Numerical Results for Snaking of Patterns over Patterns in Some 2D Selkov–Schnakenberg Reaction-Diffusion Systems, SIAM J. Appl. Dyn. Syst., Volume 13 (2014) no. 1, pp. 94-128 | DOI | MR | Zbl

[60] Andre Vanderbauwhede Local Bifurcation and Symmetry, Ph. D. Thesis, Rijksuniversiteit Gent (1980)

[61] Nicolás Vattuone; Thomas Wachtler; Inés Samengo Perceptual spaces and their symmetries: The geometry of color space, Math. Neurosci. Appl., Volume 1 (2021) https://mna.episciences.org/7680 | DOI

[62] Romain Veltz Nonlinear analysis methods in neural field models, Ph. D. Thesis, Université Paris Est (2011) https://pastel.archives-ouvertes.fr/tel-00686695

[63] Romain Veltz BifurcationKit.jl, 2020 | HAL

[64] Romain Veltz; Pascal Chossat; Olivier Faugeras On the Effects on Cortical Spontaneous Activity of the Symmetries of the Network of Pinwheels in Visual Area V1, J. Math. Neurosci., Volume 5 (2015), 11, 28 pages | DOI | MR | Zbl

[65] Romain Veltz; Olivier Faugeras Local/Global Analysis of the Stationary Solutions of Some Neural Field Equations, SIAM J. Appl. Dyn. Syst., Volume 9 (2010) no. 3, pp. 954-998 | DOI | MR | Zbl

[66] Colin Ware; William B. Cowan Changes in perceived color due to chromatic interactions, Vision Res., Volume 22 (1982) no. 11, pp. 1353-1362 | DOI

[67] Hugh R. Wilson; Jack D. Cowan Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J., Volume 12 (1972) no. 1, pp. 1-24 | DOI

[68] Hugh R. Wilson; Jack D. Cowan A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Biol. Cybern., Volume 13 (1973) no. 2, pp. 55-80 | Zbl

[69] Günther Wyszecki; W. S. Stiles Color Science: Concepts and Methods, Quantitative Data and Formulas, John Wiley & Sons, 1967

[70] Youping Xiao; Alexander Casti; Jun Xiao; Ehud Kaplan Hue maps in primate striate cortex, NeuroImage, Volume 35 (2007) no. 2, pp. 771-786 | DOI

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