Spatial and color hallucinations in a mathematical model of primary visual cortex

Faugeras, Olivier D.; Song, Anna; Veltz, Romain

Comptes Rendus. Mathématique, Tome 360 (2022) no. G1, 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 $O\left(2\right)$, 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”.

Classification : 37L10, 35B06, 35B32, 35G25, 37M20, 45B05, 45K05, 46E35, 47G20, 47H30, 65J15, 65R0, 9208, 9210, 92B20

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

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

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

[4] Avitabile, Daniele; Lloyd, David J. B.; Burke, John; Knobloch, Edgar; Sandstede, Björn 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] Barrett, Richard; Berry, Michael; Chan, Tony F.; Demmel, James; Donato, June; Dongarra, Jack; Eijkhout, Victor; Pozo, Roldan; Romine, Charles; van der Vorst, Henk Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Society for Industrial and Applied Mathematics, 1993

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

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

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

[10] Bressloff, Paul C.; Cowan, Jack D.; Golubitsky, Martin; Thomas, Peter J.; Wiener, Matthew C. 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] Busse, Friedrich H. Patterns of convection in spherical shells, J. Fluid Mech., Volume 72 (1975) no. 1, pp. 67-85 | DOI | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[26] Farrell, Patrick E.; Birkisson, Ásgeir; Funke, Simon W. 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] Faugeras, Olivier; Veltz, Romain; Grimbert, Francois 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] Funahashi, Shintaro; Bruce, Charles J.; Goldman-Rakic, Patricia S. Mnemonic coding of visual space in the monkey’s dorsolateral prefrontal cortex, J. Neurophys., Volume 61 (1989), pp. 331-349 | DOI

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[49] Potthast, Roland; Beim Graben, Peter 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] Provenzi, Edoardo 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] Rabinowitz, Paul H. Some global results for nonlinear eigenvalue problems, J. Funct. Anal., Volume 7 (1971), pp. 487-513 | DOI | MR | Zbl

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

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

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

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

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

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

[59] Uecker, Hannes; Wetzel, Daniel 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] Vanderbauwhede, Andre Local Bifurcation and Symmetry, Ph. D. Thesis, Rijksuniversiteit Gent (1980)

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

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

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

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

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

[68] Wilson, Hugh R.; Cowan, Jack D. 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] Wyszecki, Günther; Stiles, W. S. Color Science: Concepts and Methods, Quantitative Data and Formulas, John Wiley & Sons, 1967

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