Abstract
The aim of this work is to investigate the effect of the shift-twist symmetry on pattern formation processes in the visual cortex. First, we describe a generic set of Riemannian metrics of the feature space of orientation preference that obeys properties of the shift-twist, translation, and reflection symmetries. Second, these metrics are embedded in a modified Swift–Hohenberg model. As a result we get a pattern formation process that resembles the pattern formation process in the visual cortex. We focus on the final stable patterns that are regular and periodic. In a third step we analyze the influences on pattern formation using weakly nonlinear theory and mode analysis. We compare the results of the present approach with earlier models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- G(x, y, v):
-
Elongated Gaussian distribution
- \({\hat{R}(\Phi), \hat{T}(\Psi)}\) :
-
2 × 2 rotation matrices
- \({d_{s}^{2}}\) :
-
Distance between stimuli, between receptive fields and stimuli
- x, y, z1, z2:
-
Real-valued features
- x, z :
-
Complex features
- v :
-
Feature vector
- \({\mathcal{V}}\) :
-
Feature space – manifold
- \({\hat{g}, g_{ij}}\) :
-
Metric tensor
- a, b, c, h:
-
Parameter functions of the metric tensor
- β, γ, μ, ν :
-
Parameter functions in complex coordinates
References
Cross MC, Hohenberg PC (1993) Pattern formation outside equilibrium. Rev Mod Phys 65(3): 851–1112
Durbin R, Willshaw D (1987) An analogue approach to the traveling salesman problem using an elsatic net algorithm. Nature 326: 689–691
Ernst U, Pawelzik K, Sahar-Pikielny C, Tsodyks M (2001) Intracortical origin of visual maps. Nat Neurosci 4: 431–436
Freeman RD, et al (1997) Clustering of response properties of neurons in the visual cortex. In: Social Neuroscience, 23:227.1
Jones J, Palmer L (1987) The two-dimensional spatial structure of simple receptive fields in cat striate cortex. J Neurophys 58: 1187–1211
Kaschube M, Wolf F, Geisel T, Loewel S (2001) The prevalance of colinear contours in the real world. Neurocomputing 38(40): 1335–1339
Kohonen T (2001) Self-organizing maps. Springer, Heidelberg
Liu Z, Gaska JP, Jacobson LD, Pollen DA (1991) Interneural interaction between members od quadrature pairs in the cat’s visual cortex. Vision Res 32: 1193–1198
Löwel S (1998) The layout of orientation and ocular dominance domains in area 17 of strabismic cats. Eur J Neurosci 10(8): 2629–43
Mayer NM, Herrmann MJ, Theo Geisel (1998) A cortical interpretation of ASSOMs. In: Proceedings of International Conference on Artificial Neural Networks (ICANN), vol 2, pp 961–966
Mayer NM, Herrmann JM, Geisel T (2002) Curved feature metrices in models of visual cortex. Neurocomputing 44–46(C): 533–539
Mayer NM, Herrmann JM, Geisel T (2003) Shaping of receptive fields in visual cortex during retinal maturation. J Comp Neurosci 15(3): 307–320
Pollen D, Ronner S (1981) Phase relationships betweeen adjacent simple cells in the visual cortex. Science 212: 1409–1411
Riesenhuber M, Bauer H-U, Brockmann D, Geisel T (1998) Breaking rotational symmetry in a self-organizing map model for orientation map development. Neural Comput 10: 717–730
Ritter H (1999) Self-organizing maps in non-Euclidean spaces. In: Oja E, Kaski S (eds) Kohonen Maps, pp 97–108
Ritter H, Martinetz T, Schulten K (1992) Neuronale netze. Addison Wesley, Bonn
Sengpiel F, Bonhoeffer T, Stawinski P (1999) Influence of experience on orientation maps in cat visual cortex. Nat Neurosci 2: 727–732
Swindale NV (1996) The development of topography in the visual cortex: a review of models. Network 7: 161–247
Swindale NV, Shoham D, Grinvald A, Bonhoeffer T, Hübener M (2000) Optimizing coverage in the cortex. Nat Neurosci 3(8): 750–751
Thomas PJ, Cowan JD (2003) Symmetry induced coupling of cortical feature maps. Phys Rev Lett 92(18810)
Malsburg Cvd (1973) Self-organization of orientation sensitive cells in striate cortex. Kybernetik 14: 49–54
Wolf F (2005) Symmetry, multistability, and long-range interactions in brain development. Phys Rev Lett 95(208701)
Wolf F, Geisel T (1998) Spontaneous pinwheel annihilation during visual development. Nature 395: 73–78
Wolf F, Geisel T (2003) Universality in visual cortical pattern formation. J Phys 97: 253–264
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Michael Mayer, N., Browne, M., Herrmann, J.M. et al. Symmetries, non-Euclidean metrics, and patterns in a Swift–Hohenberg model of the visual cortex. Biol Cybern 99, 63–78 (2008). https://doi.org/10.1007/s00422-008-0238-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00422-008-0238-9