Journal of Statistical Physics

, Volume 125, Issue 5–6, pp 1243–1266

Patterns and Symmetries in the Visual Cortex and in Natural Images

Article

Abstract

As borders between different entities, lines are an important element of natural images. Indeed, the neurons of the mammalian visual cortex are tuned to respond best to lines of a given orientation. This preferred orientation varies continuously across most of the cortex, but also has vortex-like singularities known as pinwheels. In attempting to describe such patterns of orientation preference, we are led to consider underlying rotation symmetries: Oriented segments in natural images tend to be collinear; neurons are more likely to be connected if their preferred orientations are aligned to their topographic separation. These are indications of a reduced symmetry requiring joint rotations of both orientation preference and the underlying topography. This is verified by direct statistical tests in both natural images and in cortical maps. Using the statistics of natural scenes we construct filters that are best suited to extracting information from such images, and find qualitative similarities to mammalian vision.

Keywords

Visual cortex Orientational preference map Pinwheel structure Joint rotational symmetry Transversality Information optimization 

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References

  1. M. Cross and P. Hohenberg, Rev. Mod. Phys. 65:851 (1993).CrossRefADSGoogle Scholar
  2. J. P. Gollub and J. S. Langer, Rev. Mod. Phys. 71:S396 (1999).CrossRefGoogle Scholar
  3. E. R. Kandel, J. H. Schwartz, and T. M. Jessell, Principles of Neural Science (Appleton & Lange, Norwalk, CT, 1991).Google Scholar
  4. D. H. Hubel and T. N. Wiesel, J. Physiol. 160:215 (1962).Google Scholar
  5. G. G. Blasdel and G. Salama, Nature 321:579 (1986).CrossRefADSGoogle Scholar
  6. H. Y. Lee, M. Yahyanejad, and M. Kardar, Proc. Nat. Acad. Sci. USA 100:16036 (2003).CrossRefADSGoogle Scholar
  7. N. V. Swindale, Proc. R. Soc. Lond. B 215:211 (1982).ADSGoogle Scholar
  8. N. V. Swindale, Biol. Cybern. 66:217 (1992).CrossRefGoogle Scholar
  9. F. Wolf and T. Geisel, Nature 395:73 (1998).CrossRefADSGoogle Scholar
  10. Constant stirring by sufficiently strong external noise can also lead to dynamic creation and annihilation of pinwheels, but our focus is on evolving fields where the only randomness is in the choice of initial conditions.Google Scholar
  11. A. A. Koulakov and D. B. Chklovskii, Neuron 29:519 (2001).CrossRefGoogle Scholar
  12. F. Wolf, PhD thesis, Univeritt Göttingen, 2000.Google Scholar
  13. In fact (as we also found in our analysis of monkey map), not all orientations are equally represented. This type of anisotropy indicates the absence of any form of rotation symmetry, and should not be confused with the distinction between full and joint rotation symmetries which is the subject of this article. The former is compatible with rainbow patterns and does not appear to play a role in the stability of pinwheels. We verified this explicitly by numerical simulations in models with a preference for the horizontal direction.Google Scholar
  14. J. B. Swift and P. C. Hohenberg, Phys. Rev. A 15:319 (1977).CrossRefADSGoogle Scholar
  15. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65:851 (1993).CrossRefADSGoogle Scholar
  16. W. H. Bosking, Y. Zhang, B. Schofield, and D. Fitzpatrick, J. Neurosci. 17:2112 (1997).Google Scholar
  17. P. C. Bressloff et al., Phil. Trans. R. Soc. Lond. B 356:299 (2001).CrossRefGoogle Scholar
  18. The vectorial representation may in fact be appropriate for a more detailed description of cortical maps which includes other aspects of visual input. For example, it is known that V1 cells respond also to the motion of oriented bars. Including the direction of motion leads to a more vectorial representation.Google Scholar
  19. D. Whitney, H. C. Goltz, C. G. Thomas, J. S. Gati, R. S. Menon, and M. A. Goodale, Science 31:878 (2003).CrossRefADSGoogle Scholar
  20. B. T. Halperin, in Physics of Defects, Les Houches Session XXXV, 1980, R. Balian, M. Klèman, and J.-P. Poirir, eds. (North Holland, Amsterdam, 1981), pp. 813–857.Google Scholar
  21. M. Sigman, G. A. Cecchi, C. D. Gilbert, and M. O. Magnasco, Proc. Natl. Acad. Sci. USA 98:1935 (2001).CrossRefADSGoogle Scholar
  22. D. J. Field, J. Opt. Soc. Am. A 4:2379 (1987).ADSCrossRefGoogle Scholar
  23. D. Ruderman and W. Bialek, Phys. Rev. Lett. 73:814 (1994).CrossRefADSGoogle Scholar
  24. A. S. Monin and A. M. Yaglom, Statistical Mechanics, Vol. 2 (MIT, Cambridge, 1971), pp. 1–58.Google Scholar
  25. I. Arad et al, Phys. Rev. Lett. 81:5330 (1998).CrossRefADSGoogle Scholar
  26. J. H. van Hateren and A. Van der Schaaf, Proc. R. Soc. London B 265:359 (1998).Google Scholar
  27. W. T. Freeman and E. H. Adelson, IEEE Trans. Patt. Anal. Mach. Intell. 13:891 (1991).CrossRefGoogle Scholar
  28. E. Switkes, M. J. Mayer, and J. A. Sloan, Vision Res. 18:1393 (1978).CrossRefGoogle Scholar
  29. J. D. Pettigrew, T. Nikara, and P. O. Bishop, Exp. Brain Res. 6:373 (1968).Google Scholar
  30. B. Chapman and T. Bonhoeffer, Proc. Natl. Acad. Sci. USA 95:2609 (1998).CrossRefADSGoogle Scholar
  31. V. Dragoi, C. M. Turch, and M. Sur, Neuron 32:1181 (2001).CrossRefGoogle Scholar
  32. We confirmed that the spectra become more isotropic as we average over more rotated images. Note that with a matrix S αβ obtained from an orientation field, there is no a priori reason for the cross correlations S lt(k) and S tl(k) to be zero. We do find that these correlations are small, and also decrease as we average over rotated images.Google Scholar
  33. Additional pictures and data are available online from http://www.mit.edu/∼kardar/research/transversality/ModernArt/.Google Scholar
  34. C. D. Gilbert and T. N. Wiesel, J. Neurosci. 9:2432 (1989).Google Scholar
  35. R. Malach, Y. Amir, M. Harel, and A. Grinvald, Proc. Natl. Acad. Sci. USA 90:10469 (1993).CrossRefADSGoogle Scholar
  36. J. I. Nelson and B. J. Frost, Exp. Brain Res. 61:54 (1985).CrossRefGoogle Scholar
  37. P. Buzás, U. T. Eysel, P. Adorján, and Z. F. Kisvárday, J. Comp. Neurol. 437:259 (2001).CrossRefGoogle Scholar
  38. Z. Kourtzi et al., Neuron 37:333 (2003).CrossRefGoogle Scholar
  39. S. B. Laughlin, Z. Naturf. 36c:910 (1981).Google Scholar
  40. J. J. Atick and A. N. Redlich, Neural Comput. 2:308 (1990).Google Scholar
  41. J. J. Atick, Network: Comput. Neural Sys. 3:213 (1992).CrossRefMATHGoogle Scholar
  42. Y. Dan, J. J. Atick, and R. C. Reid, J. Neurosci. 16:3351 (1996).Google Scholar
  43. W. Bialek, D. L. Ruderman, and A. Zee, in Advances in Neural Information Processing Systems, R. P. Lippman, ed. (Morgan Kaufmann, San Mateo, CA, 1991), p. 363.Google Scholar
  44. M. Kardar and A. Zee, Proc. Natl. Acad. Sci. USA 99:15894 (2002).CrossRefADSGoogle Scholar
  45. For a small patch of the cortex, we can assume a locally linear relation between the visual and cortical coordinates, X and x. At a global level, the map is certainly non-linear. The non-linearity could itself impose rotations in the coordinate frames which complicate the notion of colinearity. Such complications are ignored in the present analysis.Google Scholar
  46. C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbaba, IL, 1962).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Theoretical Biology and Biophysics GroupLosAlamos National LaboratoryLos AlamosUSA
  2. 2.Department of PhysicsMassachusetts Institute of TechnologyCambridgeUSA

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