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The European Physical Journal Special Topics

, Volume 145, Issue 1, pp 137–157 | Cite as

Random waves in the brain: Symmetries and defect generation in the visual cortex

  • M. Schnabel
  • M. Kaschube
  • S. Löwel
  • F. Wolf
Article

Abstract.

How orientation maps in the visual cortex of the brain develop is a matter of long standing debate. Experimental and theoretical evidence suggests that their development represents an activity-dependent self-organization process. Theoretical analysis [1] exploring this hypothesis predicted that maps at an early developmental stage are realizations of Gaussian random fields exhibiting a rigorous lower bound for their densities of topological defects, called pinwheels. As a consequence, lower pinwheel densities, if observed in adult animals, are predicted to develop through the motion and annihilation of pinwheel pairs. Despite of being valid for a large class of developmental models this result depends on the symmetries of the models and thus of the predicted random field ensembles. In [1] invariance of the orientation map's statistical properties under independent space rotations and orientation shifts was assumed. However, full rotation symmetry appears to be broken by interactions of cortical neurons, e.g. selective couplings between groups of neurons with collinear orientation preferences [2]. A recently proposed new symmetry, called shift-twist symmetry [3], stating that spatial rotations have to occur together with orientation shifts in order to be an appropriate symmetry transformation, is more consistent with this organization. Here we generalize our random field approach to this important symmetry class. We propose a new class of shift-twist symmetric Gaussian random fields and derive the general correlation functions of this ensemble. It turns out that despite strong effects of the shift-twist symmetry on the structure of the correlation functions and on the map layout the lower bound on the pinwheel densities remains unaffected, predicting pinwheel annihilation in systems with low pinwheel densities.

Keywords

Correlation Function Visual Cortex Orientation Preference European Physical Journal Special Topic Visual Experience 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. F. Wolf, T. Geisel, Nature 395, 73 (1998) Google Scholar
  2. W.H. Bosking, Y. Zhang, B. Schofield, D. Fitzpatrick, J. Neurosci. 17, 2112 (1997) Google Scholar
  3. P.C. Bressloff, J.D. Cowan, M. Golubitsky, P.J. Thomas, M.C. Wiener, Philos. Trans. R. Soc. Lond. B. Biol. Sci. 356, 299 (2001) Google Scholar
  4. V. Braitenberg, A. Schüz, Cortex: statistics and geometry of neuronal connectivity (Springer, Berlin, 1998) Google Scholar
  5. M.B. Luskin, C.J. Shatz, J. Comp. Neurol. 242, 611 (1985) Google Scholar
  6. B.G. Cragg, J. Comp. Neurol. 160, 147 (1975) Google Scholar
  7. N. Daw, Visual Development (Plenum Press, New York, 1995) Google Scholar
  8. M. Stryker, Activity-dependent reorganization of afferents in the developing mammalian visual system (MIT Press, Cambridge, Mass, 1991) Vol. 3, Proceedings of the Retina Resarch Foundation Symposia, Chap. 16, p. 267 Google Scholar
  9. W. Singer, Science 270, 758 (1995) Google Scholar
  10. L.C. Katz, C.J. Shatz, Science 274, 1133 (1996) Google Scholar
  11. D. Maurer, T.L. Lewis, H.P. Brent, A.V. Levin, Science 286, 108 (1999) Google Scholar
  12. A. Antonini, M.P. Stryker, Science 260, 1819 (1993) Google Scholar
  13. J.T. Trachtenberg, M.P. Stryker, J. Neurosci. 21, 3476 (2001) Google Scholar
  14. M.C. Crair, Curr. Opin. Neurobiol. 9, 88 (1999) Google Scholar
  15. M. Kaschube, F. Wolf, T. Geisel, S. Löwel, J. Neurosci. 22, 7206 (2002) Google Scholar
  16. K.D. Miller, E. Erwin, A. Kayser, J. Neurobiol. 41, 44 (1999) Google Scholar
  17. K.D. Miller, J. Neurosci. 14, 409 (1994) Google Scholar
  18. O. Creutzfeldt, Cortex Cerebri: performance, structural and functional organization of the cortex (Oxford University Press, 1995) Google Scholar
  19. S. LeVay, S. Nelson, Columnar Organization of the visual cortex (Macmillan, Houndsmill, 1991), Chap. 11, p. 266 Google Scholar
  20. D.H. Hubel, T.N. Wiesel, J. Physiol. 160, 106 (1962) Google Scholar
  21. N.V. Swindale, D. Shoham, A. Grinvald, T. Bonhoeffer, M. Hübener, Nat. Neurosci. 3, 822 (2000) Google Scholar
  22. T. Bonhoeffer, A. Grinvald, Nature 353, 429 (1991) Google Scholar
  23. B. Chapman, M.P. Stryker, T. Bonhoeffer, J. Neurosci. 16, 6443 (1996) Google Scholar
  24. M.C. Crair, D.C. Gillespie, M.P. Stryker, Science 279, 566 (1998) Google Scholar
  25. L.E. White, D.M. Coppola, D. Fitzpatrick, Nature 411, 1049 (2001) Google Scholar
  26. J. Sharma, A. Angelucci, M. Sur, Nature 404, 841 (2000) Google Scholar
  27. N.V. Swindale, Proc. R. Soc. Lond. B. Biol. Sci. 208, 243 (1980) Google Scholar
  28. N.V. Swindale, Proc. R. Soc. Lond. B. Biol. Sci. 215, 211 (1982) Google Scholar
  29. N.V. Swindale, Network 7, 161 (1996) Google Scholar
  30. R. Durbin, G. Mitchison, Nature 343, 644 (1990) Google Scholar
  31. Obermayer, Blasdel, Schulten, Phys. Rev. A 45, 7568 (1992) Google Scholar
  32. E. Erwin, K. Obermayer, K. Schulten, Neural. Comput. 7, 425 (1995) Google Scholar
  33. F. Wolf, T. Geisel, Lecture Notes Phys. 527, 174 (1991) Google Scholar
  34. H.Y. Lee, M. Yahyanejad, M. Kardar, Proc. Natl. Acad. Sci. USA 100, 16036 (2003) Google Scholar
  35. F. Wolf, Phys. Rev. Lett. 95, 208701 (2005) Google Scholar
  36. P.C. Bressloff, Biol. Cybern. 93, 256 (2005) Google Scholar
  37. F. Wolf, Les Houches 2003 Lecture Notes, in Methods and Models in Neurophysics (Elsevier, Amsterdam, 2005) Google Scholar
  38. R. Adler, The Geometry of Random Fields (Wiley, New York, 1981) Google Scholar
  39. M. Schnabel, M. Kaschube, L. White, D. Coppola, S. Löwel, F. Wolf, Shift-twist Symmetry in natural images and orientation maps, in Society for Neuroscience Abstracts (2005), Vol. 31 Google Scholar
  40. I.S. Gradshtein, I. Ryzhik, Table of Integrals, Series, and Products, 6th edn. (Academic Press, New York, 2000) ISBN 0-12-294760-6 Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Max-Planck-Institute for Dynamics and Self-Organization and Bernstein-Center for Computational NeuroscienceGoettingenGermany
  2. 2.Physics Department and Lewis-Sigler Institute, Princeton UniversityPrincetonUSA
  3. 3.Institute of General Zoology and Animal Physiology, University JenaJenaGermany

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