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.
Similar content being viewed by others
References
F. Wolf, T. Geisel, Nature 395, 73 (1998)
W.H. Bosking, Y. Zhang, B. Schofield, D. Fitzpatrick, J. Neurosci. 17, 2112 (1997)
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)
V. Braitenberg, A. Schüz, Cortex: statistics and geometry of neuronal connectivity (Springer, Berlin, 1998)
M.B. Luskin, C.J. Shatz, J. Comp. Neurol. 242, 611 (1985)
B.G. Cragg, J. Comp. Neurol. 160, 147 (1975)
N. Daw, Visual Development (Plenum Press, New York, 1995)
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
W. Singer, Science 270, 758 (1995)
L.C. Katz, C.J. Shatz, Science 274, 1133 (1996)
D. Maurer, T.L. Lewis, H.P. Brent, A.V. Levin, Science 286, 108 (1999)
A. Antonini, M.P. Stryker, Science 260, 1819 (1993)
J.T. Trachtenberg, M.P. Stryker, J. Neurosci. 21, 3476 (2001)
M.C. Crair, Curr. Opin. Neurobiol. 9, 88 (1999)
M. Kaschube, F. Wolf, T. Geisel, S. Löwel, J. Neurosci. 22, 7206 (2002)
K.D. Miller, E. Erwin, A. Kayser, J. Neurobiol. 41, 44 (1999)
K.D. Miller, J. Neurosci. 14, 409 (1994)
O. Creutzfeldt, Cortex Cerebri: performance, structural and functional organization of the cortex (Oxford University Press, 1995)
S. LeVay, S. Nelson, Columnar Organization of the visual cortex (Macmillan, Houndsmill, 1991), Chap. 11, p. 266
D.H. Hubel, T.N. Wiesel, J. Physiol. 160, 106 (1962)
N.V. Swindale, D. Shoham, A. Grinvald, T. Bonhoeffer, M. Hübener, Nat. Neurosci. 3, 822 (2000)
T. Bonhoeffer, A. Grinvald, Nature 353, 429 (1991)
B. Chapman, M.P. Stryker, T. Bonhoeffer, J. Neurosci. 16, 6443 (1996)
M.C. Crair, D.C. Gillespie, M.P. Stryker, Science 279, 566 (1998)
L.E. White, D.M. Coppola, D. Fitzpatrick, Nature 411, 1049 (2001)
J. Sharma, A. Angelucci, M. Sur, Nature 404, 841 (2000)
N.V. Swindale, Proc. R. Soc. Lond. B. Biol. Sci. 208, 243 (1980)
N.V. Swindale, Proc. R. Soc. Lond. B. Biol. Sci. 215, 211 (1982)
N.V. Swindale, Network 7, 161 (1996)
R. Durbin, G. Mitchison, Nature 343, 644 (1990)
Obermayer, Blasdel, Schulten, Phys. Rev. A 45, 7568 (1992)
E. Erwin, K. Obermayer, K. Schulten, Neural. Comput. 7, 425 (1995)
F. Wolf, T. Geisel, Lecture Notes Phys. 527, 174 (1991)
H.Y. Lee, M. Yahyanejad, M. Kardar, Proc. Natl. Acad. Sci. USA 100, 16036 (2003)
F. Wolf, Phys. Rev. Lett. 95, 208701 (2005)
P.C. Bressloff, Biol. Cybern. 93, 256 (2005)
F. Wolf, Les Houches 2003 Lecture Notes, in Methods and Models in Neurophysics (Elsevier, Amsterdam, 2005)
R. Adler, The Geometry of Random Fields (Wiley, New York, 1981)
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
I.S. Gradshtein, I. Ryzhik, Table of Integrals, Series, and Products, 6th edn. (Academic Press, New York, 2000) ISBN 0-12-294760-6
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schnabel, M., Kaschube, M., Löwel, S. et al. Random waves in the brain: Symmetries and defect generation in the visual cortex. Eur. Phys. J. Spec. Top. 145, 137–157 (2007). https://doi.org/10.1140/epjst/e2007-00152-5
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2007-00152-5