M. Cross and P. Hohenberg,
Rev. Mod. Phys.
65:851 (1993).
CrossRefADSJ. P. Gollub and J. S. Langer,
Rev. Mod. Phys.
71:S396 (1999).
CrossRefE. R. Kandel, J. H. Schwartz, and T. M. Jessell, Principles of Neural Science (Appleton & Lange, Norwalk, CT, 1991).
D. H. Hubel and T. N. Wiesel, J. Physiol.
160:215 (1962).
G. G. Blasdel and G. Salama,
Nature
321:579 (1986).
CrossRefADSH. Y. Lee, M. Yahyanejad, and M. Kardar,
Proc. Nat. Acad. Sci. USA
100:16036 (2003).
CrossRefADSN. V. Swindale,
Proc. R. Soc. Lond. B
215:211 (1982).
ADSN. V. Swindale,
Biol. Cybern.
66:217 (1992).
CrossRefF. Wolf and T. Geisel,
Nature
395:73 (1998).
CrossRefADSConstant 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.
A. A. Koulakov and D. B. Chklovskii,
Neuron
29:519 (2001).
CrossRefF. Wolf, PhD thesis, Univeritt Göttingen, 2000.
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.
J. B. Swift and P. C. Hohenberg,
Phys. Rev. A
15:319 (1977).
CrossRefADSM. C. Cross and P. C. Hohenberg,
Rev. Mod. Phys.
65:851 (1993).
CrossRefADSW. H. Bosking, Y. Zhang, B. Schofield, and D. Fitzpatrick, J. Neurosci.
17:2112 (1997).
P. C. Bressloff
et al.,
Phil. Trans. R. Soc. Lond. B
356:299 (2001).
CrossRefThe 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.
D. Whitney, H. C. Goltz, C. G. Thomas, J. S. Gati, R. S. Menon, and M. A. Goodale,
Science
31:878 (2003).
CrossRefADSB. 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.
M. Sigman, G. A. Cecchi, C. D. Gilbert, and M. O. Magnasco,
Proc. Natl. Acad. Sci. USA
98:1935 (2001).
CrossRefADSD. J. Field,
J. Opt. Soc. Am. A
4:2379 (1987).
ADSCrossRefD. Ruderman and W. Bialek,
Phys. Rev. Lett.
73:814 (1994).
CrossRefADSA. S. Monin and A. M. Yaglom, Statistical Mechanics, Vol. 2 (MIT, Cambridge, 1971), pp. 1–58.
I. Arad
et al,
Phys. Rev. Lett.
81:5330 (1998).
CrossRefADSJ. H. van Hateren and A. Van der Schaaf, Proc. R. Soc. London B
265:359 (1998).
W. T. Freeman and E. H. Adelson,
IEEE Trans. Patt. Anal. Mach. Intell.
13:891 (1991).
CrossRefE. Switkes, M. J. Mayer, and J. A. Sloan,
Vision Res.
18:1393 (1978).
CrossRefJ. D. Pettigrew, T. Nikara, and P. O. Bishop, Exp. Brain Res.
6:373 (1968).
B. Chapman and T. Bonhoeffer,
Proc. Natl. Acad. Sci. USA
95:2609 (1998).
CrossRefADSV. Dragoi, C. M. Turch, and M. Sur,
Neuron
32:1181 (2001).
CrossRefWe 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.
Additional pictures and data are available online from http://www.mit.edu/∼kardar/research/transversality/ModernArt/.
C. D. Gilbert and T. N. Wiesel, J. Neurosci.
9:2432 (1989).
R. Malach, Y. Amir, M. Harel, and A. Grinvald,
Proc. Natl. Acad. Sci. USA
90:10469 (1993).
CrossRefADSJ. I. Nelson and B. J. Frost,
Exp. Brain Res.
61:54 (1985).
CrossRefP. Buzás, U. T. Eysel, P. Adorján, and Z. F. Kisvárday,
J. Comp. Neurol.
437:259 (2001).
CrossRefZ. Kourtzi
et al.,
Neuron
37:333 (2003).
CrossRefS. B. Laughlin, Z. Naturf.
36c:910 (1981).
J. J. Atick and A. N. Redlich, Neural Comput.
2:308 (1990).
J. J. Atick,
Network: Comput. Neural Sys.
3:213 (1992).
CrossRefMATHY. Dan, J. J. Atick, and R. C. Reid, J. Neurosci.
16:3351 (1996).
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.
M. Kardar and A. Zee,
Proc. Natl. Acad. Sci. USA
99:15894 (2002).
CrossRefADSFor 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.
C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbaba, IL, 1962).