{"title": "Simple Spin Models for the Development of Ocular Dominance Columns and Iso-Orientation Patches", "book": "Advances in Neural Information Processing Systems", "page_first": 26, "page_last": 31, "abstract": null, "full_text": "Simple Spin Models \n\nfor the Development of Ocular Dominance \n\nColumns and Iso-Orientation Patches \n\nJ.D. Cowan & A.E. Friedman \n\nDepartment of Mathematics. Committee on \nNeurobiology. and Brain Research Institute. \n\nThe University of Chicago. 5734 S. Univ. Ave .\u2022 \n\nChicago. Illinois 60637 \n\nAbstract \n\nSimple classical spin models well-known to physicists as the ANNNI \nand Heisenberg XY Models. in which long-range interactions occur in \na pattern given by the Mexican Hat operator. can generate many of the \nstructural properties characteristic of the ocular dominance columns \nand iso-orientation patches seen in cat and primate visual cortex. \n\n1 INTRODUCTION \n\nIn recent years numerous models for the formation of ocular dominance columns \n(Malsburg, 1979; Swindale. 1980; Miller. Keller, & Stryker. 1989) and of iso-orientation \npatches (Malsburg 1973; Swindale 1982 & Linsker 1986)have been published. Here we \nshow that simple spin models can reproduce many of the observed features. Our work is \nsimilar to, but independent of a recent study employing spin models (Tanaka. 1990). \n\n26 \n\n\fSimple Spin Models \n\n27 \n\n1.1 OCULAR DOMINANCE COLUMNS \n\nWe use a one-dimensional classical spin Hamiltonian on a two-dimensional lattice with \nlong-range interactions. Let O'i be a spin vector restricted to the orientations i and J, in \nthe lattice space, and let the spin Hamiltonian be: \n\nHoD = -L. L. Wij O'i \u2022 O'j , \n\ni j;ci \n\nwhere Wij is the well-known \"Mexican Hat\" distribution of weights: \n\nWij = a+ exp(- li-jI2/ 0':) - a_ exp(-li-jI2/ cr) \n\n0 \nHO~ = -L L w.. - L. L. w\u00b7\u00b7 \n. . . IJ . . . IJ \n1 J;Cl \n\n1 J;CI \n\ns \n\n(1) \n\n(2) \n\n(3) \n\n:i \n\nFigure 1. Pattern of Ocular Dominance which \nresults from simulated annealing of the energy \nfunction HOD. Light and dark shadings correspond \nrespectively to the two eyes. \n\nLet s denote retinal fibers from the same eye and 0 fibers from the opposite eye. Then \nHOD represents the \"energy\" of interactions between fibers from the two eyes. It is \nrelatively easy to find a configuration of spins which minimizes HO~ by simulated \nannealing (Kirkpatrick, Gelatt & Vecchi 1983). The result is shown in figure 1. It will \nbe seen that the resulting pattern of right and left eye spins O'R and O'L is disordered, but \nat a constant wavelength determined in large part by the space constants 0'+ and 0'_ . \n\n\f28 \n\nCowan and friedman \n\nBreaking the symmetry of the initial conditions (or letting the lattivce grow \nsystematically) results in ordered patterns. \n\nIf HOD is considered to be the energy function of a network of spins exhibiting gradient \ndynamics (Hirsch & Smale. 1974). then one can write equations for the evolution of spin \npatterns in the form: \n\nddt (Jl~ = -_a_ Hoo = L w~~ (J~ \nJ \n\n. . IJ \nJ\u00a2l \n\na a \n(J. \n1 \n\na \n= L w .. (J\u00b7 + L w .. (J\u00b7 = L w .. (J\u00b7 \nIJ 1 \n\ns ao ~ \nIJ 1 \nIJ 1 \n\nj;ti \n\nj;ti \n\nj;ti \n\n~ \n- L w .. (J. \u2022 \nj;ti \nIJ 1 \n\n(4) \n\nwhere a = R or L. ~ = L or R respectively. Equation (4) will be recognized as that \nproposed by Swindale in 1979. \n\n1.2 ISO-ORIENTATION PATCHES \n\nNow let (Ji represent avec tor in the plane of the lattice which runs continuously from i \nto J, without reference to eye class. It follows that \n\n(5) \n\nwhere 9i is the orientation of the ith spin vector. The appropriate classical spin \nHamiltonian is: \n\nHIO = - L L Wij (Ji \u2022 erj = \n\ni j;ti \n\n-L L Wij leri I leri I cos(9i - 9j). \ni j;ti \n\n(6) \n\nPhysicists will recognize HOD as a form of the Ising Lattice Hamiltonian with long-range \nalternating next nearest neighbor interactions. a type of ANNNI model (Binder. 1986) \nand HIO as a similar form of the Heisenberg XY Model for antiferromagnetic materials \n(Binder 1986). \n\nAgain one can find a spin configuration that minimizes HIO by simulated annealing. The \nresult is shown in figure 2 in which six differing orientations are depicted. corresponding \nto 300 increments (note that 9 + 1t is equivalent to 9). It will be seen that there are long \nstretches of continuously changing spin vector orientations, with intercalated \ndiscontinuities and both clockwise and counter-clockwise singular regions around which \nthe orientations rotate. A one-dimensional slice shows some of these features, and is \nshown in figure 3. \n\n\fSimple Spin Models \n\n29 \n\nFigure 2. Pattern of orientation patches obtained by \nsimulated annealing of the energy function RIO. Six \ndiffering orientations varying from 00 to 1800 are \nrepresented by the different shadings. \n\n180 \n\n9. 90 \n\nI \n\no \n\no \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\nCell Number \n\nFigure 3. Details of a one-dimensional slice through \nthe orientation map. Long stretches of smoothly \nchanging orientations are evident. \n\nThe length of O'i is also correlated with these details. Figure 4 shows that 100i I is large in \nsmoothly changing regions and smallest in the neighborhood of a singularity. In fact this \nmodel reproduces most of the details of iso-orientation patches found by Blasdel and \nSalama (1986). \n\n\f30 \n\nCowan and friedman \n\n10 \n\n5 \n\no \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\nCell Number \n\nFigure 4. Variation of leri I along the same one-dim. \nslice through the orientation map shown in figure 3. \nThe amplitude drops only near singular regions. \n\nFor example, the change in orientation per unit length, Igrad9il is shown in figure 5. It \nwill be seen that the lattice is \"tiled\", just as in the data from visual cortex, with max \nIgrad9illocated at singularities. \n\n:.- . \n\n:: .. -\n\n. ;;:.:::: .... \n\nFigure S. Plot of Igrad9i I corresponding to the \norientation map of figure 2. Regions of maximum \nrate of change of 9i are shown as shaded. These \ncorrespond with the singular regions of figure 2. \n\n\fSimple Spin Models \n\n31 \n\nOnce again, if HIO is taken to be the energy of a gradient dynamical system, there results \nthe equation: \n\n(7) \n\nd \ndt 0'1' = -- HIO = L w\u00b7\u00b7(1\u00b7 \n.. IJ J \nJ~1 \n\na \n\":}_ \nau\u00b7 \n1 \n\nwhich is exactly that equation introduced by Swindale in 1981 as a model for the \nstructure of iso-orientation patches. There is an obvious relationship between such \nequations, and recent similar treatments (Durbin & Mitchison 1990; Schulten, K. 1990 \n(preprint); Cherjnavsky & Moody, 1990). \n\n2 CONCLUSIONS \n\nSimple classical spin models well-known to physicists as the ANNNI and Heisenberg \nXY Models, in which long-range interactions occur in a pattern given by the Mexican \nHat operator, can generate many of the structural properties characteristic of the ocular \ndominance columns and iso-orientation patches seen in cat and primate visual cortex. \n\nAcknowledgements \n\nThis work is based on lectures given at the Institute for Theoretical Physics (Santa \nBarbara) Workshop on Neural Networks and Spin Glasses, in 1986. We thank the \nInstitute and The University of Chicago Brain Research Foundation for partial support of \nthis work. \n\nReferences \n\nMalsburg, Ch.v.d. (1979), BioI. Cybern., 32, 49-62. \nSwindale, N.V. (1980), Proc. Roy. Soc. Lond. B, 208, 243-264. \nMiller, K.D., Keller, J.B. & Stryker, M. P. (1989), Science, 245,605-611. \nMalsburg, Ch.v.d. (1973), BioI. Cybern., 14,85-100. \nSwindale, N.V. (1982), Proc. Roy. Soc. Lond. B, 215, 211-230. \nLinsker, R. (1986), PNAS, 83, 7508-7512; 8390-8394; 8779-8783. \nTanaka, S. (1990), Neural Networks, 3, 6, 625-640. \nKirkpatrick, S., Gelatt, C.D. Jr. & Vecchi, M.P. (1983), Science, 229, 671-679. \nHirsch, M.W. & Smale, S. (1974), Differential Equations. Dynamical Systems. \nand Linear Algebra. (Academic Press, NY). \nBinder, K. (1986), Monte Carlo Methods in Statistical Physics, (Springer, NY.). \nBlasdel, G.G. & Salama, G. (1986), Nature, 321,579-587. \nDurbin, R. & Mitchison, G. (1990), Nature, 343, 6259, 644-647. \nSchulten, K. (1990) (preprint). \nCherjnavsky, A. & Moody, J. (1990), Neural Computation, 2, 3, 334-354. \n\n\f", "award": [], "sourceid": 350, "authors": [{"given_name": "J.D.", "family_name": "Cowan", "institution": null}, {"given_name": "A.", "family_name": "Friedman", "institution": null}]}