{"title": "A Critical Comparison of Models for Orientation and Ocular Dominance Columns in the Striate Cortex", "book": "Advances in Neural Information Processing Systems", "page_first": 93, "page_last": 100, "abstract": null, "full_text": "A Critical Comparison of Models for \nOrientation and Ocular Dominance \n\nColumns in the Striate Cortex \n\nE. Erwin \n\nBeckman Institute \nUniversity of Illinois \n\nUrbana, IL 61801, USA \n\nK. Obermayer \n\nTechnische Fakultat \nU niversitat Bielefeld \n33615 Bielefeld, FRG \n\nK. Schulten \n\nBeckman Institute \nUniversity of Illinois \n\nUrbana, IL 61801, USA \n\nAbstract \n\nMore than ten of the most prominent models for the structure \nand for the activity dependent formation of orientation and ocu(cid:173)\nlar dominance columns in the striate cort(>x have been evaluated. \nWe implemented those models on parallel machines, we extensively \nexplored parameter space, and we quantitatively compared model \npredictions with experimental data which were recorded optically \nfrom macaque striate cortex. \nIn our contribution we present a summary of our results to date. \nBriefly, we find that (i) despite apparent differences, many models \nare based on similar principles and, consequently, make similar pre(cid:173)\ndictions, (ii) certain \"pattern models\" as well as the developmental \n\"correlation-based learning\" models disagree with the experimen(cid:173)\ntal data, and (iii) of the models we have investigated, \"competitive \nHebbian\" models and the recent model of Swindale provide the \nbest match with experimental data. \n\n1 Models and Data \n\nThe models for the formation and structure of orientation and ocular dominance \ncolumns which we have investigated are summarized in table 1. Models fall into \ntwo categories: \"Pattern models\" whose aim is to achieve a concise description of \nthe observed patterns and \"developmental models\" which are focussed on the pro-\n\n\f94 \n\nE. Erwin, K. Obermayer, K. Schulten \n\nClass \nPattern \nModels Models \n\nType \nStructural \n\nSpectral \nModels \n\nDevelop. Correlation \nModels \n\nBased Learning \nCompetl bve \nHebbian \n\nOther \n\nReference \nHubel and Wiesel 1977 [~I \n\nLinsker 1986c J12] \nMiller 1989, 1994 113, 14) \n\nGotz 1987 (8) \nBaxter and Dow 1989 11) \nROJer and Schwartz 1990 J20) \nNiebur and Worgotter 1993 (15) \n\nModel \n1. Icecube \n2. Pinwheel Braitenberg and Braitenberg 1979 161 \n3. Gotz \n4. Baxter \n5. ROJer \n6. Niebur \n7. Swindale Swindale 1992a (21) \n8. Linsker \n9. Miller \n10. ~UM-h Ubennayer, et. al. 1990 P~J \nObermayer, et. al. 1992(17) \n11. SOM-I \nDurbin and Mitchison 1990 (7) \n12. EN \n13. Tanaka \nTanaka 1991 [22J \n14. Yuille \nYuille, et. al. 1992 (23) \n\nTable 1: Models of visual cortical maps which have been evaluated. \n\ncesses underlying their formation. Pattern models come in two varieties, \"structural \nmodels\" and \"spectral models\", which describe orientation and ocular dominance \nmaps in real and in Fourier space, respectively. Developmental models fall into the \ncategories \"correlations based learning\", \"competitive Hebbian\" learning and a few \nmiscellaneous models. \n\nModels are compared with data obtained from macaque striate cortex through opti(cid:173)\ncal imaging [2, 3, 4, 16]. Data were recorded from the representation of the parafovea \nfrom the superficial layers of cortex. In the following we will state that a particular \nmodel reproduces a particular feature of the experimental data (i) if there exists a \nparameter regime where the model generates appropriate patterns and (ii) if the \nphenomena are robust. We will state that a particular model does not reproduce a \ncertain feature (i) if we have not found an appropriate parameter regime and (ii) if \nthere exists either a proof or good intuitive reasons that a lllodel cannot reproduce \nthis feature. \n\nOne has to keep in mind, though, that model predictions are compared with a fairly \nspecial set of data. Ocular dominance patterns, e.g., are known to vary between \nspecies and even between different regions within area 17 of an individual. Con(cid:173)\nsequently, a model which does not reproduce certain featurE'S of ocular dominance \nor orientation colulllns in the macaque may well describE' those patterns in other \nspecies. Interspecies differences, however, are not. the focus of this contribution; \nresults of corresponding modelling studies will be reported E'lsewhere. \n\n2 Examples of Organizing Principles and Model Predictions \n\nIt has been suggested t.hat the most important principles underlying the pattern of \norientation and ocular dominance are \"continuity\" and \"diversity\" [7. 19, 21]. Con(cid:173)\ntinuity, because early image processing is often local in fE'atnre space, and diversity, \nbecause, e.g., the visual system may want to avoid perceptual scotomata. The con(cid:173)\ntinuity and diversity principles underlie almost all dE'scriptive and developmental \n\n\fA Critical Comparison of Models for Orientation and Ocular Dominance Columns \n\n95 \n\nFigure 1: Typical patterns of orientation preferences as they are predicted by six \nof the models list.ed in Table 1. Orientation preferences are coded by gray values, \nwhere black - whit.e denotes preferences for vertical _ horizont.al - vertical. Top \nrow (left to right): Models 7, 11, 9. Bottom row (left to right) Models 5, 12, 8. \n\nmodels, but. maps which comply with t.hese principles often differ in qualitat.ive ways: \nThe icecube model, e.g., obeys bot.h principles but. contains no singularities in the \norient.ation preference map and no branching of ocular dominance bands. Figure 1 \nshows orientat.ion maps generated by six different. algorithms taken from Tab. 1. \nAlthough all pat.t.erns are consist.ent. wit.h the continuit.y and diversity const.raints, \ncloser comparison reveals differences. Thus additional element.s of organization must \nbe considered. \n\nIt has been suggested that maps are characterized by local correlations and global \ndisorder. Figure 2 (left) shows as an exam pIe two- point correlation functions of \norientation maps. The autocorrelation function [17] of one of the Cartesian coor(cid:173)\ndinat.es of t.he orientation vector is plotted as a function of cortical distance. The \nfact. that all correlation functions decay indicates that the orientation map exhibits \nglobal disorder. Global disorder is predicted by all models except. the early pat(cid:173)\ntern models 6, 8 and 9. Figure 2 (right) shows the corresponding power spectra. \nBandpass-like spectra which are typical for the experiment.al data [16] are well pre(cid:173)\ndicted by models 10- 12. Interestingly, they are not predicted by model 9, which \nalso fails reproducing the Mexican-hat shaped correlation functions (bold lines), \nand model 13. \n\nBased on the fact that. experimental maps are characterized by a bandpass-like \npower spectrum it has been suggested that orientation maps may be organized \n\n\f96 \n\nE. Erwin, K. Obermayer, K. Schulten \n\n1.0 . . ._ - - - - - - - - - - - - - , \n\n-0.5 ~----:.\"_ ................. - _ __....-_ _4 \n40 \n\n30 \n\no \n\n10 \n\n20 \n\ndistance (normalized) \n\n1,0 -\"'0 \n.~ 1,8 -\u00a7 1,6 \nS 0,4 \n... \n~ \n& \n\n0,2 \n\n0,0 \n\n0 \n\n5 \n\n10 \n\n15 \n\n20 \n\ndistance (normalized) \n\nFigure 2: Left: Spatial autocorrelation functions for one of the cartesian coordi(cid:173)\nnates of the orientat.ion vector. Aut.ocorrelation functions were averaged over all \ndirections. Right: Complex power spectra of orientation maps. Power was aver(cid:173)\naged over all directions of the wave vector. Modelnumhers as in Tab. 1. \n\naccording to four principles [15]: continuity, diversity, homogeneity and isotropy. \nIf those principles are implemented using bandpass filtered noise the resulting \nmaps [15, 21] indeed share many properties with the experimental data. Above \nprinciples alone, however, are not sufficient: (i) There are models such as model \u00b75 \nwhich are based on those principles but generate different patterns, (ii) homogene(cid:173)\nity and isotropy are hardly ever fulfilled ([16] and next paragraph), and (iii) those \nprinciples cannot. account for correlations between maps of various response prop(cid:173)\nerties [16]. \n\nMaps of orientation and ocular dominance in the macaque are anisotropic, i.e., \nthere exist preferred directions along which orientation and ocular dominance slabs \nalign [16]. Those anisotropies can emerge due to different mechanisms: (i) sponta(cid:173)\nneous symmetry breaking, (ii) model equations, which are not rotation invariant, \nand (iii) appropriately chosen boundary conditions. Figure 3 illustrates mecha(cid:173)\nnisms (ii) and (iii) for model 11. Bot.h mechanisms indeed predict anisotropic \npat.terns, however, preferred directions of orientation and ocular dominance align in \nboth cases (fig. 3, left and center). This is not true for the experimental data, where \npreferred directions tend to be orthogonal [16]. Ort.hogonal preferred directions can \nbe generated by llsing different neighborhood funct.ions for different components of \nthe feature vector (fig. 3, right). However, this is not a satisfactory solution, and \nthe issue of anisotropies is still unsolved. \n\nThe pattern of orientation preference in the area 17 of the macaque exhibits four \nlocal elements of organization: linear zones, singularit.ies, saddle point.s and frac(cid:173)\ntures [16]. Those element.s are correctly predict.ed by most. of the pat.t,ern models, \nexcept models 1- 3, and they appear in the maps generated by models 10- 14. In(cid:173)\nterestingly' models 9 and 13 predict very few linear zones, which is related to the \nfact. that those models generate orientat.ion maps with lowpass-like power spect.ra. \n\nAnother important property of orientation maps is that orientation preferences and \ntheir spatial layout across cortex are not correlated which each other. One conse-\n\n\fA Critical Comparison of Models for Orientation and Ocular Dominance Columns \n\n97 \n\n.~; \n\n:.~ . \n\n+ \n\n+ \n\n+ \n\n' .~,!-. \n\n+ \n\n..4-.. \n\n+ \n\nFigure 3: Anisotropic orientation and ocular dominance maps generated by model \n11. The figure shows Fourier spectra [17] of orientation (top row) and ocular dom(cid:173)\ninance maps (bottom row). Left: Maps generated with an elliptic neighborhood \nfunction (case (ii), see text); Center: Maps generated using circular input lay(cid:173)\ners and an elliptical cortical sheet (case (iii), see text), Right: Maps generated \nwith different, elliptic neighborhood functions for orientation preference and ocular \ndominance. '+' symbols indicate the locations of the origin. \n\nquence is that there exist singularities, near which the curl of the orientation vector \nfield does not vanish (fig. 4, left). This rules out a class of pattern models where the \norientation map is derived from the gradient of a potential function, model 5. Fig(cid:173)\nure 4 (right) shows another consequence of this property. In those figures cortical \narea is plotted against the angular difference between the iso-orientation lines and \nthe local orientation preference. The even distribution found in the experimental \ndata is correctly predicted by models 1,6, 7 and 10-12. Model 8, however, predicts \npreference for large difference angles while model 9 - over a wide range of parameters \n- predicts preference for small difference angles (bold lines). \n\nFinally, let us consider correlations between the patterns of orientation preference \nand ocular dominance. Among the more prominent relationships present in macaque \ndata are [3, 16,21]: (i) Singularities are aligned with the centers of ocular dominance \nbands, (ii) fractures are either aligned or run perpendicular, and (iii) iso-orientation \nbands in linear zones intersect ocular dominance bands at approximately right an(cid:173)\ngles. Those relationships are readily reproduced only by models 7 and 10- 12. For \nmodel 9 reasonable orientation and ocular dominance patterns have not been gen(cid:173)\nerated at the same time. It would seem as if the parameter regime where reasonable \norientation columns emerge is incompatible with the parameter regime where ocular \ndominance patterns are formed. \n\n\f98 \n\nE. Erwin, K. Obermayer, K. Schulten \n\n0J.5 \n\n0.10 \n\n'\" e \n'\" C+-I \n0 \nu \nbO \n'\" = O. \n~ \n8. \n\n0 .. +----.--__ -....-_--1 \n03060 \n90 \ndifference angle ( degrees) \n\nFigure 4: Left: This singularity is an example of a feature in the experimental \ndata which is not allowed by model 5. The arrows indicat.e orientation vectors, \nwhose angular component is twice the value of the local orientation preference. \nRight: Percentage of area as a function of the angular difference bet.ween preferred \norient.ation and t.he local orientation gradient vector. Model numbers as in Table 1. \n\n3 The Current Status of the Model Comparison Project \n\nLack of space prohibit.s a detailed discussion of our findings hut we have summarized \nthe current status of our project in Tables 2 and 3. Given the models list.ed in \nTab. 1 and given the properties of t.he orientation and ocular dominance patt.erns \nin macaque striate cortex listed in Tables 2 and 3 it is models 7 and 10-12 which \ncurrently are in best agreement with the data. Those models, however, are fairly \nabstract. and simplified, and they cannot easily be extended to predict receptive \nfield structure. Biological realism and predictions about. receptive fields are the \nadvantages of models 8 and 9. Those models, however, cannot account for the \nobserved orientation patterns. It. would, therefore, be of high interest, if elements \nof both approaches could be combined to achieve a better description of the dat.a. \n\nThe main conclusion, however, is that there are now enough data available to allow \na better evaluation of model approaches than just by visual comparison of the \ngenerated pat.terns. It. is our hope, that future studies will address at least those \npropert.ies of t.he patterns which are known and well described, some of which are \nlist.ed in Tables 2 and 3. \nIn case of developmental models more stringent tests \nrequire experiments which (i) monitor the actual time-course of pattern formation, \nand which (ii) study pattern development under experimentally modified conditions \n(deprivation experiments). Currently there is not enough data available to constrain \nmodels but the experiments are under way [5, 10, 11, 18]. \n\nAcknowledgements \n\nVVe are very much indebted to Drs. Linsker, Tanaka and Yuille for sharing modelling \ndata. E.E. thanks t.he Beckman Institute for support.. K.O. thanks ZiF (Universitat \nBielefeld) for it.s hospitality. Computing time on a CM-2 and a CM-5 was made \navailable by NCSA. \n\n\fA Critical Comparison of Models for Orientation and Ocular Dominance Columns \n\n99 \n\nno. \n\n1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n9 \n10 \n11 \n12 \n13 \n14 \n\ndis-\norder \n\n-\n-\n-\n+2 \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n\nband-\npass \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n-\n+ \n+ \n+ \n-\n? \n\nlinear \nzones \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n-\n-\n-\n+ \n+ \n+ \n? \n\nProperties of OR Maps \n\nsing. \nsaddle \npoints \u00b11/2 \n\nfracto \n\nindep. \ncoord. \n\nhigh \nspec. \n\namso- UR-\ntropy \nbias \n\n-\n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n\n-\n-\n+ \n+2 \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n\n-\n-\n-\n-\n+1 \n+1 \n+1 \n+ \n+ \n+1 \n+1 \n+1 \n+1 \n+ \n\n+ \n-\n-\n-\n-\n+ \n+ \n-\n-/+ \n+ \n+ \n+ \n+ \n? \n\nn \nn \nn \nn \n-\n-\n+ \nn \n+ \n+ \n+ \n+ \n+ \nn \n\n+ \nn \nn \n+ \n+ \n+ \n+ \nn \nn \n+ \n+ \n+ \nn \nn \n\nn \nn \nn \nn \nn \nn \nn \nn \nn \n+ \n+ \n+ \nn \nn \n\nTable 2: Evaluation of orientation (OR) map models. Properties of the experimen(cid:173)\ntal maps include (left to right): global disorder; bandpass-like power spectra; the \npresence of linear zones in roughly 50% of the map area; the presence of saddle \npoints, singularities (\u00b11/2 with equal densities), and fractures; independence be(cid:173)\ntween cortical and orientation preference coordinates; a distribution favoring high \nvalues of orientation specificity; global anisotropy; and a possible orientation bias. \nSymbols: '+': There exists a parameter regime in which a model generates maps \nwith this property; '-': The model cannot reproduce this property; \"n': The model \nmakes no predictions; \"?': Not enough data available. 1 Models agree with the data \nonly if one assumes that fractures are loci of rapid orientation change rather than \nreal discontinuities. 20ne of several cases. \n\nReferences \n\n[1] W. T. Baxter and B. M. Dow. Bioi. Cybern. , 61:171-182, 1989. \n[2] G. G. Blasdel. J. Neurosci., 12:3115-3138, 1992. \n[3] G. G. Blasdel. J. Neurosci., 12:3139-3161,1992. \n[4] G. G. Blasdel and G. Salama. Nature, 321:579- 585, 1986. \n[5] T. Bonhoeffer, D. Kim, and W. Singer. Soc. Neurosci. Abs., 19:1800, 1993. \n[6] V. Braitenberg and C. Braitenberg. Bioi. Cybern., 33:179- 186, 1979. \n[7] R. Durbin and G. Mitchison. Nature, 343:341-344, 1990. \n[8] K. G. Gotz. Bioi. Cybern., 56:107-109, 1987. \n[9] D. Rubel and T. N. Wiesel. Proc. Roy. Soc. Lond. B, 198:1-59, 1977. \n[10] D. Rubel, T. N. Wiesel, and S. LeVay. Phil. Trans. Roy. Soc. Lond. B, 278:377-\n\n409, 1977. \n\n[11] D. Kim and T. Bonhoeffer. Soc. Neurosci. Abs., 19:1800, 1993. \n[12] R. Linsker. Proc. Nat. Acad. Sci., USA, 83:8779-8783, 1986. \n\n\fE. Erwin, K. Obermayer, K. Schulten \n\nProperties of OD Maps \n\ndis-\norder \n\nCorrelations Between OR and OD \nspec. \nsing. \nlocal \nvs.OD vs.OD \northog. \n\nglobal \northog. \n\n100 \n\nno. \n\n1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n9 \n10 \n11 \n12 \n13 \n14 \n\nsegre-\ngation \n\n+ \nn \n+ \nn \n+ \nn \n+ \nn \n+ \n+ \n+ \n+ \n+ \n+ \n\nani so- OD-\ntropy \nbias \n+ \n+ \nn \nn \n+ \nn \nn \nn \n-\n+ \nn \nn \n+ \n+ \nn \nn \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n+ \n\nstra-\nbismus \n\nn \nn \nn \nn \nn \nn \nn \nn \n+ \n+ \n+ \n+ \n+ \nn \n\n-\nn \n-\nn \n+ \nn \n+ \nn \n+ \n+ \n+ \n+ \n+ \n+ \n\n+M \nn \n+ \nn \n+1 \nn \n-\nn \n?1 \n+ \n+2 \n+1,2 \nn \nn \n\n+:.l \nn \nn \nn \n_I \nn \n+ \nn \n?1 \nn \nn \nn \nn \nn \n\n-\nn \n+2 \nn \n+I ,O! \nn \n+ \nn \n?1 \n+ \n+2 \n+1,2 \nn \nn \n\nn \nn \nn \nn \n_I \nn \n+ \nn \n?1 \n+ \n+2 \n+1,2 \nn \nn \n\nTable 3: Left: Evaluation of ocular dominance (OD) map models. Properties of \nthe experimental maps include (left to right): Segregated bands of eye dominance; \nglobal disorder; bandpass-like power spectra; global anisotropy; a bias to the repre(cid:173)\nsentation of one eye; and OD-patterns in animals with strabismus. Right: Evalu M \nation of correlations between OD and OR. Experimental maps show (left to right): \nLocal and global orthogonality between OR and OD slabs; singularities preferably \nin monocular regions, and lower OR specificity in monocular regions. 1 Authors \ntreated OD and OR in independent models, but we consider a combined version. \n2 Correlations are stronger than in the experimental data. \n\n[13] K. D. Miller. J. Neurosci., 14:409- 441, 1994. \n[14] K. D. Miller, J. B. Keller, and M. P. Stryker. Science, 245:605-615, 1989. \n[15] E. Niebur and F. Worgotter. In F. H. Eeckman and J. M. Bower, Computation \n\nand Neura.l Systems, pp. 409-413. Kluwer Academic Publishers, 1993. \n\n[16] K. Obermayer and G. G. Blasdel. J. Neurosci., 13:4114-4129, 1993. \n[17] K. Obermayer, G. G. Blasdel, and K. Schulten. Phys. Rev. A, 45:7568-7589, \n\n1992. \n\n[18] K Obermayer, L. Kiorpes, and G. G. Blasdel. In J. D. Cowan at al., Advances \nin Neural Information Processing Systems 6. Morgan Kaufmann, 1994. 543-\n550. \n\n[19] K. Obermayer, H. Ritter, and K. Schulten. Proc. Nat. Acad. Sci., USA, \n\n87:8345- 8349, 1990. \n\n[20] A. S. Rojer and E. L. Schwartz. Bioi. Cybern., 62:381- 391, 1990. \n[21] N. V. Swindale. Bioi. Cybern., 66:217-230, 1992. \n[22] S. Tanaka. Bioi. Cybern., 65:91- 98, 1991. \n[23] A. L. Yuille, J. A. Kolodny, and C. W. Lee. TR 91-3, Harvard Robotics \n\nLaboratory, 1991. \n\n\f", "award": [], "sourceid": 1017, "authors": [{"given_name": "E.", "family_name": "Erwin", "institution": null}, {"given_name": "K.", "family_name": "Obermayer", "institution": null}, {"given_name": "K.", "family_name": "Schulten", "institution": null}]}