{"title": "Interaction Among Ocularity, Retinotopy and On-center/Off-center Pathways During Development", "book": "Advances in Neural Information Processing Systems", "page_first": 18, "page_last": 25, "abstract": null, "full_text": "INTERACTION AMONG OCULARITY, \nRETINOTOPY AND ON-CENTER/OFF(cid:173)\n\nCENTER PATHWAYS DURING \n\nDEVELOPMENT \n\nFundamental Research Laboratories, NEC Corporation, \n\nShigeru Tanaka \n\n34 Miyukigaoka, Tsukuba, Ibaraki 305, Japan \n\nABSTRACT \n\nThe development of projections from the retinas to the cortex is \nmathematically analyzed according to the previously proposed \nthermodynamic formulation of the self-organization of neural networks. \nThree types of submodality included in the visual afferent pathways are \nassumed in two models: model (A), in which the ocularity and retinotopy \nare considered separately, and model (B), in which on-center/off-center \npathways are considered in addition to ocularity and retinotopy. Model (A) \nshows striped ocular dominance spatial patterns and, in ocular dominance \nhistograms, reveals a dip in the binocular bin. Model (B) displays \nspatially modulated irregular patterns and shows single-peak behavior in \nthe histograms. When we compare the simulated results with the observed \nresults, it is evident that the ocular dominance spatial patterns and \nhistograms for models (A) and (B) agree very closely with those seen in \nmonkeys and cats. \n\n1 INTRODUCTION \n\nA recent experimental study has revealed that spatial patterns of ocular dominance columns \n(ODes) observed by autoradiography and profiles of the ocular dominance histogram \n(ODH) obtained by electrophysiological experiments differ greatly between monkeys and \ncats. ODes for cats in the tangential section appear as beaded patterns with an irregularly \nfluctuating bandwidth (Anderson, Olavarria and Van Sluyters 1988); ODes for monkeys are \nlikely to be straight parallel stripes (Hubel, Wiesel and LeVay, 1977). The typical ODH for \ncats has a single peak in the middle of the ocular dominance corresponding to balanced \nresponse in ocularity (Wiesel and Hubel, 1974). In contrast to this, the ODH for monkeys \nhas a dip in the middle of the ocular dominance (Hubel and Wiesel, 1963). Furthermore, \nneurons in the input layer of the cat's primary visual cortex exhibit orientation selectivity, \nwhile those of the monkey do not \nThrough these comparisons, we can observe distinct differences in the anatomical and \nphysiological properties of neural projections from the retinas to the visual cortex in \nmonkeys and cats. To obtain a better understanding of these differences, theoretical analyses \nof interactions among ocularity, retinotopy and on-center/off-center pathways during visual \n\n18 \n\n\fInteraction Among Ocularity, Retinotopy and On-center/Off-center Pathways \n\n19 \n\ncortical development were performed with computer simulation based on the previously \nproposed thermodynamic formulation of the self-organization of neural networks (fanaka, \n1990). \nTwo models for the development of the visual afferent pathways are assumed: model (A), in \nwhich the development of ocular dominance and retinotopic order is laken into account, and \nmodel (B), in which the development of on-center/off-center pathway terminals is \nconsidered in addition to ocular dominance and retinotopic order. \n\n2 MODEL DESCRIPTION \n\nThe synaptic connection density of afferent fibers from the lateral geniculate nucleus (LGN) \nin a local equilibrium state is represented by the Potts spin variables C1,i.J\"S because of their \nstrong winner-lake-all process (Tanaka, 1990). The following function nq<{ C1,l,J'P gives the \ndistribution of the Potts spins in equilibrium: \n\n1req ( (aj, l'll\u00bb = .1 exp( _ H( (OJ,l,ll}) ) \n\nZ \n\nT \n\nwith Z = L \n\nexp( _ H( (OJ,l,Il}) \n\n. \n\n{q,l,Jl=l,O} \n\nT \n\n(1) \n\n(2) \n\nThe Hamiltonian H in the argument of the exponential function in (1) and (2) determines \nthe behavior of this spin system at the effective temperature T, where H is given by \n\n(3) \n\nJ,J \n\nFunction V ~~ represents the interaction between synapses at positions j and j' in layer 4 \nof the primary visual cortex; function r~k,~ represents the correlation in activity between \nLGN neurons at positions k and k' of cell types 11- and 11-'. The set Hj represents a group \nof LGN neurons which can project their axons to the position j in the visual cortex; \ntherefore, the magnitude of this set is related to the extent of afferent terminal arborization \nin the cortex A.A. \n\nTaking the above formulation into consideration, we have only to discuss the \nthermodynamics in the Potts spin system described by the Hamiltonian H at the \ntemperature T in order to discuss the activity-dependent self-organization of afferent neural \nconnections during development. \nNext, let us discuss more specific descriptions on the modeling of the visual afferent \npathways. We will assume that the LGN serves only as a relay nucleus and that the signal \nis transferred from the retina to the cortex as if they were directly connected. Therefore, the \ncorrelation function r~k,~ can be treated as that in the retinas r:}I;Ic',~, This function is \ngiven by using the lateral interaction function in the retina Vl~'c' and the correlation function \n\n\f20 \n\nTanaka \n\nof stimuli to RGCs Gq Jl:12: in the following: \n\nFor simplicity. the stimuli are treated as white noise: \n\n(4) \n\n(5) \n\nNow. we can obtain two models for the formation of afferent synaptic connections between \nthe retinas and the primary visual cortex: model (A). in which ocularity and retinotopy are \ntaken into account: \n\ntIE (left. right). K = [1 '1] . \n\n'1 1 \n\n(6) \n\nwhere 11 (0 ~ n ~ 1 ) is the correlation of activity between the left and right retinas; and \nmodel (B). in which on-center and off-center pathways are added to model (A): \n\ntIE {(left. on-center). (left. Off-center). (right. on-center). (right. off-center)} \u2022 \n\nK= \n\n1 \n\n'1 +1'2 \n\n'1 +1'2 1 \n\nn \nn \n\n'1 \n'1 \n\n'1 \n'1 \n\n'1 \n'1 \n\n1 n +1'2 \n\n'1 +1'2 1 \n\n(7) \n\nwhere 1'2 (- 1 ~ '2 ~ 1 ) is the correlation of activity between the on-center and off-center \nRGCs in the same retina when there is no correlation between different retinas. A negative \nvalue of 1'2 means out-of-phase firings between on-center and off-center neurons. \n\n3 COMPUTER SIMULATION \n\nComputer simulations were carried out according to the Metropolis algorithm (Metropolis. \n1953; Tanaka. 1991). A square panel consisting of 80x80 grids was assumed to be the \ninput layer of the primary visual cortex. where the length of one grid is denoted by a. The \nPotts spin is assigned to each grid. Free boundary conditions were adopted on the border of \nthe panel. One square panel of 20><20 grids was assumed to be a retina for each submodality \nJL The length of one grid is given as 4a so that the edges for the square model cortex and \nmodel retinas are of the same length. \n\nThe following form was adopted for the interactions V1\n\nv: i IS (v= VC or R): \n\nV k . k' = \n\nv \n\u2022 \n\nqV \na \nv 2 \n2nlt a \n\n~ d! k' ) \n-\nex -\n\n. \nv2 \n21t a \n\nqV \n\ninh \nv 2 \n2nlt inh \n\n~ c(k') \n. \nex -\n\nv 2 \n21t illh \n\n(8) \n\n\fInteraction Among Ocularity, Retinotopy and On-center/Off-center Pathways \n\n21 \n\nqVcw. = 5.0, \n\nAll results reported in this paper were obtained with parameters whose values are ~s \nfollows: qVCu = 1.0, \nA. u = 0.5, \nA. R iIJI = 1.0, A. A = 1.6. a = 0.1. T = 0.001. n = 0, and r2 = -0.2. It is assumed that qRw. = 0 \nfor model (A) while qRiIJI = 0.5 for model (B). By considering that the receptive field (RF) \nof an RGC at position k is represented by JlVl~ l\" RGCs for model (A) and (B) have low(cid:173)\npass and high-pass filtering properties, respectively. Monte Carlo simulation for model (A) \nwas carried out for 200,000 steps; that for model (B) was done for 760,000 steps. \n\nA. w. = 1.0, qRu = I, \n\nVC \n\nvc \n\nu = 0.15, \n\nA. \n\nR \n\n(a) \n\n(b) \n\nL \n\nR \n\n(c) \n\n(d) \n\n(e) \n\nL \n\nR \n\n(f) \n\n(g) \n\nFig. 1 Simulated results of synaptic terminal and neuronal distributions and ocular \n\ndominance histograms for models (A) and (B). \n\n\f22 \n\nTanaka \n\n4 RESULTS AND DISCUSSIONS \n\nJ.J \n\nThe distributions of synaptic terminals and neurons, and ocular dominance histograms are \nshown in Fig. I, where (a), (b) and (c) were obtained from model (A); (d), (e), (f) and (g) \nwere obtained from model (B). The spatial distribution of synaptic terminals originating \nfrom the left or right retina (Figs. 1a and 1d) is a counterpart of an autoradiograph of the \none by the eye-injection of radiolabeled amino acid. The bandwidth of the simulated one \n(Fig. 1a) is almost constant as well as the observed bandwidth for monkeys (Hubel and \nWiesel, 1974). The distribution of ocularity in synaptic terminals shown in Fig. 1d is \nirregular in that the periodicity seen in Fig. 1a disappears even though a patchy pattern can \nbe seen. This pattern is quite similar to the ODe for cats (Anderson, Olavarria and Van \nSluyters 1988). \nBy calculating the convolution of the synaptic connections q.l.Il'S with the cortical \ninteraction function v.~, the ocular dominance in response of cortical cells to monocular \nstimulation and the spatial pattern of the ocular dominance in activity (Figs. 1b and Ie) \nwere obtained. Neurons specifically responding to stimuli presented in the right and left \neyes are, respectively, in the black and white domains. This pattern is a counterpart of an \nelectrophysiological pattern of the ODe. The distributions of ocularity in synaptic \nterminals correspond to those of ocular dominance in neuronal response to monocular \nstimulation (a to b; d to e in Fig. 1). This suggests that the borders of the autoradiographic \nODe pattern coincide with those of the electrophysiological ODe pattern. This \ncorrespondence is not trivial since strong lateral inhibition exerts in the cortex. \nReflecting the narrow transition areas between monocular domains in Fig. 1b, a dip appears \nin the binocular bin in the corresponding ODH (Fig. 1c). In contrast, the profile of the \nODH (Fig. If) has a single peak in the binocular bin since binocularly responsive neurons \nare distributed over the cortex (Fig. Ie). \nIn model (B), on-center and off-center terminals are also segregated in the cortex in \nsuperposition to the ODe paUern (Fig. 19). No correlation can be seen between the spatial \ndistribution of on-center/off-center terminals and the one pattern (Fig.1d). \n\n(a) \n\n(b) \n\n(c) \n\nFig. 2 A visual stimulation pattern (a) and the distributions of active synaptic \n\nterminals in the cortex [(b) for model (A) and (c) for model (B)]. \n\nFigures 2b and 2c visualize spatial patterns of active synaptic terminals in the cortex for \nmodel (A) and model (B), when the light stimulus shown by Fig.1d is presented to both \n\n\fInteraction Among Ocularity, Retinotopy and On-center/Off-center Pathways \n\n23 \n\nretinas. A pattern similar to the stimulus appears in the cortex for model (A) (Fig. Ie). \nThis supports the observation that retinotopic order is almost achieved. In other \nsimulations for model (A), the retinotopic order in the final pattern was likely to be \nachieved when initial patterns were roughly ordered in retinotopy. In model (B), the \nretinotopic order seems to be broken at least in this system size even though the initial \npattern has a well-ordered retinotopy (Fig. lc). There is a tendency for retinotopy to be \nharder to preserve in model (B) than in model (A). \n\nL \n\nR \n\n(a) \n\nL \n\n(b) \n\nR \n\n(c) \n\n(d) \n\n(e) \n\nFig. 3 Representative receptive fields obtained from simulations. \n\nModel (A) reproduced only concentric RFs for both eyes. The dominant RFs of monocular \nneurons were of the on-center/off-surround type (right in Fig. 3a); the other RFs of the \nsame neurons were of the type of the low-pass filter which has only the off response (left \nin Fig. 3a). In Model (B), RFs of cortical neurons generally had complex structures (Fig. \n3b). It can barely be recognized that the dominant RFs of monocular neurons showed \nsimple-cell-like RFs. \nTo determine why model (B) produced complex structures in RFs, another simulation of \nRF formation was carried out based on a model where retinotopy and on-center/off-center \npathways are considered. Various types of RFs emerged in the cortex (bottom row in Fig. \n3). The difference in structures between Figures. 3c and 3e shows the difference in the \norientation and the phase (the deviation of the on region from the RF center) in the simple(cid:173)\ncell-like RFs. Fig. 3d shows an on-center concentric RF. Such nonoriented RFs were \nlikely to appear in the vicinity of the singular points around which the orientation rotates \nby 180 degrees. \n\nSimulations for model (A) with different values of parameters such as qVcw., A. A and qRw. \nwere also carried out although the results are not visualized here. When qvcjM takes a small \n\n\f24 \n\nTanaka \n\nvalue, the OOC bandwidth fluctuates). However large the fluctuation may be, the left-eye \nor right-eye dominant domains are well connected, and the pattern does not become an \nirregular beaded pattern as seen in the cat OOC. When afferent axonal arbors were widely \nspread in the cortex (,t A \u00bb I), segregated OOC stripe patterns had only small fluctuation in \nthe bandwidth. qRw. = 0 corresponds to a monotonically decreasing function Vl~l' with \nrespect to the radial distance dk.)'. When qRw. was increased from zero, the number of \nmonocular neurons was decreased. Therefore, the profile of the ODH changes from that in \nFig. 1c. \nIn model (B), as the value of T'}. became smaller, on-center and off-center terminals were \nmore sharply segregated, and the average size of the OOC patches became smaller. The \nsegregation of on-center and off-center terminals seems to interfere strongly with the \ndevelopment of the ODe and the retinotopic organization. This may be attributed to the \ncompetition between ocularity and on-center/off-center pathways. We have seen that only \nconcentric or simple-cell-like RFs can be obtained (Fig. 3b) unless both the ocularity and \nthe on-center/off-center pathways are taken into account in simulations. However, in model \n(B) in which the two types of submodality are treated, neurons have complex separated RF \nstructures (Fig. 3b). This also seems to be due to the competition among the ocularity and \nthe on-center/off-center pathways. The simulation of model (B) was performed with no \ncorrelation in activity between the left and right eyes 1'l. This condition can be realized for \nbinocularly deprived kittens (Tanaka, 1989). By considering this, we may conclude that the \nformation of normal RFs needs cooperative binocular input \nIn this research, we did not consider the effect of color-related cell types on OOC formation. \nActually, there are varieties of single-opponent cells in the retina and LGN of monkeys \nsuch as four types of red-green opponent cells: a red on-center cell with a green inhibitory \nsurround; a green on-center cell with a red inhibitory surround; a red off-center cell with a \ngreen excitatory surround; and a green off-center cell with a red excitatory surround. The \ncorrelation of activity between red on-center and green on-center cells or green off-center and \nred off-center cells may be positive in view of the fact that the spectral response functions \nbetween three photoreceptors overlap on the axis of the wavelength. However, the red on(cid:173)\ncenter and green on-center cells antagonize the red off-center and green off-center cells, \nrespectively. Therefore, the former two and latter two can be looked upon as the on-center \nand off-center cells seen in the retina of cats. This implies that the model for monkeys \nshould be model (B); thereby, the ODC pattern for monkeys should be an irregular beaded \npattern despite the fact that the OOC and ODH in model (A) resemble those for monkeys. \nTo avoid such contradiction, the on-center and off-center cells must separately send their \naxons into different sublayers within layer 4e~, as seen in the visual cortex for Tree shrews \n(Fitzpatrick and Raczkowski, 1990). \n\n5 CONCLUSION \n\nIn model (A), the OOC showed the striped pattern and the ODH revealed a dip in the \nbinocular bin. In contrast to this, model (B) reproduced spatially modulated irregular OOC \npatterns and the single-peak behavior of the ODH. From comparison of these simulated \nresults with experimental observations, it is evident that the OOCs and ODHs for models \n(A) and (B) agree very closely with those seen in monkeys and cats, respectively. Therefore, \nthis leads to the conclusion that model (A) describes the development of the afferent fiber \nterminals of the primary visual cortex of monkeys, while model (B) describes that of the \n\n\fInteraction Among Ocularity, Retinotopy and On-center/Off-center Pathways \n\n25 \n\ncat. In fact. the assumption of the negative correlation (7'2 < 0) between the on-center and \noff-center pathways in model (B) is consistent with the experiments on correlated activity \nbetween on-center and off -center RGCs for cats (Mastronarde. 1988). \nFinally. we predict the following with regard to afferent projections for cats and monkeys. \n[1] In the input layer of the visual cortex for cats. on-center/off-center pathway terminals are \nsegregated into patches. superposing the ocular dominance patterns. \n[2] In monkeys, the axons from on-center/off-center cells in the LGN terminate in different \nsublayers in layer 4C~ of the primary visual cortex. \n\nAcknowledgment \n\nThe author thanks Mr.Miyashita for his help in performing computer simulations of \nreceptive field formation. \n\nReferences \n\nP.A. Anderson, J. Olavarria & R.C. Van Sluyters. (1988) The overall pattern of ocular \ndominance bands in the cat visual cortex. J. Neurosci.. 8: 2183-2200. \nD.H. Hubel, T.N. Wiesel and S. LeVay. (1977). Plasticity of ocular dominance columns in \nmonkey striate cortex. Philos. Trans. R. Soc. Lond .\u2022 B278: 377-409. \nT.N. Wiesel and D.H. Hubel. (l974).Ordered arrangement of orientation columns in \nmonkeys lacking visual experience. J. Compo Neurol. 158: 307-318 \nD.H. Hubel and T.N. Wiesel. (1963). Receptive fields, binocular interaction and functional \narchitecture in the cat's visual cortex. J. Physiol.. 160: 106-154. \nS. Tanaka. (1990) Theory of self-organization of cortical maps: Mathematical framework. \nNeural Networks, 3: 625-640. \nN. Metropolis, A. W. Rosenbluth. M. N. Rosenbluth, A. H. Teller and E. Teller. (1953) \nEquation of state calculations by fast computing machines. J. Chern. Phys., 21: 1087-\n1092. \nS. Tanaka. (1991) Theory of ocular dominance column formation: Mathematical basis and \ncomputer simulation. BioI. Cybem., in press. \nS. Tanaka. (1989) Theory of self-organization of cortical maps. In D. S. Touretzky (ed.), \nAdvances in Neural Information Processing Systems 1.451-458, San Mateo, CA: Morgan \nKaufmann. \nD. Fitzpatrick and D. Raczkowski. (1990) Innervation patterns of single physiologically \nidentified geniculocortical axons in the striate cortex of the tree shrew. Proc. Natl. Acad. \nSci. USA, 87: 449-453. \nD. N. Mastronarde. (1989) Correlated firing of retinal ganglion cells. Trends in Neurosci. \n12: 75-80. \n\n\f", "award": [], "sourceid": 322, "authors": [{"given_name": "Shigeru", "family_name": "Tanaka", "institution": null}]}