{"title": "Further Studies of a Model for the Development and Regeneration of Eye-Brain Maps", "book": "Advances in Neural Information Processing Systems", "page_first": 3, "page_last": 10, "abstract": null, "full_text": "Further Studies of a Model for the \n\nDevelopment and Regeneration \n\nof Eye-Brain Maps \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., \n\nChicago, Illinois 60637 \n\nAbstract \n\nWe describe a computational model of the development and regenera(cid:173)\ntion of specific eye-brain circuits. The model comprises a self-organiz(cid:173)\ning map-forming network which uses local Hebb rules, constrained by \n(genetically determined) molecular markers. Various simulations of \nthe development and regeneration of eye-brain maps in fish and frogs \nare described, in particular successful simulations of experiments by \nSchmidt-Cicerone-Easter; Meyer; and Y oon. \n\n1 INTRODUCTION \n\nIn a previous paper published in last years proceedings (Cowan & Friedman 1990) we \noutlined a new computational model for the development and regeneration of eye-brain \nmaps. We indicated that such a model can simulate the results of a number of the more \ncomplicated surgical manipulations carried out on the visual pathways of goldfish and \nfrogs. In this paper we describe in more detail some of these experiments, and our \nsimulations of them. \n\n1.1 EYE-BRAIN MAPS \n\nWe refer to figure 1 from the previous paper which shows the retinal map found in the \noptic lobe or tectum of a fish or frog. The map is topological, i.e.; neighborhood \n\n3 \n\n\f4 \n\nCowan and friedman \n\nrelationships in the retina are preserved in the optic tectum. As is well-known nearly 50 \nyears ago Sperry (1944) showed that such maps are quite precise and specific, in that \nmaps (following optic nerve sectioning and eye rotation) regenerate in such a way that \noptic nerve fibers reconnect, more or less, to their previous tectal sites. Some 20 years \nago Gaze and Sharma (1970) and Yoon (1972) found evidence for plasticity in the \nexpanded and compressed \"maps\" which regenerate following eye and brain lesions in \ngoldfish. There are now many experiments which indicate that the regeneration of \nconnections involves both specificity and plasticity. \n\n1. 2. EXPANDED MAPS \n\nSuch properties are seen in a series of more complicated experiments involving the \nexpansion of a half-eye map to a whole tectum. These experiments were carried out by \nSchmidt, Cicerone and Easter (1978) on goldfish, in which following the expansion of \nretinal fibers from a half-eye over an entire (contralateral) tectum, and subsequent \nsectioning of the fibers, diverted retinal fibers from the other (intact) eye are found to \nexpand over the tectum, as if they were also from a half-eye. This has been interpreted to \nimply that the tectum has no intrinsic positional markers to provide cues for incoming \nfibers, and that all its subsequent markers come from the retina (Chung & Cooke, 1978). \nHowever Schmidt et.al. also found that the diverted fibers also map normally. Figure 4 \nof the previous paper shows the result. \n\n1. 3. COMPRESSED MAPS \n\nCompression is found in maps from entire eyes to ablated half tecta (Gaze & Sharma, \n1970; Sharma & Gaze, 1971; Y oon, 1972). There has been considerable controversy \nconcerning the results. Recently Meyer (1982) has shown that although \nelectrophysiological techniques seem to provide evidence for smoothly expanded and \ncompressed maps, autoradiographic techniques do not. Instead of a smooth map there \nare patches, and in many cases no real expansion or compression is seen in irradiated \nsections, at least not initially. An experiment by Yoon (1976) is relevant here. Yoon \nnoticed that in the early stages of map formation under such conditions, the map is \nnormal. Only after some considerable time does a compressed map form. However if \nthe fibers are sectioned (cut) and allowed to regenerate a second time, compression is \nimmediate. This result has been challenged (Cook, 1979), but it was subsequently \nconfirmed by Schmidt (1983). \n\n1.4. MISMATCHED MAPS \n\nIn mismatch experiments, a half retina is confronted with an inappropriate half tectum. In \nYoon's classic \"mismatch\" experiment (Yoon, 1972) fibers from a half-eye fragment are \nconfronted with the \"wrong\" half-tectum: the resulting map is normally oriented, even \nthough this involves displacement of retinal fibers from near the tectal positions they \nnormally would occupy. \n\n\fStudies of a Model for the Development and Regeneration of Eye-Brain Maps \n\n5 \n\nAbout 12 years ago Meyer (1979) carried out another important mismatch experiment in \nwhich the left half of an eye and its attached retinal fibers were surgically removed, \nleaving an intact normal half-eye map. At the same time the right half the other eye and \nits attached fibers were removed, and the fibers from the remaining half eye were \nallowed to innervate the tectum with the left-half eye map. The result is shown in figure \n5 of our previous paper. Fibers from the right half-retina, labelled I through 5, would \nnormally make contact with the corresponding tectal neurons. Instead they make contact \nwith neurons 6 through 10, but in a reversed orientation. Meyer interprets this result to \nmean that optic nerve fibers show a tendency to aggregate with their nearest retinal \nneighbors. \n\n2 THE MODEL \n\nWe introduced our model in last year's NIPS proceedings (Cowan & Friedman 1990). We \nhere repeat some of the details. Let Sij be the strength or weight of the synapse made by \nthe ith retinal fiber with the jth tectal cell. Then the following system of differential \nequations expresses the changes in Sij: \nSij = ~j + cij [j.1ij + (ri - ol)tjJ Sij \n\n- k s .. (T-I ~. + R -1 ~ . ) { A' + c .. [I I ., + (r' \n\n:2 \n\nIJ \n\nLol \n\nLoJ \n\nJ \n\nIJ \n\nI\"\"IJ \n\n...J\\t\u00b7J s .. } \n\nI---'J IJ \n\n(1) \n\nwhere i = 1, 2, .... , Nr, the number of retinal ganglion cells and j = 1, 2, .... , Nt, the \nnumber of tectal neurons, Cij is the \"stickiness\" of the ijth contact, fj denotes retinal \nactivity and tj = LiSijri is the corresponding tecta! activity, and eX is a constant measuring \nthe rate of receptor destabilization (see Whitelaw & Cowan (1981) for details). In \naddition both retinal and tectal elements have fixed lateral inhibitory contacts. The \ndynamics described by eqn.l is such that both LiSij and LjSij tend to constant values T \nand R respectively, where T is the total amount of tectal receptor material available per \nneuron, and R is the total amount of axonal material available per retinal ganglion cell: \nthus if sij increases anywhere in the net, other synapses made by the ith fiber will \ndecrease, as will other synapses on the jth tectal neuron. In the current terminology, this \nprocess is referred to as \"winner-take-all\". \n\nIn addiiton Aj represents a general nonspecific growth ofretinotectal contacts, presumed \nto be controlled and modulated by nerve growth factor (Campenot, 1982). Recent \nobservations (Davies et.al., 1987) indicate that the first fibers to reach a given target \nneuron stimulate it to produce NGF, which in tum causes more fiber growth. We \ntherefore set Aj = T-l LiSijA where A is a constant. LiSij is the instantaneous value of \nreceptor material used to make contacts, and T is the total amount available, so A j --> A \nas the jth neuron becomes innervated. The coefficient j.1ij represents a postulated random \ndepolarization which occurs at synapses due to the quantal release of neurotransmitter-(cid:173)\nthe analog of end-plate potentials (Walmsley et.al., 1987). Thus even if ri = 0, map \nformation can still occur. However the resulting maps are not as sharp as those formed in \n\n\f6 \n\nCowan and friedman \n\nthe presence of retinal activity. Of course if J.lij = 0, as might be the case if ol(cid:173)\nbungarotoxin is administered, then Sij = Aj(1- Sij) and Sij --> I, i.e.; all synapses of \nequal strength. \n\nIt is the coefficients cij- which determine the nature of the solution to eqn.1. These \ncoefficients express the contact adhesion strengths of synapses. We suppose that such \nadhesions are generated by fixed distributions of molecules embedded in neural surface \nmembranes. We postulate that the t.il2S. of retinal axons and the surfaces of tectal cells \ndisplay at least two molecular species, labelled a and b, such that Cij = L~abaibj and the \nsum is over all possible combinations aa, ab etc. A number of possibilities exist in the \nchoice of ~ab and of the spatial distribution of a and b. One possibility that is consistent \nwith most of the assays which have been carried out (Trisler & Collins (1987), \nBonhoffer and Huff (1980), Halfter, Claviez & Schwarz (1981), Boenhoffer & Huff \n(1985\u00bb is ~aa = ~bb > 0 > ~ab = ~ba in which each species prefers itself and repels the \nother, the so-called homophilic case, with ai and bi as shown in figure 1. \n\n2 \n\nt\u00b7 1 \n1 \n\no \n\n1 \n\ni \n\n2 \n\n1 b. 1 \n\no \n\nFigure 1: Postulated distribution of sticky molecules \nin the retina. A similar distribution is supposed to \nexist in the tectum. \n\nThe mismatch and compound eye experiments indicate that map formation depends in \npart on a tendency for fibers to stick to their retinal neighbors, in addition to their \ntendency to stick to tectal cell surfaces. We therefore append to Cij the term L'k Skj fik \nwhere Skj is a local average of Skj and its nearest tectal neighbors, where fik measures \nthemutual stickiness of the ith and kth retinal fibers, and where L'k means Lk -:f; i. Fig. 2 \nshows the postulated form of fik' {Again we suppose this stickiness is produced by the \ninteraction of two molecular species etc.; specifically theneural contact adhesion \nmolecules (nCAM) of the sort discovered by Edelman (I983)which seem to mediate the \nfiber-fiber adhesion observed in tissue cultures by Boenhoffer & Huff (1985), but we do \nnot go into the details). \n\n\fStudies of a Model for the Development and Regeneration of Eye-Brain Maps \n\n7 \n\ni \n\nN. \n1 \n\n1 \n\ni \n\n~ \n\n0 \n\nN. \n1 \n\nFigure 2: The file surface. Retinal fibers are attracted \nonly to themselves or to their immediate retinal \nneighbors. \n\nMeyer's mismatch experiment also indicate that existing fiber projections tend to exclude \nother fibers, especially inappropriate ones, from innervating occupied areas. One way to \nincorporate such geometric effects is to suppose that each fiber which establishes contact \nwith a tectal neuron occludes tectal markers there by a factor proportional to its synaptic \nweight siJ Thus we subtract from the coefficient Cij a fraction proportional to T-l L:'kSkJ \n\nWith the introduction of occlusion effects and fiber-fiber interactions, it becomes ap(cid:173)\nparent that debris in the fonn of degenerating fiber fragments adhering to tectal cells, \nfollowing optic nerve sectioning, can also influence map formation. Incoming nerve \nfibers can stick to debris, and debris can occlude markers. There are in fact four possi(cid:173)\nbilities: debris can occlude tectal markers, markers on other debris, or on incoming fibers; \nand incoming fibers can occlude markers on debris. All these possibilities can be in(cid:173)\ncluded in the dependence of Cij on Sij' Skj etc. Note that such debris is supposed to \ndecay, and eventually disappear. \n\n3 SIMULATIONS \n\nThe model which results from all these modifications and extensions is much more com(cid:173)\nplex in its mathematical structure than any of the previous models. However computer \nsimulation studies show it to be capable of correctly reproducing the observed details of \nalmost aU the experiments cited above. For purposes of illustration we consider the \nproblem of connecting a line of Nr retinal cells to a line of Nt tectal cells. The resulting \nmaps can then be represented by two-dimensional matrices, in which the area of the \nsquare at the ijth intersection represents the weight of the synapse between the ith retinal \nfiber and the jth tecta! cell. The nonnal retino-tecta! map is represented by large squares \nalong the matrix diagonal., (see Whitelaw & Cowan (1981) for terminology and further \ndetails). \n\n\f8 \n\nCowan and friedman \n\n3.1 THE SCHMIDT ET. AL. EXPERIMENT \n\nFigure 3, for example shows a simulation of the retinal \"induction\" experiments of \nSchmidt et.al. This simulation generated both an expanded map and a nearly normal \npatch, interacting to form patches. These effects occur because some incoming retinal \nfibers stick to debris left over from the previous expanded map, and other fibers stick to \nnon-occluded tectal markers. The fiber-fiber markers control the regeneration of the \nexpanded map, whereas the retino-tectal markers control the formation of the nearly \nnormal map. \n\n1 \n\n1 \n\ni \n\nNr \n\n-Do \n\nNt \n\nFigure 3: Simulation of the Schmidt et.al. retinal in(cid:173)\nduction experiment. A nearly normal map is interca(cid:173)\nlated into an expanded map. \n\n, \u2022 \u2022 I aD \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 I \u2022 \u2022 \u2022 \u2022 \u2022 I \n\n\u2022 \n\n. C.\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7 \n-a DCa \u2022 \u2022 \u2022 aa \u2022 \u2022 \u2022 \u2022 . . \u2022 \n\u2022 al:!'OCa- aaClaaa-. I \nI \u2022 \u2022 acaa.acaaa \u2022\u2022 \u2022\u2022 \n. - -\u00b7aaa\u00b7\u00b7caaaaac a \n- - . \u2022 a c I:! \u2022\u2022\u2022\u2022 a a CCI:! \n\u2022 \u2022 \u2022 \u2022 \n\na D a \n\n1200 \n\n'\n\n5000 \n\na \n\na \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \n\na \u2022\u2022\u2022 \nCc. I \n\n- _\u00b7\u00b7COc\u00b7\u00b7\u00b7\u00b7\u00b7.\u00b7 __ \u00b7-(cid:173)\n\u2022 \u2022 I'-aaaaaaaa\u00b7\u00b7\u00b7 \u2022\u2022\u2022 \nI \u2022 \u2022 \u2022 \u2022 aaacca \u2022\u2022\u2022\u2022 \n- - \u2022\u2022\u2022 aaccaa a \u2022 -. \n. . - - - . - . - - \u2022 \u2022\u2022 a 00 c \n.......... \u2022 anne \n\n\u2022 \n\nI \n\n16000 \n\nFigure 4: Simulation of the Yoon second \ncompression experiment (see text for details). \n\n\fStudies of a Model for U\"le Development and Regeneration of Eye-Brain Maps \n\n9 \n\n3.2 THE YO ON SECOND COMPRESSION EXPERIMENT \n\nYoon's demonstration of immediate second compression can also be simulated. Figure 4 \nshows details of the simulation. At an early stage just after the first cut, both a normal \nand a compressed map are forming. The normal map eventually disappears, leaving only \na compressed map. After the second cut however, a compressed map forms immediately. \nAgain it is the debris which carries fiber-fiber markers that control map formation. \n\n3.3 THE MEYER MISMATCH EXPERIMENT \n\nIt is evident that fiber-fiber interactions are important in controlling map formation. The \nMeyer mismatch experiment shows this quite clearly. A simulation of this experiment \nalso shows the effect. If fik, the mutual stickiness of neighboring fibers is not strong \nenough, retino-tectal markers dominate, and the mismatched map forms with normal \npolarity. However if fik is large enough, Meyer's result is found, the mismatched map \nforms with a reversed polarity. Figure 5 shows the details. \n\nleft uIt eye ~1It UU eye \n12345678910 10 9 8 7 654 321 \n\n!IIII \n\n711J \n\nldt UU eye ~1It UU eye \n12345678910 10987654321 \nD.Q \n.. _-\n~ \n\nI I -\n\n, \n\n- l a a l I I l a l I I \n\u2022 a \u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\n.. :\" \n. . . \n\u00b7 , . \n\u00b7 ... \n\u00b7 .. --- .... \n......\u2022... \n\u00b7 ........ . \n\u00b7 ........ . \n\u00b7 ........ . \n\u00b7 ........ . \n\u00b7 ........ . \n\u00b7 ........ . \n\n\u2022 \u2022\u2022\u2022\u2022\u2022\u2022 \u2022 aa \n\nFigure 5: Simulation of the Meyer mismatch \nexperiment (see text for details). \n\n4 CONCLUSIONS \n\nThe model we have outlined generates correctly oriented retinotopic maps. It permits the \nsimulation of a large number of experiments, and provides a consistent explanation of \nalmost all of them. In particular it shows how the apparent induction of central markers \nby peripheral effects, as seen in the Schmidt et. aI., can be produced by the effects of \ndebris, as can Yoon's observations of immediate second compression. Affinity markers \nare seen to play a key role in such effects, as they do in the polarity reversal seen in \nMeyer's experiment. \n\n\f10 \n\nCowan and friedman \n\nIn summary much of the complexity of the many regeneration experiments which have \nbeen carried out in the last fifty years can be understood in terms of the effects produced \nby contact adhesion molecules with differing affinities, acting to control an activity(cid:173)\ndependent self-organizing mechanism. \n\nAcknowledgements \n\nWe thank The University of Chicago Brain Research Foundation for partial support of \nthis work. \n\nReferences \n\nBoenhoffer, F. & Huf, J. (1980), Nature, 288, 162-164.; (1985), Nature, 315,409-411. \nCampenot, R.B. (1982), Develop. Biol., 93, 1. \nChung, S.-H. & Cooke, J.E. (1978), Proc. Roy. Soc. Lond. B 201, 335-373. \nCowan, J.D. & A.E. Friedman (1990) Advances in NIPS, 2, Ed. D.S. Touretzky, Morgan(cid:173)\nKaufmann, 92-99. \nCook, J.E. (1979), J. Embryol. expo Morphol., 52, 89-103. \nDavies, A.M., Bandtlow, C., Heumann, R, Korsching, S., Rohrer, H. & Thoenen, H. \n(1987), Nature, 326,353-358. \nEdelman, G.M., (1983), Science, 219, 450-454. \nGaze, R.M. & Sharma, S.c. (1970), Exp. Brain Res., 10, 171-181. \nHalfter, W., Claviez, M. & Schwarz, U. (1981), Nature, 292,67- 70. \nMeyer, R.L. (1979), Science, 205, 819-821; (1982), Curro Top. Develop. Biol., 17, 101-\n145. \nSchmidt, J.T. (1983), J. Embryol. expo Morphol., 77, 39-51. \nSchmidt, J.T., Cicerone, C.M. & Easter, S.S. (1978), J. Compo Neurol., 177, 257-288. \nSharma, S.C. & Gaze, R.M. (1971), Arch. Ital. Biol., 109, 357-366. \nSperry, R.W. (1944), J. Neurophysiol., 7, 57-69. \nTrisler, D. & Collins, F. (1987), Science, 237, 1208-1210. \nWalmsley, B., Edwards, F.R. & Tracey, DJ. (1987), J. Neurosci., 7,4, 1037-1046. \nWhitelaw, V.A. & Cowan, J.D. (1981), J. Neurosci., 1,12, 1369-1387. \nYoon, M. (1972), Amer. Zool., 12, 106.; Exp. Neurol., 37, 451-462; (1976) J. Physiol. \nLond., 257,621-643. \n\n\f", "award": [], "sourceid": 437, "authors": [{"given_name": "Jack", "family_name": "Cowan", "institution": null}, {"given_name": "A.", "family_name": "Friedman", "institution": null}]}