{"title": "Development and Regeneration of Eye-Brain Maps: A Computational Model", "book": "Advances in Neural Information Processing Systems", "page_first": 92, "page_last": 99, "abstract": null, "full_text": "92 \n\nCowan and Friedman \n\nDevelopment and Regeneration of Eye-Brain \n\nMaps: A  Computational Model \n\nJ.D. Cowan  and 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\nWe outline 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 \nmolecular markers.  Various  simulations of the development of eye(cid:173)\nbrain maps in fish and frogs are described. \n\n1  INTRODUCTION \n\nThe brain is a biological computer of immense complexity comprising highly specialized \nneurons and neural circuits.  Such neurons  are interconnected with  high specificity  in \nmany regions of the brain. if not in all. There are also many observations which indicate \nthat there is also considerable circuit plasticity.  Both specificity and plasticity are found \nin  the development and regeneration  of eye-brain connections in  vertebrates.  Sperry \n(1944) frrst demonstrated specificity in the regeneration of eye-brain connections in frogs \nfollowing  optic nerve section and eye rotation; and Gaze and Sharma (1970) and Y oon \n(1972)  found  evidence  for  plasticity  in  the  expanded  and  compressed  maps  which \nregenerate following eye and brain lesions in goldfish.  There are now many experiments \nwhich indicate that the formation of connections involves both specificity and plasticity. \n\n\fDevelopment and Regeneration of Eye-Brain Maps:  A Computational Model \n\n93 \n\n1.1  EYE-BRAIN MAPS AND MODELS \n\nFig. 1 shows the retinal map found in the optic lobe or  tectum of fish and frog.  The map \nis topologicalt Le.;  neighborhood relationships in  the retina  are  preserved  in the  optic \ntectum.  How does such a map develop?  Initially there is considerable disorder in the \n\n1.\",pol'Jl. 0  usa!. G lm.pol'Jl. \n\nr. retina. \n\n1.  retina. \n\nrosll'Jl.  X  roS11'Jl. \n\n1.  optic \ntect'um. \n\nr.  optic \ntect'um. \n\nFigure 1:  The normal  retino-tectal  map in  fish  and frog.  Temporal \nretina  projects  to  (contralateral)  rostral  tectum;  nasal  retina  to \n(contralateral) caudal tectum. \n\npathway: retinal ganglion cells make contacts with many widely dispersed tectal neurons. \nHowever the mature pathway shows a high degree of topological order. How is such an \norganized map achieved?  One answer  was provided by Prestige &  Wills haw  (1975): \nretinal axons and tectal neurons are polarized by contact adhesion molecules distributed \nsuch that axons from one end of the retina are stickier than those from  the other end, and \nneurons at one end of the tectum  are (correspondingly) stickier than  those at the other \nend.  Of course this means that isolated retinal axons will all tend to stick to one end of \nthe tectum.  However if such axons compete with each other for tectal terminal sites (and \nif tectal sites compete for retinal axon terminals)t less sticky axons will be displacedt and \neventually a topological map will form. The Prestige-Willshaw theory explains many ob(cid:173)\nservations indicating neural specificity.  It does not provide for plasticity:  the ability of \nretino-tectal systems to adapt to changed target conditionst and vice-versa.  Willshaw and \nvon  der  Malsburg  (1976t  1977)  provided  a  theory  for  the  plasticity  of  map \nreorganizationt by postulating the synaptic growth in development is Hebbian.  Such a \nmechanism provides self-organizing properties in retino-tectal map formation and reor(cid:173)\nganization.  Whitelaw & Cowan (1981) combined both sticky molecules and Hebbian sy(cid:173)\nnaptic  growth to provide a  theory  which explains both the specificity and plasticity of \nmap formation and reorganization in a reasonable fashion. \n\nThere are many experiments, however t which indicate that such theories are too simple. \nSchmidt & Easter (1978) and Meyer (1982) have shown that retinal axons interact with \n\n\f94 \n\nCowan and Friedman \n\nIt is  our view  that  there  are \neach  other in  a  way  which  influences  map  formation. \n(probably) at least two different types of sticky molecules in the system: those described \nabove which  mediate retino-tectal  interactions. and an additional class which mediates \naxo-axonal interactions in a different way.  In what follows  we describe a model which \nincorporates such interactions.  Some aspects of our model are similar to those introduced \nby Willshaw &  von  der Malsburg (1979)  and Fraser (1980).  Our model can simulate \nalmost all experiments in the literature. and provides a way to titrate the relative strenghts \nof intrinsic polarity markers mediating retino-tectal interactions, (postulated) positional \nmarkers  mediating  axo-axonal  interactions,  and  stimulus-driven  Hebbian  synaptic \nchanges. \n\n2  MODELS  OF MAP FORMATION AND REGENERATION \n\n2.1. THE WHITELAW-COWAN MODEL \n\nLet Sij  be the strength or weight of the synapse made by the ith retinal axon with the jth \ntectal cell.  Then the following differential equation expresses the changes in siJ \n\ns\u00b7\u00b7 - c\"  (r\u00b7 - ol) t\u00b7  - It. (N -1  ~. + Nt-l ~. )(c\" (r\u00b7  ol) t\u00b7) \nJ \n\nJ.~  r \n\nIJ  1 -\n\n\u00a3..1 \n\nIJ  -\n\nIJ \n\n1 \n\n\u00a3.. J \n\n(1) \n\nwhere Nr is the number of retinal ganglion cells and Nt the number of tectal neurons. Cij \nis the \"stickiness\" of the ijth contact, ri denotes retinal activity and tj = l:iSijfi is the corre(cid:173)\nsponding tectal activity, and ol  is a constant measuring the rate of receptor destabiliza(cid:173)\ntion (see Whitelaw &  Cowan (1981)  for details). In addition both retinal and tectal ele(cid:173)\nments have fixed  lateral inhibitory contacts.  The dynamics described by eqn.l  is such \nthat both l:jsij and l:jSij tend to constant values T and R respectively, where T is the total \namount of tectal receptor material available per neuron, and R is the total amount of ax(cid:173)\nonal material available per retinal ganglion cell:  thus if sij increases anywhere in the net, \nother synapses made by the ith axon will decrease, as will other synapses on the jth tectal \nneuron.  In the current terminology, this process is referred to as \"winner-take-all\". \n\nFor purposes  of illustration  consider  the  problem  of connecting  a  line of Nr  retinal \nganglion cells  to a line of Nt tectal cells.  The resulting maps can then be represented by \ntwo-dimensional  matrices,  in  which  the  area  of the  square  at  the  ijth  intersection \nrepresents the weight of the synapse between the ith retinal axon and the jth tectal cell. \nThe normal retino-tectal map is represented by large squares along the matrix diagonal., \n(see Whitelaw &  Cowan (1981) for terminology and further details).  It is fairly obvious \nthat the only solutions to eqn.  (1)  lie along the matrix diagonal, or the anti-diagonal. as \nshown  in  fig.  2.  These  solutions  correspond,  respectively, \nto  normal  and  inverted \ntopological maps.  It follows that if the affmity Cij of the ith retinal ganglion cell for the \njth tectal neuron  is constant,  a  map will form  consisting of normal  and  inverted  local \npatches.  To obtain a globally normal map itis necessary to bias  the system.  One way to \ndo this is to suppose that Cij = ;aiaj, where ai and aj are respectively. the concentrations \n\n\fDevelopment and Regeneration of Eye-Brain Maps:  A Computational Model \n\n95 \n\nFigure  2:  Diagonal  and  anti-diagonal  solutions  to  eqn.1.  Such \nsolutions correspond.  respectively.  to normal  and inverted maps. \n\nof sticky molecules on the tips of retinal axons and on the surfaces of tectal neurons, and \n~ is  a  constant.  A  good  candidate  for  such  a  molecule  is  the  recently  discovered \ntoponymic or TOP molecule found in chick retina and tectum (Trisler &  Collins, 1987). \nIf ai and aj are  distributed in  the graded  fashion  shown  in  fig.  3,  then  the system is \nbiased in favor of the normally oriented map. \n\no \n\n1 \n\ni \n\nFigure 3: Postulated distribution of sticky molecules in the retina.  A \nsimilar distribution is supposed to exist in the tectum. \n\n2.2  INADEQUACIES \n\nThe Whitelaw-Cowan model simulates the normal development of monocular retinotec(cid:173)\ntal maps. starting from either diffuse or scrambled initial maps, or from no map.  In addi(cid:173)\ntion  it simulates the compressed. expanded, translocated. mismatched and rotated maps \nwhich  have  been  described in  a  variety  of surgical contexts.  However it fails  in  the \nfollowing respects: a.  Although tetrodotoxin (TTX) blocks the refinement of retinotopic \nmaps  in  salamanders. a  coarse  map can  still develop  in  the  absence of retinal activity \nHarris (1980).  The model will not simulate this effect.  b. Although the model  simulates \nthe formation of double maps in \"classical\" compound eyes {made from a half-left and a \nhalf  right  eye}  (Gaze.  Jacobson.  &  Szekely.  1963).  it  fails  to  account  for  the \nreprogramming  observed  in  \"new\"  compound eyes  {made by cutting  a  slit down  the \nmiddle of a tadpole eye}  (Hunt & Jacobson. 1974). and fails  to simulate the forming of a \n\n\f96 \n\nCowan and Friedman \n\nnormal  retinotopic  map  to  a  compound  tectum  (made  from  two  posterior  halves} \n(Sharma, 1975). \n\nl'ii'ht n tinA \n\nl'ii'ht  retinA \n\n10 9  8 7  6 5  4 3  2 1 \n\n10 9  8 7  6 5  4 3  2 1 \n\n1 2  3 4  5 6  78  910 \n\n1 2  3 4  5 6  7 8  910 \n\nright tectum. \nJLOrm.tl m.a.p \n\nright tectum. \n\nexp~a.m.a.p \n\nFigure 4: The normal and expanded maps which form  after the prior \nexpansion ofaxons from  a contralateral  half-eye.  The two  maps are \nactually superposed, but for ease of exposition are shown separately. \n\nleft nti\u00bb. \n1 2  3  4  5 \n\nright  nti\u00bb. \n5  4 3  2 1 \n\n1 2  3 4  5  6  7 8  9 10 \n\nl'ii'ht  tectum. \n\nFigure 5:  Results of Meyer's experiment. Fibers from  the right half(cid:173)\nretina fail to contact their normal targets and instead make contact with \navailable targets, but with reversed polarity. \n\nc.  More significantly, it fails to account for the apparent retinal induction  reported by \nSchmidt, Cicerone  &  Easter (1978) in  which  following  the expansion of retinal axons \nfrom  a goldfish half-eye over an entire (contralateral) tectum, and subsequent sectioning \nof the axons, diverted retinal axons from  the other (intact) eye are found to expand over \nthe  tectum, as if they were also from  a half-eye.  This has been interpreted to imply that \nthe tectum has no intrinsic markers, and that all its markers come from the retina (Chung \n&  Cooke,  1978).  However Schmidt et.al.  also  found  that the diverted axons also  map \nnormally.  Fig. 4 shows the result.  d.  There is also an important mismatch experiment \n\n\fDevelopment and Regeneration or Eye-Brain Maps:  A Computational Model \n\n97 \n\ncarried out by Meyer (1979)  which the model  cannot simulate.  In this experiment the \nleft half of an eye and its attached retinal axons are surgically removed, leaving an intact \nnormal half-eye map.  At the same time the right half the other eye and its attached axons \nare removed, and the  axons from  the  remaining  half eye are allowed to  innervate the \ntectum with the left-half eye map.  The result is shown in  fig.  5. \ne.  Finally. there are \nnow a variety of chemical assays of the nature of the affinities which retinal axons have \nfor each other. and for tectal  target sites.  Thus  Bonhoffer and Huff (1980)  found  that \ngrowing retinal axons stick preferentially to rostral tectum.  This is consistent with the \nmodel.  However, using a different assay Halfter, Claviez &  Schwarz (1981) also found \nthat  tectal  fragments  tend  to  stick  preferentially  to  that  part  of  the  retina  which \ncorresponds to caudal tectum, i.e.; to nasal  retina.  This appears to contradict the model, \nand the first assay. \n\n3  A NEW MODEL FOR MAP FORMATION \n\nThe Whitelaw-Cowan model can be modified and extended to replicate much of the data \ndescribed above.  The first modification is to replace eqn.1 by a more nonlinear equation. \nThe reason for this is that the above equation has no threshold below which contacts can(cid:173)\nnot get established. In practice Whitelaw and I  modified the equations to incorporate a \nsmall threshold effect.  Another way is to make synaptic growth and decay exponential \nrather than linear.  An equation expressing this can be easily formulated, which also in(cid:173)\ncorporates  axo-axonal  interactions,  presumed  to  be  produced  by  neural  contact \nadhesion molecules  (nCAM) of the sort discovered by Edelman (1983) which seem  to \nmediate  the  axo-axonal  adhesion  observed  in  tissue  cultures by  Boenhoffer  &  Huff \n(1985). The resulting equations take the form: \n\nSij = Aj  + Cij  [J,lij + (ri - oi)tj] Sij \n\n- ks\"(T-1\"+R-1\")(A' +c\u00b7\u00b7[\"\u00b7\u00b7+(r\u00b7-oi)t\u00b7]s\u00b7 \u00b7} \n\n(2) \n\n\u00a3..J \n\nJ \n\nIJ  \"\"IJ \n\n1 \n\nJ \n\nIJ \n\n-~  IJ \n\n\u00a3..1 \n\nwhere  A j represents a general nonspecific growth of retinotectal contacts, presumed to \nbe  controlled  and  modulated  by  nerve  growth  factor  (Campenot,  1982).  The  main \ndifference between eqns.  1 and 2  however, lies  in the coefficients Cij'  In eqn.  1, Cij  = \n<;aiaj.  In eqn. 2, Cij expresses several different effects: (a). Instead of just one molecular \nspecies on the tips of retinal axons and on corresponding tectal cell surfaces, as in eqn.l, \ntwo molecular species or two states of one species can be  postulated to  exist on these \nsites.  In  such a  case the term <;aiaj  is replaced by  L<;abaibj  where a  and b  are the \ndifferent species, and the sum is over all possible combinations aa, ab etc. A number of \npossibilities exist in the choice of <;ab'  One possibility that is consistent with most of the \nbiochemical assays described earlier is <;aa = <;bb  < <;ab = <;ba  in  which each species \nprefers the other, the so-called heterophilic  case.  (b) The mismatch experiment cited \nearlier (Meyer, 1979) indicates that existing axon projections tend to exclude other axons, \nespecially inappropriate ones, from  innervating occupied areas.  One way to incorporate \nsuch  geometric  effects  is  to  suppose  that each axon  which  establishes contact with  a \ntectal  neuron  occludes  tectal  markers  there by a  factor proportional  to  its  synaptic \n\n\f98 \n\nCowan and Friedman \n\nweight Sij' Thus we subtract from the coefficient Cij a fraction proportional to '11 L' kSkj \nwhere L k means Lk #:- i'  (c)  The mismatch experiment also indicates that  map for(cid:173)\nmation depends in part on a tendency for axons to stick to their retinal neighbors, in addi(cid:173)\ntion to their tendency to stick to tectal cell surfaces.  We therefore append to Cij  the term \nL'k Skj fik where Skj is a local average of Skj and its nearest tectal neighbors, and where \nfik  measures  the  mutual stickiness of the ith and kth  retinal  axons:  non-zero only  for \nnearest  retinal  neighbors. \n(Again  we  suppose  this  stickiness  is  produced  by  the \ninteraction of two molecular species etc.; specifically the neuronal CAMs discovered by \nEdelman,  but we do not go  into  the details).  (d) With  the  introduction of occlusion \neffects  and  axo-axonal  interactions,  it becomes  apparent  that  debris  in  the  form  of \ndegenerating axon  fragments  adhering to  tectal cells, following  optic nerve sectioning, \ncan also influence map formation.  Incoming nerve axons can stick to debris, and debris \ncan occlude  markers.  There are  in  fact  four  possibilities:  debris  can  occlude  tectal \nmarkers,  markers  on  other debris,  or  on  incoming  axons;  and  incoming  axons  can \nocclude markers on debris.  All these possibilities can be included in the dependence of \nci j on Sij' Skj etc. \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 all  the experiments cited above.  Fig.  6, for  example shows a  simulation of the \nretinal \"induction\" experiments of Schmidt el.al. \n\ni \n\nNr \n\n1 \n\n1 \n\nj \n\nFigure  6:  Simulation  of  the  Schmidt  et.al.  retinal  induction \nexperiment. A nearly normal map is intercalated into an expanded map. \n\nThis simulation  generated  both  a  patchy  expanded and  a  patchy  nearly  normal  map. \nThese effects occur because some incoming retinal axons stick to debris left over from \n\n\fDevelopment and Regeneration of Eye-Brain Maps:  A Computational Model \n\n99 \n\nthe previous expanded map, and other axons stick to non-occluded tectal markers.  The \naxo-axonal positional markers control the formation of the expanded map, whereas the \nretino-tectal polarity markers control the formation of the nearly normal map. \n\n4  CONCLUSIONS \n\nThe model we have outlined combines Hebbian  plasticity  with  intrinsic,  genetic  eye(cid:173)\nbrain and axo-axonic markers, to generate correctly oriented retinotopic maps. It permits \nthe simulation 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-Cicerone-Easter experiment (Schmidt et.al. \n1978), can be produced by the effects of debris; and the polarity reversal seen in Meyer's \nexperiment (Meyer 1979), can be produced by axo-axonal interactions. \n\nAcknowledgements \n\nWe  thank  the  System  Development  Foundation,  Palo  Alto,  California,  and  The \nUniversity of Chicago Brain Research Foundation for partial support of this 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. \nEdelman, G.M., (1983), Science, 219,450-454. \nFraser, S. (1980), Develop. BioI., 79, 453-464. \nGaze, R.M. & Sharma, S.C. (1970), Exp. Brain Res., 10,  171-181. \nGaze, R.M., Jacobson, M. & Szekely, T. (1963). J. Physiol. (Lond.), 165,484-499. \nHalfter, W., Claviez. M.  & Schwarz, U. (1981), Nature. 292.67-70. \nHarris, W.A. (1980), J. Compo Neurol., 194, 303-323. \nHubel, D.H. & Wiesel, T.N. (1974), J. Compo Neurol. 158,295-306. \nHunt, R.K. & Jacobson. M.  (1974), Devel. BioI. 40, 1-15. \nMalsburg, Ch.v.d. & Willshaw, DJ. (1977), PNAS, 74.5176-5178. \nMeyer, R.L. (1979), Science, 205. 819-821;  (1982). Curro Top. Develop. BioI., 17,  101-\n145. \nPrestige, M. & Wills haw , DJ. (1975), Proc. Roy. Soc. B,  190, 77-98. \nSchmidt, J.T. & Easter, S.S. (1978), Exp. Brain Res., 31, 155-162. \nSchmidt, J.T., Cicerone, C.M. & Easter, S.S. (1978), J. Compo Neurol., 177,257-288. \nSharma, S.C. (1975), Brain Res., 93, 497-501. \nSperry, R.W. (1944), J. Neurophysiol., 7. 57-69. \nTrisler, D.  & Collins, F. (1987). Science, 237, 1208-1210. \nWhitelaw, V.A. & Cowan, J.D. (1981), J. Neurosci .\u2022  1,12, 1369-1387. \nWillshaw, D.J.  &  Malsburg, Ch.v.d.  (1976). Proc. Roy. Soc.  B,  194,431-445; (1979), \nPhil. Trans. Roy. Soc. (Lond.). B, 287,  203-254. \nYoon, M. (1972), Amer. Zool., 12, 106. \n\n\f", "award": [], "sourceid": 221, "authors": [{"given_name": "Jack", "family_name": "Cowan", "institution": null}, {"given_name": "A.", "family_name": "Friedman", "institution": null}]}