{"title": "Morphogenesis of the Lateral Geniculate Nucleus: How Singularities Affect Global Structure", "book": "Advances in Neural Information Processing Systems", "page_first": 133, "page_last": 140, "abstract": null, "full_text": "Morphogenesis  of the Lateral  Geniculate \nNucleus:  How  Singularities Affect  Global \n\nStructure \n\nSvilen Tzonev \nBeckman Institute \nUniversity of Illinois \n\nUrbana, IL 61801 \nsvilen@ks.uiuc.edu \n\nKlaus  Schulten \nBeckman Institute \nUniversity  of Illinois \n\nUrbana, IL  61801 \n\nkschulte@ks.uiuc.edu \n\nJoseph G.  Malpeli \n\nPsychology Department \n\nUniversity of Illinois \nChampaign, IL 61820 \n\njmalpeli@uiuc.edu \n\nAbstract \n\nThe macaque lateral geniculate nucleus (LGN) exhibits an intricate \nlamination pattern, which changes midway through the nucleus at a \npoint coincident with small gaps due to the blind spot in the retina. \nWe  present a  three-dimensional model of morphogenesis in  which \nlocal cell interactions cause a  wave  of development of neuronal re(cid:173)\nceptive fields  to propagate through the nucleus  and establish two \ndistinct lamination patterns.  We  examine the interactions between \nthe wave  and the localized singularities due  to the gaps,  and find \nthat the gaps induce the change in lamination pattern.  We  explore \ncritical factors which  determine general LGN organization. \n\n1 \n\nINTRODUCTION \n\nEach side of the mammalian brain contains a structure called the lateral geniculate \nnucleus  (LGN), which receives visual input from both eyes and sends projections to \n\n\f134 \n\nSvilen  Tzonev,  Klaus  Schulten,  Joseph  G.  Malpeli \n\nthe primary visual  cortex.  In primates the LG N  consists  of several  distinct  layers \nof neurons separated by  intervening  layers  ofaxons and dendrites.  Each layer  of \nneurons maps the opposite visual hemifield in a topographic fashion.  The cells com(cid:173)\nprising these layers  differ  in  terms of their type (magnocellular and parvocellular) , \ntheir input (from ipsilateral (same side) and contralateral (opposite side) eyes), and \ntheir receptive field  organization (ON and OFF center polarity).  Cells  in one layer \nreceive input from one eye only (Kaas et al.,  1972), and in most parts of the nucleus \nhave  the same functional  properties  (Schiller  &  Malpeli,  1978).  The maps  are in \nregister, i.e., representations of a point in the visual field are found in all layers, and \nlie in a  narrow column roughly perpendicular to the layers  (Figure 1).  A prominent \n\na projection \n\ncolumn \n\nFigure  1:  A  slice  along  the  plane  of symmetry of  the  macaque LGN.  Layers  are \nnumbered ventral  to dorsal.  Posterior is  to the left,  where foveal  (central)  parts of \nthe retinas are mapped; peripheral visual fields are mapped anteriorly (right).  Cells \nin different layers have different morphology and functional properties:  6-P /C/ON; \n5-P /I/ON;  4-P /C/OFF;  3-P /I/OFF;  2-M/I/ON&OFF;  1-M/C/ON&OFF,  where \nP  is  parvocellular,  M  is  magnocellular,  C  is  contralateral, I  is  ipsilateral,  ON  and \nOFF refer to polarities of the receptive-field centers.  The gaps in layers 6,  4, and 1 \nare images of the blind spot in the contralateral eye.  Cells in columns perpendicular \nto the layers  receive input from the same point in the visual field. \n\nfeature in this laminar organization is  the presence of cell-free gaps in some layers. \nThese gaps are representations of the blind spot  (the hole in  the retina where the \noptic nerve exits)  of the opposite retina.  In the LGN  of the rhesus macaque mon(cid:173)\nkey (Macaca  mulatta)  the pattern of laminar organization drastically changes at the \nposition of the gaps -\nfoveal  to the gaps there are six distinct layers,  peripheral to \nthe gaps there are four  layers.  The layers  are extended two-dimensional structures \nwhereas the gaps are essentially localized.  However,  the laminar transition occurs \nin a  surface that extends far  beyond the gaps,  cutting completely across the main \naxis  of the LGN  (Malpeli & Baker.,  1975). \n\nWe propose a  developmental model of LGN laminar morphogenesis.  In particular, \nwe investigate the role of the blind-spot gaps in the laminar pattern transition, and \ntheir extended influence over  the global  organization of the nucleus.  In this  model \na  wave  of development caused by local cell interactions sweeps through the system \n(Figure 2).  Strict enforcement of retinotopy maintains and propagates an initially \nlocalized foveal  pattern.  At  the position of the gaps, the system is  in a  metastable \n\n\fMorphogenesis of the Lateral Geniculate Nucleus \n\n135 \n\nmaturing cells \n\nwavefront \n\nimmature cells \n\n.  ,  . \n~;r ..... . \nF ...... I \u2022\u2022\u2022\u2022 ~ \n\n\u2022 \n: \n. \n\n1 \nI \n~/!.. \n\nI \n\n.-\n.\".,. .... ~ ... \n\n.,Y \n\nI \nI \n..  -->-\n\nX \n\nblind spot gap \n(no cells) \n\nFigure 2:  Top view  of a  single layer.  As a  wave  of development sweeps through the \nLGN the foveal  part matures first and the more peripheral parts develop later.  The \nshape  of  the  developmental  wave  front  is  shown  schematically  by  lines  of \"equal \ndevelopment\" . \n\nstate, and the perturbation in retinotopy caused by the gaps is  sufficient to change \nthe state of the system to its preferred four-layered  pattern.  We study the critical \nfactors in this model, and make some predictions about LGN morphogenesis. \n\n2  MODEL  OF LGN MORPHOGENESIS \n\nWe  will  consider only the upper four  (parvocellular)  layers since the laminar tran(cid:173)\nsition does not involve  the other two  layers.  This transition results simply from a \nreordering of the four parvocellular strata (Figure 1).  Foveal to the gaps, the strata \nform  four  morphologically  distinct  layers  (6,  5,  4  and  3)  because adjacent  strata \nreceive  inputs  from  opposite  eyes,  which  \"repel\"  one  another.  Peripheral  to  the \ngaps, the reordering of strata reduces the number of parvocellular eye  alternations \nto one, resulting in two  parvocellular layers  (6+4 and 5+3). \n\n2.1  GEOMETRY AND VARIABLES \n\nLGN  cells  Ci  are  labeled  by  indices  i  =  1,2, ... , N.  The  cells  have  fixed, \nquazi-random  and  uniformly  distributed  locations  ri  EVe  R3,  where \nV  =  {(x,y,z) 10  < x < Sz,O < y < Sy,O  < z < Sz},  and  belong  to  one  projection \ncolumn  Cab,  a  =  1,2, ... ,A  and  b  =  1,2, ... ,B,  (Figure 3). \nFunctional \nproperties  of  the  neurons  change  in  time  (denoted  by  T),  and  are  described \nby  eye  specificity  and  receptive-field  polarity,  ei(T),  and  Pi(T),  respectively: \nei (T),  pdT)  E  [-1,1]  C  R,  i = 1,2, ... ,N,  T =  0, 1, ... , Tmaz \u2022 \nThe values of eye specificity and polarity represent the proportions of synapses from \ncompeting  types  of retinal ganglion  cells  (there are four  type  of ganglion  cells  -\nfrom different  eyes  and with ON or  OFF polarity).  ei = -1 (ei  = 1)  denotes that \nthe i-th cell is  receiving input solely from the opposite (same side) retina.  Similarly, \nPi  =  -1 (Pi  = 1)  denotes that the cell input is pure ON (OFF) center.  Intermediate \nvalues  of ei  and Pi  imply  that the cell  does  not  have  pure  properties  (it  receives \n\n\f136 \n\nSvilen  Tzonev,  Klaus  Schulten,  Joseph  G.  Malpeli \n\nFigure 3:  Geometry of the model.  LGN cells Ci  (i  =  1,2, ... , N)  have fixed random, \nand uniformly-distributed locations Ti  within a  volume VcR , and belong to one \nprojection column Cab. \n\ninput from retinal ganglion cells of both eyes  and with different  polarities).  Initial(cid:173)\nly,  at  r  =  0,  all  LGN  cells  are  characterized by  ei,  Pi  =  O.  This  corresponds  to \ntwo  possibilities:  no  retinal ganglion cells synapse on any LGN cell,  or proportions \nof synapses  from  different  ganglion  cells  on  all  LGN  neurons  are  equal,  i.e.,  neu(cid:173)\nrons possess completely undetermined functionality because of competing inputs of \nequal strength.  As  the neurons mature and acquire functional properties, their eye \nspecificity and polarity reach their asymptotic values,  \u00b1l. \n\nEven when cells  are not completely mature, we  will  refer  to them as  being  of four \ndifferent  types,  depending  on  the  signs  of  their  functional  properties.  Following \naccepted anatomical  notation,  we  will  label  them  as  6,  5,  4,  and  3.  We  denote \neye  specificity of cell  types 6 and 4 as  negative, and cell  types 5 and 3  as  positive. \nPolarity of cell types 6 and 5 is negative, while polarity of types 4 and 3 is  positive. \n\nCell  functional  properties  are  subject  to  the dynamics  described in  the following \nsection.  The process of LGN development  starts from  its foveal  part, since in  the \nretina it  is  the fovea  that  matures  first.  As  more  peripheral  parts  of the  retina \nmature,  their  ganglion cells  start to  compete to  establish  permanent synapses  on \nLGN cells.  In this sorting process, each LGN cell gradually emerges with permanent \nsynapses  that  connect only  to several  neighboring ganglions  of the same  type.  A \nwave  of  gradual  development  of  functionality  sweeps  through  the  nucleus.  The \ndriving force  for  this  maturation process is  described by  localized cell  interactions \nmodulated by  external influences.  The particular pattern of the foveal  lamination \nis shaped by  external forces,  and later serves as a starting point for  a  \"propagation \nof sameness\" of cell properties.  Such a sameness propagation produces clustering of \nsimilar cells  and formation of layers.  It should be stressed that cells  do  not move, \nonly their characteristics change. \n\n2.2  DYNAMICS \n\nThe variables  describing cell functional  properties are subject to the following  dy(cid:173)\nnamics \n\nedr + 1) \nPi  (r + 1) \n\n-\n\nei (r) + ~ei (r) + 1Je \nPi (r) + ~Pi (r) + 1Jp, \n\n1,2, ... ,N. \n\n(1) \n\n\fMorphogenesis of the Lateral Geniculate Nucleus \n\n137 \n\nIn Eq.  (1), there are  two  contributions to the change of the intermediate variables \nei ( T)  and Pi ( T).  The first  is  deterministic, given  by \n\n6.e;( r)  = \n\n,,(r,) [ (t. e; (r)! (Ie; - r;ll) + E\", (r,)]  (1- c; (r)) f3;. \n\n6.p,(r) \n\n-\n\n,,(r,) [(t,p;(r)!(lr,_r;ll) +P\", (r,)]  (I-p;(r)) f3; \u2022. (2) \n\nThe second is  a stochastic contribution corresponding to fluctuations in the growth \nof  the  synapses  between  retinal  ganglion  cells  and  LGN  neurons.  This  noise  in \nsynaptic growth plays  both a  driving  and a  stabilizing role to be explained below. \nWe explain the meaning of the variables in Eq. (2) only for the eye specificity variable \nei.  The corresponding parameters for  polarity Pi  have similar interpretations. \nThe parameter a  (ri)  is  the rate of cell development.  This rate is  the same for  eye \nspecificity and polarity.  It depends on  the position ri  of the cell  in order to allow \nfor  spatially non-uniform development.  The functional form of a  (ri)  is  given in the \nAppendix. \nThe term Eint (ri)  =  2:7=1 ejl (h - rjl)  is  effectively  a  cell force  field.  This  field \ninfluences  the  development  of nearby  cells  and  promotes  clustering  of same  type \nof  cells.  It  depends  on  the  maturity  of  the generating  cells  and  on  the  distance \nbetween cells  through the interaction function 1(8).  We  chose for  1(8)  a  Gaussian \nform, i.e.,  1 (8)  =  exp ( _6 2 / (T2),  with characteristic interaction distance (T. \nThe external influences on cell development are incorporated in the term for the ex(cid:173)\nternal field Eext(ri).  This external field plays two roles:  it launches a particular lam(cid:173)\ninar configuration of the system (in the foveal  part of the LGN), and determines its \nperipheral pattern.  It has, thus, two contributions Eext (ri)  =  E!xt (ri) + E~xt (ri). \nThe exact forms  of E!xt(ri)  and E~xt(ri) are provided in  the Appendix. \nThe  nonlinear  term  (1  - e~) in  Eq.  (2)  ensures  that  \u00b11  are  the only  stable fixed \npoints  of  the  dynamics.  The  neuronal  properties gradually  converge  to  either  of \nthese fixed  points  capturing the maturation process.  This term also stabilizes the \ndynamic variables and prevents them from diverging. \nThe last term f3~b ( T)  reflects the strict columnar organization of the maps.  At each \nstep of the development the proportion of all four  types of LGN cells  is  calculated \nwithin a single column Cab,  and f3~b(T) for different types t  is adjusted such that all \ntypes are equally represented.  Without this term, the cell organization degenerates \nto  a  non-laminar  pattern  (the system  tries  to  minimize  the  surfaces  between  cell \nclusters of different type).  The exact form of f3!b(T)  is  given in the Appendix. \nAt  each stage of LGN  development,  cells  receive  input from  retinal  ganglion  cells \nof  particular  types.  This  means  that  eye  specificity  and  polarity  of  LGN  cells \nare  not independent variables.  In fact,  they  are tightly  coupled in  the sense  that \nlei ( T) I =  IPi ( T) I should  hold  for  all  cells  at  all  times.  This  gives  rise  to  coupled \ndynamics described by \n\n\f138 \n\nSvilen  Tzonev,  Klaus  Schulten,  Joseph  G.  Malpeli \n\nmi(7+1) \nei (7 + 1) \nPi (7 + 1) \n\n- min ( I ei (7 + 1) I,  IPi  (7 + 1) I ) \n=  mi (7 +  1) sgn ( ei (7 +  1) ) \n- md7+ 1)  sgn(Pi(7+ 1)),  i  =  1,2, ... ,N. \n\n(3) \nThe blind spot gaps are modeled by not allowing cells in certain columns to acquire \ntypes of functionality for  which retinal projections do  not exist,  e.g., from the blind \nspot of the opposite eye.  Accordingly,  ei  is  not allowed  to become negative.  Thus, \nsome cells never reach a pure state ei, Pi  =  \u00b1 1.  It is assumed that in reality such cells \ndie  out.  Of all  quantities  and parameters,  only  variables  describing  the  neuronal \nreceptive fields  (ei  and Pi)  are time-dependent. \n\n3  RESULTS \n\nWe  simulated the dynamics described by Eqs.  (1,  2,  3), typically for  100,000 time \nsteps.  Depending on  the rate of cell  development,  mature states were  reached in \nabout 10,000 steps.  The maximum value of Q:  was  0.0001.  We used an interaction \nfunction  with u = 1. \nFirst, we  considered a  two-dimensional LGN, V  = {(x,z)IO < x < Sx,O  < z < Sz} \nwith Sx  =  10 and Sz  =  6.  There were ten projection columns (with equal size) along \nthe x  axis.  An initial  pattern was  started in the foveal  part by  the external field. \nThe size of the gaps 9  measured in  terms of the interaction distance u  was  crucial \nfor  pattern development.  When the developmental wave  reached the gaps,  layer  6 \ncould\" jump\" its gap and continued to spread peripherally if the gap was sufficiently \nnarrow (g/u < 1.5).  If its gap was  not too narrow  (g/u > 0.5),  layer  4 completely \nstopped (since cells in the gaps were not allowed to acquire negative eye specificity) \nand so layers 5 and 3 were able to merge.  Cells of type 4 reappeared after the gaps \n(Figure 4,  right side, shows behavior similar to the two-dimensional model) because \nof the required equal representation of all cell types in the projection columns, and \nbecause  of  noise  in  cell  development.  Energetically,  the most  favorable  position \nof cell  type 4  would  be on top  of type 6,  which  is  inconsistent  with  experimental \nobservations.  Therefore,  one must assume the existence of an external field  in the \nperipheral part that will  drive the system away from its otherwise preferred state. \nIf the gaps were  too large  (g/u  > 1.5),  cells  of type 6  and 4  reappeared after the \ngaps in  a  more or less  random vertical position and caused transitions of irregular \nnature.  On the other hand,  if the gaps  were  too narrow  (g/u < 0.5),  both layers \n6 and 4 could continue to grow past their gaps,  and no  transition between laminar \npatterns occurred at all.  When g/u was  close to the above limits, the pattern after \nthe gaps  differed  from  trial  to  trial.  For  the  two-dimensional  system,  a  realistic \nperipheral pattern always  occurred for  0.7 < g/u < 1.2. \nWe simulated a  three-dimensional system with size Sx  = 10,  Sy = 10,  and Sz  = 6, \nand  projection columns  ordered in  a  10  by  10  grid.  The topology  of the  system \nis  different  in two  and three dimensions:  in two  dimensions the gaps interrupt the \nlayers  completely and,  thus,  induce perturbations which  cannot  be by-passed.  In \nthree dimensions the gaps are just holes  in  a  plane and generate localized pertur(cid:173)\nbations:  the layers can, in  principle,  grow  around the gaps maintaining the initial \nlaminar pattern.  Nevertheless,  in  the three-dimensional case,  an  extended transi-\n\n\fMorphogenesis  of the Lateral Geniculate Nue/ells \n\n139 \n\n6 \n5 \n4~~UIIii \n3 \n\nFigure 4:  Left:  Mature state of the macaque LG~ -\nresult of the three-dimensional \nmodel with 4,800  cells.  Spheres with different  shades represent cells  with different \nproperties.  Gaps in strata 6 and 4 (this gap is not visible)  are coded by the darkest \ncolor,  and  coincide with the transition surface between 4- and  2-layered  patterns. \nRight:  A  cut  of the three-dimensional structure along  its  plane  of symmetry.  A \ntwo-dimensional system exhibits  similar  organization.  Compare with upper layers \nin Figure 1.  Spatial segregation between layers is  not modeled explicitly. \n\ntion  was  triggered by  the gaps.  The transition surface, which passed through the \ngaps and was oriented roughly perpendicularly to the x  axis,  cut completely across \nthe nucleus  (Figure 4). \n\nSeveral factors were critical for the general behavior of the system.  As in two dimen(cid:173)\nsions, the size of the gap~ must be within certain limits:  typically 0.5  <  g / (J\"  <  1.0. \nThese limits depend on the curvature of the wavefront.  The gaps must lie in a certain \n\"inducing\" interval along the x  axis.  If they were too close to the origin, the foveal \npattern was  still  more stable, so  no  transition could be induced there.  However,  a \nspontaneous transition might occur downstream.  If the gaps were too far from the \norigin,  a  ~pontaneou~ transition might occur before them.  The occurrence and  lo(cid:173)\ncation of a spontaneous transition, (therefore, the limits of the \"inducing\" interval) \ndepended on the external-field parameters.  A realistic transition was observed only \nwhen the front  of the developmental wave  had sufficient  curvature when it reached \nthe  gaps.  Underlying anatomical reasons for  a  sufficiently  curved front  along  the \nmain axis  could  be the curvature of the nucleus,  differences  in  layer  thickness,  or \ndifferences in ganglion-cell densities in the retinas. \n\nPropagation of the developmental wave away from the gaps was quite stable.  Before \nand  after the  gaps,  the wave  simply  propagated the  already established  patterns. \nIn a  system without gaps, transitions of variable shape and location occurred when \nthe  peripheral contribution  to  the  external  fields  was  sufficiently  large;  a  weaker \ncontribution allowed  the foveal  pattern to propagate through the entire nucleus. \n\n4  SUMMARY \n\n\\Ve  present  a  model  that  successfully  captures  the  most  important  features  of \nmacaque LG~ morphogenesis.  It produces realistic laminar patterns and supports \n\n\f140 \n\nSvilen  Tzonev,  Klaus  Schulten,  Joseph  G.  Malpeli \n\nthe hypothesis (Lee &  Malpeli, 1994)  that the blind spot gaps trigger the transition \nbetween patterns.  It predicts that critical factors in LGN development are the size \nand location  of  the gaps,  cell  interaction distances,  and shape of the front  of the \ndevelopmental  wave.  The model may  be general  enough to incorporate the LGN \norganizations of other primates.  Small singularities, similar to the blind spot gaps, \nmay have an extended influence on global organization of other biological systems. \n\nAcknowledgements \n\nThis work has been supported by a Beckman Institute Research Assistantship, and \nby  grants PHS 2P41 RR05969 and NIH EY02695. \n\nReferences \n\nJ.H.  Kaas,  R.W.  Guillery  &  J.M.  Allman.  (1972)  Some principles  of organization \nin the dorsal lateral geniculate nucleus,  Brain  Behav.  Evol.,  6:  253-299. \nD.Lee  &  J.G.Malpeli.  (1994)  Global  Form  and  Singularity:  Modeling  the  Blind \nSpot's Role in Lateral Geniculate Morphogenesis,  Science,  263:  1292-1294. \nJ.G.  Malpeli  &  F.H.  Baker.  (1975)  The  representation  of the  visual  field  in  the \nlateral geniculate nucleus of Macaca mulatta, 1.  Comp.  Neural.,  161:  569-594. \nP.H. Schiller & J.G. Malpeli.  (1978) Functional specificity ofLGN ofrhesus monkey, \n1.  Neurophysiol.,  41:  788-797. \nAPPENDIX \nThe form of 0  (x, y, z)  (with 00  = 0.0001)  was chosen as \n\no(x,y,z)  = 00(0.1+exp(-(y-Sy/2)2)). \n\n(4) \n\nFoveal external fields  of the following  form were used: \nE!:z:t(x,y,z)  =  lO[O(z-d)-20(z-2d)+20(z-3d)-O(d-z)] exp(-x) \np!:z:t(x,y,z)  =  10[20(z-2d)-1]exp(-x), \n(5) \nwhere  d  = Sz/4  is  the  layers'  thickness  and  the  \"theta\" function  is  defined  as \nO( x)  =  1, x  > 0  and O( x)  = 0, x  < O.  Peripheral  external  fields  (in  fact  they are \npresent  everywhere  but  determine  the  pattern in  the  peripheral part  only)  were \nchosen as \n\nE::z:t(x,y,z)  =  5[20(z-2d)-1] \nP::z:t(x,y,z)  =  5 [O(z-d)-20(z-2d)+20(z-3d)-O(d-z)]. \n\n(6) \nf3!b ( 1')  was  calculated in the following way:  at any given time 1',  within the column \nCab,  we  counted the number N~b( 1')  of cells,  that could be classified as  one of the \nfour  types  t  =  3,4,5,6.  Cells  with  ei( 1')  or Pi( 1')  exactly  zero  were  not  counted. \nThe total number of classified cells is then Nab( 1')  = 2:~=3 N~b( 1').  If there were no \nclassified cells (Nab( 1')  =  0), then f3~b( 1')  was set to one for all t.  Otherwise the ratio \nof different  types was  calculated:  n~b = N~b(1')/Nab(1'). In this way  we calculated \n(7) \n\nf3~b (1')  =  4 - 12 nab,  t  = 3,4,5,6. \n\nIf f3~b ( 1')  was  negative it was replaced by  zero. \n\n\f", "award": [], "sourceid": 891, "authors": [{"given_name": "Svilen", "family_name": "Tzonev", "institution": null}, {"given_name": "Klaus", "family_name": "Schulten", "institution": null}, {"given_name": "Joseph", "family_name": "Malpeli", "institution": null}]}