{"title": "Theory of Self-Organization of Cortical Maps", "book": "Advances in Neural Information Processing Systems", "page_first": 451, "page_last": 458, "abstract": null, "full_text": "451 \n\nTHEORY OF SELF-ORGANIZATION OF \n\nCORTICAL MAPS \n\nFundamental Research Laboratorys, NEC Corporation \n\n1-1 Miyazaki 4-Chome, Miyamae-ku, Kawasaki, Kanagawa 213, Japan \n\nShigeru Tanaka \n\nABSTRACT \n\nWe  have  mathematically  shown  that  cortical  maps  in  the \nprimary sensory  cortices  can  be  reproduced  by  using  three \nhypotheses  which  have  physiological  basis  and  meaning. \nHere, our main focus is on ocular.dominance column formation \nin the primary visual cortex.  Monte Carlo simulations on the \nsegregation of ipsilateral and contralateral afferent terminals \nare carried out.  Based on  these,  we  show that almost all  the \nphysiological  experimental  results  concerning  the  ocular \ndominance patterns of cats and monkeys reared under normal \nor various abnormal visual conditions can be explained from a \nviewpoint of the phase transition phenomena. \n\nROUGH SKETCH OF OUR THEORY \n\nIn order to describe the use-dependent self-organization of neural connections \n{Singer,1987 and Frank,1987},  we  have  proposed  a  set of coupled  equations \ninvolving  the  electrical  activities  and  neural  connection  density  {Tanaka, \n1988}, by using the following physiologically based hypotheses: (1) Modifiable \nsynapses grow or collapse due to  the competition among themselves for  some \ntrophic factors,  which are secreted retrogradely from  the postsynaptic side to \nthe  presynaptic  side.  (2)  Synapses  also  sprout or  retract  according  to  the \nconcurrence  of presynaptic  spike  activity  and  postsynaptic  local  membrane \ndepolarization.  (3)  There already exist lateral connections within the  layer, \ninto  which  the  modifiable  nerve  fibers  are  destined  to  project,  before  the \nsynaptic modification begins.  Considering this set of equations, we  find  that \nthe  time  scale  of electrical  activities  is  much  smaller  than  time  course \nnecessary  for  synapses  to  grow  or  retract.  So  we  can  apply  the  adiabatic \napproximation to the equations.  Furthermore, we identify the input electrical \nactivities,  i.e.,  the  firing  frequency  elicited  from  neurons  in  the  projecting \nneuronal layer, with the stochastic process which is specialized by the spatial \ncorrelation function  Ckp;k'  p'.  Here,  k  and  k'  represent  the  positions  of the \nneurons  in  the  projecting  layer. \nII  stands  for  different  pathways  such  as \nipsilateral  or  contralateral,  on-center  or  off-center,  colour  specific  or \nnonspecific  and  so  on.  From  these  approximations,  we  have  a  nonlinear \n\n\f452 \n\nTanaka \n\nstochastic differential equation for  the connection density, which describes a \nsurvival  process  of  synapses  within  a  small  region,  due  to  the  strong \ncompetition.  Therefore,  we  can  look  upon  an  equilibrium  solution  of this \nequation as a  set of the  Potts  spin variables 0jk11'S  {Wu,  1982}.  Here,  if the \nneuron k in the projecting layer sends the axon to  the position j  in the target \nlayer,  0jk11 = 1 and  if not, 0jk11 = O.  The Potts spin  variable  has  the  following \nproperty: \n\nIf we  limit the discussion  within such equilibrium solutions,  the  problem  is \nreduced  to  the  thermodynamics  in  the  spin  system.  ' The  details  of  the \nmathematics are  not argued here because  they are beyond  the  scope  of this \npaper {Tanaka}.  We  find  that equilibrium behavior of the  modifiable  nerve \nterminals  can  be  described  in  terms  of thermodynamics  in  the  system  in \nwhich Hamiltonian H and fictitious temperature T are given by \n\nwhere k  and Ck11  ;k' 11'  are  the  averaged firing frequency  and  the  correlation \nfunction,  respectively.  Vii'  describes  interaction  between  synapses  in  the \ntarget layer.  q  is the  ratio of the  total averaged membrane potential  to  the \na veraged membrane potential induced through the modifiable synapses from \nthe  projecting  layer. \n\"tc  and  \"ts  are  the  correlation  time  of the  electrical \nactivities and the time course necessary for synapses to grow or collapse. \n\nAPPLICATION TO  THE  OCULAR  DOMINANCE \n\nCOLUMN  FORMATION \n\nA specific cortical map structure is determined by the choice of the correlation \nfunction  and  the  synaptic  interaction  function.  Now,  let  us  neglect  k \ndependence of the correlation function and take into account only ipsilateral \nand contralateral pathways denoted by p, for mathematical simplicity.  In this \ncase, we can reduce the Potts spin variable into the Ising spin one through the \nfollowing transformation: \n\n\fTheory of Self-Organization of Cortical Maps \n\n453 \n\nwhere j  is the position in the layer 4 of the primary visual cortex, and Sj  takes \nonly  + 1 or  -1, according to the ipsilateral or contralateral dominance.  We \nfind that this system can be described by Hamiltonian: \n\nH = -h \"  S .- - \"\"  \"\"  V .. , S . S., \nJ \n\nJJ \n\nJ \n\nL.  J \nj \n\nJ \n2 L. L. \nj':;ej \n\nj \n\n(3) \n\nThe first term of eq.(3)  reflects the ocular dominance shift, while  the second \nterm is essential to the ocular dominance stripe segregation. \n\nHere, we adopt the following simplified function as Vii': \n\nV . . , = -2  8  (A  -d .. ,) - -2- 8  (A .  h- d . . ,)  , \n\n(4) \n\nJ J \n\nex \n\nJ J \n\nm \n\nJ J \n\nqex \nIIA \n\nex \n\nqinh \nIIA .  h \ntn \n\nwhere djj' is the distance between j  and j'.  Aex and Ainh are determined by the \nextent of excitatory and inhibitory lateral connections, respectively.  8  is the \nstep  function.  q.\"  and  q i\"~  are  propotional  to  the  membrane  potentials \ninduced by excitatory and inhibitory neurons {Tanaka}.  It is not essential to \nthe qualitative discussion whether the interaction function is given by the use \nof the step function, the Gaussian function, or others. \n\nNext, we define 11+1  and 11-1  as the average firing frequencies of ipsilateral \nand contralateral retinal ganglion cells (RGCs),  and  ~\u00b1 1 v and  ~\u00b1 18  as  their \nfluctuations which originate in the  visually stimulated and the spontaneous \nfirings  of  RGCs,  respectively.  These  are  used  to  calculate  two  new \nparameters, r and a: \n\n(5) \n\n(6) \n\na= \n\n\f454 \n\nTanaka \n\nr is related to  the correlation of firings elicited from  the left and right RGCs. \nIf there are only spontaneous firings,  there is no correlation between the left \nand  right  RGCs'  firings.  On  the  other  hand,  in  the  presence  of visual \nstimulation, they will  correlate, since the  two  eyes receive  almost the same \nimages in normal animals.  a  is a function of the imbalance of firings of the left \nand right RGCs.  Now, J and h in eq.(3) can be expressed in terms ofr and a: \n\nl_a2 ) \nJ=b  l - r - -\n1 + a2 \n\n( \n1 \n\n' \n\n(8) \n\nwhere  b1  is  a  constant  of the  order  of 1,  and  b2  is  determined  by  average \nmembrane potentials. \n\nUsing the  above  equations,  it will  now  be  shown  that patterns such  as  the \nones observed for  the  ocular  dominance  column  of new-world  monkeys  and \ncats  can  be  explained.  The  patterns  are  very  much  dependent  on  three \nparameters r, a  and K  which is the ratio of the membrane potentials (qinh/qex) \ninduced by the inhibitory and excitatory neurons. \n\nRESULTS AND DISCUSSIONS \n\nIn the subsequent analysis by  Monte  Carlo simulations, we  fix  the values of \nparameters:  qex=I.O,  Aex =O.25,  Ainh=l.O,  T=O.25,  bl=l.O,  b2=O.I,  and \ndx=O.l.  dx  is the diameter ofa small area which is occupied by one spin.  In \nthe computer simulations of Fig. 1, we can see that the stripe patterns become \nmore segregated as the correlation strength r decreases.  The similarity of the \npattern in Fig.lc to the well-known experimental evidence {Hubel and Wiesel, \n1977} is striking.  Furthermore, it is known that if the animal has been reared \nunder  the  condition  where  the  two  optic  nerves  are  electrically  stimulated \nsynchronously, stripes in the primary visual cortex are not formed {Stryker}. \nThis condition corresponds to r values close to  I  and again our theory predicts \nthese experimental results as can be seen in Fig.la.  On the contrary, if the \nstrabismic animal  has  been reared  under  the  normal  condition {Wiesel  and \nHubel,  1974},  r  is  effectively  smaller than that of a  normal  animal.  So  we \nexpect that the ocular dominance stripe has very sharp delimitations as it is \nobserved experimentally.  In the case of a binocularly deprived animal {Wiesel \nand  Hubel,  1974},i.e.,  ~+lv=~_lv=O, it  is  reasonable  to  expect  that  the \nsituation is similar to the strabismic animal. \n\n\fTheory of Self-Organization of Cortical Maps \n\n455 \n\nFigure 1.  Ocular dominance patterns given by  the computer \nsimulations  in  the  case  of the  large  inhibitory  connections \n(K= 1.0)  and  the  balanced  activities  (a= 0).  The  correlation \nstrength r  is given in each case:  r=0.9 for  (a),  r=0.6 for  (b), \nand r= 0.1 for (c). \n\nIn  the  case  of a* 0,  we  can  get  asymmetric  stripe  patterns  such  as  one  in \nFig.2a.  Since  this  situation  corresponds  to  the  condition  of the  monocular \ndeprivation,  we  can  also  explain  the  experimental  observation  {Hubel  et \na1.,1977} successfully.  There are other patterns seen in Fig.2b, which we call \nblob lattice patterns.  The existence of such patterns has not been confirmed \nphysiologically,  as  far  as  we  know.  However,  this  theory  on  the  ocular \ndominance column  formation  predicts  that the  blob  lattice  patterns  will  be \nfound  if  appropriate  conditions,  such  as  the  period  of  the  monocular \n\nFigure 2.  Ocular dominance patterns given by  the computer \nsimulations  in  the  case  of the  large  inhibitory  connections \n(K=1.0)  and  the  imbalanced  activities:  a=0.2  for  (a)  and \na= 0.4 for (b).  The correlation strength r is given by r= 0.1 for \nboth (a) and (b). \n\n\f456 \n\nTanaka \n\ndeprivation, are chosen. \nWe  find  that  the  straightness  of  the  stripe  pattern  is  controlled  by  the \nparameter  K.  Namely,  if K is  large,  i.e.  inhibitory  connections  are  more \neffective than excitatory ones,  the pattern is straight.  However if K is small \nthe pattern has many branches and ends.  This is illustrated in Fig.  3c.  We \ncan  get  a  pattern similar  to  the  ocular  dominance  pattern  of normal  cats \n{Anderson et al., 1988}, ifK is small and r~rc (Fig.3b).  The meaning of rc will \nbe discussed in the following paragraphs.  We further get a labyrinth pattern \nby means of r smaller than r c and the same K.  We can think K val ue is specific \nto  the animal  under consideration because  of its definition.  Therefore,  this \ntheory also predicts that the ocular dominance  pattern of the strabismic cat \nwill be sharply delimitated but not a straight stripe in contrast to  the pattern \nof monkey. \n\nFigure 3.  Ocular dominance patterns given by  the computer \nsimulations  in  the  case  of  the  small  inhibitory  connections \n(K=0.3)  and  the  balanced  activities(a=O).  The  correlation \nstrength r is given in each case: r= 0.9 for (a), r=0.6 for (b) and \nr=O.l for (c). \n\nHaving  seen  specific  examples,  let  us  now  discuss  the  importance  of \nparameters  r  and  a,  which  stand  for  the  correlation  strength  and  the \nimbalance of firings.  According to  qualitative difference of patterns obtained \nfrom our simulations, we classify the parameter space (r, !l)  into three regions \nin Fig.4: In region (S), stripe patterns appear.  The left-eye dominance and the \nright-eye dominance bands are equal in width, for  a=O.  On  the other hand, \nthey are not equal for non-zero value.  In region (B), patterns are blob lattices. \nIn  region  (U),  the  patterns  are  uniform  and  we  do  not  see  any  spatial \nmodulation.  A  uniform  pattern  whose  a  val ue  is  close  to  0  is  a  random \npattern, while if a is close to  1 or  -1 either ipsilateral or contralateral nerve \nterminals are present.  On the horizontal axis, (S) and (U)  regions are devided \nby  the critical  point rc.  In practice if we  define  the order  parameter as  the \n\n\fTheory of Self-Organization of Cortical Maps \n\n457 \n\nensemble-averaged amplitude of the dominant Fourier component of spatial \npatterns, and the susceptibility as the variance of the amplitude, then we can \nobserve their singular behavior near r = r c' \n\nVarious conditions  where  animals have been reared correspond positions in \nthe parameter space of Fig.4: normal (N), synchronized electrical stimulation \n(SES),  strabismus  (S),  binocular  deprivation  (BD),  long-term  monocular \ndeprivation  (LMD)  and  short-term  monocular  deprivation  (SMD).  If an \nanimal is kept under the monocular deprivation for a long period, the absolute \nvalue of is close to 1 and r  value is 0,  considering eqs.(5) and (6).  For a short(cid:173)\nterm monocular deprivation, the corresponding point falls on anywhere on the \nline from N to LMD,  because relaxation from  the symmetric stripe pattern to \nthe open-eye  dominant uniform pattern is  incomplete.  The position on  this \nline is, therefore, determined by this relaxation period, in which the animal is \nkept under the monocular deprivation. \n\n1 \n\nUiD \n\na \n\n(5) \n\n(U) \n\nBD~~ ______ ~ ____ ~_SE_S~ \no  S \n1 \n\nre \n\nN \nr \n\nFigure 4.  Schematic phase diagram for  the pattern of ocular \ndominance columns.  The parameter space (r, a)  is devided into \nthree regions: (S) stripe region, (B) blob lattice region, and (U) \nuniform  region.  N,  SES,  S,  BD,  LMD,  and  SMD  stand  for \nconditions:  normal,  synchronized  electrical  stimulation, \nstrabismus,  binocular  deprivation,  long-term  monocular \ndeprivation,  and  short-term  monocular  deprivation, \nrespectively.  We  show  only  the  diagram  on  the  upper  half \nplane, because the diagram is symmetrical with respect to the \nline of a=O. \n\n\f458 \n\nTanaka \n\nCONCLUSION \n\nIn this report, a new theory has been proposed  which is able  to explain such \nuse-dependent self-organization as  the ocular dominance  column formation. \nWe have compared the theoretical results with various experimental data and \nexcellent  agreement  is  observed.  We  can  also  explain  and  predict  self(cid:173)\norganizing process  of other cortical  map  structures such  as  the  orientation \ncolumn, the retinotopic organization, and so on.  Furthermore, the three main \nhypotheses of this theory are not confined to  the primary visual cortex.  This \nsuggests  that  the  theory  will  have  a  wide  applicability  to  the  formation  of \ncortical map structures seen in the somatosensory cortex {Kaas et al.,1983}, \nthe auditory cortex {Knudsen et al.,1987}, and the cerebellum {Ito,1984}. \n\nReferences \nP.A.Anderson, J.Olavarria, RC.Van Sluyter, J.Neurosci. 8,2184 (1988). \nE.Frank, Trends in Neurosci. 10,188 (1987). \nD.H.Hubel and T.N.Wiesel, Proc.RSoc.Lond.B198,1(1977). \nD.H.Hubel, T.N.Wiesel, S.LeVay, Phil.Trans.RSoc. Lond. B278, 131 (1977). \nM.Ito, The Cerebellum and Neural Control (Raven Press, 1984). \nJ.H.Kaas, M.M.Merzenich, H.P.Killackey, Ann. Rev. Neurosci. 6,325 (1983). \nE.I.Knudsen, S.DuLac, S.D.Esterly, Ann. Rev. Neurosci. 10,41 (1987). \nW.Singer,in The Neural and Molecular Bases of Learning (Hohn Wiley & \n\nSons Ltd.,1987) pp.301-336; \n\nM.P.Stryker, in Developmental Neurophysiology (Johns Hopkins Press), in \n\npress. \n\nS.Tanaka, The Proceeding ofSICE'88, ESS2-5, p.  1069 (1988). \nS. Tanaka, to be submitted. \nT.N.Wiesel and D.H.Hubel, J.Comp.Neurol.158, 307 (1974). \nF.Y.Wu, Rev.  Mod.  Phys.  54,235 (1982). \n\n\f", "award": [], "sourceid": 132, "authors": [{"given_name": "Shigeru", "family_name": "Tanaka", "institution": null}]}