{"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}]}