{"title": "Phase-coupling in Two-Dimensional Networks of Interacting Oscillators", "book": "Advances in Neural Information Processing Systems", "page_first": 123, "page_last": 129, "abstract": null, "full_text": "Phase-coupling in Two-Dimensional \nNetworks of Interacting Oscillators \n\nErnst Niebur, Daniel M. Kammen, Christof Koch, \n\nDaniel Ruderman! & Heinz G. Schuster2 \n\nComputation and Neural Systems \n\nCaltech 216-76 \n\nPasadena, CA 91125 \n\nABSTRACT \n\nCoherent oscillatory activity in large networks of biological or artifi(cid:173)\ncial neural units may be a useful mechanism for coding information \npertaining to a single perceptual object or for detailing regularities \nwithin a data set. We consider the dynamics of a large array of \nsimple coupled oscillators under a variety of connection schemes. \nOf particular interest is the rapid and robust phase-locking that \nresults from a \"sparse\" scheme where each oscillator is strongly \ncoupled to a tiny, randomly selected, subset of its neighbors. \n\nINTRODUCTION \n\n1 \nNetworks of interacting oscillators provide an excellent model for numerous physical \nprocesses ranging from the behavior of magnetic materials to models of atmospheric \ndynamics to the activity of populations of neurons in a variety of cortical locations. \nParticularly prominent in the neurophysiological data are the 40-60 Hz oscillations \nthat have long been reported in the rat and rabbit olfactory bulb and cortex on the \nbasis of single-and multi-unit recordings as well as EEG activity (Freeman, 1978). \nIn addition, periodicities in eye movement reaction times (Poppel and Logothetis, \n1986), as well as oscillations in the auditory evoked potential in response to single \nclick or a series of clicks (Madler and Poppel, 1987) all support a 30 - 50 Hz \nframework for aspects of cortical activity. Two groups (Eckhorn et al., 1988, Gray \n\n1 Permanent address: Department of Physics, University of California, Berkeley, CA 94720 \n2 Permanent address: Institut fiir Theoretische Physik, Universitat Kiel, 2300 Kiell, Germany. \n\n123 \n\n\f124 \n\nNiebur, Kammen, Koch, Ruderman, and Schuster \n\nand Singer, 1989; Gray et al., 1989) have recently reported highly synchronized, \nstimulus specific oscillations in the 35 - 85 Hz range in areas 17, 18 and PMLS of \nanesthetized as well as awake cats. Neurons with similar orientation tuning up to \n7 mm apart show phase-locked oscillations with a phase shift of less than 1 msec \nthat have been proposed to play a role in the coding of visual information (Crick \nand Koch, 1990, Niebur et al. 1991). \nThe complexity of networks of even relatively simple neuronal units - let alone \n\"real\" cortical cells - warrants a systematic investigation of the behavior of two \ndimensional systems. To address this question we begin with a network of mathe(cid:173)\nmatically simple limit-cycle oscillators. While the dynamics of pairs of oscillators are \nwell understood (Sakaguchi, et al. 1988, Schuster and Wagner, 1990a,b), this is not \nthe case for large networks with nontrivial connection schemes. Of general interest \nis the phase-coupling that results in networks of oscillators with different coupling \nschemes. We will summarize some generic features of simple nearest-neighbor cou(cid:173)\npled models, models where each oscillator receives input from a large neighborhood, \nand of \"sparse\" connection geometries where each cell is connected to only a tiny \nfraction of the units in its neighborhood, but with large coupling strength. The \nnumerical work was performed on a CM-2 Connection Machine and involved 16,384 \noscillators in a 128 by 128 square grid. \n\n2 The Model \nThe basic unit in our networks is an oscillator whose phase (Jij is 21r periodic and \nwhich has the intrinsic frequency Wij. The dynamics of an isolated oscillator are \ndescribed by: \n\nd(J\u00b7 . d;) = Wij. \n\n(1) \n\nThe influence of the network can be expressed as an additional interaction term, \n\n(2) \n\nThe coupling function, lij we used is expressed as the sum of terms, each one \nconsisting of the product of a coupling strength and the sine of a phase difference \n(see below, eq. 3). The sinusoidal form of the interaction is, of course, linear for \nsmall differences. \n\nThis system, and numerous variants, has received a considerable amount of attention \nfrom solid state physicists (see, e.g. Kosterlitz and Thouless 1973, and Sakaguchi et \nal. 1988), although primarily in the limit of t -\n00. With an interest in the possible \nrole of networks of oscillators in the parsing or segregating of incident signals in \nnervous systems, we will concentrate on short time, non-equilibrium, properties. \n\nWe shall confine ourselves to two generic network configurations described by \n\nd(Jij L ' \n- = w .. + a \nJ .. 1:lszn((J\u00b7\u00b7 -\ndt \n&) \n&), \n\n&) \n\n(J1:I) \n\n, \n\n1:1 \n\n(3) \n\n\fPhase-coupling in Two-Dimensional Networks of Interacting Oscillators \n\n125 \n\nwhere 0' designates the global strength of the interaction, and the geometry of the \ninteractions is incorporated in Jij ,kl' \n\nThe networks are all defined on a square grid and they are characterized as follows: \n\n1: Gaussian Connections. The cells are connected to every oscillator within a \nspecified neighborhood with Gaussian weighted connections. Hence, \n\nJ \nij ,kl = 271'0' exp \n\n1 \n\n( (i - k) 2 + (j - I) 2 ) \n\n20'2 \n\n. \n\n(4) \n\nWe truncate this function at 20', i.e. Jij,kl = 0 if (i - k)2 + (j _1)2 > (20')2. While \nthe connectivity in the nearest neighbor case is 4, the connectivity is significantly \nhigher for the Gaussian connection schemes: Already 0' = 2 yields 28 connections \nper cell, and the largest network we studied, with 0' = 6, results in 372 connections \nper cell. \n\n2: Sparse Gaussian Connections. In this scheme we no longer require sym(cid:173)\nmetric connections, or that the connection pattern is identical from unit to unit. A \ngiven cell is connected to a fixed number, n, of neighboring cells, with the probability \nof a given connection determined by \n\n'\" \nrij,kl = 271'0' exp \n\n1 \n\n( (i - k? + (j - I? ) \n\n20'2 \n\n. \n\n(5) \n\nJij,kl is unity with probability Pij,kl and zero otherwise. This connection scheme \nis constructed by drawing for each lattice site n coordinate pairs from a Gaussian \ndistribution, and use these as the indices of the cells that are connected with the \noscillator at location (i,j). Therefore, the probability of making a connection de(cid:173)\ncreases with distance. If a connection is made, however, the weight is the same as \nfor all other connections. We typically used n = 5, and in all cases 2 ~ n < 10. \nFor all networks, the sum of the weights of all connections with a given oscillator i, j \nwas conserved and chosen as 0' 2::kl Jij,kl = 10 * w, where w is the average frequency \nof all N oscillators in the system, w = k 2::ij Wij. By this procedure, the total \nimpact of the interaction term is identical in all cases. \n\n3 RESULTS \n\nPerhaps the most basic, and most revealing, comparison of the behavior of the \nmodels introduced above is the two-point correlation function of phase-coupling, \nwhich is defined as \n\n(6) \n\nwhere R is defined as the separation between a pair of cells, R = hj - nIl. We \ncompute and then average C(R, t) over 10,000 pairs of oscillators separated by R in \nthe array. In all cases, the frequencies Wij are chosen randomly, with a Gaussian dis(cid:173)\ntribution with mean 0.5 and variance 1. In Figure 1 we plot C(R, t) for separations \n\n\f126 \n\nNiebur, Kammen, Koch, Ruderman, and Schuster \n\nof R = 20, 30, 40, 50, 6, and 70 oscillators. Time is measured in oscillation peri(cid:173)\nods of the mean oscillator frequency, w. At t = 0, phases are distributed randomly \nbetween 0 and 27r with a uniform distribution. The case of Gaussian connectivity \nwith u = 6 and hence 372 connection per cell is seen in Figure l(a), and the sparse \nconnectivity scheme with u = 6 and n = 5 is presented in Figure l(b). The most \nstriking difference is that correlation levels of over 0.9 are rapidly achieved in the \nsparse scheme for all cases, even for separations of 70 oscillators (plotted as aster(cid:173)\nisks, *), while there are clear separation-dependent differences in the phase-locking \nbehavior of the Gaussian model. In fact, even after t = 10 there is no significant \nlocking over the longer distances of R = 50,60, or 70 units. For local connectivity \nschemes, like Gaussian connectivity with u = 2 or nearest neighbors connections, \nno long-range order evolves even at larger times (data not shown). \nData in Fig. 1 were computed with a uniform phase distribution for t = o. An in(cid:173)\nteresting and robust feature of the dynamics emerges when the influence of different \ntypes of initial phase distributions are examined. In Figure 2 we plot the probabil(cid:173)\nity distribution of phases at different early times. In Figure 2( a) the distribution \nof phases is plotted at t = 0 (diamonds), t = 0.2 (\"plus signs, +) and at t = 0.4 \n(squares) for the sparse scheme with a uniform initial distribution. In Figure 2(b), \nthe evolution of a Gaussian initial distribution centered at () = 7r of the phases \nis plotted. Note the slight curve in the distribution at t = 0, indicating that the \nGaussian initial seeding is rather slight (variance u = 27r) . Remarkably, however, \nthis has a dramatic impact on the phase-locking as after two-tenth of an average \ncycle time (\" plus\" signs) there is already a pronounced peak in the distribution . At \nt = 0.4 (squares) the system that started with the uniform distribution begins to \nonly exhibit a slight increase in the phase-correlation while the system with Gaus(cid:173)\nsian distributed initial phases is strongly peaked with virtually no probability of \nencountering phase values that differ significantly from the mean. \n\n4 DISCUSSION \nThe power of the sparse connection scheme to rapidly generate phase-locking through(cid:173)\nout the network that is equivalent, or superior, to that of the massively intercon(cid:173)\nnected Gaussian scheme highlights a trade-off in network dynamics: massive av(cid:173)\neraging versus strong, long-range, connections. With n = 5, the sparse scheme \neffectively \"tiles\" a two-dimensional lattice and tightly phase-locks oscillators even \nat opposite corners of the array. Similar results are obtained even with n = 2 (data \nnot shown). \n\nIn many ways the Gaussian and sparse geometries reperent opposing avenues to \nachieve global coherence: exhaustive local coupling or distributed, but powerful \nlong-range coupling. The amount of wiring necessary to implement these schemes \nis, however, radically different. \n\n\fPhase-coupling in Two-Dimensional Networks of Interacting Oscillators \n\n127 \n\nAcknowledgement \n\nEN is supported by the Swiss National Science Foundation through Grant No. 8220-\n25941. DMK is a recipient of a Weizman Postdoctoral Fellowship. CK acknowledges \nsupport from the Air Force Office of Scientific Research, a NSF Presidential Young \nInvestigator Award and from the James S. McDonnell Foundation. HGS is sup(cid:173)\nported by the Volkswagen Foundation. \n\nReferences \n\nCrick, F. and Koch, C. 1990. Towards a neurobiological theory of consciousness. \nSeminars Neurosci., 2, 263 - 275. \n\nEckhorn, R., Bauer, R., Jordan, W., Brosch, M., Kruse, W., Munk, M. and Re(cid:173)\nitboeck, H. J. 1988. Coherent oscillations: A mechanism of feature linking in the \nvisual cortex? Bioi. Cybern., 60, 121 - 130. \n\nFreeman, W. J. 1978. Spatial properties of an EEG event in the olfactory bulb and \ncortex. Elect. Clin. Neurophys., 44, 586 - 605. \n\nGray, C. M., Konig, P., Engel, A. K. and Singer, W. 1989. Oscillatory responses \nin cat visual cortex exhibit inter-columnar synchronization which reflects global \nstimulus properties. Nature, 338, 334 -337. \nKosterlitz, J. M. and Thouless, D. J. 1973. Ordering, metastability and phase \ntransitions in two-dimensional systems. J. Physics C., 6, 1181 - 1203. \n\nMadler, C. and Poppel, E. 1987. Auditory evoked potentials indicate the loss of \nneuronal oscillations during general anaesthesia. Naturwissenschaften, 74, 42 - 43. \n\nNiebur, E., Kammen, D. M., and Koch, C. 1991. Phase-locking in I-D and 2-D \nnetworks of oscillating neurons. \nIn Nonlinear dynamics and neuronal networks, \nSinger, W., and Schuster, H. G. (eds.). VCH Verlag: Weinheim, FRG. \nPoppel, E. and Logothetis, N. 1986. Neuronal oscillations in the human brain. \nNaturwissenschaften, 73, 267 - 268. \nSakaguchi, H., Shinomoto, S. and Kuramoto, Y. 1988. Mutual entrainment in \noscillator lattices with nonvariational type interaction. Prog. Theor. Phys., 79, \n1069 - 1079. \n\nSchuster, H. G. and Wagner, P. 1990a. A model for neuronal oscillations in the \nvisual cortex 1: Mean-field theory and derivation of phase equations. Biological \nCybernetics, 64, 77 - 82. \n\nSchuster, H. G. and Wagner, P. 1990b. A model for neuronal oscillations in the \nvisual cortex 2: Phase description of feature dependant synchronization. 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I \n\nFigure 1: Tw~point correlation functions, C(R, t), for various separations, R, in \n(a) the (T = 6 Gaussian scheme with 372 connections per cell and (b) the spar~e \nconnection scheme with (T = 6 and n = 5 connections per cell. Separations of \nR = 20 (diamonds), R = 30 (\"Plus\" signs, +), R = 40 (squares), R = 50 (crosses, \nx), R = 60 (triangles), and R = 70 (asterisks, *) are shown. Note the rapid locking \nfor all lengths in the sparse scheme (b) while the Gaussian scheme (a) appears far \nmore \"diffusive,\" with progressively poorer and slower locking as R increases. \n\n\fPhase-coupling in Two-Dimensional Networks of Interacting Oscillators \n\n129 \n\n14~----~----~----~-----r----~--~--~----~----~----~ \n\n(A I \n\n10 \n\nPI' 1 \n\n8 \n\nr. \n\nFigure 2: Snapshots of the distribution of phases in the sparse scheme \n(n = 5,0' = 6) when the system begins from (a) uniform and (b) a Gaussian \n\"biased\" initial distribution. The figures show the probability P(O) to find a phase \nbetween 0 and 0 + dO (bin size 'KIlO). At t = 0, the distribution is fiat (a) or very \nslightly curved (b); see text. The difference in the time evolution can clearly be \nseen in the state of the system after t = 0.2 (\"plus\" signs, +) and t = 0.4 (squares). \n\n\f", "award": [], "sourceid": 302, "authors": [{"given_name": "Ernst", "family_name": "Niebur", "institution": null}, {"given_name": "Daniel", "family_name": "Kammen", "institution": null}, {"given_name": "Christof", "family_name": "Koch", "institution": null}, {"given_name": "Daniel", "family_name": "Ruderman", "institution": null}, {"given_name": "Heinz", "family_name": "Schuster", "institution": null}]}