{"title": "Synchrony and Desynchrony in Neural Oscillator Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 199, "page_last": 206, "abstract": null, "full_text": "Synchrony and Desynchrony \nin Neural Oscillator Networks \n\nDeLiang Wang \n\nDepartment of Computer and Information Science \n\nand Center for Cognitive Science \n\nThe Ohio State University \n\nColumbus, Ohio 43210, USA \n\ndwang@cis.ohio-state.edu \n\nDavid Terman \n\nDepartment of Mathematics \nThe Ohio State University \n\nColumbus, Ohio 43210, USA \nterman@math.ohio-state.edu \n\nAbstract \n\nAn novel class of locally excitatory, globally inhibitory oscillator \nnetworks is proposed. The model of each oscillator corresponds to a \nstandard relaxation oscillator with two time scales. The network \nexhibits a mechanism of selective gating, whereby an oscillator \njumping up to its active phase rapidly recruits the oscillators stimulated \nby the same pattern, while preventing others from jumping up. We \nshow analytically that with the selective gating mechanism the network \nrapidly achieves both synchronization within blocks of oscillators that \nare stimulated by connected regions and desynchronization between \ndifferent blocks. Computer simulations demonstrate the network's \npromising ability for segmenting multiple input patterns in real time. \nThis model lays a physical foundation for the oscillatory correlation \ntheory of feature binding, and may provide an effective computational \nframework for scene segmentation and figure/ground segregation. \n\n1 INTRODUCTION \n\nA basic attribute of perception is its ability to group elements of a perceived scene into \ncoherent clusters (objects). This ability underlies perceptual processes such as \nfigure/ground segregation, identification of objects, and separation of different objects, and \nit is generally known as scene segmentation or perceptual organization. Despite the fact \n\n\f200 \n\nDeLiang Wang. David Tennan \n\nthat humans perform it with apparent ease, the general problem of scene segmentation \nremains unsolved in the engineering of sensory processing, such as computer vision and \nauditory processing. \n\nFundamental to scene segmentation is the grouping of similar sensory features and the \nsegregation of dissimilar ones. Theoretical investigations of brain functions and feature \nbinding point to the mechanism of temporal correlation as a representational framework \n(von der Malsburg, 1981; von der Malsburg and Schneider, 1986). In particular, the \ncorrelation theory of von der Malsburg (1981) asserts that an object is represented by the \ntemporal correlation of the fIring activities of the scattered cells coding different features \nof the object. A natural way of encoding temporal correlation is to use neural \noscillations, whereby each oscillator encodes some feature (maybe just a pixel) of an \nobject. In this scheme, each segment (object) is represented by a group of oscillators that \nshows synchrony (phase-locking) of the oscillations, and different objects are represented \nby different groups whose oscillations are desynchronized from each other. Let us refer to \nthis form of temporal correlation as oscillatory correlation. The theory of oscillatory \ncorrelation has received direct experimental support from the cell recordings in the cat \nvisual cortex (Eckhorn et aI., 1988; Gray et aI., 1989) and other brain regions. The \ndiscovery of synchronous oscillations in the visual cortex has triggered much interest \nfrom the theoretical community in simulating the experimental results and in exploring \noscillatory correlation to solve the problems of scene segmentation. While several \ndemonstrate synchronization in a group of oscillators using local (lateral) connections \n(Konig and Schillen, 1991; Somers and Kopell, 1993; Wang, 1993, 1995), most of these \nmodels rely on long range connections to achieve phase synchrony. It has been pointed \nout that local connections in reaching synchrony may play a fundamental role in scene \nsegmentation since long-range connections would lead to indiscriminate segmentation \n(Sporns et aI., 1991; Wang, 1993). \n\nThere are two aspects in the theory of oscillatory correlation: (1) synchronization within \nthe same object; and (2) desynchronization between different objects. Despite intensive \nstudies on the subject, the question of desynchronization has been hardly addressed. The \nlack of an effIcient mechanism for de synchronization greatly limits the utility of \noscillatory correlation to perceptual organization. In this paper, we propose a new class \nof oscillatory networks and show that it can rapidly achieve both synchronization within \neach object and desynchronization between a number of simultaneously presented objects. \nThe network is composed of the following elements: (I) A new model of a basic \noscillator; (2) Local excitatory connections to produce phase synchrony within each \nobject; (3) A global inhibitor that receives inputs from the entire network and feeds back \nwith inhibition to produce desynchronization of the oscillator groups representing \nIn other words, the mechanism of the network consists of local \ndifferent objects. \ncooperation and global competition. This surprisingly simple neural architecture may \nprovide an elementary approach to scene segmentation and a computational framework for \nperceptual organization. \n\n2 MODEL DESCRIPTION \n\nThe building block of this network, a single oscillator i, is defined in the simplest form \nas a feedback loop between an excitatory unit Xi and an inhibitory unit y( \n\n\fSynchrony and Desynchrony in Neural Oscillator Networks \n\n201 \n\n, \n, dxldt= 0 \n\n8 \n\no . \n\n-2 \n\n, , , , \n\" \" \n\" :' \n. \n\n\" \n\no \nx \n\n2 \n\nFigure 1: Nullclines and periodic orbit of \na single oscillator as shown in the phase \nplane. When the oscillator starts at a \nrandomly generated point in the phase \nplane, it quickly converged to a stable \ntrajectory of a limit cycle. \n\ndXi \n-=3x\u00b7- X \u00b7 +2-y \u00b7 +p+I\u00b7+S\u00b7 \ndt \n\n3 \nI \n\nl \n\n, \n\n\" \n\ndy \u00b7 \ndt' = e (y(1 + tanh(xl/3) - Yi) \n\nFigure 2: Architecture of a two dimensional \nnetwork with nearest neighbor coupling. \nThe global inhibitor is indicated by the \nblack circle. \n\n(la) \n\n(lb) \n\nwhere p denotes the amplitude of a Gaussian noise term. \nIi represents external \nstimulation to the oscillator, and Si denotes coupling from other oscillators in the \nnetwork. The noise term is introduced both to test the robustness of the system and to \nactively desynchronize different input patterns. The parameter e is chosen to be small. \nIn this case (1), without any coupling or noise, corresponds to a standard relaxation \noscillator. The x-nullcline of (1) is a cubic curve, while the y-nullc1ine is a sigmoid \nfunction, as shown in Fig. 1. If I > 0, these curves intersect along the middle branch of \nthe cubic, and (1) is oscillatory. The periodic solution alternates between the silent and \nactive phases of near steady state behavior. The parameter yis introduced to control the \nrelative times that the solution spends in these two phases. If I < 0, then the nullc1ines \nof (1) intersect at a stable fixed point along the left branch of the cubic. In this case the \nsystem produces no oscillation. The oscillator model (1) may be interpreted as a model of \nspiking behavior of a single neuron, or a mean field approximation to a network of \nexcitatory and inhibitory neurons. \nThe network we study here in particular is two dimensional. However, the results can \neasily be extended to other dimensions. Each oscillator in the network is connected to \nonly its four nearest neighbors, thus forming a 2-D grid. This is the simplest form of \nlocal connections. The global inhibitor receives excitation from each oscillator of the \ngrid, and in turn inhibits each oscillator. This architecture is shown in Fig. 2. The \nintuitive reason why the network gives rise to scene segmentation is the following. \nWhen multiple connected objects are mapped onto the grid, local connectivity on the grid \nwill group together the oscillators covered by each object. This grouping will be reflected \n\n\f202 \n\nDeLiang Wang, David Tennan \n\nby phase synchrony within each object. The global inhibitor is introduced for \ndesynchronizing the oscillatory responses to different objects. We assume that the \ncoupling term Si in (1) is given by \n\nSi = L W ik Soo(xk, 9x) - Wz Soo(z, 9xz) \n\nkEN(i) \n\nS (x 9' = __ -=-1 __ \n1+ exp[-K(x-e,] \n\n00 ,J \n\n(2) \n\n(3) \n\nwhere Wik is a connection (synaptic) weight from oscillator k to oscillator i, and N(i) is \nthe set of the neighoring oscillators that connect to i. In this model, NO) is the four \nimmediate neighbors on the 2-D grid, except on the boundaries where N(i) may be either \n2 or 3 immediate neighbors. 9x is a threshold (see the sigmoid function of Eq. 3) above \nwhich an oscillator can affect its neighbors. Wz (positive) is the weight of inhibition \nfrom the global inhibitor z, whose activity is defined as \n\n(4) \n\nwhere Goo = 0 if Xi < 9zx for every oscillator, and Goo = 1 if Xi ~ 9zx for at least one \noscillator i. Hence 9zx represents a threshold. If the activity of every oscillator is below \nthis threshold, then the global inhibitor will not receive any input. In this case z ~ 0 \nand the oscillators will not receive any inhibition. If, on the other hand, the activity of at \nleast one oscillator is above the threshold 9zx then, the global inhibitor will receive \ninput. In this case z ~ 1, and each oscillator feels inhibition when z is above the \nthreshold 9zx. The parameter l/J determines the rate at which the inhibitor reacts to such \nstimulation. \n\nIn summary, once an oscillator is active, it triggers the global inhibitor. This then \ninhibits the entire network as described in Eq. 1. On the other hand, an active oscillator \nspreads its activation to its nearest neighbors, again through (1), and from them to its \nfurther neighbors. In the next section, we give a number of properties of this system. \n\nBesides boundaries, the oscillators on the grid are basically symmetrical. Boundary \nconditions may cause certain distortions to the stability of synchrous oscillations. \nRecently, Wang (1993) proposed a mechanism called dynamic normalization to ensure \nthat each oscillator, whether it is in the interior or on a boundary, has equal overall \nconnection weights from its neighbors. The dynamic normalization mechanism is \nadopted in the present model to form effective connections. For binary images (each pixel \nbeing either 0 or 1), the outcome of dynamic normalization is that an effective connection \nis established between two oscillators if and only if they are neighbors and both of them \nare activated by external stimulation. The network defined above can readily be applied \nfor segmentation of binary images. For gray-level images (each pixel being in a certain \nvalue range), the following slight modification suffices to make the network applicable. \nAn effective connection is established between two oscillators if and only if they are \nneighbors and the difference of their corresponding pixel values is below a certain \nthreshold. \n\n\fSynchrony and Desynchrony in Neural Oscillator Networks \n\n203 \n\n3 ANALYTICAL RESULTS \n\nWe have formally analyzed the network. Due to space limitations, we can only list the \nmajor conclusions without proofs. The interested reader can find the details in Terman \nand Wang (1994). Let us refer to a pattern as a connected region, and a block be a subset \nof oscillators stimulated by a given pattern. The following results are about singular \nsolutions in the sense that we formally set E = O. However, as shown in (Terman and \nWang, 1994), the results extend to the case E> 0 sufficiently small. \nTheorem 1. (Synchronization). The parameters of the system can be chosen so that all \nof the oscillators in a block always jump up simultaneously (synchronize). Moreover, \nthe rate of synchronization is exponential. \n\nTheorem 2. (Multiple Patterns) The parameters of the system and a constant T can be \nchosen to satisfy the following. If at the beginning all the oscillators of the same block \nsynchronize with each other and the temporal distance between any two oscillators \nbelonging to two different blocks is greater than T, then (1) Synchronization within each \nblock is maintained; (2) The blocks activate with a fixed ordering; (3) At most one block \nis in its active phase at any time. \nTheorem 3. (Desynchronization) If at the beginning all the oscillators of the system lie \nnot too far away from each other, then the condition of Theorem 2 will be satisfied after \nsome time. Moreover, the time it takes to satisfy the condition is no greater than N \ncycles, where N is the number of patterns. \n\nThe above results are true with arbitrary number of oscillators. In summary, the network \nexhibits a mechanism, referred to as selective gating, which can be intuitively interpreted \nas follows. An oscillator jumping to its active phase opens a gate to quickly recruit the \noscillators of the same block due to local connections. At the same time, it closes the \ngate to the oscillators of different blocks. Moreover, segmentation of different patterns is \nachieved very rapidly in terms of oscillation cycles. \n\n4 COMPUTER SIMULATION \n\nTo illustrate how this network is used for scene segmentation, we have simulated a 2Ox20 \noscillator network as defined by (1)-(4). We arbitrarily selected four objects (patterns): \ntwo O's, one H, and one I ; and they form the word OHIO . These patterns were \nsimultaneously presented to the system as shown in Figure 3A. Each pattern is a \nconnected region, but no two patterns are connected to each other. \n\nAll the oscillators stimulated (covered) by the objects received an external input 1=0.2, \nwhile the others have 1=-0.02. The amplitude p of the Gaussian noise is set to 0.02. \nThus, compared to the external input, a 10% noise is included in every oscillator. \nDynamic normalization results in that only two neighboring oscillators stimulated by a \nsingle pattern have an effective connection. The differential equations were solved \nnumerically with the following parameter values: E = 0.02, l/J = 3.0; Y= 6.0, f3 = 0.1, K \n= 50, Ox = -0.5, and 0zx = 0xz = 0.1. The total effective connections were normalized to \n6.0. The results described below were robust to considerable changes in the parameters. \nThe phases of all the oscillators on the grid were randomly initialized. \n\n\f204 \n\nDeLiang Wang, David Terman \n\nFig. 3B-3F shows the instantaneous activity (snapshot) of the network at various stages \nof dynamic evolution. The diameter of each black circle represents the normalized x \nactivity of the corresponding oscillator. Fig. 3B shows a snapshot of the network a few \nsteps after the beginning of the simulation. \nIn Fig. 3B, the activities of the oscillators \nwere largely random. Fig. 3C shows a snapshot after the system had evolved for a short \ntime period. One can clearly seethe effect of grouping and segmentation: all the \noscillators belonging to the left 0 were entrained and had large activities. At the same \ntime, the oscillators stimulated by the other three patterns had very small activities. Thus \nthe left 0 was segmented from the rest of the input. A short time later, as shown in Fig. \n3D, the oscillators stimulated by the right 0 reached high values and were separated from \nthe rest of the input. Fig. 3E shows another snapshot after Fig. 3D. At this time, \npattern I had its turn to be activated and separated from the rest of the input. Finally in \nFig. 3F, the oscillators representing H were active and the rest of the input remained \nsilent. This successive \"pop-out\" of the objects continued in a stable periodic fashion. \nTo provide a complete picture of dynamic evolution, Fig. 30 shows the temporal \nevolution of each oscillator. Since the oscillators receiving no external input were \ninactive during the entire simulation process, they were excluded from the display in Fig. \n30. The activities of the oscillators stimulated by each object are combined together in \nthe figure. Thus, if they are synchronized, they appear like a single oscillator. In Fig. \n30, the four upper traces represent the activities of the four oscillator blocks, and the \nbottom trace represents the activity of the global inhibitor. The synchronized oscillations \nwithin each object are clearly shown within just three cycles of dynamic evolution. \n\n5 DISCUSSION \n\nBesides neural plausibility, oscillatory correlation has a unique feature as an \ncomputational approach to the engineering of scene segmentation and figure/ground \nsegregation. Due to the nature of oscillations, no single -object can dominate and \nsuppress the perception of the rest of the image permanently. The current dominant \nobject has to give way to other objects being suppressed, and let them have a chance to be \nspotted. Although at most one object can dominant at any time instant, due to rapid \noscillations, a number of objects can be activated over a short time period. This intrinsic \ndynamic process provides a natural and reliable representation of multiple segmented \npatterns. \n\nThe basic principles of selective gating are established for the network with lateral \nconnections beyond nearest neighbors. Indeed, in terms of synchronization, more distant \nconnections even help expedite phase entrainment. In this sense, synchronization with \nall-to-all connections is an extreme case of our system. With nearest-neighbor \nconnectivity (Fig. 2), any isolated part of an image is considered as a segment. In an \nnoisy image with many tiny regions, segmentation would result in too many small \nfragments. More distant connections would also provide a solution to this problem. \nLateral connections typically take on the form of Gaussian distribution, with the \nconnection strength between two oscillators falling off exponentially. Since global \ninhibition is superimposed to local excitation, two oscillators positively coupled may be \ndesynchronized if global inhibition is strong enough. Thus, it is unlikely that all objects \nin an image form a single segment as the result of extended connections. \n\n\fSynchrony and Desynchrony in Neural Oscillator Networks \n\n205 \n\nDue to its critical importance for computer vision, scene segmentation has been studied \nquite extensively. Many techniques have been proposed in the past (Haralick and Shapiro, \n1985; Sarkar and Boyer, 1993). Despite these techniques, as pointed out by Haralick and \nShapiro (1985), there is no underlying theory of image segmentation, and the techniques \ntend to be adhoc and emphasize some aspects while ignoring others. Compared to the \ntraditional techniques for scene segmentation, the oscillatory correlation approach offers \nmany unique advantages. The dynamical process is inherently parallel. While \nconventional computer vision algorithms are based on descriptive criteria and many adhoc \nheuristics, the network as exemplified in this paper performs computations based on only \nconnections and oscillatory dynamics. The organizational simplicity renders the network \nparticularly feasible for VLSI implementation. Also, continuous-time dynamics allows \nreal time processing, desired by many engineering applications. \n\nAcknowledgments \n\nDLW is supported in part by the NSF grant IRI-9211419 and the ONR grant NOOOI4-93-\n1-0335. DT is supported in part by the NSF grant DMS-9203299LE. \n\nReferences \n\nR. Eckhorn, et aI., \"Coherent oscillations: A mechanism of feature linking in the visual \n\ncortex?\" Bioi. Cybem., vol. 60, pp. 121-130, 1988. \n\nC.M. Gray, P. Konig, A.K. Engel, and W. Singer, \"Oscillatory responses in cat visual \ncortex exhibit inter-columnar synchronization which reflects global stimulus \nproperties,\" Nature, vol. 338, pp. 334-337, 1989. \n\nR.M. 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Wang, \"Global competition and local cooperation in a network of \n\nneural oscillators,\" Physica D, in press, 1994. \n\nC. von der Malsburg, \"The correlation theory of brain functions,\" Internal Report 81-2, \n\nMax-Planck-Institut for Biophysical Chemistry, Gottingen, FRG, 1981. \n\nC. von der Malsburg and W. Schneider, \"A neural cocktail-party processor,\" Bioi. \n\n\u2022 \n\nCybern., vol. 54, pp. 29-40, 1986. \n\nD.L. Wang, \"Modeling global synchrony in the visual cortex by locally coupled neural \n\noscillators,\" Proc. 15th Ann. Conf. Cognit. Sci. Soc., pp. 1058-1063, 1993. \n\nD.L. Wang, \"Emergent synchrony in locally coupled neural oscillators,\" IEEE Trans. on \n\nNeural Networks, in press, 1995. \n\n\f206 \n\nA \n\nDeLiang Wang, David Terman \n\n. , \n\nB \u00b7 .... ,\n..... \nc \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \u2022\u2022\u2022\u2022\u2022\u2022 \n.... ~ \u2022....\u2022\u2022.\u2022.\u2022.... \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022 \u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u00b7 .\u2022.. ........\u2022...... \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u00b7 .......\u2022.... ...... . \n\u2022 \u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \u2022\u2022\u2022 \n\u2022. .\u2022. \u2022.\u2022. \u2022 \u00b7 \u2022\u2022\u2022 \u00b7\u00b7e\u00b7 .\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \u2022 \u2022\u2022\u2022\u2022\u2022 \u2022\u2022\u2022 \u2022\u2022\u2022 \n. . . .\u2022.. . . . ...\u2022.. . . \n... ...... .\u2022.. \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\n. .... ... .. .... ... ... \n\nD \n\nE \n\nF \n\n. . . . \" \n\n. ........ . ... . \n\n\u00b7 ....... .. ... . ..... . \n\n. .. ....... .... . . .. . . \n\nG \n\nFigure 3. A An image composed of four patterns which were presented (mapped) to a \n20x20 grid of oscillators. B A snapshot of the activities of the oscillator grid at the \nbeginning of dynamic evolution. C A snapshot taken shortly after the beginning. D \nShortly after C. E Shortly after D. F Shortly after E. G The upper four traces show the \ncombined temporal activities of the oscillator blocks representing the four patterns, \nrespectively, and the bottom trace shows the temporal activity of the global inhibitor. \nThe simulation took 8,000 integration steps. \n\n\f", "award": [], "sourceid": 944, "authors": [{"given_name": "Deliang", "family_name": "Wang", "institution": null}, {"given_name": "David", "family_name": "Terman", "institution": null}]}