{"title": "Collective Oscillations in the Visual Cortex", "book": "Advances in Neural Information Processing Systems", "page_first": 76, "page_last": 83, "abstract": null, "full_text": "76 \n\nKammen, Koch and Holmes \n\nCollective Oscillations in the \n\nVisual Cortex \n\nDaniel Kammen & Christof Koch \n\nPhilip J. H oImes \n\nComputation and Neural Systems \n\nDept. of Theor. & Applied Mechanics \n\nCaltech 216-76 \n\nPasadena, CA 91125 \n\nCornell University \nIthaca, NY 14853 \n\nABSTRACT \n\nThe firing patterns of populations of cells in the cat visual cor(cid:173)\ntex can exhibit oscillatory responses in the range of 35 - 85 Hz. \nFurthermore, groups of neurons many mm's apart can be highly \nsynchronized as long as the cells have similar orientation tuning. \nWe investigate two basic network architectures that incorporate ei(cid:173)\nther nearest-neighbor or global feedback interactions and conclude \nthat non-local feedback plays a fundamental role in the initial syn(cid:173)\nchronization and dynamic stability of the oscillations. \n\nINTRODUCTION \n\n1 \n40 - 60 Hz oscillations have long been reported in the rat and rabbit olfactory \nbulb and cortex on the basis of single- and multi-unit recordings as well as EEG \nactivity (Freeman, 1972; Wilson & Bower 1990). Recently, two groups (Eckhorn et \nai., 1988 and Gray et ai., 1989) have reported highly synchronized, stimulus specific \noscillations in the 35 - 85 Hz range in areas 17, 18 and PMLS of anesthetized as \nwell as awake cats. Neurons with similar orientation tuning up to 7 mm apart show \nphase-locked oscillations, with a phase shift of less than 3 msec. We address here \nthe computational architecture necessary to subserve this process by investigating \nto what extent two neuronal architectures, nearest-neighbor coupling and feedback \nfrom a central \"comparator\", can synchronize neuronal oscillations in a robust and \nrapid manner. \n\n\fCollective Oscillations in the Visual Cortex \n\n77 \n\nIt was argued in earlier work on central pattern generators (Cohen et al., 1982), that \nin studying coupling effects among large populations of oscillating neurons, one can \nignore the details of individual oscillators and represent each one by a single periodic \nvariable: its phase. Our approach assumes a population of neuronal oscillators, \nfiring repetitively in response to synaptic input. Each cell (or group of tightly \nelectrically coupled cells) has an associated variable representing the membrane \npotential. In particular, when (Ji = 7r, an action potential is generated and the \nphase is reset to its initial value (in our case to -7r). The number of times per unit \ntime (Ji passes through 7r, i.e. d(Ji/dt, is then proportional to the firing frequency of \nthe neuron. For a network of n + 1 such oscillators, our basic model is \n\n(1) \n\nwhere Wi represents the synaptic input to neuron i and I, a function of the phases, \nrepresents the coupling within the network. Each oscillator i in isolation (i.e. with \nIi = 0), exhibits asymptotically stable periodic oscillations; that is, if the input \nis changed the oscillator will rapidly adjust to a new firing rate. In our model Wi \nis assumed to derive from neurons in the lateral geniculate nucleus (LG N) and is \npurely excitatory. \n\n2 FREQUENCY AND PHASE LOCKING \nAny realistic model of the observed, highly synchronized, oscillations must account \nfor the fact that the individual neurons oscillate at different frequencies in isolation. \nThis is due to variations in the synaptic input, Wi, as well as in the intrinsic prop(cid:173)\nerties of the cells. We will contrast the abilities of two markedly different network \narchitectures to synchronize these oscillations. The \"chain\" model (Fig. 1 top) con(cid:173)\nsists of a one-dimensional array of oscillators connected to their nearest neighbors, \nwhile in the alternative \"comparator\" model (Fig. 1 middle), an array of neurons \nproject to a single unit, where the phases are averaged (i.e. (lin) L~=o Oi(t)). This \naverage is then feed back to every neuron in the network. In the continuum limit \n(on the unit interval) with all Ii = I being identical, the two models are \n\n80(x, t) \n\n8(J(x, t) \n\n8t \n\n8t \n\n(2) \n\nn x \n\n(Chain Model) \n\n(Comparator Model) \n\nW(x) + .!.. 88f (4)) \nw(x) + 1((J(x, t) -10 1 (J(s, t)ds), (3) \nwhere 0 < x < 1 and 4> is the phase gradient, 4> = ~M. In the chain model, we \nrequire that I be an odd function (for simplicity of analysis only) while our analysis \nof the comparator model holds for any continuous function I. We use two spatially \nseparated \"spots\" of width 6 and amplitude Q' as visual input (Fig. 1 bottom). This \npattern was chosen as a simple version of the double-bar stimulus that (Gray et al. \n1989) found to evoke coherent oscillatory activity in widely separated populations \nof visual cortical cells. \n\n\f78 \n\nKammen, Koch and Holmes \n\n-\u2022 \n-~ \u2022 \n\n\u2022 \n\n\u2022\u2022\u2022 \n\n\u2022 \n\n\u2022 \n\n00(0) \n\n8i=n(t) \n\n+ \nm(n) \n\nmen) \n\n00(0) \nm(x) \nI \n\nI ta \n\nIta \n\nx \nFigure 1: The linear chain (top) and comparator (middle) architectures. The \nspatial pattern of inputs is indicated by Wj(x). See equs. 2 & 3 for a mathematical \ndescription of the models. The \"two spot\" input is shown at bottom and represents \ntwo parts of a perceptually extended figure. \n\nWe determine under what circumstances the chain model will develop frequency(cid:173)\nnecessarily at the same time), i.e. 8 2 (} /8x8t = O. We prove (Kammen, et al. 1990) \nlocked solutions, such that every oscillator fires at the same frequency (but not \nthat frequency-locked solutions exist as long as In(wx- fo:17 w(s)ds)1 does not exceed \nthe maximal value of I, Imax (with w = f; w(s)ds the mean excitation level). \nThus, if the excitation is too irregular or the chain too long (n \u00bb 1), we will not \nfind frequency-locked solutions. Phase coherence between the excited regions is not \ngenerally maintained and is, in fact, strongly a function of the initial conditions. \nAnother feature of the chain model is that the onset of frequency locking is slow \nand takes time of order Vii. \nThe location of the stimulus has no effect on phase relationships in the comparator \nmodel due to the global nature of the feedback. The comparator model exhibits \ntwo distinct regimes of behavior depending on the amplitude of the input, a. In the \ncase of the two spot input (Fig. 1 bottom ), if a is small, all neurons will frequency(cid:173)\nlock regardless of location, that is units responding to both the \"figure\" and the \nbackground (\"ground\") will oscillate at the same frequency. They will, however, \nfire at different times, with () Jig 1= () gnd. If a is above a critical threshold, the units \nresponding to the \"figure\" will decouple in frequency as well as phase from the \nbackground while still maintaining internal phase coherency. Phase gradients never \nexist within the excited groups, no matter what the input amplitude. \n\n\fCollective Oscillations in the Visual Cortex \n\n79 \n\nWe numerically simulated the chain and comparator models with the two spot input \nfor the coupling function fCf}) = sin(f}). Additive Gaussian noise was included in the \ninput, Wi. Our analytical results were confirmed; frequency and phase gradients were \nalways present in the chain model (Fig. 2A) even though the coupling strength was \nten times greater than that of the comparator modeL In the comparator network \nsmall excitation levels led to frequency-locking along the entire array and to phase(cid:173)\ncoupled activity within the illuminated areas (Fig. 2B), while large excitation levels \nled to phase and frequency decoupling between the \"figure\" and the \"background\" \n(Fig. 2C). The excited regions in the comparator settle very rapidly - within 2 to \n3 cycles - into phase-locked activity with small phase-delays. The chain model, on \nthe other hand, exhibits strong sensitivity to initial conditions as well as a very slow \napproach to coherence that is still not complete even after 50 cycles (See Fig. 2). \n\nA \n\nB \n\nc \n\nFigure 2: The phase portrait of the chain (A), weak (B) and strongly (C) excited \ncomparator networks after 50 cycles. The input, indicated by the horizontal lines, \nis the two spot pattern. Note that the central, unstimulated, region in the chain \nmodel has been \"dragged along\" by the flanking excited regions. \n\n3 STABILITY ANALYSIS \nPerhaps the most intriguing aspect of the oscillations concerns the role that they \nmay play in cortical information processing and the labeling of cells responding to a \nsingle perceptual object. To be useful in object coding, the oscillations must exhibit \nsome degree of noise tolerance both in the input signal and in the stability of the \npopulation to variation in the firing times of individual cells. \n\n. d b \n\nh \n\n. \n\nII \n\nI \u00b7 \u00b7 d \n\ne popu atlOn IS etermme \n\nThe degree to which input noise to individual neurons disrupts the synchronization \ny t e ratio coupling strength = irT. or sma per-\nf th \no \nturbations, wet) = Wo + f(t), the action of the feedback, from the nearest neighbors \nin the chain and from the entire network in the comparator, will compensate for \nthe noise and the neuron will maintain coherence with the excited population. As \nf is increased first phase and then frequency coherence will be lost. \n\ninput noise ~ F \n\nIn Fig. 3 we compare the dynamical stability of the chain and comparator models. \nIn each case the phase, (J, of a unit receiving perturbated input is plotted as the \ndeviation from the average phase, (Jo, of all the excited units receiving input WOo The \nchain in highly sensitive to noise: even 10% stochastic noise significantly perturbs \nthe phase of the neuron. In the comparator model (Fig. 3B) noise must reach the \n\n\f80 \n\nKammen, Koch and Holmes \n\n40% level to have a similar effect on the phase. As the noise increases above 0.30wo \neven frequency coherence is lost in the chain model (broken error bars). Frequency \ncoherence is maintained in the comparator for f = 0.60wo. \n\ne \nA) \n\n+0.10 ~ \n\n0.00 \n-G.OS \n-G.IO ~ \n\n0.0 \n\n20 \n\nB) \n\n............. ~ .... 1t .... :1.... ..~ \n\n\u2022 \n\nl \n\n\"I \n\n40 \n\n60 \n\n0.0 \n\u00a3 (% of 000) \n\n20 \n\n40 \n\n60 \n\nFigure 3: The result of a perturbation on the phase, 0, for the chain (A) and \ncomparator (B) models. The terminus of the error bars gives the resulting devia(cid:173)\ntion from the unperturbed value. Broken bars indicate both phase and frequency \ndecoupling. \n\nThe stability of the solutions of the comparator model to variability in the activity \nof individual neurons can easily be demonstrated. For simplicity consider the case \nof a single input of amplitude WI superposed on a background of amplitude Woo The \nsolutions in each region are: \n\n(4) \n\ndOo \ndt \ndOl \ndt \n\n2 \n\nI ( 00 - 01) \n1(01 - 00) \n\nwo+ \n\nWI + \n\n(5) \nWe define the difference in the solutions to be \u00a2(t) = 01(t)-00(t) and Aw = W1-WO. \nWe then have an equation for the rate the solutions converge or diverge: \n\n2 \n\n. \n\n