{"title": "Modeling the Olfactory Bulb - Coupled Nonlinear Oscillators", "book": "Advances in Neural Information Processing Systems", "page_first": 402, "page_last": 409, "abstract": null, "full_text": "402 \n\nMODELING THE OLFACTORY BULB \n\n- COUPLED NONLINEAR OSCILLATORS \n\nZhaoping Lit \n\nJ. J. Hopfield\u00b7 \n\nt Division of Physics, Mathematics and Astronomy \n\n\u00b7Division of Biology, and Division of Chemistry and Chemical Engineering \n\nt\u00b7 California Institute of Technology, Pasadena, CA 91125, USA \n\n\u2022 AT&T Bell Laboratories \n\nABSTRACT \n\nThe olfactory bulb of mammals aids in the discrimination of \nodors. A mathematical model based on the bulbar anatomy and \nelectrophysiology is described. Simulations produce a 35-60 Hz \nmodulated activity coherent across the bulb, mimicing the observed \nfield potentials. The decision states (for the odor information) \nhere can be thought of as stable cycles, rather than point stable \nstates typical of simpler neuro-computing models. Analysis and \nsimulations show that a group of coupled non-linear oscillators are \nresponsible for the oscillatory activities determined by the odor in(cid:173)\nput, and that the bulb, with appropriate inputs from higher centers, \ncan enhance or suppress the sensitivity to partiCUlar odors. The \nmodel provides a framework in which to understand the transform \nbetween odor input and the bulbar output to olfactory cortex. \n\n1. INTRODUCTION \n\nThe olfactory system has a simple cortical intrinsic structure (Shepherd 1979), \n\nand thus is an ideal candidate to yield insight on the principles of sensory informa(cid:173)\ntion processing. It includes the receptor cells, the olfactory bulb, and the olfactory \ncortex receiving inputs from the bulb (Figure [1]). Both the bulb and the cortex \nexhibit similar 35-90 Hz rhythmic population activity modulated by breathing. Ef(cid:173)\nforts have been made to model the bulbar information processing function (Freeman \n1979b, 1979c; Freeman and Schneider 1982; Freeman and Skarda 1985; Baird 1986j \nSkarda and Freeman 1987), which is still unclear (Scott 1986). The bulbar position \nin the olfactory pathway, and the linkage of the oscillatory activity with the sniff \ncycles suggest that the bulb and the oscillation play important roles in the olfactory \ninformation processing. We will examine how the bulbar oscillation pattern, which \ncan be thought of as the decision state about odor information, originates and how \nit depends on the input odor. We then show that with appropriate inputs from the \nhigher centers, the bulb can suppress or enhance the its sensitivity to particular \nodors. Much more details of our work are described in other two papers (Li and \nHopfield 1988a, 1988b). \n\n\fModeling the Olfactory Bulb-Coupled Nonlinear Oscillators \n\n403 \n\nThe olfactory bulb has mainly the excitatory mitral and the inhibitory granule \ncells located on different parallel lamina. Odor receptors effectively synapse on the \nmitral cells which interact locally with the granule cells and carry the bulbar outputs \n(Fig 1, Shepherd 1979). A rabbit has about 50,000 mitral, and \"'-J 10,000,000 granule \ncells (Shepherd 1979). With short odor pulses, the receptor firing rate increases in \ntime, and terminates quickly after the odor pulse terminates (Getchell and Shepherd \n1978). Most inputs from higher brain centers are directed to the granule cells, and \nlittle is know about them. The surface EEG wave (generated by granule activities, \nFreeman 1978j Freeman and Schneider 1982), depending on odor stimulations and \nanimal motivation, shows a high amplitude oscillation arising during the inhalation \nand stopping early in the exhalation. The oscillation is an intrinsic property of the \nbulb itself, and is influenced by central inputs (Freeman 1979aj Freeman and Skarda \n1985). It has a peak frequency (which is the same across the bulb) in the range of \n35-90 Hz, and rides on a slow background wave phase locked with the respiratory \nwave. \n\n2. MODEL ORGANIZATION \n\nFor simplicity, we only include ( N excitatory) mitral and ( M inhibitory ) \ngranule cells in the model. The Receptor input I is Ii = Iodor\" + Ibaekgrotmd,i, \nfor 1 ... , N, a superposition of an odor signal Iodor and a background input \nIodor > 0 increases in time during inhalation, and return expo(cid:173)\nIbaekgrov,nd. \nnentially during exhalation toward the ambient. The central input to the granule \ncells is vector Ie with components Ie,j for 1 < j < M. For now, it is assumed that \nIe = 0.1 and Ibaekgrov,nd = 0.243 do not change during a sniff (Li and Hopfield \n1988a). \n\nEach cell is one unit with its internal state level described by a single vari(cid:173)\nable, and its output a continuous function of the internal state level. The inter(cid:173)\nnal states and outputs are respectively X = {Xl' X2, ... , X N} and G z ( X) = \n{gz(xd, gz(x2),\u00b7.\u00b7 ,gz(XN)} (Y = {YI' Y2,\u00b7\u00b7\u00b7 ,YM} and Gy{Y) = {gy(Yd, \ngy(Y2), ... ,gy(YM)}) for the mitral (granule) cells, where gz > 0 and gy > 0 are \nthe neurons' non-linear sigmoid output functions essential for the bulbar oscillation \ndynamics (Freeman and Skarda 1985) to be studied. \n\nother \nbrain \n\nReceptor inputs \n... \n\n... \n\n... \n\n... \n\n... \n\n... \n... \n(t)(t)(t)(t)(t)(t)(t)(t) \n\n... \nIt' t ~ , + )If Int~~:~lion \n22222222 \nIt. \n\n... \n\nIt. \n\nIt. \n\nIt. \n\nIt. \nIt. \nCentral inputs \n\nIt. \n\nFig.l. Left: olfactory system; Right: bulbar structure \nCells marked \"+\" are mitral cells, \"_\" are granule cells \n\nThe geometry of bulbar structure is simplified to a one dimensional ring. Each \ncell is specified by an index, e.g. ith mitral cell, and jth granule cell for all i, i \n\n\f404 \n\nLi and Hopfield \n\nindicating cell locations on the ring (Fig 1). N X M matrix Ho and M X N \nmatrix Wo are used respectively to describe the synaptic strengths (postsynaptic \ninput: presynaptic output) from granule cells to mitral cells and vice versa. The \nbulb model system has equations of motion: \n\nX = -HoGy(Y) - O:zX + I, \nY = WoGz(X) - O:yY + Ie. \n\n(2.1) \n\nwhere O:z = 1/'rz , O:y = 1/'ry, and 'rz = 'ry = 7 msec are the time constants \nof the mitral and granule cells respectively (Freeman and Skarda 1985; Shepherd \n1988). In simulation, weak random noise is added to I .and Ie to simulate the \nfluctuations in the system. \n\n3. SIMULATION RESULT \n\nComputer simulation was done with 10 mitral and granule cells, and show \nthat the model can capture the major effects of the real bulb. The rise and fall \nof oscillations with input and the baseline shift wave phase locked with sniff cycles \nare obvious (Fig.2). The simulated EEG (calculated using the approximation by \nFreeman (1980)) and the measured EEG are shown for comparison. During a sniff, \nall the cells oscillate coherently with the same frequency as physiologically observed. \n\n7\\ \n.'\"\"1 \n\nB ~EEGW'\" \n\nGrwIule Output ~J1.0 \n\nlOOms \nH \n\nEEG Wave \n\nRespiratory Wave \n\nlooms \n~ \n\nFig.2. A: Simulation result; B: measured result from Freeman and Schneider 1982. \n\nThe model also shows the capability of a pattern classifier. During a sniff, \nsome input patterns induce oscillation, while others do not, and different inputs \ninduce different oscillation patterns. We showed (Li and Hopfield 1988a) that the \nbulb amplifies the differences between the different inputs to give different output \npatterns, while the responses to same odor inputs with different noise samples differ \nnegligibly. \n\n4. MATHEMATICAL ANALYSIS \n\nA (damped) oscillator with frequency w can be described by the equations \n\nX = -wy - o:x \niI = wx - o:y \n\nor \n\n(4.1) \n\n\fModeling the Olfactory Bulb-Coupled Nonlinear Oscillators \n\n405 \n\nThe solution orbit in (x, y) space is a circle if a = 0 (non-damped oscillator), \nand spirals into the origin otherwise (damped oscillator). \nIT a mitral cell and a \ngranule cell are connected to each other, with inputs i(t) and ie(t) respectively, \nthen \n\nx = -h . gy(y) - azx + i(t), \ny = w . gz(x) - ayy + ie(t). \n\n(4.2) \n\nThis is the scalar version of equation (2.1) with each upper case letter representing \na vector or matrix replaced by a lower case letter representing a scalar. It is as(cid:173)\nsumed that i(t) has a much slower time course than X or y (frequency of sniffs ~ \ncharacteristic neural oscillation frequency). Use the adiabatic approximation, and \ndefine the equilibrium point (xo, Yo) as \n\nXo ~ 0 = -h . gy(yo) - azxo + i, \nYo ~ 0 = w . gz(xo) - ayyo + ie' \n\nDefine x' = x - Xo, y' - y - Yo' Then \n\nx' = -h(gy(y) - gy(yo)) - azx', \nil = w(gz(x) - gz(xo)) - ayy'. \n(cf. equation (4.1)). IT a z = ay = 0, then the solution orbit \n\n(4.3) \n\nzo+z' \n\nR = f w(gz(s) - gz(xo))ds + f h(gy(s) - gy(Yo))ds = constant \n\nyo+Y' \n\nZo \n\nYo \n\nis a closed curve in the original (x, y) space surrounding the point (xo, Yo), i.e., \n(x, y) oscillates around the point (xo, Yo). When the dissipation is included, \ndR/ dt < 0, the orbit in (x, y) space will spiral into the point (xo, Yo). Thus \na connected pair of mitral and granule cells behaves as a damped non-linear oscilla(cid:173)\ntor, whose oscillation center (xo, Yo) is determined by the external inputs i and ie' \nFor small oscillation amplitudes, it can be approximated by a sinusoidal oscillator \nvia linearization around the (xo, Yo): \n\nx = -h . g~(yo)Y - azx \niI = w . g~(xo)x - a 1l y \n\n(4.4) \n\nwhere (x, y) is the deviation from (xo, Yo . The solution is X = Toe-at sin(wt+ 0, [G~(Yo)lii = \ng~(Yilo) > 0, for all i,j. Eliminating Y, \n\n(4.5) \n\n(4.6) \n\nwhere A = HW = HoG~(Yo)WoG~(Xo). The ith oscillator (mitral cell) follows \n\nthe equation \n\nXi + (o.z + o.y)Xi + (Aii + o.zo.y)Xi + L AijXj = 0 \n\njt.i \n\n(4.7) \n\n(cf.equation (4.1)), the the last term describes the coupling between oscillators. \nNon-linear effect occurs when the amplitude is large, and make the oscillation wave \nform non-sinusoidal. \n\nIf X k is one of the eigenvectors of A with eigenvalue Ak, equation (4.6) has \n\nkth oscillation mode \n\nComponents of Xk indicate oscillators' relative amplitudes and phases (for each \nk = 1,2, ... , N independent mode). For simplicity, we set 0.2: = 0.1/ = 0., \nthen X ex: Xke-at\u00b1i../X,.t. Each mode has frequency Re~k' where Re means \nIf Re( -0. \u00b1 i~k) > 0 is satisfied for \nthe real part of a complex number. \nsome k, then the amplitude of the kth mode will increase with time, i.e. growing \noscillation. Starting from an initial condition of arbitrary small amplitudes in linear \nanalysis, the mode with the fastest growing amplitude will dominate the output, \nand the whole bulb will oscillate in the same frequency as observed physiologically \n(Freeman 1978; Freeman and Schneider 1982) as well as in the simulation. With the \nnon-linear effect, the strongest mode will suppress the others, and the final activity \noutput will be a single \"mode\" in a non-linear regime. \n\n\fModeling the Olfactory Bulb-Coupled Nonlinear Oscillators \n\n407 \n\nBecause of the coupling between the (damped) oscillators, the equilibrium point \n(Xo, Yo) of a group of oscillators is no longer always stable with the possibility \nof growing oscillation modes. Ak must be complex in order to have kth mode \ngrow. For this, a necessary (but not sufficient) condition is that matrix A is non(cid:173)\nsymmetric. Those inputs that make matrix A less symmetric will more likely induce \nthe oscillatory output and thus presumpably be noticed by the following olfactory \ncortex (see Li and Hopfield 1988a for details). \n\nThe consequences (also observed physiologically) of our model are (Freeman \n1975,1978; Freeman and Schneider 1982; Li and Hopfield 1988a): 1): local mitral \ncells' oscillation phase leads that of the local granule cells by a quarter cycle; 2): \noscillations across the bulb have the same dominant frequency whose range possible \nshould be narrow; 3): there should be a non-zero phase gradient field across the \nbulb; 4): the oscillation activity will rise during the inhale and fall at exhale, and \nrides on a slow background baseline shift wave phase locked with the sniff cycles. \nThis model of the olfactory bulb can be generalized to other masses of inter(cid:173)\n\nacting excitatory and inhibitory cells such as those in olfactory cortex, neocortex \nand hippocampus (Shepherd 1979) etc. where there may be connections between \nthe excitatory cells as well as the inhibitory cells (Li and Hopfield 1988a). Suppose \nthat Bo and Co are excitatory-to-excitatory and inhibitory-to-inhibitory connec(cid:173)\ntion matrices respectively, then equation (4.6) becomes \n\nx + (az - B + a y + C)k + (A + (az - B)(ay + C))X = 0 \n\n(4.9) \n\n5. COMPUTATIONS IN THE OLFACTORY BULB \nReceptor input I influences (Xo, Yo) as follows \n\ndXo ~ (a2 + HW)-l(adI + di) \ndYo ~ (a2 + W H)-l(W dI - aH-1di) \n\n(5.1) \n\nThis is how the odor input determines the bulbar output. Increasing Iodor not only \nraises the mean activity level (Xo, Yo) (and thus the gain (G~(Xo), G~(Yo))), but \nalso slowly changes the oscillation modes by structurally changing the oscillation \nequation (4.6) through matrix A = HoG~(Yo)WoG~(Xo). If (Xo, Yo) is raised \nto such an extent that Re( -a\u00b1 iv'Ak) > 0 is satisfied for some mode k, the equi(cid:173)\nlibrium point (Xo, Yo) becomes unstable and this mode emerges with oscillatory \nbursts. Different oscillation modes that emerge are indicative of the different odor \ninputs controlling the system parameters (Xo, Yo), and can be thought of as the \ndecision states reached for odor information, i.e., the oscillation pattern classifies \nodors. When (Xo, Yo) is very low (e.g. before inhale), all modes are damped, and \nonly small amplitUde oscillations occur, driven by noise and the weak time variation \nof the odor input. The absence of oscillation can be interpreted by higher processing \n\n\f408 \n\nLi and Hopfield \n\ncenters as the absence of an odor (Skarda and Freeman 1987). Detailed analysis \nshows how the bulb selectively responds (or not to respond) to certain input pat(cid:173)\nterns (Li and Hopfield 1988a) by choosing the synaptic connections appropriately. \nThis means the bulb can have non-uniform sensitivities to different odor receptor \ninputs and achieve better odor discriminations. \n\n6. PERFORMANCE OPTIMIZATION IN THE BULB \n\nWe discussed (Li and Hopfield 1988a) how the olfactory bulb makes the least \ninformation contamination between sniffs and changes the motivation level for odor \ndiscrimination. We further postulate with our model that the bulb, with appropriate \ninputs from the higher centers, can enhance or suppress the sensitivity to particular \nodors (details in Li and Hopfield 1988b). When the central input Ie is not fixed, it \ncan control the bulbar output by shifting (Xo, Yo), just as the odor input I can, \nequation (5.1) becomes: \n\ndXo ~ (a2 + HW)-l(adI + di - HdIe + aW-ldic) \ndYo ~ (a2 + W H)-l(W dI - aH-ldi + adIe + die) \n\n(6.1) \nSuppose that Ie = Ie,ba.ekground + Ie,eontrol where Ie,eontrol is the control sig(cid:173)\nnal which changes during a sniff. Olfactory adaptation is achieved by having an \nIe,eontrol = Iga.neel which cancels the effect of Iodor on Xo -\ncancelling. This \nkeeps the mitral cells baseline output Gz(Xo) and gain G~(Xo) low, and thus \nmakes the oscillation output impossible as if no odor exists. We can then expect \nthat reversing the sign of Iga.neel will cause the bulb to have an enhanced, instead \nof reduced (adapted), response to Iodor -\nanti-cancelling, and achieve the olfac(cid:173)\ntory enhancement. We can derive further phenomena such as recognizing an odor \ncomponent in an odor mixture, cross-adaptation and cross-enhancement (Li and \nHopfield 1988b). Computer simulations confirmed the expected results. \n\n7. DISCUSSION \n\nOur model of the olfactory bulb is a simplification of the known anatomy \n\nand physiology. The net of the mitral and granule cells simulates a group of cou(cid:173)\npled non-linear oscillators which are the sources of the rhythmic activities in the \nbulb. The coupling makes the oscillation coherent across the bulb surface for each \nsniff. The model suggests, in agreement with Freeman and coworkers, that stability \nchange bifurcation is used for the bulbar oscillator system to decide primitively on \nthe relevance of the receptor input information. Different non-damping oscillation \nmodes emerged are used to distinguish the different odor input information which \nis the driving source for the bifurcations, and are approximately thought of as the \n(unitary) decision states of the system for the odor information. With the extra \ninformation represented in the oscillation phases of the cells, the bulb emphasizes \nthe differences between different input patterns (section 4). Both the analysis and \nsimulation show that the bulb is selectively sensitive to different receptor input pat(cid:173)\nterns. This selectivity as well as the motivation level of the animal could also be \n\n\fModeling the Olfactory Bulb-Coupled Nonlinear Oscillators \n\n409 \n\nmodulated from higher centers. This model also successfully applies to bulbar abil(cid:173)\nity to use input from higher centers to suppress or enhance sensitivity to particular \ntarget or to mask odors. \n\nThis model does not exclude the possibility that the information be coded in \nthe non-oscillatory slow wave Xo which is also determined by the odor input. The \nchief behaviors do not depend on the number of cells in the model. The model can \nbe generalized to olfactory cortex, hippocampus and neocortex etc. where there are \nmore varieties of synaptic organizations. \n\nAcknowledgements \nThis research was supported by ONR contract NOO014-87-K-0377. We would also \nlike to acknowledge discussions with J.A. Bower. \n\nReferences \nBaird B. Nonlinear dynamics of pattern formation and pattern recognition in rabbit \n\nolfactory bulb. Physica 22D, 150-175 (1986) \n\nFreeman W.J. Mass action in the nervous system. New York: Academic Press 1975 \nFreeman W.J. 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Behavioral and Brain Sciences 10, 161-195 (1987) \n\n\f", "award": [], "sourceid": 138, "authors": [{"given_name": "Zhaoping", "family_name": "Li", "institution": null}, {"given_name": "John J.", "family_name": "Hopfield", "institution": null}]}