{"title": "Oscillation Onset in Neural Delayed Feedback", "book": "Advances in Neural Information Processing Systems", "page_first": 130, "page_last": 136, "abstract": null, "full_text": "Oscillation Onset \n\n\u2022 In \n\nNeural Delayed Feedback \n\nAndre Longtin \nComplex Systems Group and Center for Nonlinear Studies \nTheoretical Division B213, Los Alamos National Laboratory \nLos Alamos, NM 87545 \n\nAbstract \n\nThis paper studies dynamical aspects of neural systems with delayed neg(cid:173)\native feedback modelled by nonlinear delay-differential equations. These \nsystems undergo a Hopf bifurcation from a stable fixed point to a sta(cid:173)\nble limit cycle oscillation as certain parameters are varied. It is shown \nthat their frequency of oscillation is robust to parameter variations and \nnoisy fluctuations, a property that makes these systems good candidates \nfor pacemakers. The onset of oscillation is postponed by both additive \nand parametric noise in the sense that the state variable spends more time \nnear the fixed point than it would in the absence of noise. This is also the \ncase when noise affects the delayed variable, i.e. when the system has a \nfaulty memory. Finally, it is shown that a distribution of delays (rather \nthan a fixed delay) also stabilizes the fixed point solution. \n\n1 \n\nINTRODUCTION \n\nIn this paper, we study the dynamics of a class of neural delayed feedback models \nwhich have been used to understand equilibrium and oscillatory behavior in recur(cid:173)\nrent inhibitory circuits (Mackey and an der Heiden, 1984; Plant, 1981; Milton et \nal., 1990) and brainstem reflexes such as the pupil light reflex (Longtin and Milton, \n1989a,b; Milton et al., 1989; Longtin et al., 1990; Longtin, 1991) and respiratory \ncontrol (Glass and Mackey, 1979). These models are framed in terms of first-order \nnonlinear delay-differential equations (DDE's) in which the state variable may rep(cid:173)\nresent, e.g., a membrane potential, a mean firing rate of a population of neurons or \n\n130 \n\n\fOscillation Onset in Neural Delayed Feedback \n\n131 \n\na muscle activity. For example, the negative feedback dynamics of the human pupil \nlight reflex have been shown to be appropriately modelled by the following equa(cid:173)\ntion for pupil area (related to the activity of the iris muscles through the nonlinear \nmonotonically decreasing function g(A) ) (see Longtin and Milton, 1989a,b): \n\ndg(A) dA(t) \n\ndA \n\ndt + o:g \n\n(A) = \n\nI [let - T)A(t - T)] \n\n, n \n\n\u00a2 \n\n(I) \n\nlet) is the external light intensity and \u00a2 is the retinal light flux below which no \npupillary response occurs. The left hand side of Eq.(I) governs the response of the \nsystem to the state-dependent forcing (i.e. stimulation) embodied in the term on \nthe right-hand side. The delay T is essential to the understanding of the dynamics \nof this reflex. It accounts for the fact that the iris muscles move in response to the \nretinal light flux variations occurring'\" 300 msec earlier. \n\n2 FOCUS AND MOTIVATION \n\nFor the sake of discussion, we shall focus on the following prototypical model of \ndelayed negative feedback \n\nd~~t) + o:x(t) = f(jj; x(t - T\u00bb \n\n(2) \n\nwhere jj is a vector of parameters and f is a monotonically decreasing function. \nThis equation typically exhibits a Hopf bifurcation (i.e. a qualitative change in \ndynamics from a stable equilibrium solution to a stable limit cycle oscillation) as \nthe slope of the feedback function or the delay are increased passed critical values. \nAutonomous (as opposed to externally forced) oscillations are frequently observed \nin real neural delayed feedback systems which suggests that these systems may \nexhibit a Hopf bifurcation. Further, it is clear that these systems operate despite \nnoisy environmental fluctuations. A clear understanding of the properties of these \nsystems can reveal useful information about their structure and the origin of the \n\"noisy\" sources, as well as enable us to extract general functioning principles for \nsystems organized according to this scheme. \n\nWe now focus our attention on three different dynamical aspects of these systems: \nI) the stability of the oscillation frequency and amplitude to parameter variations \nand to noise; 2) \nthe \nstabilization of the equilibrium behavior in the more realistic case involving a dis(cid:173)\ntribution of delays rather than a single fixed delay. \n\nthe postponement of oscillation onset due to noise; and 3) \n\n3 FREQUENCY AND AMPLITUDE \n\nUnder certain conditions, the neural delayed feedback system will settle onto equi(cid:173)\nlibrium behavior after an initial transient. Mathematically, this corresponds to the \nfixed point solution x\u00b7 of Eq.(2) obtained by setting z = O. A supercritical Hopf \nbifurcation occurs in Eq.(2) when the slope of the feedback function at this fixed \npoint ~ I exceeds some value /co called the bifurcation value. It can also occur \n\nz\u00b7 \n\n\f132 \n\nLongtin \n\nwhen the delay exceeds a critical value. The case where the parameter a increases \nis particularly interesting because the system can undergo a Hopf bifurcation at \na = al followed by a restabilization of the fixed point through a reverse Hopf \nbifurcation at a = a2 > al (see also Mackey, 1979). \nNumerical simulations of Eq.(2) around the Hopf bifurcation point ko reveal that \nthe frequency is relatively constant while the amplitude Ampl grows as Jk - k o . \nHowever, in oscillatory time series from real neural delayed feedback systems, the \nfrequency and amplitude fluctuate near the bifurcation point, with relative ampli(cid:173)\ntude fluctuations being generally larger than relative frequency fluctuations. This \npoint has been illustrated using data from the human pupil light reflex whose feed(cid:173)\nback gain is under experimental control (see Longtin, 1991; Longtin et al., 1990). \nIn the case of the pupil light reflex, the variations in the mean and standard devia(cid:173)\ntion of amplitude and period accompanying increases in the bifurcation parameter \n(the external gain) have been explained in the hypothesis that \"neural noise\" is \naffecting the deterministic dynamics of the system. This noise is strongly amplified \nnear the bifurcation point where the solutions are only weakly stable (Longtin et \nal., 1990). Thus the coupling of the noise to the system is most likely responsible \nfor the aperiodicity of the observed data. \nThe fact that the frequency is not significantly affected by the noise nor by variation \nof the bifurcation parameter (especially in comparison to the amplitude fluctua(cid:173)\ntions) suggests that neural delayed feedback circuits may be ideally suited to serve \nas pacemakers. The frequency stability in regulatory biological systems has previ(cid:173)\nously been emphasized by Rapp (1981) in the context of biochemical regulation. \n\n4 STABILIZATION BY NOISE \n\nIn the presence of noise, oscillations can be seen in the solution of Eq.(2) even \nwhen the bifurcation value is below that at which the deterministic bifurcation \noccurs. This does not mean however that the bifurcation has occurred, since these \noscillations simply become more and more prominent as the bifurcation parameter is \nincreased, and no qualitative change in the solution can be seen. Such a qualitative \nchange does occur when the solution is viewed from a different standpoint. One \ncan in fact construct a histogram of the values taken on by the solution of the \nmodel differential equation (or by the data: see Longtin, 1991). The value of this \n(normalized) histogram at a given point in the state space (e.g. of pupil area values) \nprovides a measure of the fraction of the time spent by the system in the vicinity \nof this point. The onset of oscillation can then be detected by a qualitative change \nin this histogram, specifically when it goes from unimodal to bimodal (Longtin et \nal., 1990). The distance between the two humps in the bimodal case is a measure \nof the limit cycle amplitude. For short time series however (as is often the case in \nneurophysiology), it is practically impossible to resolve this distance and thus to \nascertain whether a Hopf bifurcation has occurred. \nIntensive simulations of Eq.(2) with either additive noise (i.e. added to Eq.(2)) or \nparametric noise (e.g. on the magnitude of the feedback function) reveal that the \nstatistical limit cycle amplitude (the distance between the two humps or \"order \nparameter\") is smaller than the amplitude in the absence of noise (Longtin et al., \n1990). The bifurcation diagram is similar to that in Figure 1. This implies that the \n\n\fOscillation Onset in Neural Delayed Feedback \n\n133 \n\nsolution spends more time near the fixed point, i.e. that the fixed point is stabilized \nby the noise (i.e . in the absence of noise, the limit cycle is larger and the system \nspends less time near the unstable fixed point). In other words, the onset of the \nHopf bifurcation is postponed in the presence of these types of noise. Hence the \nnoise level in a neural system, whatever its source, may in fact control the onset of \nan oscillation. \n\nl2 \n\nII \n\nZ ll . 5 \n\u2022 .-ro1 \nt \nJ \n~ LO.5 \n~ \nZ \n9 \n~ 9 . 5 \n~ \n;:) r.. \nIi \n\nLO \n\n' . 5~ __ - - -\n\n2.S \n\n7. S \n\nLO \n\n12.5 \n\n... 7 . 5 \n\nORDD PAllAMI:TZJl \n\nFigure 1. Magnitude of the Order Parameter as a Function of the Bifurcation \nParameter n for Noise on the Delayed State of the System. \n\nIn Figure 1 it is shown that the Hopf bifurcation is also postponed (the bifurcation \ncurve is shifted to higher parameter values with respect to the deterministic curve) \nwhen the noise is applied to the delayed state variable x(t - T) and / in Eq.(2) is \nof the form (negative feedback): \n\n)..on \n\n/ = On + xn(t _ T)\" \n\n(3) \nFor parameter values Q = 3.21,).. = 200,0 = 50, T = 0.3, the deterministic Hopf \nbifurcation occurs at n = 8.18. Colored (Ornstein-Uhlenbeck type) Gaussian noise \nof standard deviation u = 1.5 and correlation time lsec was added to the variable \nx(t - T). This numerical calculation can be interpreted as a simulation of the \nbehavior of a neural delayed feedback system with bad memory (i.e. in which there \nis a small error on the value recalled from the past). Thus, faulty memory also \nstabilizes the fixed point. \n\n5 DISTRIBUTED DELAYS \n\nThe use of a single fixed delay in models of delayed feedback is often a good approx(cid:173)\nimation and strongly warranted in a simple circuit comprising only a small number \n\n\f134 \n\nLongtin \n\nof cells. However, neural systems often have a spatial extent due to the presence of \nmany parallel pathways in which the axon sizes are distributed according to a cer(cid:173)\ntain probability density. This leads to a distribution of conduction velocities down \nthese pathways and therefore to a distribution of propagation delays. In this case, \nthe dynamics are more appropriately modelled by an integro-differential equation \nof the form \n\n~; + ax(t) = f(~; z(t), x(t\u00bb, \n\nlet) = 1too K(t - u)x(u) duo \n\n(4) \n\nThe extent to which values of the state variable in the past affect its present evolu(cid:173)\ntion is determined by the kernel K(t). The fixed delay case corresponds to choosing \nthe kernel to be a Dirac delta distribution. \nWe have looked at the effect of a distributed delay on the Hopf bifurcation in our \nprototypical delayed feedback system Eq.(2). Specifically, we have considered the \ncase where the kernel in Eq.( 4) has the form of a gamma distribution \n\na m +1 \nm. \n\na, m > O. \n\nK(t) = ~(t) = -,- tm e- aq , \n\n(5) \nThe average delay of this kernel is T = m;l and the kernel has the property that it \nconverges to the delta function in the limit where m and a go to infinity all the while \nkeeping the ratio T constant. For a kernel of a given order it is possible to convert \nthe DDE Eq.(2) into a set of (m+2) coupled ordinary differential equations (ODE's) \nwhich approximate the DDE (an infinite set of ODE's is in this case equivalent to the \noriginal DDE) (see Fargue, 1973; MacDonald, 1978; Cooke and Grossman, 1982). \nWe have investigated the occurrence of a Hopf bifurcation in the (m + 2) ODE's as \na function of the order m of the memory kernel (keeping T equal to the fixed delay \nof the DDE being approximated). This involves doing a stability analysis around \nthe fixed point of the (m + 2) order system of ODE's and numerically determining \nthe value of the bifurcation parameter n at which the Hopf bifurcation occurs. \nThe result is shown in Figure 2, where we have plotted n versus the order m of \napproximation. Note that at least a 3 dimensional system of ODE's is required for \na Hopf bifurcation to occur in such a system. Note also the fast convergence of n \nto the bifurcation value for the DDE (5.04). These calculations were done for the \nMackey-G lass equation \n\ndx + ax(t) = ~onx(t - r) \nOn+xn(t-r) \ndt \n\n(6) \n\nwith parameters 0 = 1, a = 2, ~ = 2, r = 2 and n E (1,20). This equation is a \nmodel for mixed feedback dynamics (i.e. a combination of positive and negative \nfeedback involving a single-humped feedback function). It displays the same quali(cid:173)\ntative features as Eq.(2) with the feedback given by Eq.(3) at the Hopf bifurcation \nand was chosen for ease of computation since parameters can be chosen such that \nthe fixed point does not depend on the bifurcation parameter. \nWe can see that, for a memory kernel of a given order, the Hopf bifurcation occurs \nat a higher value of the bifurcation parameter (which is proportional to the slope \nof the feedback function at the fixed point) than for the DDE. This implies that a \nstronger nonlinearity is required to set the ODE system into oscillation compared \n\n\fOscillation Onset in Neural Delayed Feedback \n\n135 \n\nto the DDE. In other words, the distributed delay system with the same feedback \nfunction as the DDE is less prone to oscillate (see also MacDonald, 1978; Cooke \nand Grossman, 1982). \n\nn \n\n20r---~----~----~----~----~----~---. \n\n11 \n\n16 \n\n12 \n\n10 \n\n\u2022 \n\n6 \n\n~.5D4 --------------------------------------------------------~ \n\n\u2022 \n\n2 \n\n2 \n\n3 \n\n6 \n\nm \n\n7 \n\nFigure 2. Value of n at Which a Hopf Bifurcation Occurs Versus the Order m of \nthe Memory KerneL \n\n6 SUMMARY \n\nIn sununary we have shown that neural delayed negative feedback systems can \nexhibit either equilibrium or limit cycle behavior depending on their parameters \nand on the noise levels. The constancy of their oscillation frequency, even in the \npresence of noise, suggests their possible role as pacemakers in the nervous system. \nFurther, the equilibrium solution of these systems is stabilized by noise and by \ndistributed delays. We conjecture that these two effects may be related as they \nsomewhat share a conunon feature, in the sense that noise and distributed delays \ntend to make the retarded action more diffuse. This is supported by the fact that a \nsystem with bad memory (i.e. with noise on the delayed variable) also sees its fixed \npoint stabilized. \n\nAcknowledgements \n\nThe author would like to thank Mackey for useful conversations as well as Christian \nCor tis for his help with the numerical analysis in Section 5. This research was \nsupported by the Natural Sciences and Engineering Research Council of Canada \n(NSERC) as well as the Complex Systems Group and the Center for Nonlinear \nStudies at Los Alamos National Laboratory in the form of postdoctoral fellowships. \n\n\f136 \n\nLongtin \n\nReferences \n\nK.L. Cooke and Z. Grossman. (1982) Discrete delay, distributed delay and stability \nswitches. J. Math. Anal. Appl. 86:592-627. \nD. Fargue. \n(1973) Reductibilite des systemes hereditaires a des systemes dy(cid:173)\nnamiques (regis par des equations differentielles aux derivees partielles). C.R. Acad. \nSci. Paris T .277, No.17 (Serie B, 2e semestre):471-473. \nL. Glass and M.C. Mackey. (1979) Pathological conditions resulting from instabili(cid:173)\nties in physiological control systems. Ann. N. Y. Acad. Sci. 316:214. \nA. Longtin. (in press, 1991) Nonlinear dynamics of neural delayed feedback. In D. \nStein (ed.),Proceedings of the 3,.d Summer School on Complex Systems, Santa Fe \nInstitute Studies in the Sciences of Complexity, Lect. Vol. III. Redwood City, CA: \nAddison-Wesley. \n\nA. Longtin and J .G. Milton. (1989a) Modelling autonomous oscillations in the \nhuman pupil light reflex using nonlinear delay-differential equations. Bull. Math. \nBioi. 51:605-624. \n\nA. Longtin and J .G. Milton. (1989b) Insight into the transfer function, gain and \noscillation onset for the pupil light reflex using nonlinear delay-differential equations. \nBioi. Cybern. 61:51-59. \nA. Longtin, J .G. Milton, J. Bos and M.C. Mackey. (1990) Noise and critical behavior \nof the pupil light reflex at oscillation onset. Phys. Rev. A 41:6992-7005. \nN. MacDonald. (1978) Time lags in biological models. Lecture Notes in Biomathe(cid:173)\nmatics 27. Berlin: Springer Verlag. \n\nM.C. Mackey. (1979) Periodic auto-immune hemolytic anemia: an induced dynam(cid:173)\nical disease. Bull. Math. Bioi. 41:829-834. \n\nM.C. Mackey and U. an der Heiden. (1984) The dynamics of recurrent inhibition. \nJ. Math. Bioi. 19: 211-225. \nJ .G. Milton, U. an der Heiden, A. Longtin and M.C. Mackey. (in press, 1990) Com(cid:173)\nplex dynamics and noise in simple neural networks with delayed mixed feedback. \nBiomed. Biochem. Acta 8/9. \n\nJ .G. Milton, A. Longtin, A. Beuter, M.C. Mackey and L. Glass. (1989) Complex \ndynamics and bifurcations in neurology. J. Theor. Bioi. 138:129-147. \nR.E. Plant. (1981) A Fitzhugh differential-difference equation modelling recurrent \nneural feedback. SIAM J. Appl. Math. 40:150-162. \nP.E. Napp. (1981) Frequency encoded biochemical regulation is more accurate then \namplitude dependent control. J. Theor. Bioi. 90:531-544. \n\n\f", "award": [], "sourceid": 339, "authors": [{"given_name": "Andr\u00e9", "family_name": "Longtin", "institution": null}]}