{"title": "Dynamics of Analog Neural Networks with Time Delay", "book": "Advances in Neural Information Processing Systems", "page_first": 568, "page_last": 576, "abstract": null, "full_text": "568 \n\nDYNAMICS OF ANALOG NEURAL \nNETWORKS WITH TIME DELAY \n\nC.M. Marcus and RM. Westervelt \n\nDivision of Applied Sciences and Department of Physics \n\nHarvard University, Cambridge Massachusetts 02138 \n\nABSTRACT \n\nA time delay in the response of the neurons in a network can \ninduce sustained oscillation and chaos. We present a stability \ncriterion based on local stability analysis to prevent sustained \noscillation  in  symmetric  delay  networks,  and  show  an \nexample  of chaotic  dynamics  in  a  non-symmetric  delay \nnetwork. \n\nL  INTRODUCTION \n\nUnderstanding how time delay affects the dynamics of neural networks is important for \ntwo reasons:  First,  some  degree of time delay  is  intrinsic  to  any  physically  realized \nnetwork,  both in biological neural systems and in electronic artificial neural networks. \nAs  we  will  show,  it is  not obvious  what constitutes  a  \"small\"  (i.e.  ignorable)  delay \nwhich  will  not  qualitatively  change  the  network  dynamics.  For  some  network \nconfigurations, delay much smaller than the intrinsic relaxation time of the network can \ninduce collective oscillatory behavior not predicted by mathematical models which ignore \ndelay.  These  oscillations  mayor may  not  be desirable;  in  either  case,  one  should \nunderstand when and how new dynamics can appear.  The second reason to study time \ndelay is for its intentional use in parallel computation. The dynamics of neural networks \nwhich always converge to fixed points  are now fairly well understood.  Several neural \nnetwork  models  have  appeared recently  which  use  time  delay  to  produce  dynamic \ncomputation such as  associative recall of sequences  [Kleinfeld,1986; Sompolinsky and \nKanter, 1986].  It has also been suggested that time delay produces an effective noise in \nthe  network dynamics  which can yield improved recall of memories  [Conwell,  1987] \nFinally, to the extent that neural networks research is inspired by biological systems, the \nknown presence of time delays in a many real neural systems suggests their usefulness \nin parallel computation. \n\nIn  this  paper we  will  show  how  time delay  in an  analog  neural network can produce \nsustained oscillation and chaos.  In section 2  we consider  the case of a symmetrically \nconnected network.  It is known [Cohen and Grossberg, 1983; Hopfield, 1984] that in the \nabsence  of time  delay  a  symmetric  network  will  always  converge  to  a  fixed  point \nattractor. We show that adding a fixed delay to the response of each neuron will produce \nsustained oscillation when the  magnitude of the delay exceeds  a  critical value,  which \ndepends on the neuron gain and the network connection topology.  We then analyze the \n\n\fDynamics of Analog Neural Networks with Time Delay \n\n569 \n\nall-inhibitory  and  symmetric  ring  topologies  as  examples.  In  section  3,  we  discuss \nchaotic dynamics in asymmetric neural networks, and give an example of a small (N=3) \nnetwork which  shows  delay-induced  chaos.  The  analytical  results  presented here  are \nsupported by numerical  simulations  and experiments performed on  a small electronic \nneural network with controllable time.  A detailed derivation of the stability results for \nthe  symmetric network is  given in  [Marcus  and Westervelt,  1989],  and  the electronic \ncircuit used is described in described [Marcus and Westervelt,  1988]. \n\nII. STABILITY OF SYMMETRIC NETWORKS WITH DELAY \n\nThe  dynamical  system  we  consider  describes  an  electronic  circuit  of N  saturable \namplifiers (\"neurons\") coupled by a resistive interconnection matrix. The neurons do not \nrespond to  an  input  voltage  ui instantaneously,  but produce  an  output after  a delay, \nwhich  we  take  to  be  the  same  for  all  neurons.  The  neuron  input  voltages  evolve \naccording to the following equations: \n\niI.(t)  = -u.(t)  +  L  J .. f(u.(t-t\u00bb. \n\n1 \n\n1 \n\nN \nj = 1 \n\nIJ \n\nJ \n\n(1 ) \n\nThe transfer function for each neuron is  taken to be an identical sigmoidal function feu) \nwith a maximum slope df/du = ~ at u = O.  The unit of time in these equations has been \nscaled to the characteristic network relaxation time, thus t  can be thought of as the ratio \nof delay time to relaxation time. The symmetric interconnection matrix Jij  describes the \nconductance between neurons i and j  is  normalized to satisfy LjlJijl =  1  for all i. This \nnormalization assumes that each neuron sees the same conductance at its input [Marcus \nand Westervelt,  1989].  The initial conditions for this system are a set of N continuous \nfunctions defined on the interval -t ~ t ~ O.  We take each initial function to be constant \nover that interval, though possibly different for different i.  We find numerically that the \nresults do not depend on the form of the initial functions. \n\nLinear  Stability  Analysis  at  Low  Gain \nStUdying  the  stability  of the  fixed  point at  the  origin  (ui  = 0  for  all  i)  is  useful for \nunderstanding the source of delay-induced sustained oscillation and will lead to a low-gain \nstability criterion for symmetric networks.  It is  important to realize however, that for \nthe  system (1)  with a sigmoidal nonlinearity, if the origin is  stable then it is  the unique \nattractor,  which  makes for rather uninteresting dynamics.  Thus the origin will almost \ncertainly be unstable  in  any  useful  configuration.  Linear stability  analysis  about  the \norigin  will  show  that  at  't = 0,  as  the  gain  ~ is  increased,  the  origin  always  loses \nstability by a type of bifurcation which only produces other fixed  points,  but for 't > 0 \nan  alternative type of bifurcation of the origin can occur which produces the sustained \noscillatory modes.  The stability criterion derived insures that this  alternate bifurcation -\na Hopf bifurcation - does not occur. \n\nThe  natural  coordinate  system  for  the  linearized  version  of  (1)  is  the  set  of N \neigenvectors of the connection matrix Jij' defined as  xi(t), i=I, .. N.  In terms of the xi(t), \n\n\f570 \n\nMarcus and Westervelt \n\nthe linearized system can be written \n\ni  .( t)  =  - x .( t) +  ~ A.  x.( t - t  ) \n\nI I I  \n\n1 \n\n(2) \n\nwhere ~ is  the neuron gain and Ai (i=I, .. N)  are the eigenvalues of Jij' In general,  these \neigenvalues  have  both  real  and  imaginary  parts;  for  Jij  = Jji  the  A'  are  purely  real. \nAssuming  exponential  time  evolution  of  the  form  xi(t)  = Xi(O)e~it, where  si is  a \ncomplex characteristic exponent,  yields a set of N transcendental characteristic equations: \n(si + l)esit  = ~Ai'  The condition for stability of the origin, Re(si) < 0 for all i,  and the \ncharacteristic equations can be used to specify a stability region in the complex plane of \neigenvalues,  as  illustrated  in  Fig.  (la).  When  all  eigenvalues  of Jij  are  within  the \nstability  region,  the  origin  is  stable.  For t  = 0,  the  stability  region  is  defined  by \nRe(A) <  lI~, giving a half-plane stability condition familiar from  ordinary differential \nequations.  For t  >  0,  we define the  border of the  stability region A(e) at an  angle e \nfrom  the  Re(A)  axis  as  the radial  distance from the point A = 0  to the frrst point (Le. \nsmallest value of A(e\u00bb which satisfies the characteristic equation for purely imaginary \ncharacteristic exponent Sj  i5 iroj.  The delay-dependent value of A(e) is given by \n\nA(e)  =  ~.J ro2  +  1 \n\n; \n\nro  = - tan  (rot  - e) \n\n(3) \n\nwhere ro  is  in the range (e-1tI2)  ~ClYt ~ e, modulo 21t. \n\n(a) \n\nIm(A) \n\nA(O;t=l) \n\n--'\"\"\"\"\"'''-\n\n~~--~--~----~----~ \n\nReO.) \n\n(b)  100 \n\nJ3A \n10 \n\n0.1 \n\n't \n\n1 \n\n10 \n\nFigure 1. (a) Regions of Stability in the Complex Plane of Eigenvalues A of the \n\nConnection Matrix Jij' for't = 0,1,00. (b) Where Stability Region Crosses the Real-A \n\nAxis in the Negative Half Plane. \n\nNotice that for nonzero delay the stability region closes on the Re(A) axis in the negative \nhalf-plane.  It is therefore possible for negative real eigenvalues to induce an instability \nof the origin.  Specifically,  if the minimum eigenvalue of the  symmetric matrix Jij  is \nmore negative than -A(e = 1t)  then the origin is  unstable.  We define this \"back door\" \nto  the stability region along  the real  axis  as  A > 0, dropping  the argument e = 1t.  A is \ninversely proportional to the gain  ~ and depends on delay as  shown in  Fig. (lb).  For \nlarge and small delay, A can be approximated as an explicit function of delay and gain: \n\n\fDynamics of Analog Neural Networks with Time Delay \n\n571 \n\nA  _ \n\nt\u00ab l  \n\nt> >  1 \n\n(4 a) \n\n(4b) \n\nIn the  infinite-delay limit,  the delay-differential system (1) is  equivalent to an  iterated \nmap or parallel-update network of the form ui(t+l) = 1] Jij f(uj(t\u00bb where t is a discrete \niteration index.  In this  limit, the stability region is circular, corresponding to the fixed \npoint stability condition for the iterated map system. \n\nConsider the stability of the origin in a symmetrically connected delay system (1) as the \nneuron gain ~ is  increased from zero to a large value.  A  bifurcation of the origin will \noccur when the maximum eigenvalue Amax  > 0 of Jij  becomes larger than  l/~ or when \nthe  minimum eigenvalue Amin  < 0 becomes more negative than -A = _~-I(ro2+1)lJ2, \nwhere  ro  =  -tan(rot),  [1CI2  < ro  < x].  Which  bifurcation  occurs  first  depends  on  the \ndelay  and the eigenvalues of Jr.  The bifurcation at Amax  =  ~-1 is  a pitchfork (as  it is \nfor t  = 0) corresponding to a ctaracteristic exponent si crossing into the  positive real \nhalf plane along  the real  axis.  This bifurcation creates a pair of fixed points along  the \neigenvector Xi  associated with  that eigenvalue.  These  fixed  points  constitute  a single \nmemory  state of the network.  The bifurcation at Amin = - A  corresponds  to  a Hopf \nbifurcation [Marsden and McCracken, 1976] , where a pair of characteristic exponents pass \ninto the real half plane with imaginary components \u00b1ro  where ro  =  -tan(rot), [x/2 < ro \n< xl. This  bifurcation,  not present at t  =  0,  creates  an  oscillatory  attractor  along  the \neigenvector associated with ~in' \n\nA  simple  stability  criterion  can  be  constructed  by  requiring  that the  most  negative \neigenvalue of the (symmetric) connection matrix not be more negative than -A. Because \nA  is  always  larger than  its  small-delay  limit 7tI(2t~), the criterion can be stated as  a \nlimit on the size on the delay (in units of the network relaxation time.) \n\nt<-\n\nx \n\n2~A  . mIn \n\n=>  no sustained oscillation. \n\n(5) \n\nLinear stability analysis does not prove global stability, but the criterion (5) is  supported \nby considerable numerical and eXferimental evidence [Marcus  and Westervelt,  1989]. \nFor  long  delays,  where  A ==  W ,linear  stability  analysis  suggests  that  sustained \noscillation will not exist as  long  as _~-1 < Amin'  In  the  infinite-delay limit,  it can be \nshown  that  this  condition  insures  global  stability in  the  discrete-time parallel-update \nnetwork. [Marcus and Westervelt, to appear]. \n\nAt  large gain,  Eq.  (5)  does  not  provide  a  useful  stability  criterion because  the delay \nrequired for stability tends to zero as  ~ ~ 00.  The nonlinearity of the transfer function \nbecomes important at large gain and stable, fixed-point-only dynamics are found at large \ngain and nonzero delay,  indicating that Eq. (5) is overly conservative at large gain.  To \nunderstand  this,  we  must  include  the  nonlinearity  and  consider  the  stability  of the \noscillatory modes themselves.  This is described in the next section. \n\n\f572 \n\nMarcus and Westervelt \n\nStability  in  the  Large-Gain  Limit \nWe now  analyze the oscillatory mode at large gain for the  particular case of coherent \noscillation.  We find a second stability criterion which predicts a gain-independent critical \ndelay below which all  initial conditions lead to fixed points.  This result complements \nthe low gain result of the previous section for this class of network; experimentally and \nnumerically we find excellent agree in both regimes,  with a cross-over at the value of \ngain where fixed points appear away from the origin, p =  lIAmax. \nIn considering only coherent oscillation,  we not only assume that Iij is  symmetric  but \nthat  its  maximum  and  minimum eigenvalues  satisfy  0 < Amax  < -Amin  and that the \neigenvector associated with Amin points in a coherent direction, defined to be along any \nof the 2N  vectors  of the form (\u00b1I,\u00b1l,\u00b1I, ... ) in the ui  basis. For this  case, we find that \nin  the limit of infinite gain, where the nonlinearity is of the form f(u)  =  sgn(u), multiple \nfixed point attractors coexist with the oscillatory attractor and that the size of the basin \nof attraction  for  the  oscillatory mode  varies  with  the delay  [Marcus  and  Westervelt, \n1988].  At a critical value of delay 'tcrit the basin of attraction for oscillation vanishes \nand the oscillatory mode loses stability. In [Marcus and Westervelt, 1989] we show: \n\n't \n\ncnt \n\n.  = -In( 1 + A max / A \n\n.  ) \nmIn \n\n(6) \n\nFor delays less than this critical value, all initial states lead to stable fIXed points. \n\nNotice  that the critical  delay  for  coherent oscillation  diverges  as  IAmax/Aminl  ~ 1-. \nExperimentally  and  numerically  we  find  that  this  prediction  has  more  general \napplicability:  None  of  the  symmetric  networks  investigated  which  satisfied \nIAmax/Aminl ~ 1 (and  Amax  >  0)  showed  sustained  oscillation  for  't < -10.  This \nobservation is a useful criterion for electronic circuit design, where single-device delays \nare  generally  shorter  than  the  circuit  relaxation  time  ('t  <  1),  but only  the case of \ncoherent oscillation is supported by analysis. \n\nExamples \nAs  a first example, we consider the fully-connected all-inhibitory network, Eq. (1) with \nIii =  (N-IrI(~ij - 1).  This  matrix  has  N-I degenerate eigenvalues  at  +lI(N-I) and a \nsmgle eigenvalue at -1.  A similar network configuration (with delays) has been studied \nas  a model of lateral inhibition in the eye of the horseshoe crab, Limulus [Coleman  and \nRenninger,I975,I976; Hadeler and Tomiuk,I977; anderHeiden, 1980]. Previous analysis \nof sustained oscillation in  this  system has  assumed a coherent form for  the  oscillatory \nsolution,  which  reduces  the  problem  to  a  single  scalar  delay-differential  equation. \nHowever,  by  constraining  the  solution  to  lie  on  along  the  coherent  direction,  the \ninstability  of  the  oscillatory  mode  discussed  above  is  not  seen.  Because  of  this \nassumption,  fixed-point-only dynamics in the large-gain limit with finite delay are not \npredicted by previous treatments, to our knowledge. \n\n\fDynamics of Analog Neural Networks with Time Delay \n\n573 \n\nThe behavior of the network at various values of gain and delay are illustrated in Fig.2 \nfor the particular case of N=3.  The four regions labeled A,B,C and D  characterize the \nbehavior for  all  N.  At low gain  (~ < N-1)  the origin is  the  unique  attractor for  small \ndelay (region A) and undergoes a Hopf bifurcation at to sustained coherent oscillation  at \n't - 7t(~2_1)-112 for large delay (region B). At ~ = N-1  fixed points away from the origin \nappear.  In addition to these fixed points, an oscillatory attractor exists at large gain for \n't > In [(N-1)/(N-2)]  (==  liN for large N)  (region C).  Sustained oscillation does  not \nexist below this critical delay (region D). \n\nc \n\nD \n\nA \n\n10 \n\n100 \n\nFigure 2.  Stability Diagram for the All-Inhibitory Delay Network for the Case N  =  3. \n\nSee Text for a Description of A,B,C and D. \n\nAs a second example, we consider a ring of delayed neurons.  We allow the symmetric \nconnections to be of either sign - that is, connections between neighboring pairs can be \nmutually excitatory or inhibitory - but are all the same strength.  The eigenvalues for the \nsymmetric ring of size N  are Ak = cos(27t(k+<p)/N),  where k  = O,1,2 ... N-1,  <p  =  112 if \nthe produet of connection strengths around the ring is negative, <p  =  0 if the product is \npositive.  Borrowing from the language of disordered magnetic systems,  a ring which \ncontains  an  odd  number  of negative  connections  (the  case  <p  = 112)  is  said  to  be \n\"frustrated.\"  [Toulouse,  1977]. The large-gain stability analysis for the symmetric ring \ngives  a rather surprising result  Only frustrated rings with an odd number of neurons \nwill  show sustained oscillation. For this case (N odd.aru1 an odd number of negative \nconnections) the critical delay is  given by 'tcrit = -In (1  - cos(1tIN\u00bb.  This  agrees  very \nwell with experimental and numerical data, as does the conclusion that rings with even N \ndo not show sustained oscillation [Marcus and Westervelt,  1989].  The theoretical large(cid:173)\ngain critical delay for the all-inhibitory network and the frustrated ring of the same size \nare  compared  in  Fig.  3.  Note  that  the  critical  delay  for  the  all-inhibitory  network \ndecreases  (roughly  as  lIN) for  larger networks  while  the ring  becomes  less  prone to \noscillation as the network size increases. \n\n\f574 \n\nMarcus and Westervelt \n\n10 \n\n'terit  ~  II \n\nII \n\nII \n\n1 \n\n\u2022 \n\n\u2022 \n\n~. \u2022  \u2022  \u2022  \u2022 \n\n3 \n\n5 \n\nN \n\n7 \n\n9 \n\n0.1 \n\n1 \n\n\u2022 \n\n11 \n\nFigure 3. Critical Delay from Large-Gain Theory for All-Inhibitory Networks (circles) \n\nand Frustrated Rings (squares) of size N. \n\nITI. CHAOS IN NON-SYMMETRIC DELAY NETWORKS \n\nAllowing  non-symmetric  interconnections  greatly  expands  the  repertoire  of neural \nnetwork dynamics and can yield new, powerful computational properties. For example, \nseveral recent studies have shown that by using both asymmetric connections and time \ndelay,  a  neural  network  can  accurately  recall  of  sequences  of  stored  patterns \n[Kleinfeld,1986; Sompolinsky and Kanter,1986].  It has also been shown that for some \nparameter  values,  these  pattern-generating  networks  can  produce  chaotic  dynamics \n[Riedel, et ai,  1988]. \n\nRelatively  little is  known  about the  dynamics  of large  asymmetric  networks  [Amari, \n1971,1972;  KUrten  and  Clark,1986;  Shinomoto,1986;  Sompolinsky,  et ai,  1988, \nGutfreund,et al,1988].  A  recent  study  of continuous-time  networks  with  random \nasymmetric connections shows that as  N -+ 00  these systems  will be chaotic whenever \nthe  origin  is  unstable  [Sompolinsky,et al,1988].  In discrete-state  (\u00b11)  networks, with \neither parallel or sequential deterministic dynamics, oscillatory modes with long periods \nare  also  seen  for  fully  asymmetric  random connections  (Jij  and Jji uncorrelated),  but \nwhen Jij has either symmetric or antisymmetric correlations short-period attractors seem \nto predominate [Gutfreund,et al,1988].  It is not clear whether the chaotic dynamics of \nlarge  random networks  will  appear  in  small networks  with  non-symmetric,  but non(cid:173)\nrandom, connections. \n\nSmall  networks  with  asymmetric  connections  have  been  used  as  models  of central \npattern generators found in many biological neural systems.  [Cohen,et ai,  1988] These \nmodels frequently use time delay to produce sustained rhythmic output, motivated in part \nby the known presence of time delay in real central pattern generators. General theoretical \nprinciples concerning the dynamics of asymmetric network with  delay do not exist at \npresent. It has been shown, however, that large system size is not necessary to produce \nchaos in neural networks with delay [e.g. Babcock and Westervelt,  1987].  We fmd that \nsmall systems  (N~ 3) with certain asymmetric connections and time delay can produce \nsustained chaotic oscillation. An example is  shown in Fig. 4:  These data were produced \nusing  an  electronic  network  [Marcus  and  Westervelt,  1988]  of three  neurons  with \n\n\fDynamics of Analog Neural Networks with Time Delay \n\n575 \n\nsigmoidal  transfer  functions  f 1 (u(t\u00bb=3.8tanh(8u(t-tp,  f2(u(t\u00bb=2tanh(6.1u(t\u00bb, \nf3(u(t\u00bb=3.5tanh(2.5u(t\u00bb,  connection  resistances  of \u00b11O  .0  and input capacitances of \nlOnF.  Fig.  4  shows  the  network  configuration  and  output  voltages  VIand V 2  for \nincreasing delay in neuron  1.  For t  < O.64ms  a periodic attractor similar to  the upper \nleft figure is found; for t  > O.97ms both periodic and chaotic attractors are found. \n\n1.0-\n\nV2  -\n\no -\n\n1.0-\n\nV2  -\n\no -\n\n't \n\nA \n\n0 \n\nVI \n\n1.0 \n\n0 \n\nVI \n\n1.0 \n\nFigure 4. Period Doubling to Chaos as the Delay in Neuron 1 is Increased. \n\nChaos  in  the  network  of Fig.4  is  closely  related  to  a  well-known  chaotic  delay(cid:173)\ndifferential equation with a noninvertible feedback term [Mackey and Glass,1977]. The \nnoninvertible  or \"mixed\"  feedback  necessary  to  produce  chaos  in  the  Mackey-Glass \nequation  is  achieved  in  the  neural  network  - which  has  only  monotone  transfer \nfunctions - by  using  asymmetric connections. \n\nThis  association  between  asymmetry  and  noninvertible  feedback  suggests  that \nasymmetric  connections  may  be  necessary  to  produce  chaotic  dynamics  in  neural \nnetworks,  even  when  time delay  is  present.  This  conjecture is  further  supported by \nconsidering the two limiting cases of zero delay and infinite delay, neither of which show \nchaotic dynamics for symmetric connections. \n\nIV. CONCLUSION AND OPEN PROBLEMS \n\nWe have considered the effects of delayed response in a continuous-time neural network. \nWe find that when the delay of each neuron exceeds a critical value sustained oscillatory \nmodes  appear in  a symmetric network.  Stability analysis  yields  a design criterion for \nbuilding stable electronic neural networks, but these results can also be used to created \ndesired oscillatory modes in delay networks.  For example, a variation of the Hebb rule \n[Hebb,  1949],  created  by  simply  taking  the  negative  of a  Hebb  matrix,  will  give \nnegative real eigenvalues corresponding to programed oscillatory patterns.  Analyzing the \nstorage capacities and other properties of neural networks with dynamic attractors remain \n\n\f576 \n\nMarcus and Westervelt \n\nchallenging problems [see, e.g. Gutfreund and Mezard, 1988]. \n\nIn analyzing the stability of delay systems, we have assumed that the delays and gains of \nall neurons  are  identical.  This  is  quite restrictive and is certainly not justified from  a \nbiological viewpoint.  It would be interesting  to  study  the  effects  of a  wide  range  of \ndelays  in  both  symmetric  and  non-symmetric  neural  networks.  It is  possible,  for \nexample, that the coherent oscillation described above  will not persist when the delays \nare widely distributed. \n\nAcknowledgements \n\nOne  of us  (CMM)  acknowledges  support  as  an  AT&T  Bell  Laboratories  Scholar. \nResearch supported in part by JSEP contract NOOOI4-84-K-0465. \n\nReferences \n\nS.  Amari,  1971, Proc.  IEEE,  59, 35. \nS.  Amari,  1972,  IEEE Trans.  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Phys.  2,  115. \n\n\f", "award": [], "sourceid": 111, "authors": [{"given_name": "Charles", "family_name": "Marcus", "institution": null}, {"given_name": "R.", "family_name": "Westervelt", "institution": null}]}