{"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}]}