{"title": "Asynchronous Dynamics of Continuous Time Neural Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 493, "page_last": 500, "abstract": null, "full_text": "Asynchronous Dynamics of Continuous \n\nTime Neural Networks \n\nComputer Science  Department \n\nUniversity of California at Los  Angeles \n\nXin Wang \n\nLos  Angeles,  CA 90024 \n\nQingnan Li \n\nDepartment of Mathematics \n\nUniversity of Southern California \n\nLos  Angeles,  CA 90089-1113 \n\nEdward K.  Blum \n\nDepartment of Mathematics \n\nUniversity of Southern California \n\nLos  Angeles,  CA 90089-1113 \n\nABSTRACT \n\nMotivated by mathematical modeling, analog implementation and \ndistributed simulation of neural networks, we present a definition of \nasynchronous  dynamics  of general  CT dynamical systems  defined \nby  ordinary  differential equations,  based  on  notions of local  times \nand  communication  times.  We  provide  some  preliminary  results \non  globally  asymptotical  convergence  of  asynchronous  dynamics \nfor  contractive  and  monotone CT  dynamical systems.  When  ap(cid:173)\nplying  the  results  to  neural  networks,  we  obtain  some  conditions \nthat ensure  additive-type neural  networks  to be  asynchronizable. \n\n1 \n\nINTRODUCTION \n\nNeural networks are massively distributed computing systems.  A major issue in par(cid:173)\nallel and distributed computation is synchronization versus  asynchronization (Bert(cid:173)\nsekas and Tsitsiklis, 1989).  To fix our idea, we consider a much studied additive-type \nmodel  (Cohen  and  Grossberg,  1983;  Hopfield,  1984;  Hirsch,  1989) of a  continuous(cid:173)\ntime (CT)  neural network of n  neurons,  whose  dynamics is governed  by \n\nXi(t)  = -ajXi(t) + L WijO'j (Jlj Xj (t)) + Ii, \n\nn \n\ni = 1,2, ... , n, \n\nj=1 \n\n(1) \n\n493 \n\n\f494 \n\nWang. Li. and Blum \n\nwith neuron states  Xi (t)  at time t,  constant decay rates ai, external inputs h, gains \nJJj,  neuron  activation  functions  Uj  and  synaptic  connection  weights  Wij.  Simu(cid:173)\nlation  and  implementation of idealized  models  of neural  networks  such  as  (1)  on \ncentralized  computers  not  only  limit  the  size  of networks,  but  more  importantly \npreclude  exploiting  the  inherent  massive  parallelism in  network  computations.  A \ntruly  faithful  analog  implementation or  simulation of neural  networks  defined  by \n(1)  over  a  distributed  network  requires  that  neurons  follow  a  global clock  t,  com(cid:173)\nmunicate  timed  states  Xj(t)  to  all  others  instantaneously  and  synchronize  global \ndynamics precisely  all the  time (e.g., the same Xj(t)  should be used  in evolution of \nall  Xi(t)  at time t).  Clearly, hardware and software realities make it very  hard  and \nsometimes impossible to fulfill  these  requirements;  any  mechanism used  to  enforce \nsuch  synchronization  may  have  an  important  effect  on  performance  of  the  net(cid:173)\nwork.  Moreover,  absolutely  insisting  on  synchronization  contradicts  the  biological \nmanifestation of inherent asynchrony caused by delays in nerve signal propagation, \nvariability  of neuron  parameters  such  as  refractory  periods  and  adaptive  neuron \ngains.  On the other hand, introduction of asynchrony  may change network dynam(cid:173)\nics, for  example, from convergent to oscillatory.  Therefore, validity of asynchronous \ndynamics of neural networks must be assessed in order to ensure desirable dynamics \nin  a  distributed  environment. \n\nMotivated by the above issues,  we  study  asynchronous  dynamics of general  CT dy(cid:173)\nnamical systems  with  neural  networks  in  particular.  Asynchronous  dynamics has \nbeen  thoroughly studied  in  the  context of iterative maps or discrete-time  (DT) dy(cid:173)\nnamical systems;  see,  e.g.,  (Bertsekas  and Tsitsiklis,  1989)  and  references  therein. \nAmong other results are that P-contractive maps on Rn  (Baudet,  1978) and contin(cid:173)\nuous maps on partially ordered sets  (Wang and Parker,  1992)  are asynchronizable, \ni.e.,  any  asynchronous  iterations  of these  maps  will  converge  to  the  fixed  points \nunder synchronous (or parallel) iterations.  The synchronization issue  has also been \naddressed  in  the  context  of neural  networks.  In  fact,  the  celebrated  DT  Hopfield \nmodel  (Hopfield,  1982)  adopts  a  special  kind  of asynchronous  dynamics:  only one \nrandomly chosen  neuron  is  allowed  to update  its state  at each  iterative step.  The \nissue  is  also  studied  in  (Barhen  and  Gulati,  1989)  for  CT  neural  networks.  The \napproach  there  is,  however,  to  convert  the  additive  model  (1)  into  a  DT  version \nthrough  the  Euler  discretization and then  to apply the  existing  result  for  contrac(cid:173)\ntive  mappings in  (Baudet,  1978)  to  ensure  the  discretized  system  to be  asynchro(cid:173)\nnizable.  Overall, studies for  asynchronous  dynamics of CT dynamical systems  are \nstill lacking;  there  are  even  no  reasonable  definitions for  what it means,  at least  to \nour knowledge. \n\nIn  this  paper,  we  continue  our  studies  on  relationships  between  CT  and  DT  dy(cid:173)\nnamical systems and neural networks (Wang and Blum, 1992;  Wang, Blum and Li, \n1993)  and  concentrate on  their asynchronous  dynamics.  We  first  extend  a  concept \nof asynchronous dynamics of DT systems to CT systems, by identifying the distinc(cid:173)\ntion between synchronous and asynchronous dynamics as (i)  presence or absence  of \na  common global clock  used  to synchronize  the  dynamics of the  different  neurons \nand  (ii)  exclusion  or  inclusion of delay  times  in  communication between  neurons, \nand present some preliminary results for  asynchronous dynamics of contractive and \nmonotone CT systems. \n\n\fAsynchronous Dynamics of Continuous Time Neural Networks \n\n495 \n\n2  MATHEMATICAL  FORMULATION \n\nTo  be  general,  we  consider  a  CT  dynamical system  defined  by  an  n-dimensional \nsystem of ordinary differential equations, \n\n(2) \nwhere Ii : Rn  --+  R are continuously differentiable and x(t)  E Rn  for  all t  in R+  (the \nset  of all  nonnegative  real  numbers).  In  contrast  to  the  asynchronous  dynamics \ngiven  below,  dynamics of this system will  be called  synchronous.  An  asynchronous \n:  R+  --+  R+  and  rj  :  R+  --+  R+, \nscheme  consists  of two  families  of functions  Ci \ni, j  = 1, ... , n, satisfying the following constraints:  for  any t  > 0, \n(i)  Initiation:  Ci(t)  ~ 0 and rJ(t)  ~ 0; \n(ii)  Non-starvation:  Ci'S  are differentiable  and l\\(t) > 0; \n(iii)  Liveness:  limt_oo Ci(t)  =  00  and limt_oo rJ(t)  = 00; \n\n(iv)  Accessibility:  rj(t)  ~ Cj(t). \n\nGiven  an  asynchronous scheme ({cd, {rJ}),  the associated  asynchronous  dynamics \nof the system (2)  is  the solution of the following parametrized system: \n\n(3) \n\nWe shall  call this system an  asynchronized system of the original one  (2). \nThe functions Ci(t)  should be viewed as respective  \"local\"  times (or clocks)  of com(cid:173)\nponents  i,  as  compared  to  the  \"global\"  time  (or  clock)  t.  As  each  component  i \nevolves  its state according to its local  time Ci(t),  no shared  global time t  is  needed \nexplicitly; t only occurs implicitly. The functions rj(t) should be considered as time \ninstants at which  corresponding values  Xi  of components j  are  used  by component \ni;  hence  the  differences  (ci(t)  - rj(t\u00bb  ~ 0  can  be  interprated  as  delay  times  in \ncommunication between  the  components  j  and  i.  Constraint  (i)  reflects  the  fact \nthat we  are interested  in  the system dynamics after some global time instance, say \n0;  constraint (ii) states that the functions Ci  are monotone increasing and hence the \nlocal  times evolve  only forward;  constraint  (iii)  characterizes  the  live ness  property \nof the components and communication channels between  components; and, finally, \nconstraint  (iv)  precludes  the  possibility that  component  i  accesses  states  x j  ahead \nof the  local times Cj(t)  of components j  which  have  not  yet  been  generated. \n\nNotice that, under the assumption on monotonicity of Ci(t),  the inverses C;l(t)  exist \nand the asynchronized  system (3)  can  be transformed into \n\n(4) \nby letting Yi(t)  = Xi( Ci(t\u00bb  and y} (t)  =  Xj (rJ(t\u00bb =  Yj (c;l (rJ(t\u00bb  for  i, j  =  1,2, ... , n. \nThe vector  form of (4)  can  be given by \n\niJ  = Cf F[Y] \n\n(5) \n\n\f496 \n\nWang, Li,  and Blum \n\nwhere yet)  = [Yl (t), \"\"  Yn(t)]T,  C' = diag(dcl (t)/dt, \"\"  dcn(t)/dt) , F = [/1, \"\"  fn]T, \ny = [Y;]  and \n\n/1 cYi(t) , yHt), \"\"  y~(t)) 1 \n' \n\nhcYr(t), y~(t), \"\"  y~(t)) \n\n, \n\n_ \n\n[ \nF[Y] = \n\nfn (i/'l (t), y~(t), \"\"  y~(t)) \n\nNotice  that  the  complication in  the  way  F  applies  to Y ~imply means ,that  every \ncomponent  i  will  use  possibly  different  \"global\"  states  [Yi(t) , y2(t) , \"\"  y~(t)] ,  This \npeculiarity makes the equation (5) fit into none ofthe categories of general functional \ndifferential  equations  (Hale,  1977),  However,  if  rJ(t)  for  i  = 1, \"., n  are  equal, \nall  the  components  will  use  a  same  global  state  y  = [yHt) , y~(t), .. \" y~(t)]  and \nthe  asynchronized  system  (5)  assumes  a  form  of  retarded  functional  differential \nequations, \n\niJ  = c' FcY), \n\n(6) \nWe shall call this case  uniformly-delayed,  which  will be a  main consideration in the \nnext section  where  we  discuss  asynchronizable systems, \nThe system (5)  includes some special cases.  In a no communication delay situation, \nrj(t) = Cj(t)  for  all i  and  the system  (5)  reduces  to  iJ  = C' F(y),  This includes the \nsimplest  case  where  the local  times Ci(t)  are  taken  as  constant-time scalings  cit  of \nthe global time t;  specially,  when  all Ci(t)  = t  the system goes  back to the original \none  (2),  If,  on  the other hand,  all the  local  time~ are  identi~al to the global time t \nand the communication times take the form of rJ(t) = t - OJ(t)  one obtains a  most \ngeneral delayed system \n\n(7) \n\nwhere  the state  Yj(t)  of component  j  may have  different  delay  times O)(t)  for  dif(cid:173)\nferent  other components i. \n\nFinally, we should point out that the above definitions of asynchronous schemes and \ndynamics are  analogues of their counterparts for  DT dynamical systems (Bertsekas \nand  Tsitsiklis,  1989;  Blum,  1990),  Usually,  an  asynchronous  scheme  for  a  DT \nsystem defined  by a  map f  : X  -+ X, where  X = Xl  X  X2  X  '\"  X  X n ,  consists of a \nfamily {Ti  ~ N I i  = 1, , .. , n} of subset~ of discrete  times (N)  at which  components \ni  update  their  states  and  a  family  {rJ  : N -+  N I i  = 1,2\"\", n} of communication \ntimes,  Asynchronous  dynamics (or  chaotic iteration, relaxation)  is  then given  by \n\nX.(t + 1) = {  fi(xl(rt(t)), \"', xn(r~(t)))  if  t  E ~ \notherwise. \n\nXi(t) \n\nI \n\nNotice  that the sets Ti  can  be interpreted  as  local  times of components i .  In fact, \none can define  local time functions Ci  : N -+ N  as Ci(O)  = 0 and Ci(t + 1) = Ci(t) + 1 \nif t  E 11  and Ci(t)  otherwise.  The asynchronous dynamics can then be defined  by \n\nXi(t + 1) - Xi(t)  = (Ci(t + 1) - ci(t))(fi(xl(rf(t)), ... ,Xn(r~(t))) - Xi(t)), \n\nwhich  is  analogous to the definition given in (4). \n\n\fAsynchronous Dynamics of Continuous Time Neural Networks \n\n497 \n\n3  ASYNCHRONIZABLE  SYSTEMS \n\nIn general, we consider a CT dynamical system as  asynchronizable ifits synchronous \ndynamics  (limit sets  and  their  asymptotic stability)  persists  for  some set  of asyn(cid:173)\nchronous schemes.  In many cases,  asynchronous dynamics of an arbitrary CT sys(cid:173)\ntem  will  be  different  from  its synchronous  dynamics,  especially  when  delay  times \nin  communication are present.  An example can  be  given for  the  network  (1)  with \nsymmetric matrix W.  It is  well-known  that  (synchronous)  dynamics of such  net(cid:173)\nworks  is  quasi-convergent,  namely,  all  trajectories  approach  a  set  of fixed  points \n(Hirsch,  1989).  But  when  delay  times are  taken  into  consideration,  the  networks \nmay have sustained oscillation when  the delays exceed some threshold (Marcus and \nWestervelt,  1989).  A more careful  analysis on oscillation induced by  delays is given \nin  (Wu,  1993) for  the networks  with symmetric circulant  weight  matrices. \n\nHere,  we  focus on asynchronizable systems.  We  consider  CT dynamical systems on \nRn  of the following general form \n\nAx(t) =  -x(t) + F(x(t\u00bb \n\n(8) \nwhere  x(t)  ERn,  A  = diag(a1,a2, ... ,an) with  aj  > 0  and  F  = [Ji]  E G1(Rn).  It \nis  easy  to see  that  a  point  x  E Rn  is  a  fixed  point of (8)  if and  only  if x  is  a  fixed \npoint of the  map F.  Without  loss of generality,  we  assume that  0 is  a  fixed  point \nof the map F.  According  to  (5),  the  asynchronized  version  of (8)  for  an  arbitrary \nasynchronous scheme  ({ cd, { rj}) is \n\nAy = G'( -y + F[Y]), \n\n(9) \n\nwhere  jj = (jjtct), jj~(t), ... , y~(t)]. \n\n3.1  Contractive Systems \n\nOur  first  effort  attempts  to  obtain  a  result  similar  to  the  one  for  P-contractive \nmaps in  (Baudet,  1978).  We  call  the system (8)  strongly  P-contractive  if there  is  a \nsymmetric and  invertible  matrix S  such  that  IS- 1 F(Sx)1  < Ixl  for  all  x  E Rn  and \nIS- 1 F(Sx)1 = Ixl  only for  x = 0;  here  Ixl  denotes  the vector  with components Ixil \nand < is  component-wise. \n\nTheorem 1  If the  system  (8)  is  strongly  P-contractive,  then  it  is  asynchronizable \nfor  any  asynchronous  schemes  without  self time  delays  (i. e.,  rf (t)  = Ci(t)  for  all \ni=1,2, ... ,n). \n\nProof.  It is  not hard to see  that synchronous dynamics of a  strongly P-contractive \nsystem is globally convergent to the fixed  point O.  Now,  consider the transformation \nz = A- 1 y  and  the system for  z \n\nAi = G'( -z + S-1 F[SZ]) = G'( -z + G[Z]), \n\nwhere  G[Z]  = S-1 FS[Z].  This  system  has  the  same  type  of  dynamics  as  (9). \nDefine  a  function  E  : R+  x  Rn  --+  R+  by  E(t)  =  z T (t)Az(t)j2,  whose  derivative \nwith respect  to t  is \nE =  z T G' (-z + G(Z\u00bb < IIG'II (-z T z + IzlT IG(Z)!) < IIG'II( -z T z + IzlT Izl) ::;  O. \n\n\f498 \n\nWang, Li, and Blum \n\nHence  E  is  an  energy  function  and  the  asynchronous  dynamics  converges  to  the \nfixed  point O. \n0 \nOur  second  result  is  for  asynchronous  dynamics  of  contractive  systems  with  no \ncommunication delay.  The system (8)  is called  contractive if there is a  real constant \no ~ a  < 1 such that \n\nIIF(x) - F(y)1I  ~ allz - yll \n\nfor  all x, y  E Rn;  here  II  . II  denotes  the usual  Euclidean  norm on Rn. \n\nTheorem 2  If the  system  (8)  is  contractive,  then  it  is  asynchronizable  for  asyn(cid:173)\nchronous  schemes  with  no  communication  delay. \n\nProof.  The  synchronous  dynamics of contractive  systems is  known  to  be  globally \nconvergent  to a  unique fixed  point (Kelly,  1990).  For an  asynchronous scheme with \nno  communication delay,  the system  (8)  is  simplified to Ali = G'( -y + F(y\u00bb.  We \nconsider  again  the  function  E  = y T Ay/2,  which  is  an  energy  function  as  shown \nbelow. \n\nE = Y T G' (-y + F(y\u00bb  ~  IIG/II( -lIyll2 + lIyIlIlF(y)ID  <  O. \n\nTherefore,  the asynchronous dynamics converges  to the fixed  point O. \nFor  the  additive-type neural networks  (1),  we  have \n\no \n\nCorollary 1  Let the  network (1)  have  neuron  activation functions  Ui  of sigmoidal \ntype  with 0 < uHz)  ~ SUPzER ui(z) = 1.  If it satisfies  the  condition \n\n(10) \n\nwhere  M  =  diag(J-ll, ... , J-ln), \nschemes  with  no  communication  delay. \nProof.  The  condition  (10)  ensures  the  map  F(x)  = A-I Wu(M x) + A- 1 I  to  be \ncontractive. \n0 \n\nthen  it  is  asynchronizable  for  any  asynchronous \n\nNotice that the condition (10) is equivalent to many existing ones on globally asymp(cid:173)\ntotical stability based on various norms of matrix W, especially the contraction con(cid:173)\ndition given  in  (Kelly,  1990)  and some very  recent  ones  in (Matsuoka,  1992).  The \ncondition  (10)  is  also  related  very  closely  to the  condition  in  (Barhen  and  Gulati, \n1989)  for  asynchronous  dynamics of a  discretized  version  of (1)  and  the  condition \nin (Marcus and Westervelt,  1989) for  the networks  with delay. \n\nWe should emphasize that the results in Theorem 2 and Corollary 1 do not directly \nfollow  from  the result  in  (Kelly,  1990);  this is  because  local  times Ci(t)  are  allowed \nto be much more general functions  than linear ones  Ci  t. \n\n3.2  Monotone Systems \n\nA binary relation  ~ on Rn  is called a  partial order if it satisfies that, for  all x, y, z  E \nRn,  (i)  x  ~ x;  (ii)  x  ~ y  and  y  ~ x  imply  x  =  y;  and  (iii)  x  -<  y  and  y  -<  z \nimply  x  -<  z.  For  a  partial  order  ~ on  Rn,  define  ~ on  Rn  by  x  ~ y  iff  x  <  y \nand  Xi  # Yi  for  all  i  = 1, .. \"  n.  A  map  F  : Rn  -I- Rn  is  monotone  if x  ~ y  implies \n\n\fAsynchronous Dynamics of Continuous Time Neural Networks \n\n499 \n\nF(x) -<  F(y).  A CT dynamical system of the form (2) is  monotone if Xl  ~ X2  implies \nthe  trajectories  Xl(t), X2(t)  with  Xl(O)  =  Xl  and  X2(0)  =  X2  satisfy  Xl(t)  ::5  X2(t) \nfor  all t  ~ 0  (Hirsch,  1988). \n\nTheorem 3  If the  map F  in  (8)  is  monotone,  then  the  system  (8)  is  asynchroniz(cid:173)\nable  for uniformly-delayed asynchronous  schemes,  provided that  all orbits x(t)  have \ncompact  orbit  closure  and there  is  a to  > 0  with  x(to) ~ x(O)  or x(to) ~ x(O). \n\nProof.  This  is  an  application  of  a  Henry's  theorem  (see  Hirsch,  1988)  that  im(cid:173)\nplies  that  the  asynchronized  system  (9)  in  the  no  communication delay  situation \nis  monotone  and  Hirsch's  theorem  (Hirsch,  1988)  that guarantees  the  asymptotic \nconvergence  of monotone systems to fixed  points. \n0 \n\nCorollary 2  If the  additive-type neural network (1) with sigmoidal activation func(cid:173)\ntions  is  cooperative  (i.e.,  Wij  >  0  for  i  #  j  (Hirsch,  1988  and  1989)),  then  it  is \nasynchronizable for uniformly-delayed asynchronous  schemes,  provided that there  is \na to  > 0  with x(to) ~ x(O)  or x(to) ~ x(O). \n\nProof.  According  to  (Hirsch,  1988),  cooperative  systems  are  monotone.  As  the \nnetwork  has only bounded  dynamics, the result follows from the above theorem.  0 \n\n4  CONCLUSION \n\nBy  incorporating  the  concepts  of local  times  and  communication  times,  we  have \nprovided a mathematical formulation of asynchronous dynamics of continuous-time \ndynamical systems.  Asynchronized  systems in  the most general form  haven't been \nstudied  in  theories of dynamical systems and functional differential equations.  For \ncontractive  and  monotone  systems,  we  have  shown  that  for  some  asynchronous \nschemes, the systems are  asynchronizable, namely, their asynchronizations preserve \nconvergent  dynamics of the original (synchronous)  systems.  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To \nappear in  the  AMS proceedings  of Symposia  in  Applied Mathematics,  Mathematics \nof Computation  1943 - 1993,  Vancouver,  BC,  August,  1993,  edited by  W.  Gautschi. \n\nX.  Wang  and  E.  K.  Blum  (1992).  Discrete-time  versus  continuous-time  neural \nnetworks.  Computer and System Sciences,  49:1-17. \n\nX.  Wang  and  D.  S.  Parker  (1992).  Computing least  fixed  points by  asynchronous \niterations and random iterations.  Technical Report  CSD-920025,  Computer Science \nDepartment,  UCLA. \nJ .-H. Wu (1993).  Delay-Induced Discrete Waves of Large Amplitudes in Neural Net(cid:173)\nworks  with  Circulant  Connection  Matrices.  Preprint,  Department of Mathematics \nand  Statistics,  York  University. \n\n\f", "award": [], "sourceid": 804, "authors": [{"given_name": "Xin", "family_name": "Wang", "institution": null}, {"given_name": "Qingnan", "family_name": "Li", "institution": null}, {"given_name": "Edward", "family_name": "Blum", "institution": null}]}