{"title": "Noisy Spiking Neurons with Temporal Coding have more Computational Power than Sigmoidal Neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 211, "page_last": 217, "abstract": null, "full_text": "Noisy Spiking Neurons with Temporal \n\nCoding have more  Computational Power \n\nthan  Sigmoidal Neurons \n\nWolfgang Maass \n\nInstitute for  Theoretical  Computer Science \n\nTechnische Universitaet  Graz,  Klosterwiesgasse 32/2 \nA-80lO  Graz,  Austria, e-mail:  maass@igLtu-graz.ac.at \n\nAbstract \n\nWe exhibit a novel  way of simulating sigmoidal  neural  nets by  net(cid:173)\nworks  of  noisy  spiking  neurons  in  temporal  coding.  Furthermore \nit  is  shown  that  networks  of noisy  spiking  neurons  with  temporal \ncoding  have  a  strictly  larger  computational  power  than  sigmoidal \nneural  nets with  the same number of units. \n\n1 \n\nIntroduction and Definitions \n\nWe  consider  a  formal  model  SNN  for  a  \u00a7piking neuron  network  that  is  basically \na  reformulation  of  the  spike  response  model  (and  of  the  leaky  integrate  and  fire \nmodel)  without  using  6-functions  (see  [Maass,  1996a]  or  [Maass,  1996b]  for  further \nbackgrou nd). \n\nAn  SNN consists of a finite set V  of spiking neurons, a set  E  ~ V  x V  of synapses, a \nweight wu,v  2:  0 and  a  response function cu,v  :  R+  --+  R  for  each synapse {u, v}  E  E \n(where R+  := {x E R: x  2:  O})  , and  a  threshold function  8 v  :  R+  --+  R+  for  each \nneuron  v  E  V  . \n\nIf Fu  ~ R+ is  the set of firing  times of a neuron u  , then  the  potential at the trigger \nzone of neuron  v  at time t is  given  by \n\nPv(t)  := L \n\nL \n\nu:{u,v)EE \n\nsEF,,:s<t \n\nWu,v  . cu,v(t - s)  . \n\nIn  a  noise-free  model  a neuron  v  fires  at time t  as soon  as  Pv(t)  reaches  8 v(t - t'), \nwhere  t'  is  the  time  of the  most  recent  firing  of v  .  One  says  then  that  neuron  v \nsends out an  \"action  potential\"  or  \"spike\"  at time t  . \n\n\f212 \n\nW  Maass \n\nFor some specified subset Vin  ~ V  of input neurons one assumes that the firing times \n(\"spike trains\") Fu  for neurons u  E  Vin  are not defined  by the preceding convention, \nbut are given  from  the outside.  The firing  times  Fv  for  all  other neurons v  E  V  are \ndetermined  by the previously described  rule,  and the output of the network is given \nin  the form  of the spike  trains  Fv  for  a specified  set  of output neurons Vout  ~ V  . \n\ny \n\no \n\ns \n\ny \n\n\u00a311,\" (I-S) \n\nI \n\n0 \n\ns \n\nI \n\n__ ~e~ \n\nI' \n\n~ \nt \n\nFigure  1:  Typical shapes  of response functions cu,v  (EPSP and IPSP)  and threshold \nfunctions  8 v  for  biological  neurons. \n\nWe  will  assume  in  our  subsequent  constructions  that  all  response  functions  cu,v \nand  threshold  functions  8 v  in  an  SNN  are  \"stereotyped\",  i.e. \nthat  the  response \nfunctions  differ  apart  from  their  \"sign\"  (EPSP  or  IPSP)  only  in  their  delay  du,v \n(where  du,v  :=  inf {t  ~ 0  :  cu,v(t)  =J  O})  ,  and  that  the  threshold  functions  8 v \nonly  differ  by  an  additive  constant  (i.e.  for  all  u  and  v  there exists  a  constant  cu,v \nsuch  that  8 u(t)  =  8 v(t)  + Cu,v  for  all  t  ~ 0)  .  We  refer  to  a  term  of  the  form \nwu,v  . cu,v(t  - s)  as  an  ~xcitatory respectively  inhibitory  post~naptic potential \n(abbreviated:  EPSP respectively IPSP). \n\n- -\n\nSince  biological  neurons  do  not always fire  in  a  reliable  manner one also  considers \nthe related  model of noisy spiking neurons, where  Pv(t)  is  replaced  by  P  ~Oi8Y(t) := \nPv(t) +av(t) and  8 v(t-t') is  replaced  by  8~oi8Y(t_t') : = 8 v(t-t')+!3v(t-t'). \nav(t)  and  !3v(t  - t')  are  allowed  to  be  arbitrary functions  with  bounded  absolute \nvalue  (hence they can  also represent  \"systematic noise\"). \nFurthermore one allows  that the current value of the difference  D(t)  := p vnOi8Y(t) -\nt')  does  not  determine  directly  the firing  time of neuron  v,  but only  its \n8~oi8Y(t -\ncurrent  firing  probability.  We  assume  that  the  firing  probability  approaches  1  if \nD  --+  00  ,  and  0 if D  --+  -00  .  We  refer  to spiking neurons with  these  two  types  of \nnoise  as  \"noisy spiking neurons\". \n\nWe  will  explore in  this article  the  power of analog computations with  noisy spiking \nneurons,  and  we  refer  to  [Maass,  1996a]  for  results  about  digital  computations  in \nthis  model.  Details  to  the  results  in  this  article  appear  in  [Maass,  1996b]  and \n[Maass,  1997]. \n\n2  Fast  Simulation of Sigmoidal Neural  Nets with Noisy \n\nSpiking Neurons in Temporal Coding \n\nSo far  one has only considered  simulations of sigmoidal  neural  nets  by  spiking neu(cid:173)\nrons  where each  analog  variahle  in  the  sigmOidal  neural  net  is  represented  by  the \nfiring  rate  of a  spiking  neuron.  However  this  \"firing  rate interpretation\"  is  incon(cid:173)\nsistent with  a  number of empirical  results about  computations in  biological  neural \n\n\fSpiking Neurons have more Computational Power than Sigmoidal Neurons \n\n213 \n\nsystems.  For example [Thorpe &  Imbert,  1989]  have demonstrated that visual  pat(cid:173)\ntern analysis and  pattern classification can be carried out by  humans in  just 150 ms, \nin  spite of the fact  that it involvefi a  minimum of 10 synaptic stages from  the retina \nto  the  temporal  lobe.  [de  Ruyter van  Steveninck &  Bialek,  1988]  have  found  that \na  blowfly  can  produce flight  torques within  30  ms  of a  visual  stimulus  by  a  neural \nsystem  with  several  synaptic stages.  However  the firing  rates  of neurons  involved \nin  all  these  computations are  usually  below  100  Hz,  and  interspike  intervals  tend \nto  be  quite  irregular.  Hence  One  cannot  interpret  these  analog computations with \nspiking  neurons on  the  basis of an  encoding of analog variables  by  firing  rates. \n\nOn  the other hand experimental evidence has accumulated during the last few  years \nwhich  indicates that  many biological neural systems use the timing of action poten(cid:173)\ntials to encode information  (see e.g.  [Bialek  &  Rieke,  1992]'  [Bair &  Koch,  1996]). \n\nWe  will  now  describe a  new  way of simulating sigmoidal neural nets by  networks of \nspiking neurons that is  based on  temporal coding.  The key  mechanism for  this alter(cid:173)\nnative simulation  is  based on  the well  known fact  that EPSP's and  IPSP's are able \nto  shift the firing  time  of a  spiking neuron.  This mechanism  can  be demonstrated \nvery clearly in  our formal  model  if one assumes  that EPSP's rise  (and  IPSP's fall) \nlinearly during a certain initial  time period.  Hence we  assume in  the following  that \nthere exists some constant D.  > 0 such  that each  response function  eu,v(X)  is  of the \nform  Ctu,v  . (x - du,v)  with  Ctu,v  E {-I, 1} for  x  E [du,v,  du,v  + D.]  , and  eu,v(X)  =  0 \nfor  x  E  [0, du,v]  . \n\nConsider a spiking neuron v that receives postsynaptic potentials from n  presynaptic \nneurons at, ... ,an'  For  simplicity we  asfiume  that interspike intervals are so large \nthat  the  firing  time  tv  of neuron  v  depends just on  a  single firing  time  ta\u2022  of each \nneuron  ai,  and  8 v  has  returned  to  its  \"resting value\"  8 v(0)  before  v  fires  again. \nThen  if  the  next  firing  of  v  occurs  at  a  time  when  the  postsynaptic  potentials \ndescribed  by Wa;,V  . ea.,v(t - ta.)  are all  in  their initial  linear phase, its firing  time \ntv  is  determined  in  the  noise-free  model  for  Wi  :=  Wa;,v  . Cta.,v  by  the  equation \nE~=t Wi  . (tv  - tao  - da.,v)  =  8 v(0)  , or  equivalently \n\n(1) \n\nThis  equation  revealfi  the  fiomewhat  fiurprifiing  fact  that  (for  a  certain  range  of \ntheir  parameters)  spiking  neurons  can  compute  a  weighted  sum in  terms  of firing \ntimes,  i.e.  temporal  coding.  One  fihould  alfio  note  that in  the  case  where all  delays \nda.,v  have  the  same  value,  the  \"weights\"  Wi  of this  weighted  sum  are encoded  in \nthe  \"strengths\"  wa.,v  of the synapsefi  and  their  \"sign\"  Cta.,t,  , as in  the  \"firing  rate \ninterpretation\".  Finally  according  to  (1)  the  coefficients  of the  presynaptic  firing \ntimes  tao  are automatically normalized,  which  appears to  be of biological  interest. \n\nIn  the  simplest  scheme  for  temporal  coding  (which  is  closely  related  to  that  in \n[Hopfield,  1995])  an  analog variable x  E  [0,1]  is encoded  by the firing  time T  -,' x \nof a  neuron, where T  is  assumed  to  be  independent  of x  (in  a  biological  context T \nmight  be  time-locked  to  the  onset  of a  fitimulus,  or  to  some  oscillation)  and  ,  is \nsome constant that ifi  determined in  the proof of Theorem 2.1  (e.g.  ,  =  tlj2 in  the \nnoifie-free case).  In  contrafit to [Hopfield,  1995]  we afifiume  that both the inputs and \nthe  outputs  of computationfi  are encoded  in  thifi  fafihion.  This  has  the  advantage \nthat one can  compose computational  modules. \n\n\f214 \n\nW.  Maass \n\nWe  will  first  focus  in  Theorem 2.1  on  the simulation  of sigmoidal  neural  nets  that \nemploy  the  piecewise  linear  \"linear  saturated\"  activation  function  1r  :  R  -\n[0,1] \ndefined  by  1r(Y)  = 0  ify  <  0,  1r(Y)  = y  if 0  ~ y  ~ 1,  and  1r(Y)  = 1  ify  >  1  . \nThe Theorem  3.1  in  the  next  section  will  imply  that  one  can  simulate  with  spik(cid:173)\ning neurons also sigmoidal  neural  nets that employ  arbitrary continuous activation \nfunctions.  Apart  from  the  previously  mentioned  assumptions  we  will  assume  for \nthe proofs of Theorem 2.1  and  3.1  that any EPSP satisfies eu,v(X)  = 0 for  all  suffi(cid:173)\nciently  large x,  and  eu,v(X)  ~ eu,v(du,v  + ~) for  all  x  E  [du,v  + Ll, du,v  + Ll  + 1']  . \nWe  assume that each  IPSP is  continuous, and  has value 0 except for  some interval \nof R.  Furthermore  we  assume  for  each  EPSP  and  IPSP  that  jeu,v(x)1  grows  at \nmost  linearly during the interval  [du,v  + Ll, du,v  + Ll + 1']  .  In  addition  we  assume \nthat  0 v (x)  =  0 v(0)  for  sufficiently large x  , and  that  0 v(x)  is sufficiently  large for \nO<x~'Y. \n\nTheorem 2.1  For  any given  e, 8> 0  one  can  simulate  any  given  feed forward  sig(cid:173)\nmoidal neural net N  with activation function 1r  by  a network NN,e,6  of noisy spiking \nneurons  in  temporal  coding.  More  precisely,  for  any  network  input  Xl, . .\u2022 ,Xm  E \n[O,IJ  the  output  of  NN,e,6  differs  with  probability  2::  1 - 0  by  at  most e  from  that \nof N  .  Furthermore  the  computation  time  of NN,e,6  depends  neither on the  number \nof gates  in  N  nor  on  the  parameters  e, 0,  but  only  on  the  number  of layers  of the \nsigmoidal neural network N  . \n\n- -\n\nWe  refer  to  [Maass,  1997]  for  details  of  the  somewhat  complicated  proof.  One \nemploys  the  mechanism  described  by  (1)  to simulate  through  the  firing  time of a \nspiking  neuron  v  a  sigmoidal  gate with  activation  function  1r  for  those gate-inputs \nwhere 1r  operates in  its linearly rising  range.  With  the help  of an  auxiliary spiking \nneuron  that  fires  at  time  T  one  can  avoid  the  automatic  \"normalization\"  of the \nweights  Wi  that  is  provided  by  (1),  and  thereby  compute  a  weighted  sum  with \narbitrary  given  weights.  In  order  to simulate in  temporal  coding  the  behaviour of \nthe  gate  in  the  input  range  where  1r  is  \"saturated\"  (Le.  constant),  it  suffices  to \nemploy  some  auxiliary  spiking  neurons  which  make  sure  that  v  fires  exactly  once \nduring the relevant  time window (and  not  shortly before that). \n\nSince inputs and  outputs of the resulting modules for  each  single gate of N  are all \ngiven  in  temporal  coding,  one  can  compose  these  modules  to  simulate  the  multi(cid:173)\nlayer  sigmoidal  neural  net  N.  With  a  bit  of additional  work  one  can  ensure  that \nthis  construction also works with  noisy spiking neurons. \n\u2022 \n\n3  Universal  Approximation Property of Networks of Noisy \n\nSpiking  Neurons with Temporal Coding \n\nIt is  known  [Leshno et al.,  1993J  that feed forward sigmoidal neural nets whose gates \nemploy  the  activation  function  1r  can  approximate  with  a  single  hidden  layer  for \nany n, kEN any given  continuous  function  F  : [O,I]n  -\n[O,I]k  within  any  e  >  0 \nwith  regard  to  the Loo-norm  (i.e.  uniform  convE'rgence).  Hence we  can  derive  the \nfollowing  result  from  Theorem 2.1 : \n\nTheorem 3.1  Any given  continuous function  F  :  [0, l]n  _  [O,I]k  can  be  approxi(cid:173)\nmated within  any given  e  > 0  with  arbitrarily  high  reliability  in  temporal  coding  by \n\n\fSpiking Neurons have more Computational Power than Sigmoidal Neurons \n\n215 \n\na  network  of noisy  spiking  neurons  (SNN)  with  a  single  hidden  layer  (and  hence \nwithin  15 ms for  biologically  realistic  values  of their time-constants). \n\u2022 \n\nBecause of its generality this Theorem implies the same result also for  more general \nschemes  of coding  analog  variables  by  the  firing  times  of neurons,  besides  the  par(cid:173)\nticular  one  that  we  have  considered  so  far.  In  fact  it  implies  that  the  same result \nholds  for  any  other  coding  scheme  C  that  is  \"continuously  related\"  to  the  previ(cid:173)\nously considered one in  the sense that the transformation  between firing  times  that \nencode an  analog variable x  in  the here considered  coding scheme and in  the coding \nscheme C  can  be described  by  uniformly continuous functions  in  both  directions. \n\n4  Spiking Neurons  have  more  Computational Power than \n\nSigmoidal Neurons \n\nWe  consider  the  \"element  distinctness  function\"  EDn \nby \n\nEDn(Sl, ... ,Sn)  = \n\n{\n\nI, \n\n0, \n\narbitrary, \n\nif s i  =  s i  for  some i  =I=- j \nif lSi  -ail ~ 1 for  all  i,j with  i  =l=-j \nelse. \n\nIf one encodes  the value of input  variable  Si  by  a firing  of input  neuron  ai  at time \nTin  - c\u00b7 Si  ,  then  for  sufficiently  large  values  of the constant  C  > 0  a  single  noisy \nspiking  neuron  v  can  compute  EDn  with  arbitrarily high  reliability.  This holds for \nany reasonable type ofresponse functions, e.g.  the ones shown in  Fig.  1.  The binary \noutput of this computation is  assumed  to be encoded  by  the firing/non-firing of v  . \nHair-trigger situations are avoided  since no  assumptions have to be made about the \nfiring or  non-firing of v  if EPSP's arrive with  a temporal distance  between 0 and  c . \n\nOn  the  other  hand  the  following  result  shows  that  a  fairly  large  sigmoidal neural \nnet is  needed  to compute the same function.  Its proof provides the first  application \nfor  Sontag's recent  results about a  new  type of \"dimension\"  d of a  neural  network \nN  , where d is  chosen  maximal  so that  every subset of d inputs is  shattered  by  N  . \nFurthermore it expands a method due to [Koiran,  1995] for llsing the VC-dimension \nto prove lower  bounds on  network size. \n\nTheorem 4.1  Any sigmoidal neural net N  that computes EDn has at least  n2\"4  -1 \nhidden  units. \n\nProof:  Let  N  be  an  arbitrary sigmoidal  neural  net  with  k  gates  that  computes \nEDn.  Consider  any set  S  ~ R+ of size n -1.  Let  .x  > 0 be sufficiently large so that \nthe numbers in  .x  . S  have pairwise distance  ~ 2  .  Let  A  be a  set  of n  - 1 numbers \n> max (.x  . S) + 2 with  pairwise distance  ~ 2  . \nBy  assumption N  can  decide  for  n  arbitrary inputs  from  .x  . SuA whether  they \nare  all  different.  Let N>.  be  a  variation  of N  where  all  weights on  edges  from  the \nfirst  input variable are mUltiplied  with .x.  Then N>.  can  compute any function  from \n\n\f216 \n\nW Maass \n\nS  into  {O,  I}  after  one  has  assigned  a  suitable fixed  set  of n  - 1 pairwise  different \nnumbers from  ..\\  . SuA to  the last n  - 1 input  variables. \nThus if one considers a'l  programmable parameters of N  the factor  ..\\  in  the weights \non  edges  from  the  first  input  variable  and  the  ~ k  thresholds  of gates  that  are \nconnected  to some of the other n - 1 input variables,  then N  shatters S with  these \nk + 1 programmable parameters. \nSince S  ~ R + of size n - 1 was chosen  arbitrarily,  we  can now apply the result from \n[Sontag,  1996],  which  yields  an  upper  bound  of 2w + 1 for  the  maximal  number  d \nsuch  that  every set of d different  inputs  can  be shattered  by  a  sigmoidal  neural  net \nwith  w  programmable  parameters  (note  that  this  parameter  d  is  in  general  much \nsmaller  than  the  VC-dimension  of the  neural  net).  For  w  := k  + 1 this  implies  in \nour  case  that  n  - 1  ~ 2(k + 1) + 1  ,  hence  k  ~ (n - 4)/2  .  Thus N  has  at  least \n(n - 4) /2 computation nodes, and  therefore at least (n - 4)/2 -1 hidden units.  One \nshould  point  out  that  due  to  the generality of Sontag's  result  this  lower  bound  is \nvalid  for  all common activation functions of sigmoidal gates, and even  if N  employs \nheaviside gates  besides sigmoidal  gates. \n\u2022 \n\nTheorem  4.1  yields  a  lower  bound  of 4997  for  the  number  of hidden  units  in  any \nsigmoidal neural net that computes EDn for n  =  10 000 , where 10 000 is a  common \nestimate for  the number of inputs  (i.e.  synapses)  of a  biological  neuron. \n\nFinally we  would  like  to point  out  that  to  the  best  of our  knowledge Theorem  4.1 \nprovides the largest  known  lower bound  for  any concrete function  with  n  inputs on \na  sigmoidal  neural  net.  The  largest  previously  known  lower  bound  for  sigmoidal \nneural  nets wa<;  O(nl/4)  , due to  [Koiran,  1995J. \n\n5  Conclusions \n\nTheorems 2.1  and 3.1  provide a model for analog computations in network of spiking \nneurons  that  is  consistent  with  experimental  results  on  the  maximal  computation \nspeed of biological neural systems.  As explained after Theorem 3.1, this result holds \nfor  a  large variety of possible schemes for  encoding analog variables by  firing times. \n\nThese  theoretical  results  hold  rigoro'U.sly  only  for  a  rather  small  time  window  of \nlength,  for  temporal  coding.  However  a  closer  inspection  of  the  construction \nshows  that  the  actual  shape  of EPSP's and  IPSP's  in  biological  neurons  provides \nan  automatic adjustment of extreme values of the inputs tao  towards their average, \nwhich allows them to carry out rather similar computations for a substantially larger \nwindow size.  It also appears to be of interest from  the biological  point of view that \nthe synaptic weights play for temporal coding in our construction basically the same \nrole as for  rate coding,  and  hence the  .~ame network is in  principle able to compute \nclosely  related analog functions in  both coding schemes. \n\nWe  have focused  in  our constructions on  feedforward  nets,  but our  method  can  for \nexample also be used  to simulate a  Hopfield  net  with graded response by  a  network \nof noisy  spiking  neurons  in  temporal  coding.  A  stable  state  of  the  Hopfield  net \ncorresponds  then  to a  firing  pattern  of the  simulating  SNN  where  all  neurons  fire \nat the same frequency,  with  the  ((pattern\" of the stahle state encoded in  their phase \ndifferences. \n\n\fSpiking Neurons have more Computational Power than Sigmoidal Neurons \n\n217 \n\nThe  theoretical  results in  this  article  may  also  provide  additional  goals and  direc(cid:173)\ntions for  a  new  computer  technology  based  on  artificial spiking neurons. \n\nAcknowledgement \n\nI would like to thank David Haussler, Pa.')cal  Koiran, and Eduardo Sontag for  helpful \ncommunications. \n\nReferences \n\n[Bair  &  Koch,  1996]  W.  Bair,  C.  Koch,  \"Temporal  preCISIon  of  spike  trains  in \nextra.')triate  cortex  of  the  behaving  macaque  monkey\",  Neural  Computation, \nvol.  8,  pp  1185-1202, 1996. \n\n[Bialek  &  Rieke,  1992]  W. Bialek, and F. Rieke,  \"Reliability and information trans(cid:173)\n\nmission in spiking neurons\",  Trends  in Neuroscience, vol.  15, pp 428-434,1992. \n\n[Hopfield,  1995J  J. J.  Hopfield,  \"Pattern recognition  computation  using  action  po(cid:173)\ntential  timing for  stimulus  representations\",  Nature,  vol.  376,  pp 33-36, 1995. \n\n[Koiran,  1995]  P.  Koiran,  \"VC-dimension  in  circuit  complexity\",  Proc.  of the  11th \n\nIEEE Conference  on  Computational  Complexity,  pp  81-85, 1996. \n\n[Leshno et  aI.,  1993]  M. Leshno, V. Y.  Lin,  A.  Pinkus, and S.  Schocken,  \"Multilayer \n\nfeed forward  networks  with  a  nonpolynomial  activation  function  can  approxi(cid:173)\nmate any function\",  Neural  Networks,  vol.  6,  pp  861-867, 1993. \n\n[Maass,  1996a]  W. Maass,  \"On the computational power of noisy spiking neurons\", \nAdvances  in  Neural  Information  Processing  Systems,  vol.  8,  pp  211-217,  MIT \nPress, Cambridge,  1996. \n\n[Maass,  1996b]  W.  Maass,  \"Networks of spiking  neurons:  the  third  generation  of \nneural  network  models\",  FTP-host:  archive.cis.ohio-state.edu,  FTP-filename: \n/pub/neuroprose/maass.third-generation.ps.Z, Neural  Networks,  to appear. \n\n[Maass,  1997]  W.  Maa.')s,  \"Fa.')t  sigmoidal networks via spiking neurons\",  to appear \nin  Neural  Computation.  FTP-host:  archive.cis.ohio-state.edu  FTP-filename: \n/pub/neuroprose/maass.sigmoidal-spiking.ps.Z,  Neural  Computation,  to  ap(cid:173)\npear in  vol.  9,  1997. \n\n[de  Ruyter van  Steveninck  &  Bialek,  1988]  R.  de  Ruyter  van  Steveninck,  and \nW.  Bialek,  \"Real-time  performance  of a  movement  sensitive  neuron  in  the \nblowfly visual  system\",  Proc.  Roy.  Soc.  B,  vol.  234,  pp  379-414,  1988. \n\n[Sontag,  1996]  E. D.  Sontag,  \"Shattering all  sets of k  points in  'general position' re(cid:173)\nquires (k -1)/2 parameters\",  http://www.math.rutgers.edu/'''sontag/ , follow \nlinks  to FTP archive. \n\n[Thorpe &  Imbert,  1989J  S.  T.  Thorpe,  and  M.  Imbert,  \"Biological  constraints \non  connectionist  modelling\",  In:  Connectionism  in  Perspective,  R.  Pfeifer, \nZ.  Schreter,  F.  Fogelman-Soulie,  and  1.  Steels,  eds.,  Elsevier,  North-Holland, \n1989. \n\n\f", "award": [], "sourceid": 1307, "authors": [{"given_name": "Wolfgang", "family_name": "Maass", "institution": null}]}