{"title": "Spike-Based Compared to Rate-Based Hebbian Learning", "book": "Advances in Neural Information Processing Systems", "page_first": 125, "page_last": 131, "abstract": null, "full_text": "Spike-Based  Compared to  Rate-Based \n\nHebbian  Learning \n\nRichard  Kempter* \n\nInstitut fur  Theoretische Physik \nTechnische Universitat Munchen \n\nD-85747 Garching, Germany \n\nWulfram Gerstner \n\nSwiss  Federal Institute of Technology \n\nCenter of Neuromimetic Systems,  EPFL-DI \n\nCH-1015  Lausanne, Switzerland \n\nJ.  Leo  van Hemmen \n\nInstitut fur  Theoretische Physik \nTechnische  Universitat Munchen \n\nD-85747  Garching,  Germany \n\nAbstract \n\nA  correlation-based learning  rule  at the  spike  level  is  formulated, \nmathematically analyzed, and compared to learning in a firing-rate \ndescription.  A  differential  equation  for  the  learning  dynamics  is \nderived  under the assumption  that the time scales of learning and \nspiking  can  be  separated.  For  a  linear  Poissonian  neuron  model \nwhich receives time-dependent stochastic input we show that spike \ncorrelations on  a  millisecond time scale play  indeed  a  role.  Corre(cid:173)\nlations between input and output spikes tend to stabilize structure \nformation,  provided  that  the  form  of  the  learning  window  is  in \naccordance with Hebb's  principle.  Conditions for  an intrinsic nor(cid:173)\nmalization of the average synaptic weight are discussed. \n\n1 \n\nIntroduction \n\nMost learning rules are formulated in terms of mean firing  rates,  viz.,  a  continuous \nvariable reflecting  the mean  activity of a  neuron.  For example,  a  'Hebbian'  (Hebb \n1949)  learning  rule  which  is  driven  by  the  correlations  between  presynaptic  and \npostsynaptic rates may  be used to  generate neuronal receptive fields  (e.g.,  Linsker \n1986,  MacKay  and  Miller  1990,  Wimbauer  et  al.  1997)  with  properties  similar  to \nthose of real neurons.  A rate-based description, however,  neglects effects which  are \ndue to the pulse structure of neuronal signals.  During recent years experimental and \n\n* email:  kempter@physik.tu-muenchen.de (corresponding author) \n\n\f126 \n\nR.  Kempter.  W Gerstner and J  L.  van Hemmen \n\ntheoretical  evidence  has  accumulated  which  suggests  that  temporal  coincidences \nbetween  spikes  on  a  millisecond  or  even  sub-millisecond  scale  play  an  important \nrole in  neuronal information processing  (e.g.,  Bialek et al.  1991,  Carr 1993,  Abeles \n1994,  Gerstner  et  al.  1996).  Moreover,  changes  of  synaptic  efficacy  depend  on \nthe  precise  timing  of postsynaptic  action  potentials  and  presynaptic  input  spikes \n(Markram et al.  1997,  Zhang et al.  1998).  A synaptic weight is  found to increase,  if \npresynaptic firing precedes a postsynaptic spike and decreased otherwise.  In contrast \nto  the  standard  rate  models  of  Hebbian  learning,  the  spike-based  learning  rule \ndiscussed  in  this  paper takes  these  effects  into  account.  For  mathematical  details \nand numerical simulations  the reader is  referred to Kempter et al.  (1999) . \n\n2  Derivation of the Learning Equation \n\n2.1  Specification of the Hebb  Rule \nWe  consider  a  neuron that receives  input from  N  \u00bb  1 synapses with  efficacies  Ji , \n1  :::;  i  :::;  N.  We  assume  that  changes  of Ji  are induced  by  pre- and  postsynaptic \nspikes.  The learning rule consists of three parts.  (i)  Let tf be the time of the m th \ninput spike arriving at synapse i.  The arrival of the spike induces the weight  Ji  to \nchange by an amount win  which  can be positive or negative.  (ii)  Let  tn  be the nth \noutput spike of the neuron under consideration.  This event triggers the change of all \nN  efficacies by an amount wout  which  can also be positive or negative.  (iii)  Finally, \ntime  differences  between  input spikes  influence  the  change of the efficacies.  Given \na time difference  s =  tf - t n  between input and output spikes,  Ji  is  changed by  an \namount W(s)  where the learning  window W  is  a real  valued function  (Fig. 1).  The \nlearning  window  can  be  motivated  by  local  chemical  processes  at  the level  of the \nsynapse  (Gerstner et al.  1998,  Senn et al.  1999).  Here  we  simply assume that such \na  learning window  exist and take some  (arbitrary) functional  dependence  W(s) . \n\nFigure  1:  An  example of a  learning win(cid:173)\ndow  W  as  a  function  of  the  delay  s  = \ntf - tn  between a postsynaptic firing time \ntn  and  presynaptic  spike  arrival  tf  at \nsynapse i.  Note that for  s  < 0 the presy(cid:173)\nnaptic spike  precedes postsynaptic firing. \n\nStarting at time t with an efficacy  Ji(t),  the total change 6.Ji(t)  =  Ji(t + T) - Ji(t) \nin  a  time  interval  T  is  calculated  by  summing  the  contributions  of all  input  and \noutput  spikes  in  the  time  interval  [t, t + 7].  Describing  the  input  spike  train  at \nsynapse i  by  a  series  of 8 functions,  s:n(t)  =  Lm 8(t - tf),  and,  similarly,  output \nspikes by  sout(t) = Ln 8(t - tn),  we  can formulate  the rules  (i)--:(iii): \n\nb.J,(t) = ! dt'  Wi\" S;\"(t') + wont 8\"nt(t') + ! dt\" W(t\" - t') S;\"(t\") 8\"nt(t') \n\nt+T \n\nt+T \n\n[ \n\n] \n\n(1) \n\n2.2  Separation of Time Scales \n\nThe  total  change  6..Ji (t)  is  subject  to  noise  due  to  stochastic  spike  arrival  and, \npossibly,  stochastic  generation  of output  spikes.  We  therefore  study  the  expected \ndevelopment of the weights  Ji ,  denoted  by  angular brackets.  We  make the substi(cid:173)\ntution  s  =  til  -\n\nt'  on the  right-hand side  of (1),  divide  both  sides  by  T,  and take \n\n\fSpike-Based Compared to Rate-Based Hebbian  Learning \n\nthe expectation value: \n\n(tlJt\u00b7)(t) \n\nT \n\n_1 I t+Tdt' \nT t \n1 It+T \n+-\nT \n\nt \n\ndt' \n\nt-t' \n\n[win  (s!n)(t') + W out (sout) (t')] \nIt+T-t' \n\nds W(s) (s!n(t' + s) sout(t')) \n\n127 \n\n(2) \n\nWe  may  interpret  (s~n)(t)  for  1  ::;  i  ::;  Nand  (sout)(t)  as  instantaneous  firing \nrates. I  They  may  vary  on  very  short  time  scales  - shorter,  e.g.,  than  average \ninterspike intervals.  Such  a  model  is  consistent  with  the  idea of temporal  coding, \nsince it does  not rely on temporally averaged mean firing  rates. \nWe  note,  however,  that due to the integral over time on the right-hand side of (2) \ntemporal averaging is  indeed important.  If T  is much larger than typical interspike \nintervals, we  may  define  mean firing  rates v!n(t)  =  (s~n)(t) and vout(t)  =  (sout)(t) \nwhere we  have  used  the notation  f(t)  =  T- l  Itt+T dt' f(t').  The  mean  firing  rates \nmust  be  distinguished  from  the  previously  defined  instantaneous  rates  (s~n)  and \n(sout)  which  are defined  as  an expectation value  and have a  high  temporal resolu(cid:173)\ntion.  In  contrast, the mean  firing  rates  vin and  vout  vary  slowly  (time scale of the \norder of T)  as a  function of time. \nIf the  learning  time T  is  much  larger than  the width  of the  learning window,  the \nintegration over s in (2)  can be extended to run from  -00 to 00 without introducing \na  noticeable error.  With the definition of a temporally averaged correlation, \n\n(3) \nthe last term on the right of (2)  reduces to I~oo ds W(s) Ci(s; t).  Thus, correlations \nbetween  pre- and  postsynaptic spikes  enter spike-based  Hebbian learning  through \nCi  convolved  with the learning window W.  We  remark that the correlation Ci(s; t) \nmay  change as  a  function  of s  on a  fast  time scale.  Note that,  by  definition,  s  < 0 \nimplies  that  a  presynaptic  spike  precedes  the  output spike  - and  this  is  when  we \nexpect  (for excitatory synapses)  a  positive  correlation between input and output. \n\nAs usual in the theory of Hebbian learning, we require learning to be a slow process. \nThe  correlation Ci  can then  be  evaluated for  a  constant  Ji  and the left-hand side \nof (2)  can be  rewritten as a  differential on the slow  time scale of learning \n\n:t Ji(t)  ==  ji =  win v!n(t) + Wout  vout(t) + i: ds W(S) Ci(S; t) \n\n(4) \n\n2.3  Relation to Rate-Based Hebbian Learning \n\nIn neural network theory, the hypothesis of Hebb (Hebb 1949) is  usually formulated \nas a  learning rule  where the change of a  synaptic efficacy  Ji  depends on the corre(cid:173)\nlation between  the mean firing  rate vln  of the i th  presynaptic and the mean firing \nrate vout  of a  postsynaptic neuron, viz. , \n\nji =  ao + al v!n  + a2  vout  + a3  v!n  vout  + a4  (v~n)2 + a5  (vout )2 \n\n(5) \nwhere ao,  aI, a2,  a3 , a4, and a5  are proportionality constants.  Apart from the decay \nterm  ao  and the  'Hebbian'  term  vin  vout  proportional to the  product of input  and \n\n, \n\n1 An  example  of  rapidly  changing  instantaneous  rates  can  be  found  in  the  auditory \nsystem .  The auditory nerve carries noisy spike trains with a stochastic intensity modulated \nat the frequency  of the applied acoustic tone.  In the barn owl,  a  significant modulation of \nthe rates  is  seen up  to a  frequency of 8  kHz  (e.g.,  Carr 1993). \n\n\f128 \n\nR.  Kempler,  W  Gerstner and J  L.  van Hemmen \n\noutput rates, there are also synaptic changes which are driven separately by the pre(cid:173)\nand postsynaptic rates.  The parameters ao, ... , as  may depend on Ji .  Equation (5) \nis  a  general formulation  up  to second order in  the rates;  see,  e.g.,  (Linsker  1986). \n\nTo get  (5)  from  (4)  two approximations are necessary.  First, if there are no correla(cid:173)\ntions between input and output spikes apart from  the correlations contained in  the \nrates,  we  can  approximate  (SJn(t + s) sout(t))  ~ (s~n)(t + s) (SOUtHt).  Second,  if \nthese rates change slowly as compared to T, then we have Ci(s; t)  ~ v;n(t+s) vout(t). \nSince we  have assumed that the learning time  T  is  long  compared to the width of \nthe  learning  window,  we  may  simplify  further  and  set  vJn(t  + s)  ~ v!n(t),  hence \nJ~oo ds W(s) Ci(s; t)  ~ W(O) vjn(t) vout(t),  where  W(O)  = J~oo ds W(s).  We  may \nnow  identify  W(O)  with  a3.  By  further  comparison  of  (5)  with  (4)  we  identify \nwin  with  al  and  wout  with  a2,  and  we  are  able  to  reduce  (4)  to  (5)  by  setting \nao  = a4  = as = O. \nThe  above  set  of  of  assumption  which  is  necessary  to  derive  (5)  from  (4)  does, \nhowever,  not hold in general.  According to the results of Markram et aI. (1997)  the \nwidth of the learning window in cortical pyramidal cells  is  in the range of ~ 100 ms. \nA  mean  rate formulation  thus requires  that all  changes  of the activity are slow  on \na time scale of lOOms.  This is  not necessarily the case.  The existence of oscillatory \nactivity  in  the  cortex  in  the  range  of 50 Hz  implies  activity  changes  every  20 ms. \nMuch  faster  activity  changes  on  a  time  scale  of 1 ms  and  below  are  found  in  the \nauditory  system  (e.g.,  Carr  1993).  Furthermore,  beyond  the correlations  between \nmean activities additional correlations between spikes may exist; see below.  Because \nof all these reasons, the learning rule (5)  in the simple rate formulation is insufficient . \nIn the following  we  will  study the full  spike-based learning equation  (4). \n\n3  Stochastically Spiking Neurons \n\n3.1  Poisson Input  and  Stochastic Neuron Model \n\nTo proceed with the analysis of (4) we  need to determine the correlations Ci  between \ninput spikes  at synapse i  and  output  spikes.  The correlations depend  strongly on \nthe  neuron  model  under  consideration.  To  highlight  the  main  points  of learning \nwe  study  a  linear  inhomogeneous  Poisson  neuron  as  a  toy  model. \nInput  spike \ntrains arriving at the N  synapses are statistically independent and generated by  an \ninhomogeneous  Poisson  process  with  time-dependent  intensities  (Sin) (t)  = ,\\~n (t), \nwith 1 ~ i  ~ N.  A spike arriving at tf at synapse i , evokes a postsynaptic potential \n(PSP)  with  time  course E(t  - tf) which  we  assume  to  be  excitatory  (EPSP).  The \namplitude is  given by the synaptic efficacy  Ji(t)  > O.  The membrane potential u of \nthe neuron is  the linear superposition of all  contributions \n\nu(t) = Uo  + L L Ji(t) E(t  - t~) \n\nN \n\ni=l  m \n\n(6) \n\nwhere  Uo  is  the  resting  potential.  Output  spikes  are  assumed  to  be  generated \nstochastically with  a  time dependent  rate ,\\out(t)  which  depends  linearly upon the \nmembrane potential \n\n,\\out(t) = f3  [u(t)l+  =  Vo  + L L Ji(t) E(t  - tf)\u00b7 \n\nN \n\n(7) \n\ni=l  m \n\nwith  a  linear  function  f3[ul+  = f30  + f31  u  for  u  > 0  and  zero  otherwise.  After  the \nsecond  equality  sign,  we  have  formally  set  Vo  = Uo  + f30  and  f31  = 1.  vo  >  can \n\n\fSpike-Based Compared to Rate-Based Hebbian  Learning \n\n129 \n\nbe  interpreted  as  the spontaneous  firing  rate.  For  excitatory  synapses  a  negative \nu  is  impossible  and that's what  we  have  used  after  the  second  equality  sign.  The \nsums run over all spike arrival times at all synapses.  Note that the spike generation \nprocess is  independent of previous output spikes.  In particular, the Poisson model \ndoes  not include refractoriness. \n\nIn  the  context  of  (4),  we  are  interested  in  the  expectation  values  for  input  and \noutput.  The expected input is  (s~n)(t) = A~n(t).  The expected output is \n\n(sout)(t)  =  va  + L Ji(t) 10 00 d8\u20ac(s)  A~n(t - 8)  , \n\n(8) \n\nt \n\nThe expected output rate in (8) depends on the convolution of \u20ac  with the input rates. \nIn the following we  will denote the convolved rates by A~n(t) =  1000 d8 \u20ac(8)A~n(t - 8). \nNext  we  consider  the  expected  correlations  between  input  and  output,  (s!n(t + \n8)  sout(t)), which  we  need  in  (3): \n\n(s~n (t + 8) sout(t))  =  A~n (t + 8)  [Va  + Ji (t)  \u20ac(  -8) + L Jj (t) A~n(t)] \n\n(9) \n\nj \n\nThe first term inside the square brackets is the spontaneous output rate.  The second \nterm  is  the specific  contribution of an input spike at time  t + 8 to the output rate \nat t.  It vanishes for  8 > 0  (Fig. 2).  The sum in  (9)  contains the mean contributions \nof all  synapses to an output spike  at time t.  Inserting  (9)  in  (3)  and  assuming the \nweights  Jj  to be constant in the time interval  [t, t + T]  we  obtain \n\nCi(8; t)  = L Jj(t) A~n(t + 8) A~n(t) + A~n(t + 8)  [Va  + Ji(t) \u20ac(  -8)]. \n\n(10) \n\nj \n\nFor  excitatory  synapses,  the  second  term  gives  for  8  <  0  a  positive  contribution \n(Recall  that  8  <  0  means  that  a \nto  the  correlation  function  - as  it  should  be. \npresynaptic spike  precedes postsynaptic firing.) \n\n[ ... ](t') \n\n---\u00b7r ---------\n\no -'-----r----,,---'---------___. t' \n\nt+s \n\nt \n\nFigure 2:  Interpretation of the term in square \nbrackets in  (9).  The dotted line  is  the contri(cid:173)\nbution  of an  input  spike  at time  t + 8 to  the \noutput rate as a function of t', viz.,  Ji (t) \u20ac  (t' -\nt - 8).  Adding this to the mean rate contribu(cid:173)\ntion,  Va  + Lj Jj(t') A~n(t')  (dashed  line),  we \nobtain  the  rate  inside  the square  brackets of \n(9)  (full  line).  At  time t'  =  t  the contribution \nof an input spike at time t + 8 is  Ji(t) \u20ac(  -s). \n\n3.2  Learning Equation \n\nThe assumption of identical  and  constant  mean input  rates,  A~n(t)  =  v:n(t)  =  vin \nfor  all  i,  reduces the  number of free  parameters in  (4)  and eliminates all  effects  of \nrate coding.  We  introduce r~n(t) := [W(O)]-l J.~oo d8 W(8)A~n(t + 8)  and define \n\n(11) \nUsing (8),  (10),  (11) in (4)  we find for the evolution on the slow time scale oflearning \n\nji(t) =  kl  + L Jj(t) [Qij(t) + k2  + k3  bij]  ,  where \n\n(12) \n\nj \n\n\f130 \n\nR. Kempter.  W Gerstner and 1. L.  van Hemmen \n\n[w out + W(O)  vin] Vo  + win v in \n[w out + W(O)  vin] v in \n\nv in /  ds\u20ac(-s)  W(s)  . \n\n(13) \n(14) \n\n(15) \n\n4  Discussion \n\nEquation  (12),  which  is  the  central  result  of our  analysis,  describes  the  expected \ndynamics of synaptic weights for  a  spike-based Hebbian learning rule  (1)  under the \nassumption of a linear inhomogeneous Poisson neuron.  Linsker (1986)  has derived a \nmathematically equivalent equation starting from  (5)  and a linear graded response \nneuron, a rate-based model.  An equation of this type has been analyzed by MacKay \nand Miller  (1990).  The difference between Linsker's equation and (12)  is, apart from \na  slightly different  notation, the term  k3 6ij  and the interpretation of Qij. \n\n4.1 \n\nInterpretation of Qij \n\nIn  (12)  correlations  between  spikes  on  time  scales  down  to  milliseconds  or  below \ncan enter the driving term Qij for  structure formation; cf.  (11).  In contrast to that, \nLinsker 's ansatz is  based on a  firing  rate description,  where  the term  Qij  contains \ncorrelations between  mean firing  rates  only.  In his  Qij  term,  mean firing rates take \nthe place of r~n and A~n. If we use a standard interpretation of rate coding, a mean \nfiring rate corresponds to a temporally averaged quantity with an averaging window \nor a  hundred milliseconds or more. \nFormally,  we  could  define  mean  rates  by  temporal  averaging  with  either  \u20ac(  s)  or \nW(s)  as  the  averaging  window.  In  this  sense,  Linsker's  'rates'  have  been  made \nmore  precise  by  (11).  Note,  however,  that  (11)  is  asymmetric:  one  of the  rates \nshould be convolved with  \u20ac\n\n,  the other one with  W. \n\n4.2  Relevance of the k3  term \n\nThe most important difference  between  Linsker's  rate-based learning  rule  and our \nEq.  (12)  is  the existence of a  term  k3  I: O.  We  now  argue that for  a  causal chain of \nevents k3  ex:  I dx \u20ac(x)  W( -x) must be positive.  [We have set x  =  -s in (15).]  First, \nwithout  loss  of generality,  the  integral  can  be  restricted  to  x  >  0  since  \u20ac(x) \nis  a \nresponse kernel and vanishes for  x < O.  For excitatory synapses, \u20ac(x) \nis  positive for \nx  > O.  Second,  experiments on excitatory synapses show that W(s)  is  positive for \ns  < 0 (Markram et al.  1997,  Zhang et al.  1998).  Thus the integral I dx \u20ac(x)  W( -x) \nis  positive - and so is  k 3 . \nThere is also a more general argument for  k3  > 0 based on a literal interpretation of \nHebb's statement (Hebb  1949).  Let us  recall that s  < 0 in  (15)  means that a  presy(cid:173)\nnaptic spike  precedes  postsynaptic spiking.  For  excitatory  synapses, a  presynaptic \nspike which  precedes  postsynaptic firing  may  be the  cause  of the postsynaptic  ac(cid:173)\ntivity.  [As  Hebb  puts it,  it has  'contributed in  firing  the postsynaptic cell'.]  Thus, \nthe  Hebb  rul~ 'predicts'  that  for  excitatory  synapses  W(s)  is  positive  for  s  <  O. \nHence, k3  = vln I ds \u20ac(  - s) W (s)  > 0 as  claimed  above. \nA positive k3  term in  (12)  gives rise to an exponential growth of weights.  Thus any \nexisting structure in the distribution of weights is enhanced.  This contributes to the \nstability  of weight distributions,  especially when  there are few  and strong synapses \n(Gerstner et al.  1996). \n\n\fSpike-Based Compared to Rate-Based Hebbian Learning \n\n131 \n\n4.3 \n\nIntrinsic  Normalization \n\nLet  us  suppose  that  no  input  synapse is  special  and  impose  the  (weak)  condition \nthat N - 1 Li Qij  =  Qo  > 0 independent of the synapse index j .  We  find  then from \n(12)  that  the average weight  Jo  := N-l Li Ji  has  a  fixed  point  Jo =  -kd[Qo + \nk2  + N- 1 k 3 ].  The fixed  point  is  stable  if  Qo  + k2  + N- 1 k3  < O.  We  have  shown \nabove that  k3  > O.  Furthermore,  Qo  > 0  according to our  assumption.  The only \nway  to  enforce  stability  is  therefore  a  term  k2  which  is  sufficiently  negative.  Let \nus  now  turn to the  definition  of  k2  in  (14).  To  achieve  k2  < 0,  either  W(O)  (the \nintegral over  W)  must  be  sufficiently  negative;  this  corresponds to a  learning rule \nwhich  is, on the  average,  anti-Hebbian.  Or, for  W(O)  > 0,  the linear term  wout  in \n(1)  must  be sufficiently  negative.  In addition,  for  excitatory synapses  a  reasonable \nfixed  point  Jo  has  to be  positive.  For  a  stable fixed  point this  is  only  possible  for \nkl  > 0, which,  in turn, implies  win  to be sufficiently  positive;  cf.  (13). \nIntrinsic normalization of synaptic weights is  an interesting property, since it allows \nneurons  to  stay  at  an optimal  operating  point  even  while  synapses  are  changing. \nAuditory neurons may use such  a  mechanism to stay during learning in the regime \nwhere coincidence detection is  possible  (Gerstner et al.  1996, Kempter et al.  1998). \nCortical  neurons  might  use  the  same  principles  to  operate  in  the  regime  of  high \nvariability  (Abbott,  invited NIPS  talk,  this  volume). \n\n4.4  Conclusions \n\nSpike-based  learning  is  different  from  simple  rate-based  learning  rules.  A  spike(cid:173)\nbased  learning  rule  can  pick  up  correlations  in  the  input  on  a  millisecond  time \nscale.  Mathematically,  the  main  difference  to  rate-based  Hebbian  learning  is  the \nexistence  of a  k3  term  which  accounts  for  the  causal  relation  between  input  and \noutput spikes.  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