{"title": "How to Describe Neuronal Activity: Spikes, Rates, or Assemblies?", "book": "Advances in Neural Information Processing Systems", "page_first": 463, "page_last": 470, "abstract": null, "full_text": "How to Describe Neuronal Activity: \n\nSpikes, Rates, or Assemblies? \n\nWulfram Gerstner and J.  Leo van  Hemmen \n\nPhysik-Department der  TU  Miinchen \n\nD-85748 Garching bei  Miinchen,  Germany \n\nAbstract \n\nWhat  is  the  'correct'  theoretical  description  of neuronal  activity? \nThe  analysis  of the  dynamics  of a  globally  connected  network  of \nspiking neurons  (the Spike Response  Model)  shows  that a  descrip(cid:173)\ntion  by  mean firing  rates  is  possible  only if active  neurons  fire  in(cid:173)\ncoherently.  If firing  occurs  coherently  or  with spatio-temporal cor(cid:173)\nrelations,  the  spike  structure of the neural  code  becomes  relevant. \nAlternatively, neurons can  be gathered into local or distributed en(cid:173)\nsembles or 'assemblies'.  A description  based on the mean ensemble \nactivity is,  in principle, possible but the interaction  between  differ(cid:173)\nent assemblies becomes highly nonlinear.  A description with spikes \nshould  therefore  be  preferred. \n\n1 \n\nINTRODUCTION \n\nNeurons  communicate by  sequences  of short  pulses,  the so-called  action  potentials \nor spikes.  One of the most important problems in theoretical neuroscience  concerns \nthe question of how information on the environment is encoded  in such spike trains: \nIs  the exact timing of spikes with relation to earlier spikes relevant  (spike or interval \ncode  (MacKay and McCulloch 1952) or does  the mean firing rate averaged over sev(cid:173)\neral spikes contain all important information (rate  code;  see,  e.g.,  Stein  1967)?  Are \nspikes of single neurons important or do we  have to consider ensembles of equivalent \nneurons  (ensemble  code)?  If so,  can  we  find  local ensembles  (e.g.,  columns;  Hubel \nand  Wiesel  1962)  or  do  neurons form  'assemblies' (Hebb  1949)  distributed  all  over \nthe network? \n\n463 \n\n\f464 \n\nGerstner and van Hemmen \n\n2  SPIKE  RESPONSE MODEL \n\nWe consider a globally connected network of N  neurons with 1 ~ i  ~ N.  A neuron i \nfires,  if its membrane potential passes  a  threshold ().  A spike at time t{  is  described \nby  a  6-pulse;  thus  Sf (t)  =  L:~=1 6(t - t{)  is  the spike  train of neuron  i.  Spikes  are \nlabelled such  that tt  is  the most recent  spike and tf is  the  Fth  spike  going  back in \ntime. \nIn  the  Spike  Response  Model,  short SRM,  (Gerstner  1990,  Gerstner  and van  Hem(cid:173)\nmen  1992) a neuron is  characterized by two different  response  junctions,  f  and \"1ref . \nSpikes  which  neuron  i  receives  from other  neurons evoke  a  synaptic  potential \n\nwhere  the response  kernel \n\nf(S)  = \n\n{ \n\n0 \n,,_a tr \n-::-r- exp  - - - lor  s  >  u \n\nfor  s  < Ll tr \n(,,_a tr )  CA t  \nr \n\nT. \n\nT, \n\n(1) \n\n(2) \n\ndescribes  a  typical excitatory or inhibitory postsynaptic  potential;  see  Fig.  1.  The \nweight  Jij  is the synaptic efficacy of a connection from j  to i,  Ll tr  is the axonal (and \nsynaptic)  transmission time,  and  T\"  is  a  time constant  of the  postsynaptic neuron. \nThe origin S  = 0 in  (2)  denotes the firing time of a presynaptic spike.  In  simulations \nwe  usually assume  T\"  = 2 ms  and for  Lltr  a value between  1 and  4  ms \nSimilarly, spike  emission  induces  refractoriness  immediately after  spiking.  This  is \nmodelled by  a  refractory  potential \n\nwith a  refractory  function \n\nref () \n\"1 \n\n{  -00 \n\ns  =  \"1o/(s  _  ,ref) \n\n(3) \n\n(4) \n\nfor  S  ~ ,ref \nfor  S  > ,ref. \n\nFor 0 ~ s  ~ ,ref the neuron is in the absolute refractory  period and cannot spike at \nall whereas for  s  > ,ref spiking is  possible but difficult  (relative refractory  period). \nTo put it  differently,  ()  - \"1ref (s)  describes  an increased  threshold immediately after \nspiking; cf.  Fig.  1.  In  simulations, ,ref is  taken to be 4 ms.  Note that, for  the sake \nof simplicity, we  assume  that only  the  most recent  spike Sf  induces  refractoriness \nwhereas all past spikes Sf contribute to the synaptic potential; cf.,  Eqs.  (1) and (3). \n\n\fHow to  Describe Neuronal Activity: Spikes, Rates, or Assemblies? \n\n465 \n\n9-n(s) \n\nw f 0.5 \n\nCD \n\nFig  1  Response  functions. \nImmediately  after  firing  at  8  = \no the  effective  threshold  is  in(cid:173)\ncreased  to (J  - TIre! (8)  (dashed). \nThe form  of an  excitatory  post(cid:173)\nsynaptic  potential  (EPSP)  is \ndescribed  by  the  response func(cid:173)\ntion f( 8) (solid).  It is delayed by \na  time  ~ tr.  The  arrow  denotes \n20.0  the  period  Tosc  of  coherent  os-\n\ncillations;  d.  Section  5. \n\n0.0  '-........o...-_Ll------'-_L-..--'---.----l---=::::t=~ \n\n0.0 \n\n5.0 \n\n15.0 \n\n10.0 \n5 [m5] \n\nThe total membrane potential  is  the sum of both parts,  i.e. \n\nhi(t) =  h~ef (t) + h:yn(t). \n\n(5) \n\nNoise  is  included  by  introduction of a firing  probability \n\n(6) \nwhere 6t  is  an infinitesimal time interval and r(h) is a  time constant which  depends \non  the  momentary value of the membrane potential in  relation  to  the  threshold  (). \nIn analogy to  the chemical reaction  constant  we  assume \n\nPF(h; 6t)  =  r- 1 (h) 6t. \n\nr(h) =  ro exp[-,B(h - (})], \n\n(7) \nwhere ro  is the response time at threshold.  The parameter ,B determines the amount \nof noise in the system.  For,B  --+  00 we  recover the noise-free  behavior, i.e., a  neuron \nfires  immediately, if h > ()  (r  --+  0),  but it  cannot fire,  if h < ()  (r  --+  (0).  Eqs.  (1), \n(3),  (5),  and  (6)  define  the spiking dynamics in  a network  of SRM-neurons. \n\n3  FIRING STATISTICS \n\nWe start our considerations with a large ensemble of identical neurons driven by the \nsame arbitrary synaptic potential h3yn (t) .  We  assume  that all neurons  have fired  a \nfirst  spike at t  =  t{ . Thus the total membrane potential is h(t) =  hsyn(t) + 7]re f (t(cid:173)\nto.  If h(t)  slowly  approaches  (),  some of the  neurons  will  fire  again.  We  now  ask \nfor  the probability that a neuron  which  has fired  at time t{  will fire  again at a  later \ntime t.  The conditional probability p~2\\tlt{) that the  next spike of a given neuron \noccurs  at  time t > t{  is \n\np~2)(tlt{) = r-l[h(t)] exp { -1; r- 1[h(S')]dS'} . \n\n(8) \n\nThe exponential factor is  the portion of neurons  that have survived from time t{  to \ntime t  without  firing  again  and  the  prefactor  r- 1 [h(t)]  is  the  instantaneous firing \nprobability  (6)  at  time t.  Since  the  refractory  potential  is  reset  after  each  spike, \nthe  spiking statistics  does  not  depend  on  earlier  spikes,  in  other  words,  it  is  fully \ndescribed  by  p~2)(tlt{).  This will  be used  below;  cf.  Eq.  (14) . \n\n\f466 \n\nGerstner and van Hemmen \n\nAs  a  special case,  we  may consider  constant synaptic input h 3yn = hO\u2022  In this case, \n(8) yields  the distribution of inter-spike intervals in a spike train of a  neuron driven \nby  constant  input  hO\u2022  The  mean firing  rate  at an  input  level  h O  is  defined  as  the \ninverse  of the mean inter-spike interval.  Integration by  parts yields \n\nI[ho] =  {J.;dt(t-t{lP~2)(tlt{l} -I =  {J.oodsexp{-lT-I[hO+~\"f (s'l]ds'} } -I \n\n(9) \nThus both firing rate and interval distribution can be calculated for arbitrary inputs. \n\n4  ASSEMBLY FORMATION AND  NETWORK \n\nDYNAMICS \n\nWe  now  turn  to  a  large,  but  structured  network.  Structure  is  induced  by  the \nformation of different  assemblies in  the system.  Each neuronal  assembly aP.  (Hebb \n1949)  consists  of neurons  which  have  the  tendency  to  be active  at  the same  time. \nFollowing the  traditional interpretation,  active  means an elevated  mean firing  rate \nduring some reasonable  period  of time.  Later,  in  Section  5.3,  we  will  deal  with  a \ndifferent  interpretation  where  active  means a spike  within a  time window  of a  few \nms.  In  any  case,  the  notion  of simultaneous  activity  allows  to  define  an  activity \npattern  {~r, 1 :::;  i  :::;  N}  with ~r = 1 if i  E aP.  and  ~r = 0  otherwise.  Each  neuron \nmay belong  to  different  assemblies  1 :::;  I-l  :::;  q.  The  vector ei  = (a, ... ,~n is  the \n'identity  card'  of neuron  i,  e.g.,  ei  =  (1,0,0,1,0)  says  that  neuron  i  belongs  to \nassembly  1 and 4  but not  to assembly 2,3,  and  5. \n\nNote  that,  in  general,  there  are  many neurons  with  the  same identity  card.  This \ncan  be  used  to  define  ensembles  (or  sublattices)  L(x)  of equivalent  neurons,  i.e., \nL(x)  =  {ilei  =  x}  (van  Hemmen  and  Kiihn  1991).  In  general,  the  number  of \nneurons  IL(x)1  in  an  ensemble  L(x)  goes  to  infinity  if  N \n--;.  00,  and  we  write \nIL(x)1 = p(x)N.  The mean  activity of an ensemble  L(x)  can  be  defined  by \n\nA(x, t) =  lim \n\nat--+o N--+oo \n\nlim  IL(x)I- 1  L \n\niEL(X) \n\nI t+at \n\nt \n\nS[ (t)dt. \n\n(10) \n\nIn the following we assume that the synaptic efficacies have been adjusted according \nto some Hebbian learning rule in a way  that allows to stabilize the different  activity \npatterns  or  assemblies ap..  To be specific,  we  assume \n\nJij  =  ~ L  L  Qp.vpost(~r)pre(~j) \n\nJ \n\nq \n\nq \n\np.=lv=l \n\n(11) \n\nwhere  post(x)  and pre(x)  are some arbitrary functions  characterizing  the  pre- and \npostsynaptic  part  of synaptic  learning.  Note  that for  Qp.v  = fJp.v  and  post(x)  and \npre(x) linear,  Eq.  (11)  can  be reduced  to  the usual  Hebb  rule. \nWith the above definitions we  can  write the synaptic potential of a  neuron i  E L(x) \nin  the  following form \n(>0 \nh3yn (x , t) = Jo  L  L  Qp.vpost(xp.) Lpre(zV) 10 \n\nf(s')p(z)A(z, t - s')ds'.  (12) \n\nq \n\nq \n\np.=lv=l \n\nz \n\n0 \n\n\fHow to Describe Neuronal Activity: Spikes, Rates, or Assemblies? \n\n467 \n\nWe  note  that  the  index  i  and  j  has  disappeared  and  there  remains  a  dependence \nupon  x  and  z  only.  The activity  of a  typical  ensemble  is  given  by  (Gerstner  and \nvan  Hemmen 1993,  1994) \n\nA(x, t) = 100  p?)(tlt - s)A(x, t - s)ds \n\n(13) \n\nwhere \n\np~2)(tlt-s) = r- 1 [h',yn(x, t)+7]ref (s)] exp {-13r- 1 [h3 yn(x, t - s+s' )+7]ref (s')]ds' } \n\n(14) \nis  the  conditional  probability  (8)  that  a  neuron  i  E  L(x)  which  has  fired  at  time \nt-s fires  again at time t.  Equations (12) - (14)  define  the ensemble dynamics of the \nnetwork. \n\n5  DISCUSSION \n\n5.1  ENSEMBLE  CODE \n\nEquations.  (12) - (14)  show that in a  large network a description  by mean ensemble \nactivities  is,  in  principle,  possible.  A  couple  of things,  however,  should  be  noted. \nFirst,  the interaction between the activity of different ensembles is  highly nonlinear. \nIt involves three  integrations over  the  past and  one exponentiation;  cf.  (12) - (14). \nIf we  had  started theoretical  modeling with  an approach  based  on  mean  activities, \nit  would  have been  hard  to find  the correct  interaction  term. \nSecond,  L(x)  defines  an ensemble of equivalent neurons  which is  a  subset of a given \nassembly  al-'.  A  reduction  of (12)  to  pure  assembly  activities  is,  in  general,  not \npossible.  Finally, equivalent neurons that form an ensemble L(x) are not necessarily \nsituated  next  to each  other.  In  fact,  they  may be distributed  all  over  the  network; \nIn  this  case  a  local  ensemble  average  yields  meaningless  results.  A \ncf.  Fig.  2. \ntheoretical  model based  on  local ensemble  averaging is  useful  only if we  know  that \nneighboring neurons  have  the same 'identity card'. \n\na) \n\nb) \n\n~': t \n\n100 \n\nactivity \n\nl \n.....\u2022 : .. : .. :': ' .... : .. :.: .. : .. : ': ': -:  \\ , \n\ntime [ms] \n.. \n.. \n\n..  .... \n\n150 \n\n200 \n\n.. \n\n.. \n\n.. \n\n.. \n\n.. \n\n.. \n\n.. \n\n20 \n\n30  \u2022 \u2022 \u2022   .. \n\n.. \n\n.. \n\n..  -.  -...  -.  -.  .. \n\n_  20 \n.. \n~ \n.. \n.. \n.. \n~ 10  \u2022\u2022\u2022 -\n.. \" ...... ... -.- .................... \" .. \" .. \" .-\n\n.. \n\u2022\u2022  :. - ............ . \n..  ....  .. \n.. \n.. \n.- .- .-.. \n\n....  .. \n.. \n\n.- ..... - ....  -. \n\n.. \n.. \n\n.. \n.. \n\n.. \n\n.. \n\n.. \n\n.. \n\n.. \n\n.. \n\n.. \n\n.. \n\n.-\n\n.-\n\nI \n\n.. \n\nI \n\nill. \n\n10 \n\no\u00b7\u00b7\u00b7~\u00b7\u00b7\u00b7\u00b7-\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7  0 \n100 \n\n150 \n\n200 \n\n0 \n\ntime [ms] \n\nrate  [Hz] \n\n0) \n30  r:::::: \n\n:J \n\n::> \n\n:::I \n\n100 \n\n200 \n\nrate  [Hz] \n\nFig.  2 \nStationary  activity  (incoherent \nfiring).  In  this  case  a  descrip(cid:173)\ntion  by  firing  rates  is  possible. \n(a)  Ensemble  averaged  activity \n(b)  Spike  raster  of 30 \nA(x, t). \nneurons  out  of  a  network  of \n4000.  (c)  Time-averaged  mean \nfiring  rate  f.  We  have  two  dif(cid:173)\nferent  assemblies,  one  of  them \nactive (d tr = 2 ms, f3  = 5). \n\n5.2  RATE  CODE \n\nCan the system of Eqs. (12) -(14) be transformed into a rate description?  In general, \nthis  is  not  the  case  but  if we  assume  that  the  ensemble  activities  are  constant  in \n\n\f468 \n\nGerstner and van Hemmen \n\n1.0  .---~--~----~--~--------~--~---.----~--~----~--, \n\nO.B \n\nO.B \n\n0.4 \n\n0.2 \n\n0.0 \n\no \n\n~.x-2 \n\n~.)(-3.5 \n\n100 \n\n200 \n\n300 \n\n400 \n\n500 \n\nBOO \n\nZeit [rn5] \n\nFig. 3  Stability of stationary states.The postsynaptic potential h~yn is plotted as a function \nof time.  Every 100  ms the delay  Lltr  has been increased  by 0.5 ms.  In the stationary state \n(Lltr  =  1.5 ms and Ll tr  =  3.5 ms), active  neurons fire  regularly  with rate T;l  =  1/5.5 ms. \nFor  a  delay  Ll tr  >  3.5  ms,  oscillations  with  period  Wl  =  27r /Tp  build  up  rapidly.  For \nintermediate  delays  2  ~ Ll tr  ~ 2.5  small-amplitude  oscillations  with  twice  the  frequency \noccur.  Higher  harmonics  are suppressed  by  noise  (/3  = 20). \n\ntime,  i.e.,  A(x, t)  = A(x),  then  an  exact  reduction  is  possible.  The  result  IS  a \nfixed-point  equation  (Gerstner  and van  Hemmen  1992) \n\nwhere \n\nq \n\nq \n\nA(x) = f[Jo L L Q~lIpost(X~) L pre(zll)p(z)A(z)] \nf[h,yn] = {J.oo dsexp{-1.' r- 1[h,yn + ~\"J(8')]ds'}} -1 \n\n~=lll=l \n\nz \n\n(15) \n\n(16) \n\nis  the  mean firing  rate  (9)  of a  typical  neuron stimulated by  a synaptic input  h3yn. \nConstant  activities  correspond  to  incoherent,  stationary  firing  and  in  this  case  a \nrate code  is  sufficient;  cf.  Fig.  2. \n\nTwo points should, however,  be kept in mind.  First, a stationary state of incoherent \nfiring  is  not  necessarily  stable.  In  fact,  in  a  noise-free  system  the stationary state \nis  always  unstable  and  oscillations build  up  (Gerstner  and  van  Hemmen  1993).  In \na system with noise,  the stability depends  on the noise  level f3  and the delay  Ll tr  of \naxonal and synaptic transmission (Gerstner and van Hemmen  1994).  This is shown \nin  Fig.  3  where  the  delay  Lltr  has  been  increased  every  100  ms.  The frequency  of \nthe small-amplitude oscillation around  the stationary state is  approximately equal \nto  the  mean  firing  rate  (16)  in  the  stationary  state  or  higher  harmonics  thereof. \nA  small-amplitude oscillation corresponds  to  partially synchronized  activity.  Note \nthat for  Ll tr  =  4 ms a  large-amplitude oscillation builds up.  Here  all neurons fire  in \nnearly  perfect  synchrony;  cf.  Fig.  4.  In  the  noiseless  case  f3  -\n00,  the  oscillations \nperiod  of such  a  collective  or  'locked'  oscillation  can  be  found  from  the  threshold \ncondition \n\nT\", = inf {s I 0 = ~\"J (8) + Jo ~ f(nS)} . \n\n(17) \nIn most cases  the contribution with n = 1 is  dominant which allows a simple graph(cid:173)\nTJref (s)  with  the \nical  solution.  The first  intersection  of the  effective  threshold  ()  -\n\n\fHow to Describe Neuronal Activity: Spikes. Rates. or Assemblies? \n\n469 \n\nweighted  EPSP  JOf( s)  yields  the oscillation  period;  cf.  Fig 1.  An  analytical argu(cid:173)\nment shows  that  locking  is  stable only  if  ;\" dTooc  >  0  (Gerstner  and  van  Hemmen \n1993). \n\na) \n\nactivity \n\n~:lliHHHlUHHHj \n\n1~ \n\n100 \n\n200 \n\nb) \n\ntime [ms] \n\n~ ................. . \nI f  S S S S SIS ) S S S S \\ 'a  \\  \\ \n,.,  20  . . . . . . . . . . . . . . . . . . \n. . . . . . . . . . . . . . . . . . \n\n\u00b7 . \\ ' \\ \\ '111 \" . \u00b7 . \u00b7 '1 11 1  \n\\ \n\n~ \n!  10 \n\nI  1  \\  \u2022\u2022\u2022\u2022  , \n\n\\ \n\n\\ \n\n, \n\n\\ \n\n\\ \n\n\\ \n\n\\ \n\n\\ \n\n\\ \n\n\\ \n\n\\ \n\n0) \n\nrate  [Hz] \n\n] \n\n10 t==:::;;= \n\n\"1 '111  \\ \\1 \\ \\1  \" '  ..  1 \no  \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022\u2022 \n100 \n\n150 \n\n200  0  0 \n\ntime [ms] \n\n100  200 \n\nrate  [Hz] \n\nFig.  4 \nOscillatory  activity  (coherent \nfiring).  In  this  case  a  descrip(cid:173)\ntion by firing rates must be com(cid:173)\nbined  with a  description  by  en(cid:173)\nsemble activities.  (a)  Ensemble \naveraged  activity  A(x, t). \n(b) \nSpike  raster  of 30  neurons  out \nof a network  of 4000.  (c)  Time(cid:173)\naveraged  mean firing  rate f.  In \nthis  simulation,  we  have  used \nLl tr =  4  ms and f3  =  8. \n\nSecond, even if the incoherent state is stable and attractive, there is always a transi(cid:173)\ntion time before the stationary state is assumed.  During this time, a rate description \nis insufficient and we  have to go  back  to the full dynamic equations (12) - (14).  Sim(cid:173)\nilarly, if neurons are subject  to a fast  time-dependent external stimulus, a rate code \nfails . \n\n5.3  SPIKE CODE \n\nA superficial inspection of Eqs.  (12) - (14)  gives the impression that all information \nabout neuronal spiking has disappeared.  This is,  however, false.  The term A(x, t-s) \nin  (13)  denotes  all  neurons  with  'identity card'  x  that  have fired  at time t-s .  The \nintegration  kernel  in  (13)  is  the  conditional  probability that  one  of these  neurons \nfires  again  at  time  t.  Keeping  t - s  fixed  and  varying  t  we  get  the  distribution \nof inter-spike  intervals for  neurons  in  L(x).  Thus  information on  both  spikes  and \nintervals is  contained  in  (13)  and  (14). \n\nWe can make use of this fact, if we consider network states where in every time step a \ndifferent assembly is  active.  This leads to a  spatia-temporal spike  pattern as shown \nin  Fig.  5.  To  transform  a  specific  spike  pattern  into  a  stable state of the  network \nwe  can  use  a  Hebbian  learning  rule.  However,  in  contrast  to  the  standard  rule,  a \nsynapse is strenthened  only if pre- and postsynaptic activity occurs  simultaneously \nwithin  a  time window  of a  few  ms  (Gerstner  et  al.  1993).  Note  that  in  this  case, \naveraging over  time or space spoils the information contained  in  the spike pattern. \n\n5.4  CONCLUSIONS \n\n(12)  - (14)  show  that  in  our  large  and  fully  connected  network  an \nEquations. \nensemble code  with an  appropriately chosen  ensemble is  sufficient.  If,  however,  the \nefficacies  (11)  and  the  connection  scheme  become  more  involved,  the  construction \nof appropriate ensembles becomes more and more difficult.  Also,  in a finite  network \nwe  cannot make use of the law of large number in  defining the activities (10).  Thus, \nin  general,  we  should  always start with  a  network  model of spiking neurons. \n\n\f470 \n\nGerstner and van Hemmen \n\na) \n\nactivity \n\n~~C =: ] \n\n100 \n\n200 \n\n150 \n\ntime [ms] \n\nb) \n\n30 \n\n..  20 \ng \n!5 \n!  10 \n\n\u00b70 \n\n0 \n\no \n\n.. \n.. . \n\n\u2022 \n0 \n\n\u2022\u2022\u2022 \n\nrata  [Hz] \n\n0) \n30  n-\"~----' \n\n.. \n\u2022 \n. .... \n\n\u2022 \n\ne. \n\n20 \n\n10 \n\nFig.  5 \nSpatio-temporal  spike  pattern. \nIn  this  case,  neither firing  rates \nnor  locally  averaged  activities \ncontain  enough  information  to \ndescribe  the  state  of  the  net(cid:173)\n(a)  Ensemble  averaged \nwork. \nactivity A(t).  (b)  Spike raster of \n30  neurons  out  of a  network  of \n4000.  ( c)  Time-averaged  mean \nfiring  rate f. \n\no  '--__  o-\".'___---:-~---\"'----~  0 \n1()0 \n\n200 \n\n,-,-. (---''---'. \n0-\n\n100  200 \n\nrata  [Hz] \n\nAcknowledgements: This work  has been supported by  the Deutsche  Forschungs(cid:173)\ngemeinschaft  (DFG) under  grant  No.  He  1729/2-1. \n\nReferences \n\nGerstner  W  (1990)  Associative  memory in  a  network  of 'biological' neurons.  In: \nAdvances  in  Neural  Information  Processing  Systems  3,  edited  by  R.P.  Lippmann, \nJ .E.  Moody,  and  D.S.  Touretzky  (Morgan  Kaufmann, San Mateo,  CA)  pp  84-90 \n\nGerstner  Wand  van  Hemmen  JL  (1992a)  Associative  memory  in  a  network  of \n'spiking' neurons.  Network  3:139-164 \n\nGerstner  W, van  Hemmen JL (1993)  Coherence  and incoherence in  a  globally cou(cid:173)\npled  ensemble of pulse-emitting units.  Phys.  Rev.  Lett.  71:312-315 \nGerstner  W,  Ritz  R,  van  Hemmen JL  (1993b)  Why spikes?  Hebbian  learning and \nretrieval  of time-resolved excitation  patterns.  BioI.  Cybern.  69:503-515 \n\nGerstner  Wand  van  Hemmen  JL  (1994)  Coding  and  Information  processing  in \nneural  systems.  In:  Models  of neural  networks,  Vol.  2,  edited  by  E.  Domany,  J .L. \nvan  Hemmen and  K.  Schulten  (Springer-Verlag,  Berlin,  Heidelberg,  New  York)  pp \nIff \n\nHebb  DO  (1949)  The  Organization  of Behavior.  Wiley,  New York \n\nvan  Hemmen JL  and  Kiihn  R(1991)  Collective phenomena in  neural  networks.  In: \nModels  of neural networks, edited by E.  Domany, J .L.  van Hemmen and K.  Schulten \n(Springer-Verlag,  Berlin,  Heidelberg,  New  York)  pp  Iff \n\nHubel  DH,  Wiesel TN  (1962)  Receptive fields,  binocular interaction and functional \narchitecture  in the  cat's visual cortex.  J.  Neurophysiol.  28:215-243 \n\nMacKay DM, McCulloch WS (1952) The limiting information capacity of a neuronal \nlink.  Bull.  of Mathm.  Biophysics  14:127-135 \n\nStein  RB  (1967)  The  information  capacity  of nerve  cells  using  a  frequency  code. \nBiophys.  J.  7:797-826 \n\n\f", "award": [], "sourceid": 850, "authors": [{"given_name": "Wulfram", "family_name": "Gerstner", "institution": null}, {"given_name": "J.", "family_name": "van Hemmen", "institution": null}]}