{"title": "Information through a Spiking Neuron", "book": "Advances in Neural Information Processing Systems", "page_first": 75, "page_last": 81, "abstract": null, "full_text": "Information through a  Spiking Neuron \n\nCharles F.  Stevens and Anthony Zador \n\nSalk  Institute  MNL/S \n\nLa J olIa,  CA 92037 \n\nzador@salk.edu \n\nAbstract \n\nWhile  it  is  generally  agreed  that  neurons  transmit  information \nabout their synaptic inputs through spike trains, the code by which \nthis  information is  transmitted  is  not  well  understood.  An  upper \nbound  on  the  information  encoded  is  obtained  by  hypothesizing \nthat the precise  timing of each spike conveys information.  Here  we \ndevelop  a  general  approach  to quantifying the information carried \nby  spike  trains  under  this  hypothesis,  and  apply  it  to  the  leaky \nintegrate-and-fire  (IF)  model  of  neuronal  dynamics.  We  formu(cid:173)\nlate  the  problem  in  terms  of the  probability  distribution  peT)  of \ninterspike  intervals  (ISIs),  assuming that  spikes  are  detected  with \narbitrary  but finite  temporal resolution .  In  the  absence  of added \nnoise,  all  the  variability in  the  ISIs  could encode  information, and \nthe  information rate  is  simply the entropy of the lSI  distribution, \nH (T)  =  (-p(T) log2 p(T)},  times  the  spike  rate.  H (T)  thus  pro(cid:173)\nvides  an  exact  expression  for  the  information rate.  The  methods \ndeveloped  here  can be  used  to determine experimentally the infor(cid:173)\nmation carried  by  spike  trains,  even  when  the  lower  bound of the \ninformation rate  provided  by  the  stimulus reconstruction  method \nis  not  tight.  In  a  preliminary series  of experiments,  we  have  used \nthese  methods  to estimate information rates of hippocampal neu(cid:173)\nrons  in  slice  in  response  to somatic  current  injection.  These  pilot \nexperiments suggest  information rates as  high as  6.3  bits/spike. \n\n1 \n\nInformation rate of spike trains \n\nCortical neurons use  spike  trains  to communicate with other  neurons.  The output \nof each  neuron  is  a  stochastic function of its input from  the  other neurons.  It is  of \ninterest  to know  how  much each  neuron  is  telling other neurons about its inputs. \n\nHow  much information does  the spike train provide about a signal?  Consider noise \nnet)  added  to  a  signal  set)  to  produce  some  total  input  yet)  =  set)  + net).  This \nis  then  passed  through  a  (possibly  stochastic)  functional  F  to produce  the  output \nspike  train  F[y(t)]  --+  z(t).  We  assume  that  all the  information contained  in  the \nspike  train  can  be  represented  by  the  list  of spike  times;  that  is,  there  is  no extra \ninformation contained in  properties  such  as spike  height or  width.  Note,  however, \nthat many characteristics of the spike  train such  as the mean or instantaneous rate \n\n\f76 \n\nC. STEVENS, A.  ZADOR \n\ncan be  derived  from  this  representation;  if such  a  derivative  property  turns  out  to \nbe  the relevant one,  then  this formulation can be specialized  appropriately. \n\nWe  will  be  interested,  then,  in  the  mutual  information 1(S(t); Z(t\u00bb  between  the \ninput signal ensemble S(t) and the output spike train ensemble Z(t) . This is defined \nin  terms  of the  entropy  H(S)  of the  signal,  the  entropy  H(Z)  of the  spike  train, \nand  their joint entropy  H(S, Z), \n\n1(S; Z) =  H(S) + H(Z) - H(S, Z). \n\n(1) \nNote that the mutual information is  symmetric, 1(S; Z) =  1(Z; S),  since  the joint \nentropy  H(S, Z)  =  H(Z, S).  Note  also  that  if the  signal  S(t)  and  the  spike  train \nZ(t)  are  completely independent,  then  the mutual information is  0,  since  the joint \nentropy is just the sum of the individual entropies H(S, Z) = H(S) + H(Z).  This is \ncompletely in lin'e  with our intuition, since  in  this case  the  spike  train  can  provide \nno  information about the signal. \n\nInformation estimation through stimulus reconstruction \n\n1.1 \nBialek  and  colleagues  (Bialek  et  al.,  1991)  have  used  the  reconstruction  method \nto  obtain  a  strict  lower  bound  on  the  mutual information in  an experimental set(cid:173)\nting.  This method is  based on  an expression  mathematically equivalent  to eq.  (1) \ninvolving the conditional entropy  H(SIZ) of the signal given the spike train, \n\n1(S; Z) \n\nH(S) - H(SIZ) \n\n>  H(S) - Hest(SIZ), \n\n(2) \nwhere  Hest(SIZ)  is  an  upper  bound  on  the  conditional  entropy  obtained  from  a \nreconstruction sest< t) of the signal.  The entropy is estimated from the second order \nstatistics of the reconstruction error e(t) ~ s(t)-sest (t);  from the maximum entropy \nproperty of the Gaussian this is  an upper  bound.  Intuitively, the first  equation says \nthat the information gained  about the spike train by observing the stimulus is just \nthe initial uncertainty of the signal (in the absence  of knowledge of the spike  train) \nminus the uncertainty  that remains about the signal once  the spike train is known, \nand the second  equation says that this second  uncertainty  must be greater for  any \nparticular estimate than for  the optimal estimate. \n\nInformation estimation through spike train reliability \n\n1.2 \nWe have adopted a different approach based an equivalent expression for the mutual \ninformation: \n\n1(S; Z) =  H(Z) - H(ZIS). \n\n(3) \nThe  first  term  H(Z)  is  the  entropy  of the  spike  train,  while  the  second  H(ZIS) \nis  the  conditional  entropy  of the  spike  train  given  the  signal;  intuitively  this  like \nthe inverse  repeatability of the spike  train given  repeated  applications of the  same \nsignal.  Eq.  (3)  has the advantage that, if the spike  train is  a  deterministic function \nof the  input,  it  permits  exact  calculation of the  mutual information .  This follows \nfrom  an important difference  between  the  conditional entropy term here  and in eq. \n2:  whereas  H(SIZ)  has  both a  deterministic and  a  stochastic component, H(ZIS) \nhas only  a  stochastic  component.  Thus in the  absence  of added  noise,  the  discrete \nentropy H(ZIS) =  0,  and eq.  (3)  reduces  to 1(S; Z) =  H(Z). \n\nIf ISIs  are  independent,  then  the  H(Z)  can  be  simply  expressed  in  terms  of the \nentropy of the  (discrete)  lSI  distribution p(T), \n\n00 \n\nH(T) = - LP(1'i) 10g2P(1'i) \n\ni=O \n\n(4) \n\n\fInfonnation Through  a  Spiking  Neuron \n\n77 \n\nas  H(Z)  =  nH(T),  where  n  is  the  number of spikes  in  Z.  Here  p('li)  is  the  prob(cid:173)\nability that the  spike  occurred  in the  interval  (i)~t to (i + l)~t.  The  assumption \nof finite  timing precision  ~t keeps  the  potential information finite.  The advantage \nof considering  the lSI distribution peT)  rather than the full  spike  train distribution \np(Z) is  that the former  is univariate while the latter is multivariate; estimating the \nformer  requires  much less  data. \n\nUnder  what  conditions  are  ISIs  independent?  Correlations between  ISIs  can  arise \neither through the stimulus or the spike generation mechanism itself.  Below we shall \nguarantee that correlations do not arise from the spike-generator by considering the \nforgetful  integrate-and-fire  (IF)  model, in which all information about the  previous \nspike  is  eliminated by  the  next  spike.  If we  further  limit ourselves  to  temporally \nuncorrelated stimuli (i. e.  stimuli drawn from a  white  noise ensemble),  then we  can \nbe sure  that ISIs  are independent,  and eq.  (4)  can  be applied. \n\nIn  the presence  of noise,  H(ZIT)  must also  be evaluated,  to give \n\nf(S; T)  =  H(T) - H(TIS). \n\nH(TIS)  is  the conditional entropy of the lSI given the signal, \n\nH(TIS) =  - / t p(1j ISi(t)) log2 p(1j ISi(t))) \n\n\\J=1 \n\n3;(t) \n\n(5) \n\n(6) \n\nwhere  p(1j ISi(t))  is  the probability of obtaining an lSI of 1j  in  response  to a  par(cid:173)\nticular stimulus Si(t)  in the  presence  of noise net).  The conditional entropy  can be \nthought of as  a  quantification of the reliability of the spike generating mechanism: \nit is  the average  trial-to-trial variability of the spike  train generated  in  response  to \nrepeated  applications of the same stimulus. \n\n1.3  Maximum spike train entropy \nIn what follows,  it will be useful  to compare the information rate for  the IF neuron \nwith  the  limiting case  of an exponential  lSI  distribution,  which  has  the  maximum \nentropy for  any  point process  of the given  rate  (Papoulis,  1984).  This provides  an \nupper  bound on  the  information rate  possible  for  any  spike  train,  given  the  spike \nrate and the temporal precision.  Let  f(T) =  re-rr  be  an exponential  distribution \nwith  a  mean spike  rate r.  Assuming a  temporal precision of ~t, the entropy/spike \nis  H(T)  =  log2  r~t'  and  the  entropy/time  for  a  rate  r  is  rH(T)  =  rlog2 -~ . \nFor  example,  if r  = 1  Hz  and  ~t = 0.001  sec,  this  gives  (11.4  bits/second)  (1 \nspike/second)  =  11.4  bits/spike.  That  is,  if we  discretize  a  1  Hz  spike  train  into \n1  msec  bins,  it  is  nof possible  for  it  to  transmit  more  than  11.4  bits/second.  If \nwe  reduce  the  bin  size  two-fold,  the  rate  increases  by  log2 1/2  = 1  bit/spike  to \n12.4  bits/spike,  while  if we  double  it  we  lose  one  bit/s  to  get  10.4  bit/so  Note \nthat  at  a  different  firing  rate,  e.g.  r  =  2  Hz,  halving  the  bin  size  still  increases \nthe  entropy/spike  by  1 bit/spike,  but  because  the  spike  rate  is  twice  as  high,  this \nbecomes a  2 bit/second  increase  in  the information rate. \n\n1.4  The IF model \nNow  we  consider  the functional :F  describing  the forgetful  leaky  IF model of spike \ngeneration.  Suppose  we  add  some  noise  net)  to  a  signal  set),  yet)  =  net) + set), \nand  threshold  the sum  to produce  a  spike  train  z(t) =  :F[s(t) + net)].  Specifically, \nsuppose  the voltage vet)  of the neuron  obeys vet)  =  -v(t)/r + yet),  where  r  is  the \nmembrane time  constant,  both  s(t~  and  net)  have  a  white  Gaussian  distributions \nand yet)  has mean I' and variance (T \n\u2022  If the voltage reaches the threshold ()o  at some \ntime t,  the neuron  emits a spike  at that time and resets  to the initial condition Vo. \n\n\f78 \n\nc. STEVENS, A. ZAOOR \n\nIn  the language of neurobiology,  this  model can  be  thought of (Tuckwell,  1988)  as \nthe limiting case of a  neuron  with  a leaky IF spike generating  mechanism receiving \nmany excitatory  and  inhibitory synaptic  inputs.  Note  that since  the  input  yet)  is \nwhite,  there  are  no  correlations in  the spike  train  induced  by  the signal,  and  since \nthe  neuron  resets  after  each  spike  there  are  no  correlations  induced  by  the  spike(cid:173)\ngenerating  mechanism.  Thus ISIs  are independent,  and eq.  (4)  r.an  be applied. \n\nWe  will  estimate  the  mutual information  I(S, Z)  between  the  ensemble  of input \nsignals S and the ensemble of outputs Z.  Since in this model ISIs are independent by \nconstruction,  we  need  only evaluate H(T) and H(TIS); for  this we  must determine \np(T),  the  distribution  of ISIs,  and  p(Tlsi),  the  conditional  distribution of ISIs  for \nan ensemble of signals Si(t).  Note  that peT)  corresponds  to the first  passage  time \ndistribution of the Ornstein-Uhlenbeck  process  (Tuckwell,  1988). \n\nThe  neuron  model  we  are  considering  has  two  regimes  determined  by  the  relation \nof the  asymptotic  membrane  potential  (in  the  absence  of threshold)  J.l.T  and  the \nthreshold  (J.  In the  suprathreshold regime, J.l.T  > (J,  threshold  crossings occur even  if \nthe signal  variance  is  zero  (0-2 =  0).  In  the  subthreshold regime,  J.l.T  ~ (J,  threshold \ncrossings occur only if 0-2  > O.  However,  in  the limit that E{T} ~ T,  i.e.  the mean \nfiring rate is low  compared with  the integration time constant  (this can only occur \nin  the subthreshold  regime),  the  lSI  distribution is  exponential,  and  its coefficient \nof variation (CV) is unity (cf.  (Softky and Koch,  1993)).  In this low-rate regime the \nfiring  is  deterministically Poisson;  by  this we  mean to distinguish it from the more \nusual  usage of Poisson  neuron,  the stochastic situation in  which  the instantaneous \nfiring  rate  parameter  (the  probability of firing  over  some  interval)  depends  on  the \nstimulus (i.e.  f  ex:  set)).  In the present  case  the  exponential lSI distribution arises \nfrom  a  deterministic mechanism. \n\nAt the border between these regimes, when the threshold is just equal to the asymp(cid:173)\ntotic  potential,  (Jo  =  J.l.T,  we  have  an  explicit  and  exact  solution for  the  entire  lSI \ndistribution (Sugiyama et al.,  1970) \n\npeT) =  (J.l.T)(T/2)-3/2 [e 2T1T  _  1]-3/ 2exp(2T/T _ \n\n(211\")1/20-\n\n(J.l.T? \n\n(0-2T)(e 2TIT  - 1) \n\n). \n\n(7) \n\nThis is the special case where,  in the absence of fluctuations (0- 2  =  0), the membrane \npotential hovers just subthreshold.  Its neurophysiological interpretation is that the \nexcitatory inputs just balance the inhibitory inputs, so that the neuron  hovers just \non the verge  of firing. \n\nInformation rates for  noisy and noiseless signals \n\n1.5 \nHere we compare the information rate for a IF neuron at the \"balance point\"  J.l.T  =  (J \nwith the maximum entropy spike train.  For simplicity and brevity we  consider only \nthe  zero-noise  case,  i.e.  net)  =  O.  Fig.  1A  shows  the  information per  spike  as  a \nfunction  of the  firing  rate  calculated  from  eq.  (7),  which  was  varied  by  changing \nthe  signal  variance  0-2 .  We  assume  that  spikes  can  be  resolved  with  a  temporal \nresolution  of 1  msec,  i. e. \nthat  the  lSI  distribution  has  bins  1  msec  wide.  The \ndashed line shows the theoretical upper bound given by the exponential distribution; \nthis  limit  can  be  approached  by  a  neuron  operating  far  below  threshold,  in  the \nPoisson  limit.  For  both  the  IF  model  and  the  upper  bound,  the  information  per \nspike  is  a  monotonically  decreasing  function  of the  spike  rate;  the  model  almost \nachieves  the  upper  bound  when  the  mean lSI  is just equal  to the  membrane time \nconstant.  In the model the information saturates at very low firing rates,  but for the \nexponential  distribution  the  information increases  without  bound.  At  high  firing \nrates the information goes to zero  when the firing rate is too fast for  individual ISIs \nto be resolved  at the temporal resolution.  Fig.  1B  shows  that the information rate \n(information per  second)  when  the  neuron  is  at the  balance  point goes  through  a \n\n\fInfonnation Through  a Spiking  Neuron \n\n79 \n\nmaximum as  the  firing  rate  increases.  The maximum occurs  at a  lower  firing  rate \nthan for  the exponential distribution  (dashed  line). \n\n1.6  Bounding information rates by stimulus reconstruction \nBy  construction,  eq.  (3)  gives  an exact  expression  for  the  information rate in  this \nmodel.  We  can therefore  compare the lower  bound provided by the stimulus recon(cid:173)\nstruction  method  eq.  (2)  (Bialek  et  aI.,  1991).  That is,  we  can  assess  how  tight \na  lower  bound  it  provides.  Fig.  2  shows  the  lower  bound  provided  by  the  recon(cid:173)\nstruction  (solid  line)  and  the reliability  (dashed  line)  methods as  a  function  of the \nfiring  rate.  The  firing  rate  was  increased  by  increasing  the  mean  p.  of the  input \nstimulus  yet),  and  noise  was  set  to  O.  At  low  firing  rates  the  two  estimates  are \nnearly  identical,  but  at  high  firing  rates  the  reconstruction  method  substantially \nunderestimates the information rate.  The amount of the underestimate depends on \nthe model parameters,  and  decreases  as  noise  is  added  to the stimulus.  The tight(cid:173)\nness  of the  bound  is  therefore  an  empirical question.  While  Bialek  and  colleagues \n(1996) show that under the conditions of their experiments the underestimate is less \nthan a factor of two,  it is  clear that the  potential for  underestimate under  different \nconditions or in different  systems is  greater. \n\n2  Discussion \n\nWhile it is generally  agreed  that spike  trains encode  information about a  neuron's \ninputs,  it is  not  clear  how  that  information is  encoded.  One  idea is  that  it is  the \nmean firing rate alone that encodes the signal, and that variability about this mean \nis effectively noise.  An alternative view is that it is the variability itself that encodes \nthe signal,  i. e. \nthat the information is encoded in the precise  times at which spikes \noccur.  In  this  view  the  information  can  be  expressed  in  terms  of the  interspike \ninterval  (lSI)  distribution  of  the  spike  train.  This  encoding  scheme  yields  much \nhigher information rates  than one in which  only the mean rate (over  some interval \nlonger  than the typical lSI)  is considered.  Here  we  have quantified  the information \ncontent of spike  trains under  the latter hypothesis for  a  simple neuronal  model. \n\nWe  consider a  model in which by construction the ISIs  are independent,  so that the \ninformation rate  (in  bits/sec)  can  be  computed  directly  from  the  information per \nspike  (in  bits/spike)  and  the  spike  rate  (in spikes/sec).  The information per  spike \nin  turn  depends  on  the  temporal  precision  with  which  spikes  can  be  resolved  (if \nprecision were  infinite, then  the information content would be infinite as  well, since \nany  message could for  example be encoded  in the  decimal expansion of the precise \narrival time of a  single spike),  the reliability of the spike  transduction  mechanism, \nand the entropy of the lSI  distribution itself.  For low  firing  rates,  when  the neuron \nis in the subthreshold limit, the lSI distribution is close to the theoretically maximal \nexponential distribution. \n\nMuch of the recent  interest in information theoretic analyses of the neural code can \nattributed  to the seminal work  of Bialek  and colleagues  (Bialek et  al.,  1991;  Rieke \net al.,  1996), who measured the information rate for sensory neurons in a number of \nsystems.  The present results are in broad agreement with those of DeWeese (1996)  , \nwho considered  the information rate of a  linear-filtered threshold crossing!  (LFTC) \nmodel.  DeWeese developed a functional expansion, in which the first  term describes \nthe  limit in  which  spike  times  (not  ISIs)  are  independent,  and  the  second  term  is \na  correction  for  correlations.  The  LFTC  model  differs  from  the  present  IF  model \nmainly  in  that  it  does  not  \"reset\"  after  each  spike.  Consequently  the  \"natural\" \n\n1 In  the LFTC  model,  Gaussian  signal  and  noise  are  convolved  with  a  linear  filter;  the \n\ntimes  at  which  the resulting  waveform  crosses  some  threshold  are called  \"spikes\". \n\n\f80 \n\nC. STEVENS, A.  ZADOR \n\nrepresentation  of the  spike  train  in  the  LFTC  model  is  as  a  sequence  to  . . . tn  of \nfiring  times,  while  in  the  IF  model  the  \"natural\"  representation  is  as  a  sequence \nTl  .. . Tn  of ISIs.  The choice is one of convenience,  since the two representations are \nequivalent. \n\nThe two models are complementary. In the LFTC model, results can be obtained for \ncolored signals and noise, while such conditions are awkward in the IF model.  In the \nIF  model by  contrast,  a  class of highly  correlated  spike  trains can  be  conveniently \nconsidered that are awkward in the LFTC model.  That is,  the indendent-ISI condi(cid:173)\ntion required in the IF model is less restrictive than the independent-spike condition \nof the  LFTC  model-spikes are  independent  iff ISIs  are  indepenndent  and the  lSI \ndistribution  p(T)  is  exponential.  In  particular,  at  high  firing  rates  the  lSI  distri(cid:173)\nbution can be far from exponential (and therefore  the spikes far from independent) \neven  when the ISIs  themselves are  independent. \n\nBecause we  have assumed that the input s(t) is white, its entropy is infinite, and the \nmutual information can  grow without bound as  the  temporal precision  with which \nspikes  are  resolved  improves.  Nevertheless,  the  spike  train  is  transmitting only  a \nminute fraction of the total available information.  The signal thereby saturates the \ncapacity  of the  spike  train.  While  it  is  not  at  all  clear  whether  this  is  how  real \nneurons  actually behave,  it is not implausible:  a  typical cortical neuron receives  as \nmany as 104 synaptic inputs, and if the information rate of each input is the same as \nthe target,  then  the information rate  impinging upon the target is  104-fold greater \n(neglecting synaptic unreliability, which  could  decrease  this  substantially)  than its \ncapacity. \n\nIn a  preliminary series  of experiments,  we  have used  the reliability method to esti(cid:173)\nmate the information rate of hippocampal neuronal spike trains in slice in response \nto somatic current  injection  (Stevens  and  Zador,  unpublished).  Under  these  condi(cid:173)\ntions ISIs  appear to be independent,  so  the method developed  here  can be applied. \nIn  these  pilot  experiments,  an  information rates  as  high  as  6.3  bits/spike  was  ob(cid:173)\nserved. \n\nReferences \n\nBialek,  W.,  Rieke,  F.,  de  Ruyter  van  Steveninck,  R.,  and  Warland,  D.  (1991). \n\nReading a  neural code.  Science,  252:1854- 1857. \n\nDeWeese,  M.  (1996).  Optimization  principles  for  the  neural  code.  In  Hasselmo, \nM.,  editor,  Advances  in  Neural Information  Processing  Systems,  vol.  8.  MIT \nPress,  Cambridge, MA. \n\nPapoulis,  A.  (1984) .  Probability,  random  variables  and  stochastic  processes,  2nd \n\nedition.  McGraw-Hill. \n\nRieke,  F., Warland, D., de Ruyter van Steveninck, R., and Bialek, W . (1996).  Neural \n\nCoding.  MIT Press. \n\nSoftky,  W .  and  Koch,  C.  (1993).  The  highly  irregular  firing  of cortical  cells  is \ninconsistent  with  temporal  integration  of  random  epsps.  J .  Neuroscience., \n13:334-350. \n\nSugiyama, H.,  Moore,  G.,  and  Perkel,  D.  (1970).  Solutions for  a  stochastic  model \n\nof neuronal spike  production.  Mathematical  Biosciences,  8:323-34l. \n\nTuckwell, H.  (1988).  Introduction  to  theoretical neurobiology  (2  vols.).  Cambridge. \n\n\fInfonnation Through a  Spiking  Neuron \n\n81 \n\nInformation at balance point \n\n\u00a7 1000 \n~ \n15  500 \n\n/ \n\n/ \n\n/ \n\n/ \n\n/ \n\n/ \n\n, \n\n\\ \n\n\\ \n\\ \n\\ \n\\ \n\\ \n\noL-~~====~--~~~~--~~~~~~~~~~ \n1~ \n1~ \n\n1~ \n\n1~ \n\n1~ \n\nfiring rate (Hz) \n\nFigure  1:  Information  rate  at  balance  point.  (A;  top)  The  information per  spike \ndecreases  monotonically  with  the  spike  rate  (solid  line) .  It is  bounded  above  by \nthe entropy of the exponential limit (dashed  line),  which  is  the highest entropy lSI \ndistribution  for  a  given  mean  rate;  this  limit  is  approached  for  the  IF  neuron  in \nthe subthreshold  regime.  The information rate  goes  to  0 when  the firing  rate is  of \nthe  same order  as  the  temporal resolution  tit.  The  information per  spike  at  the \nbalance  point  is  nearly  optimal when  E{T}  :::::::  T.  (T  = 50  msec;  tit  = 1  msec); \n(B;  bottom)  Information  per  second  for  above  conditions.  The  information  rate \nfor  both  the  balance  point  (solid  curve)  and  the  exponential  distribution  (dashed \ncurve)  pass  through  a  maximum,  but  the  maximum is  greater  and  occurs  at  an \nhigher rate for  the latter.  For firing rates much smaller than T,  the rates are almost \nindistinguishable.  (T =  50  msec;  tit =  1 msec) \n\n~r-----~----~----~----~-----'~----~----~----, \n\n250 \n\n200 \n\n1150 \nD \n\n100 \n\n\" \" \n\n\",'\" \n\n,,-\n\n,,-\n\n~V-----\n\n0 0 \n\n10 \n\n20 \n\n30 \n\n40 \n\nspike rate (Hz) \n\n50 \n\neo \n\n70 \n\n80 \n\nFigure  2:  Estimating  information  by  stimulus  reconstruction.  The  information \nrate estimated  by  the  reconstruction  method  solid  line  and  the exact  information \nrate  dashed  line  are  shown  as  a  function  of  the  firing  rate.  The  reconstruction \nmethod  significantly  underestimates  the  actual  information,  particularly  at  high \nfiring  rates.  The firing  rate  was  varied through  the mean input p.  The parameters \nwere:  membrane time  constant  T  =  20  msec;  spike  bin  size  tit  =  1  msec;  signal \nvariance 0\";  =  0.8;  threshold  Q =  10. \n\n\f", "award": [], "sourceid": 1135, "authors": [{"given_name": "Charles", "family_name": "Stevens", "institution": null}, {"given_name": "Anthony", "family_name": "Zador", "institution": null}]}