{"title": "When is an Integrate-and-fire Neuron like a Poisson Neuron?", "book": "Advances in Neural Information Processing Systems", "page_first": 103, "page_last": 109, "abstract": null, "full_text": "When is  an Integrate-and-fire  Neuron \n\nlike a  Poisson Neuron? \n\nCharles F.  Stevens \nSalk Institute  MNL/S \n\nLa Jolla, CA  92037 \n\ncfs@salk.edu \n\nAnthony  Zador \n\nSalk Institute  MNL/S \n\nLa Jolla, CA  92037 \n\nzador@salk.edu \n\nAbstract \n\nIn the  Poisson neuron  model, the output is  a  rate-modulated Pois(cid:173)\nson  process  (Snyder  and  Miller,  1991);  the  time  varying  rate  pa(cid:173)\nrameter  ret)  is  an  instantaneous  function  G[.]  of  the  stimulus, \nret)  =  G[s(t)].  In  a  Poisson  neuron,  then,  ret)  gives  the  instan(cid:173)\ntaneous firing  rate-the instantaneous probability of firing  at  any \ninstant t-and the output is  a  stochastic function  of the input.  In \npart because of its great simplicity, this model is  widely used  (usu(cid:173)\nally with  the addition of a  refractory  period), especially  in  in  vivo \nsingle unit electrophysiological studies,  where  set)  is usually taken \nto be  the value of some sensory  stimulus.  In the  integrate-and-fire \nneuron model, by  contrast, the output is a filtered and thresholded \nfunction  of the input:  the input is  passed  through a  low-pass filter \n(determined by the membrane time constant T)  and integrated un(cid:173)\ntil the membrane potential vet)  reaches threshold 8,  at which  point \nvet)  is reset  to its initial value.  By contrast with the Poisson model, \nin the integrate-and-fire model the ouput is a deterministic function \nof the input.  Although the integrate-and-fire model is  a caricature \nof real  neural  dynamics,  it  captures  many of the  qualitative  fea(cid:173)\ntures,  and  is  often  used  as  a  starting point for  conceptualizing the \nbiophysical behavior of single neurons.  Here we show how a slightly \nmodified  Poisson model can  be  derived  from  the integrate-and-fire \nmodel with noisy inputs  yet)  =  set) + net).  In  the modified model, \nthe  transfer  function  G[.] is  a  sigmoid  (erf)  whose  shape  is  deter(cid:173)\nmined  by  the  noise  variance  /T~.  Understanding  the  equivalence \nbetween  the  dominant  in  vivo  and  in  vitro  simple neuron  models \nmay help  forge  links between  the  two levels. \n\n\f104 \n\n1 \n\nIntroduction \n\nc. F. STEVENS. A. ZADOR \n\nIn  the  Poisson  neuron  model,  the output is  a  rate-modulated  Poisson process;  the \ntime  varying  rate  parameter  ret)  is  an  instantaneous  function  G[.] of the  stimu(cid:173)\nlus,  ret)  =  G[s(t)].  In  a  Poisson  neuron,  then,  ret)  gives  the  instantaneous firing \nrate-the instantaneous probability of firing  at any instant  t-and the output  is  a \nstochastic function  of the  input.  In part because  of its great simplicity, this model \nis  widely  used  (usually  with  the  addition  of a  refractory  period),  especially  in  in \nvivo  single  unit  electrophysiological  studies,  where  set)  is  usually  taken  to  be  the \nvalue of some sensory stimulus. \n\nIn  the  integrate-and-fire  neuron  model,  by  contrast,  the  output  is  a  filtered  and \nthresholded  function  of  the  input:  the  input  is  passed  through  a  low-pass  filter \n(determined by the membrane time constant T)  and integrated until the membrane \npotential  vet)  reaches  threshold  0,  at  which  point  vet)  is  reset  to  its  initial value. \nBy  contrast  with  the  Poisson  model,  in  the  integrate-and-fire  model  the  ouput  is \na  deterministic  function  of the  input.  Although  the  integrate-and-fire  model  is  a \ncaricature of real  neural dynamics, it captures many of the qualitative features,  and \nis often used as a starting point for conceptualizing the biophysical behavior of single \nneurons  (Softky and Koch ,  1993; Amit and Tsodyks,  1991;  Shadlen and Newsome, \n1995;  Shadlen  and  Newsome,  1994;  Softky,  1995;  DeWeese,  1995;  DeWeese,  1996; \nZador and  Pearlmutter,  1996). \n\nHere  we  show  how  a  slightly  modified  Poisson  model  can  be  derived  from  the \nintegrate-and-fire  model  with  noisy  inputs  yet)  =  set)  + net). \nIn  the  modified \nmodel,  the  transfer  function  G[.]  is  a  sigmoid  (erf)  whose  shape  is  determined  by \nthe noise variance (j~ .  Understanding the equivalence between the dominant in  vivo \nand  in  vitro simple neuron  models may help forge  links between  the  two levels. \n\n2  The integrate-and-fire model \n\nHere  we  describe  the  the forgetful  leaky integrate-and-fire model.  Suppose  we  add \na  signal set)  to some noise  net), \n\nyet)  =  net) + set), \n\nand  threshold  the sum to produce  a  spike  train \n\nz(t) =  F[s(t) + net)], \n\nwhere F  is the thresholding functional and z(t)  is a  list of firing times generated by \nthe input.  Specifically, suppose  the voltage vet)  of the neuron  obeys \n\nvet)  = - vet)  + yet) \n\nT \n\n(1) \n\nwhere  T  is  the membrane time constant.  We assume that  the noise  net)  has O-mean \nand  is  white  with  variance  (j~.  Thus  yet)  can  be  thought  of as  a  Gaussian  white \nprocess  with variance  (j~  and  a  time-varying mean set) .  If the  voltage reaches  the \nthreshold  00  at  some  time t,  the  neuron  emits  a  spike  at  that  time  and  resets  to \nthe  initial  condition  Vo.  This  is  therefore  a  5  parameter  model:  the  membrane \ntime constant  T,  the  mean  input signal Il,  the  variance of the input  signal  172 ,  the \nthreshold  0,  and  the  reset  value  Vo.  Of course,  if net)  =  0,  we  recover  a  purely \ndeterministic integrate-and-fire model. \n\n\fWhen Is  an  Integrate-and-fire Neuron like a Poisson Neuron? \n\n105 \n\nIn order  to forge  the link between  the integrate-and-fire  neuron  dynamics and  the \nPoisson  model,  we  will  treat  the  firing  times T  probabilistically.  That  is,  we  will \nexpress  the  output  of the  neuron  to  some  particular  input  set)  as  a  conditional \ndistribution  p(Tls(t\u00bb,  i.e. \nthe  probability  of obtaining  any  firing  time  T  given \nsome particular input set) . \n\nUnder  these  assumptions,  peT)  is  given  by  the  first  passage  time  distribution \n(FPTD)  of the  Ornstein-Uhlenbeck  process  (Uhlenbeck  and Ornstein,  1930; Tuck(cid:173)\nwell,  1988).  This  means  that  the  time evolution  of the  voltage  prior  to  reaching \nthreshold  is given  by  the  Fokker-Planck equation  (FPE), \n\n8 \n8t g(t, v) = 2  8v2 get, v)  - av [(set)  - --;:- )g(t, v)], \n\nvet) \n\n8 \n\nu;  82 \n\nwhere  uy =  Un  and  get, v)  is  the  distribution  at  time t  of voltage  -00 <  v  ::;  (}o. \nThen  the first  passage time distribution  is  related  to g( v, t)  by \n\n(2) \n\n(3) \n\n81 90 \n\npeT)  =  - at \n\n-00 get, v)dv. \n\nThe integrand is  the fraction of all paths that p.ave not yet  crossed  threshold.  peT) \nis  therefore just  the  interspike  interval  (lSI)  distribution for  a  given  signal  set).  A \ngeneral  eigenfunction  expansion  solution  for  the  lSI  distribution  is  known,  but  it \nconverges  slowly  and  its terms offer  little insight into the  behavior (at  least  to us) . \n\nWe  now  derive  an expression  for  the  probability of crossing threshold  in some very \nshort  interval  ~t, starting  at  some  v.  We  begin  with  the  \"free\"  distribution  of g \n(Tuckwell,  1988):  the  probability of the  voltage jumping to v'  at time t' = t + ~t, \ngiven  that  it  was  at  v  at  time t,  assuming  von  Neumann  boundary  conditions  at \nplus and  minus infinity, \n\nget', v'lt, v)  = \n\n1 \n\n[ \nexp  -\n\nJ27r  q(~t;Uy) \n\n(v'  - m( ~t; u  \u00bb)2] \n\ny ,  \n\n2 q(~t;Uy) \n\n(4) \n\nwith \n\nand \n\nm(~t) = ve- at / T  + set) * T(l  _  e- at / T ), \n\nwhere  * denotes  convolution.  The  free  distribution  is  a  Gaussian  with  a  time(cid:173)\ndependent  mean m(~t) and  variance  q(~t; uy).  This expression  is valid for  all  ~t. \nThe probability of making a jump \n\nin a  short  interval ~t ~ T  depends  only on  ~v and ~t, \n\n~v = v' - v \n\nga(~t, ~v; uy) = \n\nFor small ~t, we  expand to get \n\n1 \n\n..j27r  qa(uy) \n\nexp  [_  ~~2  )]. \n\n2 qa  uy \n\n(5) \n\nwhich  is  independent of T,  showing that the leak  can  be  neglected  for  short  times. \n\nqa(uy) ::::::  2u;~t, \n\n\f106 \n\nc. F. STEVENS, A.  ZADOR \n\nNow  the probability Pt>,  that the voltage exceeds  threshold in some short Ilt, given \nthat it started at v,  depends  on how  far  v  is  from threshold;  it is \n\nPr[v + Ilv  ~ 0]  =  Pr[llv  ~ 0 - v]. \n\nThus \n\n(6) \n\n(Xl  dvgt>,(llt, v; O\"y) \nJ9-v \n1  (o-v) \n1  (o-v) \n\nJ2qt>,(O\"y) \n\nO\"yJ21lt \n\n-erfc \n2 \n\n-erfc \n2 \n\nwhere  erfc(x)  =  1 - -j; I; e-t~ dt  goes  from  [2  :  0].  This  then  is  the  key  result: \nit  gives  the  instantaneous  probability  of firing  as  a  function  of the  instantaneous \nvoltage v.  erfc  is sigmoidal with a slope determined by O\"y,  so a smaller noise yields \na steeper  (more deterministic)  transfer function;  in the limit of 0 noise,  the transfer \nfunction  is a  step  and we  recover  a  completely deterministic neuron. \n\nNote  that  Pt>,  is  actually an instantaneous function  of v(t),  not  the stimulus itself \ns(t).  If the  noise  is  large  compared  with  s(t)  we  must  consider  the  distribution \ng$ (v, t; O\"y)  of voltages reached  in response  to the input  s(t): \n\nPy(t) \n\n(7) \n\n3  Ensemble of Signals \n\nWhat if the inputs s(t) are themselves drawn from an ensemble?  If their distribution \nis  also  Gaussian  and  white  with mean  Jl  and  variance 0\";,  and  if the  firing  rate  is \nlow  (E[T] ~ T),  then  the output spike train is  Poisson.  Why is firing  Poisson  only \nin the  slow firing  limit?  The reason  is  that,  by  assumption,  immediately following \na  spike  the membrane potential resets  to 0;  it must then  rise  (assuming Jl  > 0)  to \nsome asymptotic level that is  independent of the initial conditions.  During this rise \nthe firing  rate is  lower than the asymptotic rate,  because on average the membrane \nis farther from threshold,  and its variance is lower.  The rate at which the asymptote \nis  achieved  depends  on  T.  In  the  limit as  t  ~ T,  some  asymptotic  distribution  of \nvoltage  qoo(v),  is  attained.  Note  that  if we  make  the  reset  Vo  stochastic,  with  a \ndistribution  given  by  qoo (v),  then  the  firing  probability  would  be  the  same  even \nimmediately after spiking,  and firing  would be  Poisson for  all firing  rates. \n\nA  Poisson  process  is  characterized  by  its mean alone.  We  therefore  solve  the  FPE \n(eq.  2)  for  the  steady-state  by  setting  \u00b0tg(t, v)  =  0  (we  consider  only  threshold \ncrossings  from  initial  values  t  ~ T;  negYecting  the  early  events  results  in  only  a \nsmall error, since we  have assumed E{T} ~ T).  Thus with the absorbing boundary \n\n\fWhen Is an Integrate-and-fire Neuron like a Poisson Neuron? \n\nat 0 the  distribution at time t  ~ T  (given  here  for  JJ = 0)  is \n\ng~(Vj uy) = kl  (1 - k2erfi [uyfi]) exp [~i:] , \n\n107 \n\n(8) \n\nwhere u; = u; + u~, erfi(z) = -ierf(iz), kl  determines the normalization (the sign \nof kl  determines whether  the  solution extends to positive or  negative infinity)  and \nk2  = l/erfi(O/(uy Vr))  is  determined  by  the  boundary.  The instantaneous  Poisson \nrate parameter is  then obtained through eq.  (7), \n\n(9) \n\nFig.  1 tests  the  validity of the  exponential  approximation.  The  top  graph  shows \nthe lSI distribution near the  \"balance point\" , when the excitation is in balance with \nthe  inhibition and  the  membrane potential  hovers just subthreshold.  The  bottom \ncurves  show  the  lSI  distribution  far  below  the  balance  point.  In  both  cases,  the \nexponential distribution provides a  good approximation for t  ~ T. \n\n4  Discussion \n\nThe main point of this paper  is  to make explicit  the  relation  between  the  Poisson \nand  integrate-and-fire  models  of  neuronal  acitivity.  The  key  difference  between \nthem is that the former is stochastic while the latter is deterministic.  That is, given \nexactly  the  same stimulus,  the  Poisson  neuron  produces  different  spike  trains  on \ndifferent  trials, while the integrate-and-fire neuron  produces exactly the same spike \ntrain  each  time.  It is  therefore  clear  that  if some  degree  of stochasticity  is  to  be \nobtained  in  the  integrate-and-fire  model,  it  must  arise  from  noise  in  the stimulus \nitself. \n\nThe relation  we  have  derived  here  is  purely formalj we  have intentionally remained \nagnostic about the deep issues of what is signal and what is  noise in the inputs to a \nneuron.  We observe  nevertheless  that although we  derive  a  limit (eq.  9)  where  the \nspike train of an integrate-and-fire neuron is a Poisson process-i.e.  the probability \nof obtaining a spike in any  interval is independent  of obtaining a spike in any other \ninterval  (except  for  very  short  intervals )-from the  point  of view  of information \nprocessing  it is  a  very  different  process  from  the purely  stochastic  rate-modulated \nPoisson  neuron.  In  fact,  in  this  limit the  spike  train  is  deterministically  Poisson \nif u y  = u.,  i. e.  when  n( t)  =  OJ  in  this  case  the  output  is  a  purely  deterministic \nfunction  of the input,  but the lSI  distribution is exponential. \n\n\f108 \n\nReferences \n\nC. F. STEVENS, A.  ZADOR \n\nAmit,  D.  and  Tsodyks,  M.  (1991).  Quantitative  study  of attractor  neural  net(cid:173)\n\nwork retrieving at low spike rates. i. substrate-spikes,  rates and neuronal gain. \nNetwork:  Computation  in  Neural Systems, 2:259-273 . \n\nDeWeese,  M.  (1995).  Optimization principles for the  neural code.  PhD thesis,  Dept \n\nof Physics,  Princeton  University. \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\nShadlen, M.  and Newsome, W. (1994) . Noise, neural codes and cortical organization. \n\nCurrent  Opinion  in  Neurobiology,  4:569-579. \n\nShadlen,  M.  and  Newsome,  W.  (1995) .  Is  there  a  signal  in  the  noise?  [comment]. \n\nCurrent  Opinion  in  Neurobiology,  5:248-250. \n\nSnyder,  D.  and Miller,  M.  (1991).  Random  Point  Processes  in  Time  and Space,  2nd \n\nedition.  Springer-Verlag. \n\nSoftky,  W.  (1995) .  Simple codes  versus  efficient  codes.  Current  Opinion  in  Neuro(cid:173)\n\nbiology,  5:239-247 . \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\nTuckwell,  H.  (1988).  Introduction  to  theoretical neurobiology  (2 vols.).  Cambridge. \nUhlenbeck,  G.  and Ornstein,  L.  (1930).  On  the  theory  of brownian  motion.  Phys. \n\nRev., 36:823-84l. \n\nZador, A.  M.  and Pearlmutter, B.  A.  (1996) . VC  dimension of an integrate and fire \n\nneuron  model.  Neural  Computation,  8(3).  In  press. \n\n\fWhen Is an  Integrate-and-fire Neuron like a Poisson Neuron? \n\n109 \n\nlSI distributions at balance point and the exponential limit \n\n0.02 \n\n0.015 \n\n.~ \n15 \n\n.8 e a. \n\n0.01 \n\n0.005 \n\n50 \n\n100 \n\n150 \n\n2 x 10-3 \n\n200 \n\n250 \n\nTime (msec) \n\n300 \n\n350 \n\n400 \n\n450 \n\n500 \n\n1.5 \n\n~ \n~  1 \n0 ... a. \n\n.0 \n\n0.5 \n\n0 \n0 \n\n200 \n\n400 \n\n600 \n\n800 \n\n1000 \n\nlSI  (msec) \n\n1200 \n\n1400 \n\n1600 \n\n1800 \n\n2000 \n\nFigure  1:  lSI  distributions.  (A;  top)  lSI  distribution  for  leaky  integrate-and-fire \nmodel  at the balance point, where  the asymptotic membrane potential is just sub(cid:173)\nthreshold, for  two values of the signal variance (1'2 .  Increasing  (1'2  shifts the distribu(cid:173)\ntion  to  the left .  For the left  curve,  the parameters were  chosen  so  that  E{T} ~ T, \ngiving a  nearly exponential distribution; for  the  right curve,  the distribution would \nbe  hard  to  distinguish  experimentally from  an  exponential  distribution with  a  re(cid:173)\nfractory  period.  (T  = 50  msec;  left:  E{T} = 166  msec;  right:  E{T}  =  57  msec). \n(B;  bottom)  In  the  subthreshold  regime,  the lSI  distribution (solid}  is  nearly expo(cid:173)\nnential  (dashed)  for  intervals greater  than  the  membrane time constant.  (T  =  50 \nmsec;  E{T} = 500  msec) \n\n\f", "award": [], "sourceid": 1057, "authors": [{"given_name": "Charles", "family_name": "Stevens", "institution": null}, {"given_name": "Anthony", "family_name": "Zador", "institution": null}]}