{"title": "Dynamical Constraints on Computing with Spike Timing in the Cortex", "book": "Advances in Neural Information Processing Systems", "page_first": 285, "page_last": 292, "abstract": null, "full_text": "Dynamical Constraints  on  Computing \n\nwith  Spike  Timing  in  the  Cortex \n\nArunava Banerjee and Alexandre Pouget \nDepartment of Brain and Cognitive Sciences \n\nUniversity of Rochester, Rochester, New York 14627 \n\n{arunavab, alex} @bcs.rochester.edu \n\nAbstract \n\nIf the  cortex uses  spike timing to  compute,  the  timing of the  spikes \nmust  be  robust to  perturbations.  Based on  a  recent  framework  that \nprovides  a  simple  criterion  to  determine  whether  a  spike  sequence \nproduced by a generic network is  sensitive to  initial conditions, and \nnumerical  simulations  of  a  variety  of  network  architectures,  we \nargue  within  the  limits  set  by  our  model  of the  neuron,  that  it  is \nunlikely  that  precise  sequences  of  spike  timings  are  used  for \ncomputation under conditions typically found  in the cortex. \n\n1  Introduction \n\nSeveral  models  of neural  computation  use  the  precise  timing  of spikes  to  encode \ninformation.  For  example,  Abeles  et  al.  have  proposed  synchronous  volleys  of \nspikes  (synfire  chains)  as  a  candidate for  representing  information in the  cortex  [1]. \nMore  recently,  Maass  has  demonstrated  how  spike  timing  in  general,  not  merely \nsynfire chains, can be utilized to perform nonlinear computations [6]. \n\nFor  any  of these  schemes  to  function,  the  timing  of the  spikes  must  be  robust  to \nsmall  perturbations;  i.e.,  small  perturbations  of spike  timing  should  not  result  in \nsuccessively  larger  fluctuations  in  the  timing  of  subsequent  spikes.  To  use  the \nterminology  of dynamical  systems  theory,  the  network  must  not  exhibit  sensitivity \nto  initial  conditions.  Indeed,  reliable computation would simply be  impossible if the \ntiming of spikes  is  sensitive to  the  slightest source of noise, such  as  synaptic release \nvariability, or thermal fluctuations in the opening and closing of ionic channels. \n\nDiesmann et  al.  have  recently  examined  this  issue  for  the  particular case  of synfire \nchains  in  feed-forward  networks  [4].  They  have  demonstrated  that  the  propagation \nof a synfire chain over several layers of integrate-and-fire neurons can be robust to  2 \nHz  of random  background  activity  and  to  a  small  amount  of noise  in  the  spike \ntimings.  The  question  we  investigate  here  is  whether  this  result  generalizes  to  the \npropagation  of  any  arbitrary  spatiotemporal  configuration  of  spikes  through  a \nrecurrent network of neurons.  This  question is  central  to  any  theory  of computation \nin  cortical  networks  using  spike  timing  since  it  is  well  known  that  the  connectivity \nbetween  neurons  in  the  cortex  is  highly recurrent.  Although  there  have  been  earlier \nattempts  at  resolving  like  issues,  the  applicability  of the  results  are  limited  by  the \nmodel of the neuron [8]  or the pattern of propagated spikes [5]  considered. \n\n\fBefore  we  can  address  this  question  in  a  principled  manner,  however,  we  must \nconfront a couple of confounding issues.  First stands the problem of stationarity.  As \nis  well  known,  Lyapunov characteristic exponents of trajectories are  limit quantities \nthat  are  guaranteed  to  exist  (almost  surely)  in  classical  dynamical  systems  that  are \nstationary.  In  systems  such  as  the  cortex that receive  a  constant barrage  of transient \ninputs,  it  is  questionable  whether such  a concept bears much  relevance.  Fortunately, \nour  simulations  indicate  that  convergence  or  divergence  of trajectories  in  cortical \nnetworks  can  occur  very  rapidly  (within  200-300  msec).  Assuming  that  external \ninputs  do  not  change  drastically  over  such  short  time  scales,  one  can  reasonably \napply the results from analysis under stationary conditions to  such systems. \n\nSecond,  the  issues  of how  a  network  should  be  constructed  so  as  to  generate  a \nparticular spatiotemporal  pattern of spikes as  well  as  whether a given spatiotemporal \npattern  of spikes  can  be  generated  in  principle,  remain  unresolved  in  the  general \nsetting.  It might  be  argued that  without  such  knowledge,  any classification  of spike \npatterns  into  sensitive  and  insensitive  classes  is  inherently incomplete.  However,  as \nshall  be  demonstrated  later,  sensitivity  to  initial  conditions  can  be  inferred  under \nrelatively  weak  conditions.  In  addition,  we  shall  present  simulation  results  from  a \nvariety of network architectures to  support our general conclusions. \n\nThe  remainder  of the  paper is  organized as  follows.  In  section  2,  we  briefly review \nrelevant  aspects  of  the  dynamical  system  corresponding  to  a  recurrent  neuronal \nnetwork  as  formulated  in  [2]  and  formally  define  \"sensitivity to  initial  conditions\". \nIn  Section  3,  we  present  simulation  results  from  a  variety  of network  architectures. \nIn  Section  4,  we  interpret  these  results  formally  which  in  turn  lead  us  to  an \nadditional  set of experiments.  In  Section 5,  we  draw conclusions regarding the  issue \nof computation using spike timing in cortical networks based on these results. \n\n2  Spike  dynamics \n\nA detailed exposition of an  abstract dynamical  system that models recurrent systems \nof biological  neurons  was  presented  in  [2].  Here,  we  recount  those  aspects  of the \nsystem  that  are  relevant  to  the  present  discussion.  Based  on  the  intrinsic  nature  of \nthe  processes  involved  in  the  generation  of postsynaptic  potentials  (PSP's)  and  of \nthose  involved  in  the  generation  of action  potentials  (spikes),  it  was  shown  that the \nstate of a system of neurons  can be  specified by enumerating the  temporal positions \nof all  spikes  generated in the system over a bounded past.  For example, in Figure  1, \nthe  present  state  of the  system  is  described  by  the  positions  of the  spikes  (solid \nlines)  in  the  shaded  region  at  t= 0  and  the  state  of the  system  at  a  future  time  T  is \nspecified  by  the  positions  of the  spikes  (solid  lines)  in  the  shaded  region  at  t= T. \nEach  internal  neuron  i  in  the  system is  assigned  a membrane  potential  function  PJ) \nthat  takes  as  its  input  the  present  state  and  generates  the  instantaneous  potential  at \nthe soma of neuron  i.  It is the particular instantiation of the set of functions PJ) that \ndetermines the nature of the neurons as  well as their connectivity in the network. \n\nConsider now the  network in Figure  1 initialized at the particular state  described by \nthe shaded region at t= O.  Whenever the  integration of the PSP's from all presynaptic \nspikes to  a  neuron combined  with the  hyperpolarizing  effects  of its  own spikes  (the \nprecise  nature  of the  union  specified  by  PJ))  brings  its  membrane  potential  above \nthreshold,  the  neuron  emits  a  new  spike.  If the  spikes  in  the  shaded  region  at  t= 0 \nwere perturbed in time  ( dotted lines), this  would result  in  a perturbation on the  new \nspike.  The  size  of the  new  perturbation  would  depend  upon  the  positions  of the \nspikes  in  the  shaded  region,  the  nature  of  PJ) ,  and  the  sizes  of  the  old \nperturbations.  This scenario would in  turn  repeat to  produce further perturbations on \nfuture  spikes.  In essence, any initial  set of perturbations would propagate from  spike \nto  spike to  produce a set of perturbations at any arbitrary future time t= T. \n\n\f: \nI: \n\nI: \n: I  I  : \n:  I \n\nI \nI \n\nI \nI \n\n:  I \n:  I \n\nI \nI \n\nI: \nI: \n\nI \nI \n\n: \n\n:  I \n:  I \n\nI: \nI :  \n\nI \nI \n\nI: \nI: \n\nI \nI \n\n: \n\nI \nI \n\n:  I \n: \nI \n\nPa st \n\n1==0 \n\nI==T  Future \n\nFigure  1:  Schematic  diagram  of the  spike  dynamics  of a  system  of neurons. \nInput neurons are colored gray and internal neurons black. Spikes are shown \nin  solid lines and their corresponding perturbations in dotted lines.  Note that \nspikes  generated  by  the  input  neurons  are  not  perturbed.  Gray  boxes \ndemarcate a bounded past history starting at time  t. The temporal  position of \nall spikes in the boxes specify the state of the system at times t= 0 and  t= T. \n\nIt is  of considerable  importance  to  note  at this juncture  that  while  the  specification \nof the  network architecture  and the  synaptic  weights  determine the precise temporal \nsequence  of  spikes  generated  by  the  network,  the  relative  size  of  successive \nperturbations  are  determined  by  the  temporal  positions  of the  spikes  in  successive \nstate  descriptions  at  the  instant  of the  generation  of each  new  spike.  If it  can  be \ndemonstrated  that there  are  particular classes  of state  descriptions  that  lead  to  large \nrelative  perturbations,  one  can  deduce  the  qualitative  aspects  of the  dynamics  of a \nnetwork armed  with only a  general description of its  architecture.  A formal  analysis \nin  Section 4  will bring to  light such a classification. \nLet  column  vectors  ~  and  y denote,  respectively,  perturbations  on  the  spikes  of \ninternal  neurons  at  times  t=O  and  t=T.  We  pad  each  vector  with  as  many  zeroes  as \nthere  are  input  spikes  in  the  respective  state  descriptions.  Let AT  denote  the  matrix \nsuch that y =  Ar~. Let Band  C be  the  matrices  as  described  in  [3]  that  discard  the \nrigid  translational  components  from  the  final  and  initial  perturbations.  Then,  the \ndynamics  of the  system  is  sensitive  to  initial  conditions  if  lim T_ oo  liB * AT * ell =  00  . \nIf instead,  lim T_ oo  liB * AT * ell =  0 , the dynamics is  insensitive to  initial conditions. \n\nA few  comments are  in  order here.  First, our interest lies not in  the  precise values of \nthe  Lyapunov  characteristic  exponents  of  trajectories  (where  they  exist),  but  in \nwhether the  largest exponent is  greater than  or less  than  zero.  Furthermore, the class \nof  trajectories  that  satisfy  either  of  the  above  criteria  is  larger  (although  not \nnecessarily  in  measure)  than  the  class  of trajectories  that  have  definite  exponents. \nSecond,  input spikes are  free  parameters that have to  be constrained in some manner \nif the  above  criteria are  to  be  well-defined.  By the  same  token,  we  do  not  consider \nthe effects that perturbations of input spikes have on  the  dynamics of the system. \n\n3  Simulations  and  results \n\nA  typical  column in the  cortex contains  on the  order of 10 5  neurons,  approximately \n80%  of which  are  excitatory  and  the  rest  inhibitory.  Each  neuron  receives  around \n104  synapses, approximately half of which  are  from  neurons in  the same column  and \nthe rest from  excitatory neurons  in  other columns  and the  thalamus.  These estimates \nindicate  that  even  at  background  rates  as  low  as  0.1  Hz,  a  column  generates  on \naverage  10  spikes every millisecond.  Since perturbations are propagated from  spikes \n\n\fto  generated spikes, divergence and/or convergence of spike trajectories could occur \nextremely rapidly.  We  test this hypothesis in this section through model simulations. \n\nAll  experiments  reported here  were  conducted  on a system containing  1000 internal \nneurons  (set  to  model  a  cortical  column)  and  800  excitatory  input  neurons  (set  to \nmodel  the  input into  the column).  Of the  1000 internal  neurons,  80% were chosen to \nbe  excitatory  and  the  rest  inhibitory.  Each  internal  neuron  received  100  synapses \nfrom  other (internal  as  well  as  input) neurons in the  system.  The input neurons  were \nset to  generate random uncorrelated Poisson spike trains at a fixed rate of 5 Hz. \n\nThe  membrane potential  function P/) for  each  internal  neuron  was  modeled  as  the \nsum of excitatory and inhibitory PSP ' s triggered by the arrival of spikes at synapses, \nand  afterhyperpolarization  potentials  triggered  by  the  spikes  generated  by  the \nneuron.  PSP ' s were modeled using the  function  \"'.Ji  e-\"'i e-Y,  where  v,  E and  Twere set \n\nv \n\nt \n\nto  mimic  four  kinds  of synapses,  NMDA,  AMP A,  GABA A ,  and  GABA B .  OJ  was  set \nfor  excitatory and  inhibitory synapses so  as  to  generate a mean spike rate  of 5 Hz by \nexcitatory and  15  Hz  by inhibitory internal  neurons.  The  parameters  were then  held \nconstant  over the  entire  system  leaving  the  network  connectivity and  axonal  delays \nas  the  only  free  parameters.  After  the  generation  of a  spike,  an  absolute  refractory \nperiod  of  1  msec  was  introduced  during  which  the  neuron  was  prohibited  from \ngenerating  a  spike.  There  was  no  voltage  reset.  However,  each  spike  triggered  an \nafterhyperpolarization  potential  with  a  decay  constant  of  30  msec  that  led  to  a \nrelative  refractory  period.  Simulations  were  performed  in  0.1  msec  time  steps  and \nthe time bound on the state description,  as related in  Section 2, was  set at 200 msec. \n\nThe  issue  of correlated  inputs  was  addressed  by  simulating  networks  of disparate \narchitectures.  On the  one  extreme  was  an  ordered two  layer ring  network with  input \nneurons  forming  the  lower  layer  and  internal  neurons  (with  the  inhibitory  neurons \nplaced evenly among the  excitatory neurons)  forming  the  upper layer.  Each internal \nneuron received inputs  from  a sector of internal and input neurons that was  centered \non that  neuron. As  a result,  any two  neighboring  internal neurons  shared 96  of their \n100  inputs  (albeit  with different axonal  delays  of 0.5-1.1  msec).  This  had  the  effect \nof output  spike  trains  from  neighboring  internal  neurons  being  highly  correlated, \nwith  sectors  of internal  neurons  producing  synchronized  bursts  of spikes.  On  the \nother  extreme  was  a  network  where  each  internal  neuron  received  inputs  from  100 \nrandomly  chosen  neurons  from  the  entire  population  of internal  and  input  neurons. \nSeveral  other  networks  where  neighboring  internal  neurons  shared  an  intermediate \npercentage  of their  inputs  were  also  simulated.  Here,  we  present  results  from  the \ntwo  extreme architectures.  The results from all the other networks were similar. \n\nFigure  2(a)  displays  sample  output  spike  trains  from  100  neighboring  internal \nneurons  over  a  period  of  450  msec  for  both  architectures.  In  the  first  set  of \nexperiments,  pairs  of identical  systems  driven  by  identical  inputs  and  initialized  at \nidentical  states  except for one randomly chosen  spike that was  perturbed by  1 msec, \nwere simulated.  In  all  cases, the spike trajectories diverged very rapidly.  Figure 2(b) \npresents  spike  trains  generated  by the  same  100  neighboring  internal  neurons  from \nthe two  simulations from 200 to  400 msec  after initialization, for both architectures. \n\nTo  further  explore  the  sensitivity  of  the  spike  trajectories,  we  partitioned  each \ntrajectory  into  segments  of 500  spike  generations  each.  For  each  such  segment,  we \nthen  extracted  the  spectral  norm  liB * AT * ell  after  every  100  spike  generations. \nFigure  2( c)  presents  the  outcome  of this  analysis  for  both  architectures.  Although \nsuccessive  segments  of 500  spike  generations  were  found  to  be  quite  variable  in \ntheir absolute  sensitivity,  each  such  segment was  nevertheless  found  to  be  sensitive. \nWe  also  simulated  several  other  architectures  (results  not  shown),  such  as  systems \nwith fixed axonal delays and ones with bursty behavior, with similar outcomes. \n\n\f(a) \n\n\"  : . \n\n\u2022 . '. \n\no msec \n\nRing Network (above) and Random Network (below) \n\n450 msec \n\n(b) \n\n, \n\n'.~ \n.:~ \n., \n\n', ' \n\n. : \n\n,\", \n\n200  msec \n\n(c) \n\n400 msec \n\n200 msec \n\n400 msec \n\nRing Network \n\nRandom Network \n\nlO',.-----~~-~--~--~--__, \n\n103 r--~~-~--~--~-----' \n\n200 \n\n300 \n\n400 \n\n500 \n\n400 \n\n500 \n\nFigure 2:  (a)  Spike trains of 100 neighboring neurons  for  450  msec  from  the \nring  and  the  random  networks  respectively.  (b)  Spike  trains  from  the  same \n100  neighboring  neurons  (above  and  below)  200  msec  after  initialization. \nNote  that the  trains  have  already diverged at  200  msec.  (c)  Spectral norm of \nsensitivity  matrices  of  14  successive  segments  of  500  spike  generations \neach, computed in steps of 100 spike generations for both architectures. \n\n\f4  Analysis  and  further  simulations \n\nThe  reasons  behind  the  divergence  of the  spike  trajectories  presented  in  Section  3 \ncan be found by considering how perturbations are propagated from the set of spikes \nin  the  current  state  description  to  a  newly  generated  spike.  As  shown  in  [3] ,  the \nperturbation  in  the  new  spike  can  be  represented  as  a  weighted  sum  of  the \nperturbations of those spikes in  the state description that contribute to  the  generation \nof the new spike.  The weight assigned to  a spike Xi  is  proportional to  the slope of the \nPSP or that of the  hyperpolarization  triggered by that  spike  (apo/aXi  in  the  general \ncase),  at  the  instant  of the  generation  of the  new  spike.  Intuitively,  the  larger  the \nslope  is,  the  greater  is  the  effect  that  a  perturbation  of that  spike  can  have  on  the \ntotal  potential  at  the  soma,  and  hence,  the  larger  is  the  perturbation  on  the  new \nspike.  The  proportionality  constant  is  set  so  that  the  weights  sum  to  1.  This \nconstraint  is  reflected  in  the  fact  that  if all  spikes  were  to  be  perturbed  by  a  fixed \nquantity, this  would amount to  a rigid displacement in  time causing the new spike to \nbe  perturbed  by the  same  quantity.  We  denote  the  slopes  by Pi, and  the  weights  by \nai.  Then,  a  =  p.I\" n  p., where j  ranges over all contributing spikes. \n\ni \n\nI  ~ j\"\", l \n\nJ \n\nWe  now  assume  that  at  the  generation  of  each  new  spike,  the  p,'s  are  drawn \nindependently from a stationary distribution (for both internal and input contributing \nspikes),  and  that the  ratio  of the  number of internal  to  the  total  (internal plus  input) \nspikes  in  any state  description  remains  close  to  a  fixed  quantity f-l  at  all  times.  Note \nthat  this  amounts  to  an  assumed  probability  distribution  on  the  likelihood  of \nparticular  spike  trajectories  rather  than  one  on  possible  network  architectures  and \nsynaptic  weights.  The  iterative  construction  of  the  matrix  AT,  based  on  these \nconditions,  was  described  in  detail  in  [3].  It  was  also  shown  that  the  statistic \n\\I;I~l a i\n2 )  plays  a  central  role  in the  determination of the  sensitivity of the  resultant \nspike  trajectories.  In  a  minor modification  to  the  analysis  in  [3],  we  assume  that AT \nrepresents  the  full  perturbation  (internal  plus  input)  at  each  step  of the  process. \nWhile  this  merely  entails  the  introduction  of additional  rows  with  zero  entries  to \naccount for  input spikes in each state, this alters the effect that B  has  on liB * AT * ell \nin  a way that allows  for  a  simpler as  well  as  bidirectional  bound on  the  norm.  Since \nthe analysis is  identical to  that in  [3]  and does not introduce any new techniques, we \nonly report the result.  If \\I;I~l a i\n2 )  <  ~ -I} then the \nspike  trajectories  are  almost  surely sensitive  (resp.  insensitive)  to  initial  conditions. \nm  denotes the number of internal spikes in the state description. \n\n2 )  >  (2 + ~(y\")  -1  (resp. \\I;~l  a i\n\nIf we  make  the  liberal  assumption that  input spikes  account  for  as  much  as  half the \ntotal  number  of spikes  in  state  descriptions,  noting  that  m  is  a  very  large  quantity \n(greater than  103  in all our simulations), the above constraint requires (Ian> 3  for \nspike trajectories to  be almost surely  sensitive to  initial  conditions.  From our earlier \nsimulations,  we  extracted  the  value  of L a i\n2  whenever  a  spike  was  generated,  and \ncomputed the  sample mean  (I a i\n2 )  over all  spike  generations.  The mean was  larger \nthan  3 in  all  cases (it was 69.6 for the ring and  11.3  for the random network). \n\nThe  above  criterion enables  us  to  peer into  the  nature  of the  spike  dynamics  of real \ncortical  columns,  for  although  simulating  an  entire  column  remains  intractable,  a \nsingle  neuron  can  be  simulated  under  various  input  scenarios,  and  the  resultant \nstatistic  applied to  infer the  nature  of the  spike  dynamics  of a  cortical  column most \nof whose neurons operate under those conditions. \n\n\fAn  examination  of the  mathematical  nature  of L: a i\n2  reveals  that  its  value  rises  as \nthe  size  of  the  subset  of  p;'s  that  are  negative  grows  larger.  The  criterion  for \nsensitivity  is  therefore  more  likely  to  be  met  when  a  substantial  portion  of  the \nexcitatory  PSP's  are  on  their  falling  phase  (and  inhibitory  PSP ' s  on  their  rising \nphase) at the  instant of the  generation of each  new spike.  This corresponds to  a case \nwhere  the  inputs  into  the  neurons  of  a  system  are  not  strongly  synchronized. \nConversely,  if spikes  are  generated soon after the  arrival  of a synchronized burst of \nspikes  (all  of whose  excitatory  PSP ' s  are  presumably  on  their  rising  phase),  the \ncriterion  for  sensitivity  is  less  likely  to  be  met.  We  simulated  several  combinations \nof  the  two \nidentify  cases  where  the  corresponding  spike \ntrajectories in  the system were not likely to  be  sensitive to  initial  conditions. \n\ninput  scenarios  to \n\nWe  constructed  a  model  pyramidal  neuron  with  10,000  synapses,  85%  of which \nwere  chosen  to  be  excitatory  and  the  rest  inhibitory.  The  threshold  of the  neuron \nwas  set  at  15  mV  above  resting  potential.  PSP ' s  were  modeled  using  the  function \ndescribed  earlier  with  values  for  the  parameters  set  to  fit  the  data  reported  in  [7]. \nFor  excitatory  PSP's  the  peak  amplitudes  ranged  between  0.045  and  1.2  mV  with \nthe  median  around  0.15  mY,  10-90  rise  times  ranged  from  0.75  to  3.35  msec  and \nwidths  at  half amplitude  ranged  from  8.15  to  18.5  msec.  For  inhibitory  PSP's,  the \npeak amplitudes were on average twice as  large and  the  10-90 rise times and  widths \nat  half amplitude  were  slightly  larger.  Whenever the  neuron  generated  a  new  spike, \nthe  values  of the  p;'s  were  recorded  and  L: a }  was  computed.  The  mean  (L: a i\n2 ) \nwas  then  computed  over  the  set  of  all  spike  generations.  In  order  to  generate \nconservative  estimates,  samples  with  value  above  104  were  discarded  (they \ncomprised about 0.1% of the  data).  The datasets ranged in size from 3000 to  15,000. \n\nThree  experiments  simulating  various  levels  of uncorrelated  input/output  activity \nwere  conducted.  In  particular,  excitatory  Poisson  inputs  at  2,  20  and  40  Hz  were \nbalanced by inhibitory Poisson  inputs  at  6.3,  63  and  124 Hz to  generate output rates \nof approximately 2,  20  and  40 Hz, respectively.  We  confirmed that the  output in  all \nthree  cases  was  Poisson-like  (CV=O.77,  0.74,  and  0.89,  respectively).  The  mean \n(L: a i\n\n2 )  for the three experiments was 4.37 , 5.66, and 9.52 , respectively. \n\nNext,  two  sets  of  experiments  simulating  the  arrival  of regularly  spaced  synfire \nchains  were  conducted.  In  the  first  set  the  random background activity  was  set at  2 \nHz  and  in  the  second,  at  20  Hz.  The  synfire  chains  comprised of spike  volleys  that \narrived  every  50  msec.  Four  experiments  were  conducted  within  each  set:  volleys \nwere composed of either 100 or 200 spikes (producing jolts of around  10  and 20  mV \nrespectively)  that were  either fully  synchronized or were  dispersed  over  a  Gaussian \ndistribution  with  a=1  msec.  The  mean  (Lan for  the  experiments  was  as  follows. \nAt  2  Hz  background  activity,  it  was  0.49  (200  spikes/volley,  synchronized),  0.60 \n(200  spikes/volley,  dispersed),  2.46  (100  spikes/volley,  synchronized),  and  2.16 \n(100  spikes/volley,  dispersed).  At  20  Hz  background  activity,  it  was  4.39  (200 \nspikes/volley,  synchronized),  8.32  (200  spikes/volley,  dispersed),  6.77 \n(100 \nspikes/volley, synchronized), and 6.78  (l00 spikes/volley, dispersed). \n\nFinally,  two  sets  of experiments  simulating  the  arrival  of randomly  spaced  synfire \nchains  were  conducted.  In  the  first  set  the  random  background  activity  was  set  at  2 \nHz and in  the  second, at 20  Hz.  The synfire chains comprised of a sequence of spike \nvolleys  that  arrived  randomly  at  a  rate  of 20  Hz.  Two  experiments  were  conducted \nwithin  each  set:  volleys  were  composed  of either  100  or  200  synchronized  spikes. \nThe mean  (L a i\n2 )  for  the  experiments was  as  follows.  At 2 Hz background activity, \nit  was  4.30  (200  spikes/volley)  and  4.64  (100  spikes/volley).  At  20  Hz  background \nactivity, it was  5.24 (200 spikes/volley) and 6.28  (l00 spikes/volley). \n\n\f5  Conclusion \n\nAs  was  demonstrated  in  Section  3,  senslllvlty  to  initial  conditions  transcends \nunstructured  connectivity  in  systems  of  spiking  neurons.  Indeed,  our  simulations \nindicate  that  sensitivity  is  more  the  rule  than  the  exception  in  systems  modeling \ncortical networks operating at  low to  moderate levels of activity.  Since perturbations \nare  propagated  from  spike  to  spike,  trajectories  that  are  sensitive  can  diverge  very \nrapidly  in  systems  that  generate  a  large  number  of spikes  within  a  short  period  of \ntime.  Sensitivity therefore  is  an  issue,  even  for  schemes  based on  precise sequences \nof spike timing with computation occurring over short (hundreds of msec) intervals. \n\nWithin  the  limits  set  by  our  model  of  the  neuron,  we  have  found  that  spike \ntrajectories  are  likely  to  be  sensitive  to  initial  conditions  in  all  scenarios  except \nwhere  large  (100-200)  synchronized bursts of spikes occur in the presence of sparse \nbackground  activity  (2  Hz)  with  sufficient  but  not  too  large  an  interval  between \nsuccessive bursts  (50  msec).  This  severely restricts the  possible use  of precise spike \nsequences  for  reliable  computation  in  cortical  networks  for  at  least  two  reasons. \nFirst,  un synchronized  activity  can  rise  well  above  2  Hz  in  the  cortex,  and  second, \nthe  highly constrained nature of this dynamics would show in  in  vivo recordings. \n\nAlthough  cortical  neurons  can  have  vastly  more  complex  responses  than  that \nmodeled  in  this  paper,  our  conclusions  are  based  largely  on  the  simplicity  and  the \ngenerality  of the  constraints  identified  (the  analysis  assumes  a  general  membrane \npotential function PO).  Although a more  refined model of the  cortical neuron could \nlead  to  different  values  of the  statistic  computed,  we  believe  that  the  results  are \nunlikely to  cross the noted bounds and therefore change our overall  conclusions. \n\nWe  are  however  not  arguing  that  computation  with  spike  timing  is  impossible  in \ngeneral.  There  are  neural  structures,  such  as  the  nucleus  laminaris  in  the  barn  owl \nand  the  electro sensory  array  in  the  electric  fish , which have  been shown to  perform \nexquisitely  precise  computations  using  spike  timing.  Interestingly,  these  structures \nhave very specialized neurons and network architectures. \n\nTo  conclude, computation using precise spike sequences does not appear to  be likely \nin  the cortex in the presence of Poisson-like activity at  levels typically found there. \n\nReferences \n\n[1]  Abeles, M., Bergman, H., Margalit, E.  &  Vaadia, E.  (1993)  Spatiotemporal firing patterns \nin the frontal  cortex of behaving monkeys. Journal of Neurophysiology 70, pp.  1629-1638. \n\n[2]  Banerjee,  A.  (2001)  On  the  phase-space  dynamics  of  systems  of  spiking  neurons:  I. \nmodel and experiments. Neural Computation  13, pp.  161-193. \n\n[3]  Banerjee,  A.  (2001)  On  the  phase-space  dynamics  of systems  of spiking  neurons:  II. \nformal  analysis. Neural Computation  13, pp.  195-225. \n\n[4]  Diesmann, M. , Gewaltig, M.  O.  &  Aertsen,  A.  (1999)  Stable propagation of synchronous \nspiking in  cortical  neural networks. Nature 402, pp.  529-533. \n\n[5]  Gerstner,  W. ,  van  Hemmen,  J.  L.  &  Cowan,  J.  D.  (1996)  What  matters  in  neuronal \nlocking. Neural Computation 8, pp.  1689-1712. \n\n[6]  Maass ,  W.  (1995)  On  the  computational  complexity  of networks  of spiking  neurons. \nAdvances in  Neural Information Processing Systems  7,  pp.  183-190. \n\n[7]  Mason,  A. ,  Nicoll,  A.  &  Stratford,  K.  (1991)  Synaptic  transmission  between  individual \npyramidal neurons of the rat visual  cortex in  vitro . Journal of Neuroscience 11(1), pp.  72-84. \n[8]  van  Vreeswijk,  c.,  &  Sompolinsky,  H.  (1998)  Chaotic  balanced  state  in  a  model  of \ncortical circuits. Neural Computation  10, pp.  1321-1372. \n\n\f", "award": [], "sourceid": 2337, "authors": [{"given_name": "Arunava", "family_name": "Banerjee", "institution": null}, {"given_name": "Alexandre", "family_name": "Pouget", "institution": null}]}