{"title": "Distributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly", "book": "Advances in Neural Information Processing Systems", "page_first": 129, "page_last": 135, "abstract": null, "full_text": "Distributed Synchrony of Spiking Neurons \n\nin a  Hebbian  Cell  Assembly \n\nDavid Horn  Nir Levy \n\nSchool of Physics and Astronomy, \n\nRaymond and Beverly Sackler Faculty of Exact Sciences, \n\nTel  Aviv  University,  Tel  Aviv  69978,  Israel \n\nhorn~neuron.tau.ac.il  nirlevy~post.tau.ac.il \n\nIsaac  Meilijson  Eytan Ruppin \n\nSchool of Mathematical Sciences, \n\nRaymond and  Beverly Sackler Faculty of Exact Sciences, \n\nTel  Aviv  University, Tel  Aviv  69978, Israel \n\nisaco~math.tau.ac.il \n\nruppin~math.tau.ac.il \n\nAbstract \n\nWe  investigate the behavior of a  Hebbian  cell  assembly of spiking \nneurons formed via a temporal synaptic learning curve.  This learn(cid:173)\ning  function  is  based  on  recent  experimental findings .  It  includes \npotentiation for  short time delays  between  pre- and  post-synaptic \nneuronal spiking,  and depression for  spiking events occuring in the \nreverse order.  The coupling  between the dynamics of the synaptic \nlearning and of the neuronal activation leads to interesting results. \nWe  find  that  the  cell  assembly  can  fire  asynchronously,  but  may \nalso  function  in  complete  synchrony,  or  in  distributed  synchrony. \nThe latter implies spontaneous division of the Hebbian cell  assem(cid:173)\nbly into groups of cells  that fire  in  a cyclic manner.  We  invetigate \nthe behavior of distributed synchrony both by  simulations and by \nanalytic calculations of the resulting synaptic distributions. \n\n1 \n\nIntroduction \n\nThe Hebbian paradigm that serves as the basis for  models of associative memory is \noften  conceived  as  the statement that  a  group  of excitatory neurons  (the  Hebbian \ncell  assembly)  that  are  coupled  synaptically  to  one  another  fire  together  when  a \nsubset  of the  group  is  being  excited  by  an  external  input.  Yet  the  details  of the \ntemporal  spiking  patterns of neurons  in  such an  assembly  are  still  ill  understood. \nTheoretically it seems  quite  obvious  that there  are two  general types of behavior: \nsynchronous neuronal firing,  and asynchrony where no temporal order exists in  the \nassembly  and  the  different  neurons  fire  randomly  but  with the  same  overall  rate. \nFurther  subclassifications  were  recently  suggested  by  [BruneI,  1999].  Experimen(cid:173)\ntally  this  question  is  far  from  being  settled  because  evidence  for  the  associative \n\n\f130 \n\nD.  Hom, N.  Levy, 1.  Meilijson and E.  Ruppin \n\nmemory paradigm is  quite scarce.  On one hand, one possible realization of associa(cid:173)\ntive memories in the brain was demonstrated by  [Miyashita,  1988]  in the inferotem(cid:173)\nporal  cortex.  This  area was  recently  reinvestigated  by  [Yakovlev  et  al.,  1998]  who \ncompared their experimental results with a model of asynchronized spiking neurons. \nOn  the  other  hand there  exists  experimental evidence  [Abeles,  1982]  for  temporal \nactivity patterns in  the frontal cortex that Abeles called synfire-chains.  Could they \ncorrespond to an alternative type of synchronous realization of a memory attractor? \nTo  answer  these  questions  and  study  the  possible  realizations  of  attractors  in \ncortical-like networks we investigate the temporal structure of an attractor assuming \nthe existence of a synaptic learning curve that is  continuously applied to the mem(cid:173)\nory  system.  This  learning curve  is  motivated  by  the  experimental observations  of \n[Markram  et  al.,  1997,  Zhang  et  al.,  1998]  that synaptic potentiation or depression \noccurs within a  critical time window in which both pre- and post-synaptic neurons \nhave  to fire.  If the  pre-synaptic neuron fires  first  within  30ms  or  so,  potentiation \nwill  take  place.  Depression  is  the rule  for  the reverse order. \nThe  regulatory  effects  of  such  a  synaptic  learning  curve  on  the  synapses  of \na  single  neuron  that  is  subjected  to  external  inputs  were  investigated  by \n[Abbott  and Song,  1999]  and  by  [Kempter  et  al.,  1999].  We  investigate  here  the \neffect  of  such  a  rule  within  an  assembly  of  neurons  that  are  all  excited  by  the \nsame external input throughout a training period, and are allowed  to influence one \nanother through their resulting sustained activity. \n\n2  The Model \n\nWe study a network composed of N E  excitatory and NJ  inhibitory integrate-and-fire \nneurons.  Each  neuron  in  the  network  is  described  by  its  subthreshold  membrane \npotential Vi{t)  obeying \n\n. \nVi{t)  = - - Vi{t) + R1i(t) \n\n1 \n\nTn \n\n(1) \n\nwhere Tn  is  the neuronal integration time constant.  A spike is generated when Vi{t) \nreaches the threshold Vrest + fJ,  upon which a refractory period of TRP  is  set on and \nthe membrane  potential is  reset to Vreset  where  Vrest  < Vreset  < Vrest  + fJ.  Ii{t)  is \nthe sum of recurrent and external synaptic current inputs.  The net synaptic input \ncharging the membrane of excitatory neuron i  at time t  is \n\nR1i(t)  =  L J~E{t) L 0 (t  - t~ - Td)  - L Ji~J L 0 (t  - tj - Td)  + r xt \n\n(2) \n\nj \n\nI \n\nj \n\nm \n\nsumming  over  the  different  synapses  of j  =  1, ... , NE  excitatory  neurons  and  of \nj  =  1, ... ,NJ inhibitory neurons,  with  postsynaptic efficacies  J~E{t) and  Ji~J re(cid:173)\nspectively.  The  sum  over  1 (m)  represents  a  sum  on  different  spikes  arriving  at \nsynapse j, at times  t = t;  + Td  (t  = tj + Td),  where  t~  (tj)  is  the emission time of \nthe l-th  (m-th) spike from  the excitatory (inhibitory) neuron j  and Td  is  the synap(cid:173)\ntic  delay.  Iext,  the external current, is  assumed  to be  random and independent  at \neach neuron and each time step, drawn from  a Poisson distribution with mean A ext. \nAnalogously, the synaptic input to the inhibitory neuron i  at time tis \n\nj \n\nj \n\nm \n\nWe  assume  full  connectivity  among  the  excitatory  neurons,  but  only  partial  con(cid:173)\nnectivity  between  all  other  three  types  of possible  connnections,  with  connection \n\n\fDistributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly \n\n131 \n\nprobabilities denoted by eEl, e l E  and Cl I.  In the following we  will report simula(cid:173)\ntion results in which the synaptic delays Td  were assigned to each synapse, or pair of \nneurons,  randomly,  chosen from  some finite  set of values.  Our analytic calculation \nwill  be done for  one fixed  value of this delay  parameter. \n\nThe  synaptic efficacies  between  excitatory neurons  are assumed to  be  potentiated \nor depressed according to the firing  patterns of the pre- and post-synaptic neurons. \nIn  addition  we  allow  for  a  uniform  synaptic  decay.  Thus  each excitatory  synapse \nobeys \n\n(4) \n\nwhere the synaptic decay  constant Ts  is  assumed to be very  large compared to the \nmembrane time constant Tn.  J/JE(t)  are constrained to vary in  the range [0, Jma:~]. \nThe change in synaptic efficacy  is  defined  by  Fij (t),  as \n\nFij(t)  = L [6(t  - t:)Kp(t; - t:) + 6(t - t;)KD(t; - t:)] \n\n(5) \n\nk ,l \n\nwhere Kp and KD  are the potentiation and depression branches of the kernel func(cid:173)\ntion \n\nK(6) = -cO exp [- (a6 + b)2] \n\n(6) \n\nplotted  in  Figure  1.  Following  [Zhang  et  al.,  1998]  we  distinguish between  the  sit(cid:173)\nuation where the postsynaptic spike,  at t~, appears after or before the presynaptic \nspike,  at t~, using the asymmetric kernel  that captures the essence of their experi(cid:173)\nmental observations. \n\n... \n\n-o !.'--~-~~--'o-~-~~-----' \n\n/I  =t'_tk \nI \n\n, \n\nFigure  1:  The  kernel  function  whose  left  part,  Kp,  leads  to  potentiation  of the \nsynapse, and whose right  branch,  KD,  causes synaptic depression. \n\n3  Distributed Synchrony of a  Hebbian Assembly \n\nWe  have run our system with synaptic delays chosen randomly to be either 1,  2,  or \n3ms, and temporal parameters Tn  chosen as 40ms for  excitatory neurons and 20ms \nfor  inhibitory  ones.  Turning external input  currents  off  after  a  while  we  obtained \nsustained firing activities in the range of 100-150 Hz.  We  have found,  in  addition to \nsynchronous and asynchronous realizations of this attractor,  a  mode  of  distributed \nsynchrony.  A  characteristic example of a  long  cycle  is  shown in Figure  2:  The 100 \nexcitatory neurons split into groups such that each group fires at the same frequency \nand  at a  fixed  phase difference  from  any other group.  The  J/JE  synaptic efficacies \n\n\f132 \n\nD.  Horn,  N  Levy, 1.  Meilijson and E.  Ruppin \n\n: I  :  I :  r  I \n\n':~ \n:tI  :  I :  r  I \n':f  I :  r  I \n':f  r  I \nj \n:1 \nj \n\n:1 \n: I  ] \n: I  : I  :1 \n: 1  : 1  :  1 :1 \n:  1  :  1 :  n \nI \n: I  :  I  :  I :  r  ~ \n\n, \n\nJO \n\n\" \n\n\" \n\nI \n\n\" \n\n\" \n\nFigure  2:  Distributed synchronized firing  mode.  The firing  patterns of six  cell  as(cid:173)\nsemblies  of excitatory neurons  are  displayed  vs  time  (in  ms).  These six groups  of \nneurons  formed  in  a  self-organized manner for  a  kernel function  with equal  poten(cid:173)\ntiation and depression.  The delays were  chosen randomly from  three values,  1 2 or \n3ms,  and the system is  monitored every 0.5ms . \n\nare initiated as small random values.  The learning process leads to the self-organized \nsynaptic matrix displayed in  Figure 3(a).  The block form of this matrix represents \nthe ordered couplings  that are responsible for  the fact  that each coherent group of \nneurons  feeds  the activity of groups  that follow  it.  The self-organized groups form \nspontaneously.  When the synapses are affected by some external noise, as can come \nabout  from  Hebbian  learning in which  these neurons are being coupled  with other \npools  of neurons,  the  groups  will  change  and  regroup,  as  seen  in  Figure  3(b)  and \n3(c). \n\n(a) \n\n(b) \n\n(c) \n\nFigure 3:  A synaptic matrix for  n = 6 distributed synchrony.  The synaptic matrix \nbetween  the  100 excitatory neurons of our system is  displayed  in  a  grey-level code \nwith black meaning zero efficacy and white standing for  the synaptic upper-bound. \n(a)  The  matrix  that  exists  during the  distributed synchronous  mode  of Figure  2. \nIts basis  is  ordered such that neurons that fire  together  are grouped together.  (b) \nUsing  the same  basis  as in  (a)  a  new  synaptic matrix is  shown,  one  that is  formed \nafter stopping the sustained activity of Figure 2,  introducing noise  in  the synaptic \nmatrix, and reinstituting the original memory training.  (c)  The same matrix as  (b) \nis  shown  in  a  new  basis  that exhibits  connections that lead  to  a  new  and  different \nrealization of distributed synchrony. \n\nA  stable  distributed  synchrony  cycle  can  be  simply  understood  for  the  case  of  a \nsingle synaptic delay setting the basic step, or phase difference, of the cycle.  When \nseveral delay parameters exist, a situation that probably more accurately represents \nthe a-function character of synaptic transmission in cortical networks, distributed \n\n\fDistributed Synchrony of Spiking Neurons in a Hebbian Cell Assembly \n\n133 \n\nsynchrony  may  still  be  obtained,  as  is  evident  from  Figure  2.  After  some  time \nthe  cycle  may  destabilize  and  regrouping  may  occur  by  itself,  without  external \ninterference.  The likelihood of this scenario is  increased because  different  synaptic \nconnections that have different  delays can interfere with one  another.  Nonetheless, \nover time scales of the type shown  in  Figure 2,  grouping is  stable. \n\n4  Analysis  of a  Cycle \n\nIn this section we  analyze the dynamics of the network  when  it is  in  a  stable state \nof  distributed  synchrony.  We  assume  that  n  groups  of  neurons  are  formed  and \ncalculate  the  stationary distribution of  JffE(t) .  In  this  state the  firing  pattern  of \nevery two neurons in the network can be  characterized by their frequency  l/(t)  and \nby  their  relative  phase  8.  We  assume  that  8  is  a  random  normal  variable  with \nmean  J.Lo  and standard deviation  0'0 .  Thus,  Eq.  4 can be rewritten as  the following \nstochastic differential equation \n\ndJi~E(t) =  [J.LFij(t)  - :s J!jE(t)]  dt+O'Fij(t)dW(t) \n\n(7) \n\nwhere  Fij (t)  (Eq.  5)  is  represented  here  by  a  drift  term  J.LFij (t)  and  a  diffusion \nterm O'Fij (t)  which  are its  mean  and standard deviation.  W(t)  describes  a  Wiener \nprocess.  Note that both J.LFij  (t)  and O'Fij (t)  are calculated for  a specific distribution \nof 8 and  are functions  of J.Lo  and 0'0. \nThe  stochastic  process  that  satisfies  Eq.  7  will  satisfy  the  Fokker-Plank  equation \nfor  the probability distribution f  of JIfE, \n8f(JPlE  t) \n\n1) \n\n8 [( \n\n.. (t)  _ _  JPlE \n\n(8) \n\n]  0'2 \n' J '   + \n\nf(JPlE  t) \n\n(t)  82f(JPlE  t) \nFij \n2 \n\nt J '  \n8JEE2 \n\nij \n\n. J '   =  ___ \n8JPlE \n8t \n\nJ.LF.] \n\n1J \n\nT ' J \n\nS \n\nwith  reflecting  boundary conditions  imposed  by  the synaptic bounds,  0 and  Jmax . \nSince  we  are  interested  in  the  stable  state  of  the  process  we  solve  the  stationary \nequation.  The resulting density function  is \n\n[1  ( \n\nEE  1 EE2) 1 \n\n- Ts Jij \n\n(9) \n\n(10) \n\nEE \n\nN \n\nf(Jij  ,J.Lt5, 0'15)  =  O'}ij (t)  exp  O'}ij (t) \n\n2J.LFij Jij \n\nwhere \n\nEq.  9  enables  us  to  calculate  the stationary  distribution  of the  synaptic  efficacies \nbetween  the  presynaptic neuron  i  and  the  post-synaptic  neuron  j  given  their  fre(cid:173)\nquency  l/ and the parameters J.Lo  and  0'15.  An  example of a  solution for  a  3-cycle  is \nshown in  Figure 4.  In this  case all  neurons fire  with frequency  l/ =  (3Td)-1  and  J.Lt5 \ntakes one of the values  -Td, 0, Td. \n\nSimulation results  of a  3-cycle in  a  network of excitatory and inhibitory integrate(cid:173)\nand-fire neurons  described  in  Section  2 are  given  in  Figure  5.  As  can  be  seen  the \nresults  obtained from  the analysis  match those observed in  the simulation. \n\n5  Discussion \n\ninteresting  experimental  observations  of \n\nThe \n[Markram  et  al.,  1997,  Zhang  et  al.,  1998]  have  led  us  to  study  their  implica(cid:173)\ntions  for  the firing  patterns of a  Hebbian  cell  assembly.  We  find  that,  in  addition \n\nsynaptic \n\nlearning \n\ncurves \n\n\f134 \n\nD.  Horn, N.  Levy, 1.  Meilijson and E.  Ruppin \n\n(a) \n\n(b) \n\n70 \n\n60 \n\nso \n\n40 \n\n30 \n\n20 \n\n, 0 \n\n0 \n\n0 \n\n) \n\n0.' \n\n0.2 \n\n0.3 \n\n0.4 \n\n0. 5 \n\nJEE \n\n\" \n\n01 \n\n0 .2 \n\n0.3 \n\n0 4 \n\nFigure  4:  Results  of the  analysis  for  n  =  3,  a6  =  2ms  and  Td  =  2.5ms.  (a)  The \nsynaptic matrix.  Each of the nine blocks symbolizes a group of connections between \nneurons  that  have  a  common  phase-lag  J..l6 .  The  mean  of Ji~E was  calculated for \neach cell  by Eq. 9 and its value is given by the gray scale tone.  (b)  The distribution \nof synaptic values between all excitatory neurons. \n\n(a) \n\n(b) \n\n5o0 0 , - - - - - - - - - - - - - ,  \n\n4500 \n\n4000 \n\n3500 \n\n3000 \n\n2500 \n\n2000 \n\n1500 \n\n500 \n\n0. ' \n\n0.2 \n\n0.3 \n\n0.4 \n\n0. 5 \n\no 0\" - - -0 .... \n\n'  - - '  ... 0.2:--~0.3::----::\"\"0.4c--'\"-:\"0.5 \n\nJEE , \n\nFigure  5:  Simulation  results  for  a  network  of N E  =  100  and  NJ  =  50  integrate(cid:173)\nand-fire  neurons, when  the network is  in  a  stable n  =  3 state.  Tn  =  10ms for  both \nexcitatory and inhibitory neurons.  The average frequency of the neurons is  130 Hz. \n(a)  The excitatory synaptic matrix.  (b)  Histogram of the synaptic efficacies. \n\nto  the  expected  synchronous  and  asynchronous  modes,  an  interesting  behavior \nof  distributed  synchrony  can  emerge.  This  is  the  phenomenon  that  we  have \ninvestigated both by simulations and by analytic evaluation. \nDistributed synchrony is  a mode in  which the Hebbian cell assembly breaks into an \nn-cycle.  This  cycle  is  formed  by  instantaneous symmetry  breaking, hence  specific \nclassification of neurons into one of the n groups depends on initial conditions, noise, \netc.  Thus  the  different  groups  of a  single  cycle  do  not  have  a  semantic  invariant \nmeaning of their own.  It seems perhaps premature to try and identify these cycles \nwith synfire  chains [Abeles, 1982]  that show recurrence of firing  patterns of groups \nof neurons with periods of hundreds of ms.  Note  however, that if we  make such an \nidentification , it is a different explanation from the model of [Herrmann  et  al.,  1995J , \nwhich realizes the synfire chain by combining sets of preexisting patterns into a cycle. \n\nThe  simulations  in  Figures  2  and  3  were  carried  out  with  a  learning  curve  that \npossessed  equal  potentiation  and  depression  branches,  i.e.  was  completely  anti(cid:173)\nsymmetric in  its argument.  In that case  no  synaptic decay  was  allowed.  Figure 5, \non the other hand, had stronger potentiation than depression, and a finite  synaptic \n\n\fDistributed Synchrony of Spiking Neurons  in a Hebbian Cell Assembly \n\n135 \n\ndecay  time  was  assumed.  Other  conditions  in  these  nets  were  different  too,  yet \nboth  had  a  window  of parameters where  distributed  synchrony showed  up.  Using \nthe  analytic  approach  of  section  4  we  can  derive  the  probability  distribution  of \nsynaptic  values  once  a  definite  cyclic  pattern  of distributed  synchrony  is  formed. \nAn analytic solution of the combined dynamics of both the synapses and the spiking \nneurons is still an open challenge.  Hence we have to rely on the simulations to prove \nthat  distributed  synchrony is  a  natural spatiotemporal behavior that follows  from \ncombined neuronal dynamics and synaptic learning as outlined in section 2.  To the \nextent that both types of dynamics reflect correctly the dynamics of cortical neural \nnetworks,  we  may  expect  distributed  synchrony  to  be  a  mode  in  which  neuronal \nattractors are being realized. \n\nThe mode of distrbuted synchrony is of special significance to the field of neural com(cid:173)\nputation since it forms  a  bridge between the feedback  and feed-forward  paradigms. \nNote  that whereas the attractor that is  formed  by  the Hebbian cell  assembly is  of \nglobal  feedback  nature,  i.e.  one  may regard  all  neurons  of the  assembly  as  being \nconnected  to  other  neurons  within  the  same  assembly,  the  emerging  structure  of \ndistributed  synchrony  shows  that  it  breaks  down  into  groups.  These  groups  are \nconnected to one another in a self-organized feed-forward manner, thus forming the \ncyclic  behavior we  have observed. \n\nReferences \n\n[Abbott and Song,  1999]  L.  F.  Abbott  and  S.  Song.  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Nature,  395:37 - 44,  1998. \n\n\f", "award": [], "sourceid": 1703, "authors": [{"given_name": "David", "family_name": "Horn", "institution": null}, {"given_name": "Nir", "family_name": "Levy", "institution": null}, {"given_name": "Isaac", "family_name": "Meilijson", "institution": null}, {"given_name": "Eytan", "family_name": "Ruppin", "institution": null}]}