{"title": "Spike-based Learning Rules and Stabilization of Persistent Neural Activity", "book": "Advances in Neural Information Processing Systems", "page_first": 199, "page_last": 208, "abstract": null, "full_text": "Spike-based learning rules and stabilization of \n\npersistent neural activity \n\nXiaohui Xie and H. Sebastian Seung \n\nDept.  of Brain &  Cog.  Sci., MIT, Cambridge, MA 02139 \n\n{xhxie, seung}@mit.edu \n\nAbstract \n\nWe  analyze  the  conditions  under  which  synaptic  learning  rules  based \non  action  potential timing can  be approximated by  learning rules  based \non  firing  rates.  In  particular, we  consider a form  of plasticity in  which \nsynapses depress when a presynaptic spike is followed by a postsynaptic \nspike, and potentiate with the opposite temporal ordering.  Such differen(cid:173)\ntial anti-Hebbian plasticity can be approximated under certain conditions \nby a learning rule that depends on the time derivative of the postsynaptic \nfiring rate.  Such a learning rule acts to stabilize persistent neural activity \npatterns in recurrent neural networks. \n\n1 \n\nINTRODUCTION \n\n\u00b0 \n\n1000 \n\ntime (ms) \n\n2000 \n\nexperiments \n\no \n- t \n\npre \n\nt \npost \n\nA  o~i =~=::====: \nB  0L - . i  _:3/_-----' \n\n~re 11111111111  111111111111111111 \npost 11111111111 11111111 \u2022 \u2022  11. \nt .:oLl ____ \\:;J \n__ ~ \n\nRecent \nhave \ndemonstrated types of synaptic \nplasticity  that  depend  on  the \ntemporal ordering of presynap(cid:173)\ntic and postsynaptic spiking.  At \ncortical [ I]  and  hippocampal[2] \nsynapses, \nlong-term  potenti(cid:173)\nation  is  induced  by  repeated \npairing  of  a  presynaptic  spike \nand  a  succeeding  postsynaptic \nspike,  while  long-term  depres(cid:173)\nsion  results  when  the  order \nis  reversed.  The  dependence \nin  synaptic \nof \nthe  difference \nstrength  on \nl:!..t  = \ntpre  between \npostsynaptic  and  presynaptic \nspike times  has  been  measured \nquantitatively. \nThis  pairing \nfunction,  sketched  in  Figure \nlA,  has  positive  and  negative  lobes  correspond  to  potentiation  and  depression.  and  a \nwidth  of tens  of milliseconds.  We  will  refer  to  synaptic  plasticity  associated  with  this \npairing  function  as  differential  Hebbian  plasticity-Hebbian because the  conditions  for \n\nFigure  I:  (A)  Pairing  function  for  differential  Heb(cid:173)\nbian learning.  The change in  synaptic strength is  plot(cid:173)\nted  versus  the  time  difference  between  postsynaptic \nand  presynaptic spikes.  (B)  Pairing  function  for  dif(cid:173)\nferential  anti-Hebbian  learning.  (C)  Differential anti(cid:173)\nHebbian learning is  driven  by changes in  firing  rates. \nThe synaptic learning rule of Eq.  (l) is  applied to two \nPoisson  spike  trains.  The  synaptic  strength  remains \nroughly constant in  time,  except when  the postsynap(cid:173)\ntic rate changes. \n\nthe  change \n\ntpost  -\n\n\f200 \n\nX  Xie and H.  S.  Seung \n\npotentiation  are  as  predicted  by  Hebb[3],  and  differential  because  it  is  driven  by  the \ndifference between the opposing processes of potentiation and depression. \n\nThe  pairing function  of Figure  IA is  not characteristic  of all  synapses.  For example,  an \nopposite temporal dependence has been observed at electrosensory lobe synapses of elec(cid:173)\ntric  fish[4].  As  shown  in  Figure  IB,  these  synapses depress  when  a  presynaptic spike is \nfollowed by a postsynaptic one, and potentiate when the order is  reversed.  We will refer to \nthis as  differential anti-Hebbian plasticity. \n\nAccording to these experiments, the maximum ranges of the differential Hebbian and anti(cid:173)\nHebbian pairing functions are roughly 20 and 40 ms, respectively.  This is fairly  short, and \nseems more compatible with descriptions  of neural activity  based on spike timing  rather \nthan instantaneous firing  rates[5,  6].  In fact,  we will  show  that there are some conditions \nunder which spike-based learning rules can be approximated by rate-based learning rules. \nOther people have also studied the relationship between spike-based and rate-based learn(cid:173)\ning rules[7,  8]. \n\nThe pairing functions  of Figures  IA and  IB  lead  to  rate-based  learning  rules  like those \ntraditionally  used in  neural networks,  except that they  depend on temporal derivatives of \nfiring  rates  as  well  as  firing  rates  themselves.  We  will  argue  that  the  differential  anti(cid:173)\nHebbian learning rule of Figure  IB  could be a general mechanism for tuning the strength \nof positive feedback in networks that maintain a short-term memory of an  analog variable \nin  persistent neural activity.  A  number of recurrent network  models  have  been  proposed \nto  explain  memory-related neural  activity  in  motor  [9]  and  prefrontal [ 10]  cortical  areas, \nas  well  as  the head direction  system  [11]  and  oculomotor integrator[ 12,  13,  14].  All  of \nthese models require precise tuning of synaptic strengths in order to maintain continuously \nvariable levels of persistent activity.  As a simple illustration of tuning by differential anti(cid:173)\nHebbian learning, a model of persistent activity maintained by an integrate-and-fire neuron \nwith an excitatory autapse is studied. \n\n2  SPIKE-BASED LEARNING RULE \n\nPairing functions  like those of Figure  1 have  been  measured using  repeated pairing  of a \nsingle presynaptic spike  with  a  single postsynaptic spike.  Quantitative measurements  of \nsynaptic changes due to more complex patterns of spiking activity have not yet been done. \nWe will assume a simple model in which the synaptic change due to arbitrary spike trains is \nthe sum of contributions from all possible pairings of presynaptic with postsynaptic spikes. \nThe model is  unlikely to be an exact description of real synapses, but could turn out to be \napproximately valid. \nWe will  write the spike train of the ith neuron as a series of Dirac delta functions,  Si (t)  = \nLn <5(t - Tr), where Tr is the nth spike time of the ith neuron.  The synaptic weight from \nneuron j  to i  at time t  is denoted by Wij (t).  Then the change in  synaptic weight induced \nby presynaptic spikes occurring in the time interval [0, Tj is modeled as \n\nWij(T + >.)  - Wij(>')  =  [T dtj  foo  dti  f(ti - tj)Si(ti) Sj(tj) \n\nio \n\n-00 \n\n(1) \n\nEach  presynaptic  spike  is  paired  with  all  postsynaptic  spikes  produced before and  after. \nFor each pairing,  the synaptic weight is  changed by an amount depending on the pairing \nfunction f.  The pairing function is  assumed to be nonzero inside the interval [-T, Tj,  and \nzero outside.  We will refer to T  as the pairing range. \n\nAccording to our model, each presynaptic spike results in induction of plasticity only after \na latency>..  Accordingly, the arguments T + >.  and >.  of Wij  on the left hand side of the \nequation are shifted relative to the limits T  and 0 of the integral on the right hand side.  We \n\n\fSpike-based Learning and Stabilization of Persistent Neural Activity \n\n201 \n\nwill assume that the latency>. is greater than the pairing range T, so that Wi}  at any time is \nonly influenced by presynaptic and postsynaptic spikes that happened before that time, and \ntherefore the learning rule is causal. \n\n3  RELATION TO RATE-BASED LEARNING RULES \n\nThe learning rule of Eq. (1) is driven by correlations between presynaptic and postsynaptic \nactivities.  This dependence can be made explicit by making the change of variables u = \nti  - t j in Eq. (I), which yields \n\nWij(T + >.)  - Wij (>.)  = iTT duf(u)Cij(u) \n\nwhere we have defined the cross-correlation \n\nCij(u) = !aT dt Si(t + u) Sj(t)  . \n\n(2) \n\n(3) \n\nand made use of the fact that f  vanishes outside the interval  [-T, T].  Our immediate goal \nis to relate Eq.  (2) to learning rules that are based on  the  cross-correlation between firing \nrates, \n\nCrre(u) = !aT dt Vi(t + u) Vj(t) \n\n(4) \n\nThere  are  a  number of ways  of defining  instantaneous  firing  rates.  Sometimes  they  are \ncomputed by averaging over repeated presentations of a stimulus.  In other situations, they \nare defined by temporal filtering of spike trains.  The following discussion is general, and \nshould apply to these and other definitions of firing rates. \n\nThe \"rate correlation\" is commonly subtracted from the total correlation to obtain the \"spike \ncorrelation\" C:rke  = Cij  - Cijate.  To derive a rate-based approximation to  the learning \nrule (2), we rewrite it as \n\nWij(T + >.)  - Wij(>')  = iTT du f(u)Cijate(u) + iTT du  f(u)C:r ke (u) \n\n(5) \n\nand simply neglect the second term.  Shortly we will  discuss the conditions  under which \nthis is a good approximation. But first we derive another form for the first term by applying \nthe approximation Vi(t + u)  ~ Vi(t)  + UVi(t)  to obtain \n\nj T  duf(u)Crre(u) ~ iT dt[fiovi(t) + 131Vi(t)]VJ (t) \n\n-T \n\n0 \n\nwhere we define \n\n(6) \n\n(7) \n\nThis  approximation is  good when  firing  rates  vary  slowly compared to the pairing range \nT .  The  learning  rule  depends  on  the postsynaptic  rate  through fio Vi  + 131 Vi .  When  the \nfirst  term  dominates the second,  then  the  learning rule  is  the  conventional one based  on \ncorrelations between firing rates, and the sign of fio  determines whether the rule is Hebbian \nor anti-Hebbian. \nIn the remainder of the paper,  we will  discuss the more novel  case where 130  =  O.  This \nholds for the pairing functions shown in Figures  lA and  IB, which have positive and neg(cid:173)\native lobes with areas that exactly cancel in  the definition of 130.  Then the dependence on \n\n\f202 \n\nX  Xie and H.  S.  Seung \n\npostsynaptic activity is purely on the time derivative of the firing rate.  Differential Hebbian \nlearning corresponds to /31  > 0 (Figure IA), while differential anti-Hebbian learning leads \nto /31  <  0  (Figure  IB).  To  summarize the /30  =  0 case,  the synaptic changes due to  rate \ncorrelations are approximated by \n\nWij ex:  -ViVj \n\n(diff.  anti-Hebbian) \n\n(8) \n\nfor  slowly  varying  rates.  These  formulas  imply  that a  constant  postsynaptic  firing  rate \ncauses no net change in synaptic strength.  Instead, changes in  rate are required to induce \nsynaptic plasticity. \n\nTo  illustrate  this  point,  Figure  lC shows  the  result  of applying  differential  anti-Hebbian \nlearning to two spike trains.  The presynaptic spike train was generated by a 50 Hz Poisson \nprocess,  while the postsynaptic spike train  was  generated by  an  inhomogeneous Poisson \nprocess  with  rate that shifted from  50 Hz  to 200 Hz at  1 sec.  Before and  after the  shift, \nthe synaptic strength fluctuates but remains roughly constant. But the upward shift in firing \nrate causes a downward shift in synaptic strength, in accord with the sign of the differential \nanti-Hebbian rule in Eq. (8). \n\nThe rate-based approximation works well for this example, because the second term of Eq. \n(5)  is  not so  important.  Let us  return  to the  issue  of the  general conditions  under which \n\nthis term can be neglected.  With Poisson spike trains, the spike correlations C: Pike (u)  are \nzero in  the limit T  -7  00,  but for finite T  they fluctuate about zero.  The integr~l over u in \nthe second term of (5) dampens these fluctuations.  The amount of dampening depends on \nthe pairing range T,  which sets the limits of integration.  In Figure 1 C we used a relatively \nlong pairing range of 100 ms,  which made the fluctuations small even for small T.  On the \nother hand,  if T  were short,  the fluctuations  would be small  only for large T_  Averaging \nover large T  is  relevant  when the amplitUde of f  is  small,  so  that the rate of learning  is \nslow.  In  this  case,  it takes  a  long time for significant synaptic changes to  accumulate,  so \nthat plasticity is  effectively driven by integrating over long time periods T  in Eq.  (l). \nIn the brain, nonvanishing spike correlations are sometimes observed even in  the T  -7  00 \nlimit,  unlike  with  Poisson  spike  trains.  These  correlations  are  often  roughly  symmetric \nabout zero,  in  which case they should produce little plasticity if the pairing functions  are \nantisymmetric  as  in  Figures  lA and  lB.  On the other hand,  if the  spike correlations are \nasymmetric, they could lead to substantial effects[6]. \n\n4  EFFECTS ON RECURRENT NETWORK DYNAMICS \n\nThe learning rules of Eq. (8) depend on both presynaptic and postsynaptic rates, like learn(cid:173)\ning  rules  conventionally used in  neural networks.  They  have the  special  feature  that they \ndepend  on  time  derivatives,  which  has  computational  consequences  for  recurrent  neural \nnetworks of the form \n\nXi  + Xi  = L Wiju(Xj) + bi \n\n(9) \n\nj \n\nSuch classical  neural  network equations can  be derived from  more biophysically realistic \nmodels using the method of averaging[ 15]  or a mean  field  approximation[ 16].  The firing \nrate of neuron j  is conventionally identified with Vj  = u(Xj). \nThe cost function E( {Xi}; {Wij}) =  ~ Li v; quantifies the amount of drift in firing rate at \nthe point Xl , ...  , X N  in the state space of the network. If we consider Vi  to be a function of \nXi  and Wij defined by (9), then the gradient ofthe cost function with respect to Wij is given \nby BE / BWij =  U' (Xi)ViVj.  Assuming that U  is a monotonically increasing function so that \nu' (xd  >  0, it follows that the differential Hebbian update of (8) increases the cost function, \n\n\fSpike-based Learning and Stabilization of Persistent Neural Activity \n\n203 \n\nand hence increases  the  magnitude of the drift velocity.  In  contrast, the  differential  anti(cid:173)\nHebbian update decreases the drift velocity. This suggests that the differential anti-Hebbian \nupdate could be useful for creating fixed points of the network dynamics (9). \n\n5  PERSISTENT ACTIVITY IN A SPIKING AUTAPSE MODEL \n\nThe preceding  arguments about drift velocity  were based  on  approximate rate-based de(cid:173)\nscriptions  of learning  and  network  dynamics.  It is  important to  implement spike-based \nlearning  in  a  spiking  network  dynamics,  to  check  that  our  approximations  are  valid. \nTherefore  we  have  numerically  simu(cid:173)\nlated  the  simple  recurrent  circuit  of \nintegrate-and-fire neurons shown in Fig(cid:173)\nure  2.  The  core  of  the  circuit  is  the \n\"memory neuron,\" which makes an exci(cid:173)\ntatory autapse onto itself. It also receives \nsynaptic input from three input neurons: \na  tonic  neuron,  an  excitatory burst neu(cid:173)\nron, and an  inhibitory burst neuron.  It is \nknown that this circuit can store a short(cid:173)\nterm  memory  of an  analog  variable  in \npersistent activity, if the strengths of the \nautapse and  tonic  synapse are precisely \ntuned[ 17].  Here we show  that this  tun(cid:173)\ning  can  be  accomplished  by  the  spike(cid:173)\nbased learning rule of Eq.  (1), with a  d(cid:173)\nifferential anti-Hebbian pairing function \nlike that of Figure 1 B. \n\nINHIBITORY BURST \n\u2022 \n\nFigure 2:  Circuit diagram for autapse model \n\nThe memory neuron is described by the equations \n\n= \n\nC  dV \nm  dt \ndr \n\nTsyn dt  + r \n\nn \n\n(10) \n\n(1) \n\nwhere V  is the membrane potential. When V  reaches V'thres,  a spike is considered to have \noccurred, and V  is  reset to  Vreset.  Each spike at time Tn  causes a jump in  the synaptic \nactivation  r  of size CY.r/Tsyn,  after which r  decays exponentially with time constant Tsyn \nuntil the next spike. \n\nThe synaptic conductances of the memory neuron are given by \n\n(12) \n\nThe term W r  is  recurrent excitation from the autapse, where W  is  the strength of the au(cid:173)\ntapse.  The synaptic activations ro, r +, and r _  of the tonic, excitatory burst, and inhibitory \nburst neurons are governed by equations like (10) and (1), with a few differences.  These \nneurons have no synaptic input; their firing patterns are instead determined by applied cur(cid:173)\nrents  lapp,o,  lapp,+  and  lapp,_.  The tonic  neuron has  a  constant applied current,  which \nmakes  it fire  repetitively  at roughly  20 Hz  (Figure 3).  For the excitatory  and  inhibitory \nburst neurons the applied current is  normally zero, except for brief 100 ms current pulses \nthat cause bursts of action potentials. \n\nAs shown in Figure 3, if the synaptic strengths W  and Wo  are arbitrarily set before learning, \nthe burst neurons  cause only  transient changes in  the  firing  rate  of the  memory  neuron. \nAfter applying  the  spike-based  learning  rule  (1)  to  tune  both  W  and  Wo, the  memory \n\n\f204 \n\nX  Xie and H.  S.  Seung \n\n111111111111  I \n\nIUIIIIIIII  I \n\n~IIIIIIIII  I \n/untuned \n\n111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 \nI~ ____ ~I~ ____ ~ ______ =-____ _ \n\nI \n\nI \n\nI \n\n\" \n\ntuned \n\n1 sec \n\n1IIIIIIIIIIIIIIIIIIIIIIIIIIINIIIIII'.tl \n\n1111111111'\"111111111111111111111111111 \n\nFigure 3:  Untuned and tuned autapse activity.  The middle three traces  are the  membrane \npotentials  of the  three  input neurons  in  Figure  2 (spikes  are  drawn  at  the  reset times  of \nthe integrate-and-fire neurons).  Before learning, the activity of the  memory neuron is  not \npersistent, as  shown  in  the top trace.  After the spike-based learning rule  (1) is  applied to \nthe synaptic weights Wand Wo, then the burst inputs cause persistent changes in activity. \nem  = 1 nF,  gL  = 0.025 J-lS,  VL  = -70 mY,  VE  = 0 mY,  VI  = -70 mY,  vthres  = -52 \nmY,  Vr eset  = -59 mY,  a s  = 1,  Tsyn  = 100 ms,  Iapp,o  = 0.5203 nA,  I app,\u00b1 = 0 or 0.95 \nnA,  Ts yn ,O = 100 ms,  Tsyn,+  = Tsyn,- = 5 ms,  W+  = 0.1, W_  = 0.05. \n\nneuron is  able to  maintain  persistent activity.  During the interburst intervals (from A after \none burst until  A before the  next),  we  made  synaptic changes  using  the differential  anti(cid:173)\nHebbian  pairing  function  f(t)  =  -Asin(7l'tjT)  for  spike  time  differences  in  the range \n[-T, T]  with  A  =  1.5  X  10-4  and T=A=120 ms. The resulting increase in  persistence time \ncan be seen in  Figure 4A, along with the values of the synaptic weights versus time. \n\nTo quantify the performance of the system at maintaining persistent activity, we determined \nthe relationship between dv / dt and v using a long sequence of interburst intervals, where v \nwas defined as  the reciprocal of the interspike interval. If Wand Wo  are fixed at optimally \ntuned values, there is still a residual drift, as shown in Figure 4B. But if these parameters are \nallowed to adapt continuously, even after good tuning has  been achieved, then the residual \ndrift is  even  smaller in  magnitude.  This  is  because the  learning  rule  tweaks  the  synaptic \nweights during each interburst interval, reducing the drift for that particular firing rate. \n\nAutapse  learning  is  driven  by  the  autocorrelation  of the  spike  train,  rather  than  a  cross(cid:173)\ncorrelation.  The peak  in  the  autocorrelogram at zero  lag  has  no  effect, since the  pairing \nfunction is zero at the origin.  Since the autocorrelation is  zero for small time lags, we used \na fairly  large pairing range in  our simulations.  In a recurrent network of many neurons, a \nshorter pairing range would suffice, as the cross-correlation does not vanish near zero. \n\n6  DISCUSSION \n\nWe  have shown that differential anti-Hebbian learning can tune a recurrent circuit to main(cid:173)\ntain persistent neural activity. This behavior can be understood by reducing the spike-based \nlearning rule (l) to the rate-based learning rules ofEqs. (6) and (8). The rate-based approx(cid:173)\nimations are good if two conditions are satisfied.  First, the pairing range must be large, or \nthe  rate of learning  must be slow.  Second, spike  synchrony  must  be  weak,  or have  little \neffect on learning due to the shape of the pairing function. \n\nThe differential anti-Hebbian pairing function results  in  a learning rule  that uses  -Vi as  a \nnegative feedback signal  to  reduce the amount of drift  in  firing  rate,  as  illustrated by  our \nsimulations  of an  integrate-and-fire neuron  with  an  excitatory  autapse.  More  generally, \nthe learning rule could be relevant for tuning the strength of positive feedback in network(cid:173)\ns  that  maintain  a  short-term  memory  of an  analog  variable  in  persistent  neural  activity. \n\n\fSpike-based Learning and Stabilization of Persistent Neural Activity \n\n205 \n\nA \n\n200 \n\n250  c '  6 \n\n0.16  1 \n0.12  1 ~ \n:I:  0 \n~-2 \n\n0.395  W  WO \n\n10 \n\"me Is) \n\n0385 \n\n4 \n\n2 \n\n0 \n\n20 \n\nB \n\nI \n\n~150 \n~ \ni! \n\u00a7100 \n\"\" \n\nI  ~~ -4f \n\n50 \n\n-at \n!  -8' \n\n00 \n\n5 \n\n10 \n\n15 \ntlme(s) \n\n20 \n\n25 \n\n20 \n\n40 \n\n60 \n\nrate  (Hzl \n\n.~~ \n~r\u00b7  ~~~ \n\n'0 \n\n80 \n\ni \n100 \n\nFigure 4:  Tuning the autapse.  (A) The persistence time of activity increases as the weight(cid:173)\ns  Wand  Wo  are tuned.  Each transition  is  driven  by  pseudorandom bursts  of input (B) \nSystematic relationship between drift dv/dt in firing rate and v, as measured from a long \nsequence of interburst intervals. If the weights are continuously fine-tuned ('*') the drift is \nless than with fixed well-tuned weights ('0'). \n\nFor example, the learning rule could be used to improve the robustness of the oculomotor \nintegrator[12,  13,  14]  and head direction system[l1] to mistuning of parameters.  In deriv(cid:173)\ning the differential forms of the learning rules in (8), we assumed that the areas under the \npositive and negative lobes of the pairing function  are equal, so that the integral defining \n130  vanishes.  In  reality,  this cancellation  might not be exact.  Then the ratio of 131  and 130 \nwould limit the persistence time that can be achieved by the learning rule. \n\nBoth  the  oculomotor integrator and  the head  direction  system  are  also  able  to  integrate \nvestibular inputs to produce changes in activity patterns.  The problem of finding general(cid:173)\nizations of the present learning rules that train networks to integrate is  still  open. \n\nReferences \n\n[1]  H. Markram, J.  Lubke, M. Frotscher, and B.  Sakmann.  Science, 275(5297):213-5, 1997. \n[2]  G.  Q. Bi and M.  M.  Poo.  1 Neurosci,  18(24):10464-72,1998. \n[3]  D. O . Hebb.  Organization of behavior.  Wiley, New York,  1949. \n[4]  C. C.  Bell, V.  Z.  Han, Y.  Sugawara, and K.  Grant.  Nature, 387(6630):278-81 ,  1997. \n[5]  w. Gerstner, R . Kempter, 1.  L.  van Hemmen, and H. Wagner.  Nature,  383(6595):76-81,  1996. \n[6]  L.  F.  Abbott and S. Song.  Adv. Neural Info.  Proc.  Syst., 11,  1999. \n[7]  P.  D. Roberts. 1.  Comput. Neurosci., 7:235-246,  1999. \n[8]  R.  Kempter, W.  Gerstner, and J.  L.  van Hemmen.  Phys.  Rev.  E, 59(4):4498-4514,  1999. \n[9]  A. P. Georgopoulos, M.  Taira, and A. Lukashin.  Science, 260:47-52, 1993. \n[10]  M. Camperi and X. J. Wang. 1 Comput Neurosci, 5(4):383-405,  1998. \n[11]  K. Zhang. 1.  Neurosci.,  16:2112-2126,  1996. \n[12]  S. C. Cannon, D. A.  Robinson, and S. Shamma.  Bio!.  Cybern., 49:127-136,1983. \n[13]  H. S.  Seung.  Proc.  Nat!.  A cad. Sci. USA,  93:13339-13344, 1996. \n[14]  H. S. Seung, D.  D.  Lee, B. Y.  Reis, and D.  W.  Tank.  Neuron,  2000. \n[15]  B.  Ermentrout.  Neural Comput., 6:679-695,  1994. \n[16]  O. Shriki, D.  Hansel, and H.  Sompolinsky.  Soc. Neurosci.  Abstr., 24:143, 1998. \n[17]  H.  S. Seung, D. D. Lee, B. Y.  Reis, and D.  W.  Tank.  1.  Comput. Neurosci., 2000. \n\n\f\fPART III \nTHEORY \n\n\f\f", "award": [], "sourceid": 1658, "authors": [{"given_name": "Xiaohui", "family_name": "Xie", "institution": null}, {"given_name": "H. Sebastian", "family_name": "Seung", "institution": null}]}