{"title": "Plasticity Kernels and Temporal Statistics", "book": "Advances in Neural Information Processing Systems", "page_first": 1303, "page_last": 1310, "abstract": "", "full_text": "Plasticity Kernels and Temporal Statistics \n\nPeter Dayan1  Michael Hausser2  Michael London1\u00b72 \n\n1 GCNU,  2WIBR, Dept of Physiology \nUCL,  Gower Street, London \n\ndayan@gats5y.ucl.ac.uk  {m.hausser,m.london}@ucl.ac.uk \n\nAbstract \n\nComputational  mysteries  surround  the  kernels  relating  the \nmagnitude  and  sign  of  changes  in  efficacy  as  a  function  of \nthe time difference between pre- and post-synaptic  activity at \na synapse.  One important idea34  is that kernels result from fil(cid:173)\ntering, ie an attempt by synapses to eliminate noise corrupting \nlearning.  This idea has hitherto been applied to trace learning \nrules; we apply it to experimentally-defined kernels, using it to \nreverse-engineer assumed signal statistics. We also extend it to \nconsider the additional goal for  filtering of weighting learning \naccording  to  statistical  surprise,  as  in the  Z-score  transform. \nThis provides a fresh view of observed kernels and can lead to \ndifferent, and more natural, signal statistics. \n\n1  Introduction \n\nSpeculation and  data that the  rules  governing  synaptic  plasticity  should in(cid:173)\nclude  a  very  special  role  for  timel,7,13,17,20,21,23,24,26,27,31,32,3S  was  spectac-\nularly  confirmed by  a  set  of  highly  influential  experiments4\u00b7S,ll,l6,25  show(cid:173)\ning  that  the  precise  relative  timing  of  pre-synaptic  and  post-synaptic  ac(cid:173)\ntion  potentials  governs  the  magnitude  and  sign  of the  resulting  plasticity. \nThese  experimentally-determined rules  (usually called  spike-time  dependent \nplasticity  or  STDP  rules),  which  are  constantly  being  refined,18,3o  have  in(cid:173)\nspired substantial further theoretical work on their modeling and interpreta(cid:173)\ntion.2\u00b79,l0,22\u00b728\u00b729\u00b733  Figure l(Dl-Gl)* depict some of the main STDP findings/ \nof which the best-investigated are shown in figure l(Dl;El), and are variants of \na  'standard' STDP  rule.  Earlier work considered  rate-based rather than spike(cid:173)\nbased temporal rules, and so we adopt the broader term 'time dependent plas(cid:173)\nticity' or TDP. Note the strong temporal asymmetry in both the standard rules. \n\nAlthough the theoretical studies have provided us with excellent tools for mod(cid:173)\neling the detailed consequences of different time-dependent rules, and under(cid:173)\nstanding characteristics  such as long-run stability  and the relationship  with \nnon-temporal learning  rules  such  as  BCM,6  specifically  computational  ideas \nabout TDP  are  rather thinner  on the  ground.  Two  main qualitative notions \nexplored in various of the works  cited  above  are that the temporal asymme(cid:173)\ntries in TDP  rules are associated with causality or prediction.  However, look(cid:173)\ning  specifically  at  the  standard STDP  rules,  models interested in prediction \n\n*We refer to graphs in this figure by row and column. \n\n\fconcentrate mostly on the L 1P component and have difficulty explaining the \npredsely-timed nature of the LTD. Why should it be particularly detrimental to \nthe weight of a synapse that the pre-synaptic action potential comes just  after \na post-synaptic action-potential, rather than 200ms later, for instance?.  In the \ncase  of time-difference or temporal difference  rules, 29\u202232  why might the LTD \ncomponent be  so  different from the mirror reflection of the L1P  component \n(figure 1(\u00a31)),  at least short of being tied to some particular biophysical char(cid:173)\nacteristic of the post-synaptic cell. We seek alternative computationally-sound \ninterpretations. \nWallis & Baddeley34 formalized the intuition underlying one class of TDP rules \n(the so-called trace based rules, figure  l(A1)) in terms of temporal filtering.  In \ntheir model,  the actual output is a noisy version of a 'true' underlying signal. \nThey suggested, and showed in an example, that learning proceeds more profi(cid:173)\nciently if the output is filtered by an optimal noise-removal filter (in their case, \na Wiener filter) before entering into the learning rule. This is like using a prior \nover the signal,  and performing learning based on the (mean) of the posterior \nover the  signal given the observations (ie the output).  If objects in the world \nnormally persist for substantial periods, then, under some reasonable assump(cid:173)\ntions  about noise,  it turns out to be appropriate to  apply a low-pass filter  to \nthe output. One version of this leads to a trace-like learning rule. \nOf  course,  as seen in column 1 of figure  1,  TDP  rules are generally not trace(cid:173)\nlike. Here, we extend the Wallis-Baddeley (WB) treatment to rate-based versions \nof the actual rules shown in the figure.  We consider two possibilities, which in(cid:173)\nfer optimal signal models from the rules, based on two different assumptions \nabout their computational role.  One continues to regard them as Wiener filters. \nThe other, which is closely related to recent work on adaptation and modula(cid:173)\ntion, 3\u2022 s, 15\u2022 36 has the kernel normalize frequency components according to their \nstandard deviations,  as well as removing noise.  Under this interpretation, the \nlearning signal is a Z-score-transformed veFsion of the output. \n\nIn section 2,  we  describe the WB  model.  In section 3,  we  extend this model to \nthe case of the observed rules for synaptic plasticity. \n\n2  Filtering \n\nConsider a set of pre-synaptic inputs i  E  { 1 ... n} with firing rates Xi ( t) at time \nt  to a neuron with output rate y  ( t). A general TDP plasticity rule suggests that \nsynaptic weight Wi  should change according to the correlation between input \nXi(t) and output y(t), through the medium of a temporal filter cf>(s) \n\n~wi ex: J dtxi(t) {J dt'y(t')cf>(t-t')} =  J dt'y(t'){J dtxi(t)cp(t-t')} \n\n(1) \n\nProvided  the  temporal filters for  each synapse on a  single  post-synaptic  cell \nare the same, equation 1 indicates that pre-synaptic and post-synaptic filtering \nhave essentially the same effect. \nWB34  consider the case that the output can be decomposed as y(t)  =  s(t) + \nn(t), where s(t) is a 'true' underlying signal and n(t) is  noise corrupting the \nsignal.  They suggest defining the filter so that s(t)  =  f dt' y(t')cf>(t-t') is the \noptimal least-squares estimate of the signal.  Thus, learning would be based on \nthe best available information about the signal s ( t). If signal and noise are sta(cid:173)\ntistically stationary signals,  with power spectra  IS ( w) 12  and  IN ( w) 12  respec(cid:173)\ntively at (temporal) frequency w, then the magnitude of the Fourier transform \n\n\fCD \nkernel \n\n0 \nkernel \n\nspectrum \n\n(A]~ \n;\u00a7 \n:o \n...... \n\nt \n\n[[] \nsignal \n\nspectrum \n\n!wiener I \n\n[!] \nsignal \n\nspectrum \n\n!white I \n\n~ \n\nw \n\n\u00a9j \n\nL  L  klrug~N~ \n\n0 \n\nt \n\nw \n\nw \n\nw \n\nFigure  1:  Time-dependent plastidty rules.  The  rows are for  various  suggested rules \n(A; 17 B;23 D;25  E;16 F; 2  G, 14 from Abbott & Nelson2); the columns show:  (1) the kernels in \ntime t;  (2) their temporal power spectra as a function of frequency w; (3)  signal power \nS ( w)  as function of w assuming the kernels are derived from the underlying Wiener \nfilter;  (4)  signal power S(w) assuming the kernels are derived from the noise-removal \nand whitening filter.  Different kernels  also have different phase spectra.  See text for \nmore details.  The ordinates of the plots have been individually normalized;  but the \nabscissce  for  all the temporal (t) plots and,  separately,  all  the the  spectral (w) plots, \nare the same, for the purposes of comparison.  Numerical scales are omitted to focus \non structural characteristics. In the text, we refer to individual graphs in this figure by \ntheir row letter (A-G)  and column number (1-4). \n\n\fof the (Wiener) filter is \n\nI<P(w)l  =  IS(w)l 2  + IN(w)l 2 \n\nIS(w)l 2 \n\n(2) \n\nAny filter  with this  spectral tuning will  eliminate noise  as  best  as  possible; \nthe remaining freedom lies  in choosing the phases of the various frequency \ncomponents.  Following  Foldiak,l?  WB  suggest using  a  causal filter  for y(t), \nwith  cp(t-t')  =  0  fort< t'.  This  means  that the  input Xi(t)  at timet is \ncorrelated with weighted values of y(t') for times t'::;;; t  only.  In fact, WB  derive \nthe optimal acausal filter and take its casual half, which is not necessarily the \nsame thing.  Interestingly,  the forms  of TDP  that have  commonly been used \nin reinforcement learning23\u202231\u202232  consider purely acausal filters for y(t) (such \nthat Xi(t)  is  correlated with  future  values  of the  output),  and therefore use \nexactly the opposite condition on the filter, namely that cp(t-t') =  0 fort> t'. \n\nIn the context of input coming from visually presented objects, WB  suggest us(cid:173)\ning white noise N ( w) =  N, 'if w, and consider two possibilities for S ( w), based \non the assumptions that objects persist for  either fixed,  or randomly variable, \nlengths  of time.  We  summarize their  main result in the  first  three rows  of \nfigure  1.  Figure  l(A3)  shows  the  assumed,  scale-free,  magnitude  spectrum \nIS(w)l  =  1/w for  the  signal.  Figure  l(Al) shows  the  (truly  optimal)  purely \ncausal version of the  filter  that results  - it  can be  shown to involve  exactly \nan exponential decay, with a rate constant which depends on the level of the \nnoise N.  In WB's  self-supervised setting, it is rather unclear a priori whether \nthe assumption of white noise is valid; WB's experiments bore it out to a rough \napproximation, and showed that the filter of figure l(Al) worked well on a task \ninvolving digit representation and recognition. \n\nFigure  l(Bl;B3)  repeat  the  analysis,  with the  same  signal  spectrum,  but for \nthe optimal purely acausal filter as used in reinforcement learning's synaptic \neligibility traces.  Of course,  the true TDP  kernels  (shown in figure  l(Dl-Gl)) \nare neither purely casual nor acausal; figure  l(Cl) shows the normal low pass \nfilter that results from assuming phase 0 for all frequency components. \n\nAlthough the WB  filter  of figure  l(Cl) somewhat resembles a Hebbian version \nof the anti-Hebbian rule for layer IV  spiny stellate cells shown in figure  l(Gl), \nit is clearly not a good match for the standard forms of TDP.  One might also \nquestion the relationship between the time  constants of the kernels and the \nsignal spectrum that comes from object persistence.  The next section consid(cid:173)\ners two alternative possibilities for interpreting TDP kernels. \n\n3  Signalling and Whitening \n\nThe main intent of this paper is to combine WB's idea about the role of filtering \nin synaptic plasticity with the  actual forms  of the kernels that have been re(cid:173)\nvealed in the experiments.  Under two  different models for the computational \ngoal of filtering,  we work back from the experimental kernels to the implied \nforms  of the statistics of the signals.  The first method employs WB's  Wiener \nfiltering idea. The second method can be seen as using a more stringent defin(cid:173)\ntion of statistical significance. \n\n\f~ phase for E1  ~ phase=-\u00a5 \n\n(g DoG kernel  @]  DoG signals \n\nb ,, \n\n\\ \n\\  Wiener \n\nw \n\nt \n\nt \n\nw \n\nFigure 2:  Kernel manipulation.  A)  The phase spectrum (ie kernel phase as a function \nof frequency)  for the kernel (shown in figure  l(El)) with asymmetric LTP  and LTD.l 6 \nB)  The kernel that results from the power spectrum of figure  l(E2) but constant phase \n-rr/2.  This kernel has  symmetric LTP  and LTD,  with an intermediate time constant. \nC)  Plasticity kernel that is  exactly a  difference  of two  Gaussians  (DoG;  compare fig(cid:173)\nure l(Fl)). White (solid; from equation 4)  and Wiener (dashed; from equation 3)  signal \nspectra derived from the DoG kernel in (C).  Here,  the signal  spectrum in the case  of \nwhitening has been vertically displaced so it is clearer.  Both signal spectra show dear \nperiodicities. \n\n3.1  Reverse engineering signals from Wiener filtering \n\nAccepting  equation  2  as  the  form  of the  filter  (note  that  this  implies  that \nlci>(w)l  ~ 1),  and,  with WB,  making  the  assumption that the  noise  is  white, \nso  IN(w)l  = N, Vw,  the  assumed  amplitude  spectrum of the signal process \ns(t) is \n\n(3) \nImportantly, the assumed power of the noise does  not affect the form of the \nsignal power, it only scales it. \n\nIS(w)l  = N~lci>(w)l/(1-lci>(i.o)i). \n\nFigure  1(D2-G2) shows the magnitude of the Fourier transform of the experi(cid:173)\nmental kernels (which are shown in figure 1(D1-G 1)),  and figure 1(D3-G3) show \nthe implied signal spectra. Since there is no natural data that specify the abso(cid:173)\nlute scale of the kernels (ie the maximum value of lci>(w) 1),  we set it arbitrarily \nto  0.5.  Any value  less than  ~ 0.9  leads to  similar predictions for  the  signal \nspectra.  We  can relate figure  1(D3-G3)  to  to the heuristic  criteria mentioned \nabove  for  the  signal power spectrum.  In two  cases  (D3;F3),  the  clear peaks \nin the signal power spectra imply strong periodicities.  For  layer V pyramids \n(D3),  the time constant for the kernel is  ~ 20ms, implying a peak frequency of \nw =50Hz in they band.  In the hippocampal case, the frequency may be a lit(cid:173)\ntle lower.  Certainly,  the signal power spectra underlying the different kernels \nhave quite different forms. \n\n3.2  Reverse engineering signals from whitening \n\nWB's  suggestion that the underlying signal s(t) should be extracted from the \noutput y  ( t) far from exhausts the possibilities for filtering.  In particular, there \nhave been various suggestions36  that learning should be licensed by statistical \nsurprise,  ie  according to how components of the  output differ from expecta(cid:173)\ntions.  A  simple form  of this  that has  gained recent  currency is  the  Z-score \ntransformation,8\u202215\u202236  which implies considering components of the signal in \nunits of (ie  normalized by) their standard deviations.  Mechanistically,  this is \nclosely related to whitening in the face of input noise, but with a rather differ(cid:173)\nent computational rationale. \n\n\fA simple formulation of a noise-sensitive Z-score is Dong & Atick's12 whitening \nfilter.  Under the same formulation as WB  (equation 2),  this suggests multiply(cid:173)\ning the Wiener filter by 1 I IS ( w) I,  giving \n\nI<I>(w)l  =  IS(w)II(IS(w)l 2  + N(w) 2). \n\n(4) \n\nAs  in equation 3,  it is possible to  solve for the  signal power  spectra implied \nby the various kernels.  The 4th column of figure  1 shows the result of doing \nthis for the experimental kernels. In particular, it shows that the clear spectral \npeaks suggested by the Wiener filter (in the 3rd column) may be artefactual -\nthey can arise from a form of whitening. Unlike the case of Wiener filtering, the \nsignal statistics derived from the assumption of whitening have the common \ncharacteristic of monotonically decreasing signal powers as a function of fre(cid:173)\nquency w, which is a common finding for natural scene statistics, for instance. \n\nThe case of the layer V pyramids25  (row Din figure  1) is particularly clear.  If \nthe time constants of potentiation (LTP)  and depression (LTD)  are  T,  and LTP \nand LTD  are matched, then the Fourier transform of the plasticity kernel is \n\n<I>  W)  =  -\n\n( \n\n1 \n\nVEi \n\n( \n\n1 \n\niw + 1. \n\nT \n\n+ \n\n) \n\n1 \n\niw - 1. \nT \n\n-~  w \n\n=  -t  -\n\nrr w2 + ~ \n\nT \n\n.  2~  tJ \n\n= -tT \n\n-\nrr ~ + T2 \n\nW \n\n( ) \n5 \n\nwhich is exactly the form of equation 4 for S ( w) = 1 I w  (which is duly shown \nin figure  1(D4)).  Note  the  factor  of -i in <I>(w).  This  is  determined by the \nphases of the frequency components,  and comes from the anti-symmetry of \nthe kernel.  The phase of the components (L<I>(w) = -rr 12, by one convention) \nimplies the predictive nature of the kernel:  Xi(t) is being correlated with led \n(ie future) values of noise-filtered, significance-normalized, outputs. \n\nThe other cases in figure  1 follow in a similar vein.  Row E,  from cortical layer \nII/ll,  with its  asymmetry between LTP  and LID,  has  similar  signal  statistics, \nbut with an extra falloff  constant  w0,  making S(w)  =  11(w + w 0 ).  Also,  it \nhas  a  phase spectrum  L<I>(w)  which is  not constant with w  (see  figure  2A). \nRow F, from hippocampal GABAergic cells in culture, has a form that can arise \nfrom an exponentially decreasing signal power and little assumed noise (small \nN ( w) ).  Conversely, row G,  in cortical layer IV  spiny-stellate cells,  arises from \nthe same signal statistics, but with a large noise term N(w).  Unlike the case \nof the Wiener filter (equation 3),  the form of the signal statistics, and not just \ntheir magnitude, depends on the amount of assumed noise. \n\nFigure 2B-C  show various aspects of how these results change with the param(cid:173)\neters or forms  of the kernels.  Figure  2B  shows that coupling the power spec(cid:173)\ntrum (of figure 1E2) for the rule with asymmetric LTP and LTD with a constant \nphase spectrum (-rr 12) leads to a rule with the same filtering  characteristic, \nbut with symmetric LTP and LTD. The phase spectrum concerns the predictive \nrelationship between pre- and post-synaptic frequency components; it will be \ninteresting to  consider the kernels  that result from  other temporal relation(cid:173)\nships between pre- and post-synaptic  activities.  Figure  2C  shows  the kernel \ngenerated as a difference of two Gaussians (DoG). Although this kernel resem(cid:173)\nbles that of figure  1F1, the signal spectra (figure 2D) calculated on the basis of \nwhitening (solid;  vertically displaced)  or Wiener filtering  (dashed)  are  similar \nto each other, and both involve strong periodicity near the spectral peak of the \nkernel. \n\n\f4  Discussion \n\nTemporal asymmetries in synaptic plasticity have been irresistibly alluring to \ntheoretical treatments.  We  followed  the suggestion that the kernels indicate \nthat learning is not based on simple correlation between pre- and post -synaptic \nactivity, but rather involves filtering in the light of prior information, either to \nremove noise from the signals (Wiener filtering), or to remove noise and boost \ncomponents of the signals according to their statistical significance. \n\nAdopting this view leads to new conclusions about the kernels, for instance re(cid:173)\nvealing how the phase spectrum differentiates rules with symmetric and asym(cid:173)\nmetric potentiation and depression components  (compare figures  l(El);  2B). \nMaking  some  further  assumptions  about the  characteristics  of the  assumed \nnoise, it permits us to reverse engineer the assumed statistics of the signals, ie \nto give a window onto the priors at synapses or cells (columns 3;4 of figure  1). \nStructural features in these signal statistics, such as strong periodicities, may \nbe related to experimentally observable characteristics such as oscillatory ac(cid:173)\ntivity in relevant brain regions.  Most importantly,  on this view,  the detailed \ncharacteristics of the filtering might be expected to  adapt in the light of pat(cid:173)\nterns of activity.  This  suggests the straightforward experimental test of ma(cid:173)\nnipulating the input and/or output statistics and recording the consequences. \n\nVarious characteristics of the rules bear comment. Since we wanted to focus on \nstructural features of the rules, the graphs in the figures all lack precise time \nor frequency scales. In some cases we know the time constants of the kernels, \nand they are usually quite fast (on the order of tens of milliseconds).  This can \nsuggest high frequency spectral peaks in assumed signal statistics. However, it \n\u00b7also hints at the potential inadequacy of our rate-based treatment that we have \ngiven,  and suggests the importance  of a  spike-based treatment. 22\u2022 30  Recent \nevidence that successive pairs of pre- and post-synaptic spikes do not interact \nadditively in determining the magnitude and direction of plasticity18 make the \naveraging inherent in the rate-based approximation less  appealing.  Further, \nwe commented at the outset that pre- and post-synaptic filtering have similar \neffects, provided that all the filters  on one post-synaptic cell are the same.  If \nthey are  different,  then synapses  might well be treated as  individual filters, \nascertaining important signals for learning. In our framework, it is interesting \nto speculate about the role of (pre-)synaptic depression itself as a form of noise \nfilter (since noise should be filt\u20acred  before it can affect the activity of the post(cid:173)\nsynaptic cell, rather than just its plasticity); leaving the kernel as a significance \nfilter,  as in the whitening  treatment.  Finally,  largely because  of the  separate \nroles of signal and noise, we have been unable to think of a simple experiment \nthat would test between Wiener and whitening filtering.  However, it is a quite \ncritical issue in further exploring computational accounts of plasticity. \n\nAcknowledgements \n\nWe  are very grateful to Odelia Schwartz for helpful discussions.  Funding was \nfrom the Gatsby Charitable Foundation, the Wellcome Trust (MH)  and an HFSP \nLong Term Fellowship (ML). \n\nReferences \n\u00b7 [1]  Abbott, LF,  & Blum, KI  (1996) Functional significance of long-term potentiation for sequence learning \n\nand prediction. Cerebral Cortex 6:406-416. \n\n[2]  Abbott, LF & Nelson, SB  (2000) Synaptic plasticity: taming the beast. Nature Neuroscience 3:1178-1183. \n\n\f[3]  Atick, JJ,  li, Z,  & Redlich,  AN  (1992) Understanding retinal color coding from first principles. Neural \n\nComputation 4:559-572. \n\n[4]  Bell,  CC,  Han,  VZ,  Sugawara,  Y &  Grant K  (1997)  Synaptic plasticity in a  cerebellum-like  structure \n\ndepends on temporal order. Nature 387:278-81. \n\n[5]  Bi,  GQ &  Poo,  MM  (1998)  Synaptic modifications in cultured hippocampal neurons:  dependence on \n\nspike timing, synaptic strength, and postsynaptic cell type. Journal of Neurosdence 18:10464-10472. \n[6]  Bienenstock, EL,  Cooper,  IN, & Munro, PW  (1982)  Theory for  the development of neuron selectivity: \n\nOrientation specificity and binocular interaction in visual cortex. Journal of Neuroscience 2:32-48. \n\n[7]  Blum, KI, & Abbott, LF (1996) A model of spatial map formation in the hippocampus of the rat. Neural \n\nComputation 8:85-93. \n\n[8]  Buiatti, M & Van Vreeswijk, C (2003) Variance normalisation: a key mechanism for temporal adaptation \n\nin natural vision?  VISion Research, in press. \n\n[9]  Cateau,  H  &  Fukai,  T  (2003)  A stochastic method to predict the consequence of arbitrary forms  of \n\nspike-timing-dependent plasticity. Neural Computation 15:597-620. \n\n[1 0]  Chechik, G (2003). Spike time dependent plasticity and information maximization. Neural Computation \n\nin press. \n\n[11]  Debanne,  D,  Gahwiler,  BH  &  Thompson,  SM  (1998)  Long-term  synaptic plasticity between pairs  of \n\nindividual CA3  pyramidal cells in rat hippocampal slice cultures. Journal of Physiology 507:237-247. \n\n[12]  Dong, DW,  & A tick, JJ (1995) Temporal decorrelation: A theory of lagged and nonlagged responses in \n\nthe lateral geniculate nucleus. Network:  Computation in Neural Systems 6:159-178. \n\n[13]  Edelman, S & Weinshall, D (1991) A self-organizing multiple-view representation of 3D objects. Biolog(cid:173)\n\nical Cybernetics 64:209-219. \n\n[14]  Egger, V, Feldmeyer, D & Sakmann, B (1999) Coincidence detection and changes of synaptic efficacy in \n\nspiny stellate neurons in rat barrel cortex. Nature Neurosdence 2:1098-1105. \n\n[15]  Fairhall, AL,  Lewen,  GD,  Bialek, W &  de Ruyter Van Steveninck, RR (2001) Efficiency and ambiguity in \n\nan adaptive neural code. Nature 412:787-792. \n\n[16]  Feldman, DE  (2000) Timing-based LTP  and LTD  at vertical inputs to layer II/III pyramidal cells in rat \n\nbarrel cortex. Neuron 27:45-56. \n\n[17]  Foldiflk, P (1991) Learning invariance from transformed sequences. Neural Computation 3:194-200. \n[18]  Froemke, RC  & Dan, Y (2002) Spike-timing-dependent synaptic modification induced by natural spike \n\ntrains. Nature 416:433-438. \n\n[19]  Ganguly K,  Kiss,  L & Poo,  M (2000)  Enhancement of presynaptic neuronal excitability by correlated \n\npresynaptic and postsynaptic spiking. Nature Neuroscience 3:1018-1026. \n\n[20]  Gerstner, W & Abbott, LF  (1997) Learning navigational maps through potentiation and modulation of \n\nhippocampal place cells. Journal of Computational Neurosdence 4:79-94. \n\n[21]  Gerstner,  W,  Kempter,  R,  van Hemmen,  JL  &  Wagner,  H  (1996)  A  neuronal learning rule  for  sub(cid:173)\n\nmillisecond temporal coding. Nature  383:76-81. \n\n[22]  Gerstner, W & Kistler, WM (2002) Mathematical formulations ofHebbianlearning. Biological Cybernetics \n\n87:404-15 .. \n\n[23]  Hull, CL (1943) Prindples of Behavior New York, NY: Appleton-Century. \n[24]  Levy,  WB  & Steward, D (1983)  Temporal contiguity requirements for long-term associative potentia(cid:173)\n\ntion/depression in the hippocampus. Neurosdence 8:791-797 \n\n[2 5]  Markram, H, Lubke, J, Frotscher, M, & Sakmann, B (1997) Regulation of synaptic efficacy by coincidence \n\nof postsynaptic APs and EPSPs.  Science 275:213-215. \n\n[26]  Minai, AA, & Levy,  WB  (1993) Sequence learning in a single trial. International Neural Network Society \n\nWorld Congress of Neural Networks II. Portland, OR: International Neural Network Society, 505-508. \n\n[2 7]  Pavlov, PI (192 7)  Conditioned Reflexes Oxford, England, OUP. \n[28]  Porr, B & Worgotter, F (2003) Isotropic sequence order learning. Neural Computation 15:831-864. \n[29]  Rao, RP & Sejnowski, TJ (2001) Spike-timing-dependentHebbian plasticity as temporal difference learn(cid:173)\n\ning. Neural Computation 13:2221-2237. \n\n\u00b7 \n\n[30]  Sjostrom,  PJ,  Turrigiano,  GG  &  Nelson,  SB  (2001)  Rate,  timing,  and cooperativity jointly determine \n\ncortical synaptic plasticity. Neuron 32:1149-1164. \n\n[31]  Sutton, RS  (1988) Learning to predict by the methods of temporal difference. Machine Learning 3:9-44. \n[3 2]  Sutton, RS  & Barto, AG (1981) Toward a modern theory of adaptive networks: Expectation and predic(cid:173)\n\ntion. Psychological Review 88:135-170. \n\n[33]  van Rossum, MC, Bi,  GQ & Turrigiano, GG (2000) Stable Hebbian learning from spike timing-dependent \n\nplasticity. journal of Neurosdence 20:8812-21. \n\n[34]  Wallis,  G & Baddeley, R (1997) Optimal, unsupervised learning in invariant object recognition. Neural \n\nComputation 9:883-894. \n\n[35]  Wallis,  G & Rolls,  ET  (1997).  Invariant face  and object recognition in the visual system. it Progress in \n\n[36]  Yu,  AJ & Dayan,  P (2003) Expected and unexpected uncertainty:  ACh & NE  in the neocortex. In NIPS \n\nNeurobiology 51:167-194. \n\n2002 Cambridge, MA: MIT Press. \n\n\f", "award": [], "sourceid": 2392, "authors": [{"given_name": "Peter", "family_name": "Dayan", "institution": null}, {"given_name": "Michael", "family_name": "H\u00e4usser", "institution": null}, {"given_name": "Michael", "family_name": "London", "institution": null}]}