{"title": "LTD Facilitates Learning in a Noisy Environment", "book": "Advances in Neural Information Processing Systems", "page_first": 150, "page_last": 156, "abstract": null, "full_text": "LTD Facilitates Learning In a Noisy \n\nEnvironment \n\nPaul Munro \n\nSchool of Information Sciences \n\nUniversity of Pittsburgh \nPittsburgh PA 15260 \n\npwm+@pitt.edu \n\nGerardina Hernandez \n\nIntelligent Systems Program \n\nUniversity of Pittsburgh \nPittsburgh PA 15260 \n\ngehst5+@pitt.edu \n\nAbstract \n\nLong-term potentiation (LTP) has long been held as a biological \nsubstrate for associative learning. Recently, evidence has emerged \nthat long-term depression (LTD) results when the presynaptic cell \nfires after the postsynaptic cell. The computational utility of LTD \nis explored here. Synaptic modification kernels for both LTP and \nLTD have been proposed by other laboratories based studies of one \npostsynaptic unit. Here, the interaction between time-dependent \nLTP and LTD is studied in small networks. \n\n1 \n\nIntroduction \n\nLong term potentiation (LTP) is a neurophysiological phenomenon observed under \nlaboratory conditions in which two neurons or neural populations are stimulated at a \nhigh frequency with a resulting measurable increase in synaptic efficacy between \nthem that lasts for several hours or days [1]-[2] LTP thus provides direct evidence \nsupporting the neurophysiological hypothesis articulated by Hebb [3]. \nThis increase in synaptic strength must be countered by a mechanism for weakening \nthe synapse [4]. The biological correlate, long-term depression (LTD) has also been \nobserved in the laboratory; that is, synapses are observed to weaken when low \npresynaptic activity coincides with high postsynaptic activity [5]-[6]. \n\nMathematical formulations of Hebbian learning produce weights, Wi}, (where i is the \npresynaptic unit and j is the postsynaptic unit), that capture the covariance [Eq. 1] \nbetween the instantaneous activities of pairs of units, ai and aj [7]. \n\nwij(t) = (ai (t) -ai)(a j (t)-aj) \n\n[1] \n\nThis idea has been generalized to capture covariance between acUvlUes that are \nshifted in time [8]-[9], resulting in a framework that can model systems with \ntemporal delays and dependencies [Eq. 2]. \n\nWij(t) = ffK(t\"-t')ai (tA')aj (t')dt'dt' \n\n[2] \n\n\fLTD Facilitates Learning in a Noisy Environment \n\n151 \n\nAs will be shown in the following sections, depending on the choice of the function \nK(L1t), this formulation encompasses a broad range of learning rules [10]-[12] and \ncan support a comparably broad range of biological evidence. \n\nFigure 1. Synaptic change as a function of the time difference between spikes from \nthe presynaptic neuron and the postsynaptic neuron. Note that for tpre < tpos t , LTP \nresults (L1w > 0), and for tpre > tp ost , the result is LTD. \n\nRecent biological data from [13]-[15], indicates an increase in synaptic strength \n(LTP) when presynaptic activity precedes postsynaptic activity, and LTD in the \nreverse case (postsynaptic precedes presynaptic). These ideas have started to appear \nin some theoretical models of neural computation [10]-[12], [16]-[18]. Thus, Figure \n1 shows the form of the dependence of synaptic change, Liw on the difference in \nspike arrival times. \n\n2 A General Framework \n\nGiven specific assumptions, the integral in Eq. 2 can be separated into two integrals, \none representing LTP and one representing LTD [Eq. 3]. \n\nWij(t) = \n\nt \nf Kp(t-t')ai(t')a/)dt' + \n\n{' = -00 \n\nv \n\nL~ \n\n' \n\nLID \n\nt \nf KD(t-t')ai(t)a/t')dt' \nr~'_=_-_oo __ ~v~ ____ ~ \n\n[3] \n\nThe activities that do not depend on t' can be factored out of the integrals, giving \ntwo Hebb-like products, between the instantaneous activity in one cell and a \nweighted time average of the activity in the other [Eq. 4]: \n\nwij (t) = (ai (t)) p a j (t) - ai (t)( a j (t)) D \nwhere (f(t)) X :; I ,f K X (t - t')f(t')dt' I for X E {P, D} \n\nt \n\nt =-00 \n\n[4] \n\nThe kernel functions Kp and KD can be chosen to select precise times out of the \nconvoluted function fit), or to average across the functions for an arbitrary range. The \nalpha function is useful here [Eq. 5]. A high value of a selects an immediate time, while \na small value approximates a longer time-average. \n\n-a 'r \nKx (r) = fixu X \n\nwith ap > O,aD > O,fip > O,fiD < \u00b0 \n\nfor X E {P,D} \n\n[5] \n\n\f152 \n\nP. W. Munro and G. Hernandez \n\nFor high values of ap and QD, only pre- and post- synaptic activities that are very close \ntemporally will interact to modify the synapse. In a simulation with discrete step sizes, \nthis can be reasonably approximated by only considering just a single time step [Eq. 6]. \ndWjj (t) = a j (t-l)a j(t)-aj (t)a j (t-1) \nSumming LiWi,{t) and Liwiit+l) gives a net change in the weights Li(2)Wij = wiiH1)-Wi/t-l) \nover the two time steps: \n\n[6] \n\n[7) \n\nThe first tenn is predictive in that it has the fonn of the delta rule where a/HI) acts as a \ntraining signal for aj (t-1), as in a temporal Hopfield network [9]. \n\n3 Temporal Contrast Enhancement \n\nThe computational role of the LTP term in Eq. 3 is well established, but how does the \nsecond term contribute? A possibility is that the term is analogous to lateral inhibition in \nthe temporal domain; that is, that by suppressing associations in the \"wrong\" temporal \ndirection, the system may be more robust against noise in the input. The resulting system \nmay be able to detect the onset and offset of a signal more reliably than a system not \nusing an anti-Hebbian LTD tenn. \n\nThe extent to which the LTD term is able to enhance temporal contrast is likely to depend \nidiosyncratically on the statistical qualities of a particular system. If so, the parameters of \nthe system might only be valid for signals with specific statistical properties, or the \nparameters might be adaptive. Either of these possibilities lies beyond the scope of \nanalysis for this paper. \n\n4 Simulations \n\nTwo preliminary simulation studies illustrate the use of the learning rule for predictive \nbehavior and for temporal contrast enhancement. For every simulation, kernel functions \nwere specified by the parameters a and p, and the number of time steps, np and nD, that \nwere sampled for the approximation of each integral. \n\n4.1 Task 1. A Sequential Shifter \n\nThe first task is a simple shifter over a set of 7 to 20 units. The system is trained on these \nstimuli and then tested to see if it can reconstruct the sequence given the initial input. \nThe task is given with no noise and with temporal noise (see Figure 2). Task 1 is \ndesigned to examine the utility of LTD as an approach to learning a sequence with \ntemporal noise. The ability of the network to reconstruct the noise-free sequence after \ntraining on the noisy sequence was tested for different LTD kernel functions. \n\nNote that the same patterns are presented (for each time slice, just one of the n units is \nactive), but the shifts either skip or repeat in time. Experiments were run with k = 1, 2, or \n3 of the units active. \n\n4.2 Task 2. Time series reconstruction. \n\nIn this task, a set of units was trained on external sinusoidal signals that varied according \nto frequency and phase. The purpose of this task is to examine the role of LTD in \nproviding temporal context. The network was then tested under a condition in which the \n\n\fLTD Facilitates Learning in a Noisy Environment \n\n153 \n\nexternal signals were provided to all but one of the units. The activity of the deprived \nunit was then compared with its training signal \n\nSequence \n\nReconstruction \n\nClean \n\naaaaaaa \n12::1 4 ~ 6 7 \n_ClClClaaa \nCI_aaaaa \nCla_aaaa \nClCla_aaa \naClaa_aa \naClaaa_a \naaaaaa_ \n_ aaaaaa \na_aaaaa \naa_aaaa \naaa_aaa \naaaa_aa \naaaaa_a \naaaaaa_ \n\nNoisy \n\naaaaaaa \n12.'14~67 \n_aaaaaa \na_aaaaa \naa_aaaa \nCla_aaaa \naaa_aaa \naaaa_aa \nClaaaaa_ \naaaaaa _ \n_aaaaaa \na_aaaaa \naaa_aaa \naaaa_aa \naaaa_aa \naaaaa_a \n\nT \n\nm \ne \n\nLTP alone \n\na a a a a a a \n12::14~67 \n_CICICICICICI \n______ CI \nCI ___ CICICI \n\n-------\n-------\n-------\n-------\n-------\n-------\n-------\n-------\n-------\n-------\n-------\n\nLTP&LTD \naaaaaaa \n::I 4 ~ 6 7 \n1 2 \n_aaaaaa \nCI_aaaaa \naa_aaaa \naaa_aal:l \naaaa_al:l \naaaaa_1:I \naaaaaa_ \n_aal:laaa \na_aaaaa \naa_aaaa \naaa_al:la \naaaa_aa \naaaaa_a \nl:Iaaaaa_ \n\nFigure 2. Reconstruction of clean shifter sequence using as input the noisy stimulus \nshifter sequence. For each time slice, just one of the 7 units is active. In the clean \nsequence, activity shifts cyclically around the 7 units. The noisy sequence has a random \njitter of \u00b11. \n\n5 ResultsSequential Shifter Results \n\nAll networks trained on the clean sequence can learn the task with LTP alone, but \nno networks could learn the shifter task based on a noisy training sequence unless \nthere was also an LTD term. Without an LTD term, most units would saturate to \nmaximum values. For a range of LTD parameters, the network would converge \nwithout saturating. Reconstruction performance was found to be sensitive to the \nLTD parameters. The parameters a and f3 shown in Table.l needed to be chosen \nvery specifically to get perfect reconstruction (this was done by trial and error). For \na narrow range of parameters near the optimal values, the reconstructed sequence \nwas close to the noise-free target. However, the parameters a and f3 shown in \nTable 2 are estimated from the experimental result of Zhang,et al [15]. \n\nT bl 1 R \n\na e \n\nesu ts 0 \n\nfth \n\ne sequentla s Iter tas . \nk \n\n. I h\u00b7f \n\nk n nr \n\nap \n\nf3p \n\nnp \n\naD \n\nf3D \n\nnD Time \n\n1 7 \n2 7 \n\n~ 7 \n1 10 \n\n12. 10 \n1 15 \n1 20 \n\n1 \n1 \n2 \n1 \n1 \n\n1 \n1 \n1 \n\n2.72 \n2.72 \n\n1 \n1 \n0.5 0.4 \n0.5 0.4 \n1 \n\n2.72 \n\n1 \n1 \n1 \n\n2.72 \n2.72 \n2.72 \n\n1 \n1 \n3 \n1 \n1 \n\n1 \n1 \n1 \n\n0.4 \n0.1 \n0.1 0.4 \n0.2 0.1 \n0.2 0.1 \n0.1 \n0.4 \n\n0.1 \n0.4 \n0.1 0.4 \n0.1 0.4 \n\n5 \n4 \n7 \n6 \n8 \n\n7 \n13 \n18 \n\n208 \n40 \n192 \n168 \n682 \n99 \n1136 \n4000 \n\nThe task was to shift a pattern 1 unit with each time step. A block of k of n units was \nactive. The parameters of the kernel functions (aand /3), the number of values sampled \nfrom the kernel (the number of time slices used to estimate the integral), np and nD, and \nthe number of steps used to begin the reconstruction, nr (usually n, = 1) are given in the \ntable. The last column of the table (Time) reports the number of iterations required for \nperfect reconstruction. \n\n\f154 \n\nP. W. Munro and G. Hernandez \n\nTable 2 .. Results of the sequential shifter task using as parameters: nr =1; np =1; \n\naD =0.125; a \n\np=O.5; fJI =-aD *e*O.35; \n\nfJp=ap *e*O.8. \n\n~ n \n1 7 \n~ 7 \n~ 7 \n\nnD \n6 \n5 \n4 \n\nTime \n288 \n96 \n64 \n\nFor the above results, the k active units were always adjacent with respect to the shifting \ndirection. For cases with noncontiguous active units, reconstruction was never exact. \nNetworks trained with LTP alone would saturate, but would converge to a sequence \n\"close\" to the target (Fig. 3) if an LTD term was added. \n\nSequence \n\nReconstruction \n\nClean \n\nNoisy \n\nLTP alone \n\na G a a a a a \n!'i 6 7 \n121 4 \n\na a a G G a a \n1214!'i67 \n\u2022 a.aaaa \n\u2022 a.aaaa \na a . a . a a \na . a . a a a \na a a a . a . \na a a a . a . \n\u2022 aaaa.a \na . a a a a . \n\na G G a a a a \n12:>4!'i67 \na.aaaa \u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022 \u2022\u2022\u2022\u2022\u2022\u2022 \n\nLTP&LTD \na a a a a a a \n1 2 .0 4!'i67 \na.aaaa \u2022 \n. a . a a a a \n\u2022 \u2022 \u2022 \u2022 aaa \naa \u2022 \u2022 \u2022 \u2022 a \naaa \u2022 \u2022 \u2022 \u2022 \n. a a a \u2022 \u2022 \u2022 \na.aaa \u2022 \u2022 \n\u2022 \u2022 \u2022 aaa. \n\nFigure 3. This base pattern (k=2, n=7) with noncontiguous active units was \npresented as a shifted sequence with noise. The target sequence is partially \nreconstructed only when LTP and LTD are used together. \n\n5.1 Time Series Reconstruction Results \n\nA network of just four units was trained for hundreds of iterations, the units were each \nexternally driven by a sinusoidally varying input. Networks trained with LTP alone fail \nto reconstruct the time series on units deprived of external input during testing. In these \nsimulations, there is no noise in the patterns, but L TO is shown to be necessary for \nreconstruction of the patterns (Fig. 4). \n\nI'~. \n\n!', \nI \nI \n\n= \n: . : \n\nf: \n\n\u2022 \n. \nI \n\n\u2022 II \n\n..... ' \n: ~ \n. \nI. : I \n\n. \n.:1 \n\u2022 \nI \n~. \n\n: \n\n~ \n' I ~. \nI \n~, \n~ \n.'~ ........ ~! ......... 'll ......... .,,: ........ :.;. ... u \u2022\u2022\u2022 \\ . : \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 ~,: \u2022\u2022 \u2022 \u2022\u2022 \u2022 ~ \n\nI.:': I \n~I \n\nl: \n\n'\"I \n\nI I \n\nI I \n\n: : \n\n,: \n\n.' \", \n\n:'I ~ \nIJ \n\n\u2022 \u2022 : \n\nI . \n\n\u00b7\u00b7\u00b7\u00b7\u00b7 \u00b7 \u00b7,\u00b7y \u00b7\u00b7 \u00b7\u00b7\u00b7\u00b7\u00b7rl:\u00b7\u00b7\u00b7\u00b7\u00b7 \u00b7 \u00b7 \u00b7.~\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7 \u00b7\u00b7::.\u00b7 \u00b7 \u00b7 \u00b7\u00b7\u00b7\u00b7\"!I\u00b7 \u00b7\u00b7\u00b7\u00b7 \u00b7 \u00b7\u00b7 ~~\u00b7\u00b7\u00b7 \u00b7\u00b7 \u00b7\u00b7 \u00b7~I \n~J \n\n\u2022 \" \n\n\\ \n\n: \n\nI \n\nI \n\nI \n\nI \n\nI \n\n: \n\n.; \u2022 \u2022 ! \n\ni \n\nFigure 4. Reconstruction of sinusoids. Target signals , from training (dashed) \nplotted with reconstructed signals (solid). Left: The best reconstruction using LTP \nalone. Right: A typical result with LTP and LTD together. \n\nFor high values of ap and aD, the reconstruction of sinusoids is very sensitive to the \nvalues of fJD and fJP. Figure 5 shows the results when IfJD I and fJp values are close. In \nIfJD I is slightly smaller than f3p , the first two neurons \nthe first case (top), when \n(from left to right) saturate. And, in the contrary case (bottom) the first two neurons \n\n\fLTD Facilitates Learning in a Noisy Environment \n\n155 \n\nshow almost null activation. However, the third and fourth neurons (from left to \nright) in both cases (top and bottom) show predictive behavior. \n\n:~ \n\n~ ~ ~ :~ ; '\\ \nI \n\n. ' I ' : \n\n. ' I I \n\nI: I: I: \n\" \nI. \" \n\" \nI . I ' I \n\nI . \n\u2022 \n\n\" \n\nI: !I \n'\\ \n:: ;~ :: ,::, \n\n=-\n\nII \n\n( \n\n:'~ ~ 1 \nI,. .. I \n:: ::. : ~ :: :: i \n\nI \n\n, , \n\n: \n\nI \n\u2022 I ' \u2022 '\\ \n~ . : \" I : : \u2022 \n, \n\u2022 \u2022 I \n\nI \n\n: \n\n, \n\n. : I, \n\n: : ;: \u2022 \u2022 \u2022 \u2022 \u2022 I \n\n\u2022 I , ' . \u2022 \nI : . ~ \nI \ni ~!:: .. :~::;!;~~~ \n, \n;: ;: ; ~ \n:: I: ~ : :: ~: .' : .! \n:: :: :: ~.I :: ~; ~I\u00b7~ \nII'! \n.\" \nI: I: It. \n.; \u2022 :.: \n~: \n\u2022\u2022 ~ I,' \n~I \n.. ~ \n...\u2022.\u2022\u2022\u2022\u2022 \\ .\u2022\u2022.\u2022\u2022.\u2022... 'J. .\u2022.\u2022\u2022...\u2022.\u2022..... !C, \n\nI \\ . ' . I I \n\n:. :. .' 0' \n\n. ' . ' I \n\u2022 \nl ' \u2022 \n\n.: \n\nI. \n\n~: \n\nI t \n\nI . : \n\nII \n\n\\ I! \n\n.. \n\n. ' \n\nI I \n\n' . \n\n\" \n\n\u2022 \n\n, \n\nI \n\n' \n\n, \n\n, : : \n, , , , \n, , \n., , , \n: : \n.. \n\" \n, \n\" \n:. . :: \n\nII \n\nI, \n\n. \n:' \n\" \" \n\" \" \" \n:' \n, \" . \" \n\" , ' \nI:. \n, , . , \" \n\n, \n\n\\ . . . , , \n\n, \n, ' \n\u2022 I \n\n. : \n, , \n\n, \n, , \n\n, \n\" \n, \n, \n\n, \n\n, \n, \n\nI \n\n, . ! \n\ni \nc \u2022 \n, ; \n, : t \n, \n, c \nj \n~ \n~ \n,1 \n\nI \n\n~r \n.\u2022\u2022. I1 .......... A: \u2022 \u2022 \u2022\u2022\u2022\u2022\u2022 \\ .\u2022\u2022\u2022 -....4: .... . ... l ...\u2022 \n\n~: \n\n~: \n\n~: \n\n\u00b7 \u00b7\u00b7 \u00b7\u00b7\u00b7~\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7:t\u00b7\u00b7\u00b7\u00b7 \u00b7\u00b7ft\u00b7\u00b7u .. ~ ...... ~ \nr: \n.~ \nI { \n\n. . . . \n\nI. \n\nIt \n\n.. \n\nI \n\n\u2022 I \n\nI \n\nI \n\nI \n\nI \n\n\u2022 \n\nI \n\nI \n\u2022 \n\nI \n\n, \n\nI \n\nI \n\n: \n\n: \n: . \n; I I: : : \n\n: ~ \nt \u00b7; \n:' ~ \nI \n: \nI\n\u2022 I . \u2022 ~ \nI : \nI \nI : I : I \n:\n:: \n~I ~I ~I ~I , I ; \n., I : \n\u2022 I \n\nI I : \n\nI . \" \n\n\u2022 I \n\n\u2022 I \n\n\u2022 I \n\n\u2022 \nI \n\nI \n\u2022 \n\nI \n\u2022 \n\nI \nI \n\nI \n\nI \n\n\" \n\nI \n\n~! \n\n.~! \n\n.~ : .~! \n\n't! j \n\n~ \n.... _~-. \u2022\u2022 :.-.-.~ . . .. . . ..... 0\"<\". I ._._~ .... ...? ...... j \n\n\\ \n\n.\"+ \u2022\u2022 _.-. \n\n\u2022 \n..... ...\n\n......\n\n.................. . \n\n, .. , .. . . ~: \n\n' 1\"'~\" '\" , {\u00b7\u00b7~\u00b7\u00b7l\u00b7 \u00b7 \u00b7\u00b7 : \"};' ' .~ \n.. \n'. \nI, \n_: \n\nIt \n... : . \nI \nI ' \n\n' : \n'I : \n,; \n. : \n\n\u2022 \n' \nI \nI \n\n'\\ \nI \nI \n\n\u2022 \nI \nI \n\nI \nI \nI \n\n\u2022 \n\nI \n\n' \n\n: \n\n:: .' \n~:; \n\nI:: I \n:,= \n\n::,:J \n~~'~ \n\n, \n\nI \n\n\u2022 \n\n\"; \u2022 \n\u2022 \nI \n,-\n\nI \n\u2022 \n\nI \n\n: \n\nI \n: \n\n\u2022 \n\nI \n\n\" \nI \n\n_. \n\n.. t . \n\n. .\n\n. ,. \n\n\" \n\n' I : ' : \n\np \n\nI , : ~ \n\n\u2022 \n\n\u2022 \n\nI \n\nI \n\n\u2022 \n\" \n\n: \n.: \n\n. . . . t.. .. \n\nI \n\n: \n\n. \n\nI \n\n: \nt \n\nFigure 5. Reconstruction of sinusoids . Examples of target signals from training \n(dashed) plotted with reconstructed signals (solid). Top: When IfJD kfJp. Bottom: \nWhen IfJDI> fJp . \n\n6 Discussion \n\nIn the half century that has elapsed since Hebb articulated his neurophysiological \npostulate, the neuroscience community has corne to recognize its fundamental role \nin plasticity. Hebb's hypothesis clearly transcends its original motivation to give a \nneurophysiologically based account of associative memory. \nThe phenomenon of LTP provides direct biological support for Hebb's postulate, \nand hence has clear cognitive implications. \nInitially after its discovery in the \nlaboratory, the computational role of LTD was thought to be the flip side of LTP. \nThis interpretation would have synapses strengthen when activities are correlated \nand have them weaken when they are anti-correlated. Such a theory is appealing for \nits elegance, and has formed the basis many network models [19]-[20]. However, \nthe dependence of synaptic change on the relative timing of pre- and post- synaptic \nactivity that has recently been shown in the laboratory is inconsistent with this story \nand calls for a computational interpretation. A network trained with such a learning \nrule cannot converge to a state where the weights are symmetric, for example, since \nLiWij 7:- LiWji. \nWhile the simulations reported here are simple and preliminary, they illustrate two \ntasks that benefit from the inclusion of time-dependent LTD. \nIn the case of the \nsequential shifter, an examination of more complex predictive tasks is planned in \nIt is expected that this will require architectures with unclamped \nthe near future. \n(hidden) units. The role of LTD here is to temporally enhance contrast, in a way \ninhibition for computing spatial contrast \nanalogous \nenhancement in the retina. The time-series example illustrates the possible role of \nLTD for providing temporal context. \n\nthe role of lateral \n\nto \n\n\fJ 56 \n\n7 References \n\nP. W. Munro and G. Hernandez \n\n[1] Bliss TVP & Lcllmo T (1973) Long-lasting potentiation of synaptic in the dentate area of \nthe un anaesthetized rabbit following stimulation of the perforant path.J Physiol 232:331-356 \n\n[2] Malenka RC (1995) LTP and LTD: dynamic and interactive processes of synaptic \nplasticity. The Neuroscientist 1:35-42. \n\n[3] Hebb DO (1949) The Organization of Behavior. Wiley: NY. \n\n[4] Stent G (1973) A physiological, mechanism for Hebb's postulate of learning. Proc. Natl. \nAcad. Sci. USA 70: 997-1001 \n\n[5] Barrionuevo G, Schottler F & Lynch G (1980) The effects of repetitive low frequency \nstimulation on control and \"pontentiated\" synaptic responses in the hippocampus. Life Sci \n27:2385-2391. \n\n[6] Thiels E, Xie X, Yeckel MF, Barrionuevo G & Berger TW (1996) NMDA Receptor(cid:173)\ndependent LTD in different subfields of hippocampus in vivo and in vitro. Hippocampus \n6:43-51. \n\n[7] Sejnowski T J (1977) Storing covariance '!Vith nonlinearly interacting neurons. 1. Math. \nBioL 4:303-321. \n\n[8] Sutton RS (1988) Learning to predict by the methods of temporal difference. Machine \nLearning. 3:9-44 \n\n[9] Sompolinsky H and Kanter I (1986) Temporal association in asymmetric neural networks. \nPhys.Rev.Letter. 57:2861-2864. \n\n[10] Gerstner W, Kempter R, van Hemmen JL & Wagner H (1996) A neuronal learning rule \nfor sub-millisecond temporal coding. Nature 383:76-78 . \n\n[11] Kempter R, Gerstner W & van Hemmen JL (1999) Spike-based compared to rate-based \nhebbian learnin g. Kearns , Ms. , Solla, S.A and Cohn, D.A. Eds. Advances in Neural \nInformation Processing Systems J J. MIT Press, Cambridge MA. \n\n[12] Kempter R, Gerstner W, van Hemmen JL & Wagner H (1996) Temporal coding in the \nsub-millisecond range: Model of barn owl auditory pathway. Touretzky, D.S, Mozer, M.C, \nHasselmo, M.E, Eds. Advances in Neural Information Processing Systems 8. MIT Press, \nCambridge MA pp.124-130. \n\n[13] Markram H, Lubke J, Frotscher M & Sakmann B (1997) Regulation of synaptic efficacy \nby coincidence of postsynaptic Aps and EPPSPs. Science 275 :213-215 . \n\n[14] Markram H & Tsodyks MV (1996) Redistribution of synaptic efficacy between \nneocortical pyramidal neurons. Nature 382:807-810. \n\n[15] Zhang L, Tao HW, Holt CE & Poo M (1998) A critical window for cooperation and \ncompetition among developing retinotectal synapses. Nature 35:37-44 \n\n[16] Abbott LF, & Blum KI (1996) Functional significance of long-term potentiation for \nsequence learning and prediction. Cerebral Cortex 6: 406-416. \n\n[17] Abbott LF, & Song S (1999) Temporally asymmetric hebbian learning, spike timing and \nneuronal response variability. Kearns, Ms., Solla, S.A and Cohn, D.A. Eds. Advances in \nNeural Information Processing Systems J J. MIT Press, Cambridge MA. \n\n[18] Goldman MS, Nelson SB & Abbott LF (1998) Decorrelation of spike trains by synaptic \ndepression. Neurocomputing (in press). \n\n[19] Hopfield J (1982) Neural networks and physical systems with emergent collective \ncomputational properties. Proc. Natl. Acad. Sci. USA. 79:2554-2558. \n\n. \n\n[20] Ackley DH, Hinton GE, Sejnowski TJ (1985) A learning algorithm for Boltzmann \nmachines. Cognitive Science 9:147-169. \n\n\f", "award": [], "sourceid": 1776, "authors": [{"given_name": "Paul", "family_name": "Munro", "institution": null}, {"given_name": "Gerardina", "family_name": "Hern\u00e1ndez", "institution": null}]}