{"title": "Spike timing-dependent plasticity as dynamic filter", "book": "Advances in Neural Information Processing Systems", "page_first": 2110, "page_last": 2118, "abstract": "When stimulated with complex action potential sequences synapses exhibit spike timing-dependent plasticity (STDP) with attenuated and enhanced pre- and postsynaptic contributions to long-term synaptic modifications. In order to investigate the functional consequences of these contribution dynamics (CD) we propose a minimal model formulated in terms of differential equations. We find that our model reproduces a wide range of experimental results with a small number of biophysically interpretable parameters. The model allows to investigate the susceptibility of STDP to arbitrary time courses of pre- and postsynaptic activities, i.e. its nonlinear filter properties. We demonstrate this for the simple example of small periodic modulations of pre- and postsynaptic firing rates for which our model can be solved. It predicts synaptic strengthening for synchronous rate modulations. For low baseline rates modifications are dominant in the theta frequency range, a result which underlines the well known relevance of theta activities in hippocampus and cortex for learning. We also find emphasis of low baseline spike rates and suppression for high baseline rates. The latter suggests a mechanism of network activity regulation inherent in STDP. Furthermore, our novel formulation provides a general framework for investigating the joint dynamics of neuronal activity and the CD of STDP in both spike-based as well as rate-based neuronal network models.", "full_text": "Spike timing-dependent plasticity as dynamic \ufb01lter\n\nJoscha T. Schmiedt\u2217, Christian Albers and Klaus Pawelzik\n\nInstitute for Theoretical Physics\n\nschmiedt@uni-bremen.de, {calbers, pawelzik}@neuro.uni-bremen.de\n\nUniversity of Bremen\n\nBremen, Germany\n\nAbstract\n\nWhen stimulated with complex action potential sequences synapses exhibit spike\ntiming-dependent plasticity (STDP) with modulated pre- and postsynaptic contri-\nbutions to long-term synaptic modi\ufb01cations. In order to investigate the functional\nconsequences of these contribution dynamics (CD) we propose a minimal model\nformulated in terms of differential equations. We \ufb01nd that our model reproduces\ndata from to recent experimental studies with a small number of biophysically in-\nterpretable parameters. The model allows to investigate the susceptibility of STDP\nto arbitrary time courses of pre- and postsynaptic activities, i.e. its nonlinear \ufb01lter\nproperties. We demonstrate this for the simple example of small periodic mod-\nulations of pre- and postsynaptic \ufb01ring rates for which our model can be solved.\nIt predicts synaptic strengthening for synchronous rate modulations. Modi\ufb01ca-\ntions are dominant in the theta frequency range, a result which underlines the\nwell known relevance of theta activities in hippocampus and cortex for learning.\nWe also \ufb01nd emphasis of speci\ufb01c baseline spike rates and suppression for high\nbackground rates. The latter suggests a mechanism of network activity regulation\ninherent in STDP. Furthermore, our novel formulation provides a general frame-\nwork for investigating the joint dynamics of neuronal activity and the CD of STDP\nin both spike-based as well as rate-based neuronal network models.\n\n1\n\nIntroduction\n\nDuring the past decade the effects of exact spike timing on the change of synaptic connectivity have\nbeen studied extensively. In vitro studies have shown that the induction of long-term potentiation\n(LTP) requires the presynaptic input to a cell to precede the postsynaptic output and vice versa\nfor long-term depression (LTD) (see [1, 2, 3]). This phenomenon has been termed spike timing-\ndependent plasticity (STDP) and emphasizes the importance of a causal order in neuronal signaling.\nThereby it extends pure Hebbian learning, which requires only the coincidence of pre- and postsy-\nnaptic activity. Consequently, experiments have shown an asymmetric exponential dependence on\nthe timing of spike pairs and a molecular mechanism mostly dependent on the in\ufb02ux of Ca2+ (see\n[4, 5] for reviews). Further, when induced with more complex spike trains, synaptic modi\ufb01cation\nshows nonlinearities ([6, 7, 8]) indicating the in\ufb02uence of short-term plasticity.\nTheoretical approaches to STDP cover studies using the asymmetric pair-based STDP window as\na lookup table, more biophysical models based on synaptic and neuronal variables, and sophisti-\ncated kinetic models (for a review see [9]). Recently, the experimentally observed in\ufb02uence of the\npostsynaptic membrane potential (e.g. [10]) has also been taken into account ([11]).\nOur approach is based on differential Hebbian learning ([12, 13]), which generates asymmetric\ntiming windows similar to STDP ([14]) depending on the shape of the back-propagating action\n\u2217Postal correspondence should be addressed to Universit\u00a8at Bremen, Fachbereich 1, Institut f\u00a8ur Theoretische\n\nPhysik, Abt. Neurophysik, Postfach 330 440, D-28334 Bremen, Germany\n\n1\n\n\fpotential ([15]). We extend it with a mechanism for activating learning by an increase in postsynaptic\nactivity, because both the induction of LTP and LTD require [Ca2+] to exceed a threshold ([16]).\nMoreover, we include a mechanism for adaptive suppression on both synaptic sides, similar to the\nmodel in [7]. Finally, we for simplicity assume that both the presynaptic and the postsynaptic\nside function as low-pass \ufb01lters; a spike leaves a fast increasing and exponentially decaying trace.\nTogether, we propose a set of differential equations, which captures the contribution dynamics (CD)\nof pre- and postsynaptic activities to STDP, thereby describing synaptic plasticity as a \ufb01lter.\nOur framework reproduces experimental \ufb01ndings from two recent in vitro studies in the visual cor-\ntex and the hippocampus in most details. Furthermore, it proves to be particularly suitable for the\nanalysis of the susceptibility of STDP to pre- and postsynaptic rate modulations. This is demon-\nstrated by an analysis of synaptic changes depending on oscillatory modulations of baseline \ufb01ring\nrates.\n\n2 Formulation of the model\n\nWe use a variant of the classical differential Hebbian learning assuming a change of synaptic con-\nnectivity w, which is dependent on the presynaptic activity trace ypre and the temporal derivative of\nthe postsynaptic activity trace ypost:\n\n(1)\ncw denotes a constant learning rate. An illustration of this learning rule for pairs of spikes is given\nin Figure 1B. For simplicity, we assume these activity traces to be abstract low-pass \ufb01ltered versions\nof neuronal activity x in the presynaptic and postsynaptic cells, e.g. the concentration of Ca2+ or\nthe amount of bound glutamate:\n\n\u02d9w(t) = cw ypre(t) \u02d9ypost(t) .\n\n\u02d9ypre(t) = upre(t) \u00b7 xpre(t) \u2212\n\nypre(t)\n\n\u03c4pre\n\n\u02d9ypost(t) = upost(t)z(t) \u00b7 xpost(t) \u2212\n\nypost(t)\n\n\u03c4post\n\n.\n\n(2)\n\n(3)\n\nThe dynamics of the y\u2019s are characterized by their respective time constants \u03c4pre and \u03c4post. The\ncontribution of each spike is regulated by a suppressing attenuation factor u pre- and postsynapti-\ncally. On the postsynaptical side an additional activation factor z \u201denables\u201d the synapse to learn.\nThe dynamics of u and z are discussed below. x represents neuronal activity which can be either a\ntime-continuous \ufb01ring rate or spike trains given by series of \u03b4 pulses\n\nxpre, post(t) =(cid:88)i\n\n\u03b4(t \u2212 ti\n\npre, post)\n\n,\n\n(4)\n\nwhich allows analytical investigations of the properties of our model. Note that formally x(t) has\nthen to be taken as x(t + 0). An illustrating overview over the different parts of the model with\nsample trajectories is shown in Figure 1A.\nWe de\ufb01ne the relative change of synaptic connectivity after after a period T from Equation (1) as\n\n\u2206w =\n\nw(t0 + T )\n\nw(t0) \u2212 1 =\n\ncw\n\nw(t0)(cid:90)T\n\nypre \u02d9ypost dt .\n\n(5)\n\nThe dependence on the initial synaptic strength w(t0) as observed in [3, 8] shall not be discussed\nhere, but can easily be achieved by making the learning rate cw in Equation (1) w-dependent. Here,\nw(t0) is chosen to be 1.\nIgnoring attenuation and activation, a single pair of spikes at temporal distance \u2206t analytically yields\nthe typical STDP window (see Figure 2A and 3A):\n\n\u2206w(\u2206t) =(cid:40)cw(cid:16)1 \u2212 \u03c4pre\n\n\u03c4pre+\u03c4post(cid:17)e\u2212\u2206t/\u03c4pre\n\ne\u2212\u2206t/\u03c4post\n\ncw \u00b7\n\n\u03c4pre\n\n\u03c4pre+\u03c4post\n\nfor \u2206t > 0\nfor \u2206t < 0\n\n(6)\n\n2\n\n\fFigure 1: Schematic illustration of differential Hebbian learning with contribution dynamics. A:\nPre- and postsynaptic activity (x, second column) is modulated (attenuated with u, activated with z,\n\ufb01rst column) and \ufb01ltered (y, third column) before it contributes to differential Hebbian learning (w,\nfourth column). B: Spike pair example for differential Hebbian learning. Left: a presynaptic spike\ntrace (ypre) preceding a postsynaptic spike trace (ypost, dotted line) yields a synaptic strengthening\ndue to the initially positive postsynaptic contribution ( \u02d9ypost, solid line), which is always stronger\nthan the following negative part. Right: for the reverse timing the positive presynaptic contribution\nis only multiplied with the negative postsynaptic trace (right). Areas contributing to learning are\nshaded.\n\nThe importance of adaptive suppressing mechanisms for synaptic plasticity has experimentally been\nshown by Froemke and colleagues ([7, 6]). Therefore, we down-regulate the contribution of the\nspikes to the activity traces y in Equation (2) and (3) with an attenuation factor u on both pre- and\npostsynaptic sides:\n\n\u02d9upre =\n\n\u02d9upost =\n\n1\n\u03c4 rec\npre\n1\n\u03c4 rec\npost\n\n(1 \u2212 upre) \u2212 cpreuprexpre\n\n(1 \u2212 upost) \u2212 cpost(upost \u2212 u0)xpost\n\n.\n\n(7)\n\n(8)\n\nThis should be understood as an abstract representation of for instance the depletion of transmitters\nin the presynaptic bouton ([17]) or the frequency-dependent spike attenuation in dendritic spines\n([18]), respectively. These recover with their time constants \u03c4 rec and are bound between u0 and 1.\n\n3\n\nSYNAPSEPREPOSTLow-passddt\u2206wuActivityActivity Traces(Contributions)ModulationFactorsuzxxyyLow-pass &Differential Hebbian Learning\u03a0Example for spike pairs\u2206w\u223c(cid:31)ypre(t)\u02d9ypost(t)dt>0\u2206w\u223c(cid:31)ypre(t)\u02d9ypost(t)dt<0\u03c4pre\u03c4postTimeypreypost00ypost\u2206t > 0\u2206t < 0AB\fFor the presynaptic side we assume in the following upre\nconstants cpre, post \u2208 [0, 1] denote the impact a spike has on the relaxed synapse.\nIn several experiments it has been shown that a single spike is not suf\ufb01cient to induce synaptic\nmodi\ufb01cation ([10, 8]). Therefore, we introduce a spike-induced postsynaptic activation factor z\n\n0 = 0, so we abbreviate u0 = upost\n\n. The\n\n0\n\n\u02d9z = cactxpostz \u2212 \u03b1(z \u2212 z0)2 ,\n\n(9)\n\nwhich enhances the contribution of a postsynaptic spike to the postsynaptic trace, e.g. by the removal\nof the Mg2+ block from postsynaptic NMDA receptors ([19, 5]). The nonlinear positive feedback\nis introduced to describe strong enhancing effects as for instance autocatalytic mechanisms, which\nhave been suggested to play a role in learning on several time-scales ([20, 21]). The activation\nz decays hyperbolically to a lower bound z0 and the contribution of a spike is weighted with the\nconstant cact.\n\n3 Comparison to experiments\n\nIn order to evaluate our model we implemented experimental stimulation protocols from in vitro\nstudies on synapses of the visual cortex ([7]) and the hippocampus ([8]) of rats. In both studies,\nsimple pairs of spikes and more complex spike trains were arti\ufb01cially elicited in the presynaptic and\nthe postsynaptic cell and the induced change of synaptic connectivity was recorded.\nFroemke and colleagues ([7]) focused on the effects of spike bursts on synaptic modi\ufb01cation in the\nvisual cortex. In addition to the classical STDP pairing protocol \u2013 a presynaptic spike preceding\nor following a postsynaptic spike after a speci\ufb01c time \u2206t \u2013 four other experimental protocols (see\nFigure 2B to E) were performed: (1) 5-5 bursts with \ufb01ve spikes of a certain frequency on both\nsynaptic sides, where the postsynaptic side follows the presynaptic side, (2) presynaptic 100 Hz\nbursts with n spikes following one postsynaptic spike (post-n-pre), (3) presynaptic 100 Hz bursts\nwith different numbers of spikes followed by one postsynaptic spike (n-pre-post) and (4) a post-pre\npair with varying number of following postsynaptic spikes (post-pre-n-post).\n\nFigure 2: Differential Hebbian learning with CD reproduces synaptic modi\ufb01cation induced with\nSTDP spike patterns in visual cortex. Data taken from [7], personal communication. A: experi-\nmental \ufb01t and model prediction with Equation (6) of pair-based STDP. B: dependence of synaptic\nmodi\ufb01cations on the frequency of 5-5 bursts with presynaptic spikes following postsynaptic spikes\nby 6 ms. C, D and E: synaptic modi\ufb01cation induced by post-n-pre, n-pre-post and post-pre-n-post\n100 Hz spike trains.\n\n4\n\n\u2212150\u2212100\u221250050100\u22120.500.511.5\u2206 t (ms)\u2206 w1050100\u22120.500.5Frequency (Hz)12345\u22120.4\u22120.20Presynaptic spikes1234500.20.40.60.81Presynaptic spikes12345\u22120.4\u22120.200.20.4Postsynaptic spikesprepostLTDLTPABCDEExperimentModel\fFigure 3: Differential Hebbian learning with CD reproduces synaptic modi\ufb01cation induced with\nSTDP spike patterns in hippocampus. Data taken from [8] as reported in [22]. A: experimental\n\ufb01t and model prediction with Equation (6) of pair-based STDP. B: quadruplet protocol. C and D:\npost-pre-post and pre-post-pre triplet protocol for different interspike intervals.\n\nTable 1: Parameters and evaluation results for the data sets from visual cortex ([7]) and hippocampus\n([8]). E: normalized mean-square error, S: ratio of correctly predicted signs of synaptic modi\ufb01ca-\ntion.\n\nVisual cortex\nHippocampus\n\ncpre\n0.9\n0.6\n\ncpost\n1\n0.4\n\ncact\n1.5\n3.5\n\n\u03c4 rec\npre [s]\n2\n0.5\n\n\u03c4 rec\npost [s]\n0.2\n0.5\n\n\u03b1\n1\n1\n\nu0\n0.01\n0.7\n\nz0\n1\n0.2\n\nE\n4.04\n2.16\n\nS\n18/18\n10/11\n\nIn the hippocampal study of Wang et al. ([8]) synaptic modi\ufb01cation induced by triplets (pre-post-pre\nand post-pre-post) and quadruplets (pre-post-post-pre and post-pre-pre-post) of spikes was measured\nwhile the respective interspike intervals were varied. (see Figure 3B to D).\nAs a \ufb01rst step we took the time constants from the experimentally measured pair-based STDP win-\ndows as our low-pass \ufb01lter time constants (see Equation 6). They remained constant for each data\nset: (1) \u03c4pre = 13.5 ms and \u03c4post = 42.8 ms for [7], (2) \u03c4pre = 16.8 ms and \u03c4post = 33.7 ms for [8]\n(taken from [23] since not present in the study). Next, we chose the learning rate cw in Equation (6)\nto \ufb01t the synaptic change for the pairing protocol: (1) cw = 1.56 for the visual cortex data, (2)\ncw = 0.99 for the hippocampal data set. The remaining parameters were estimated manually within\nbiologically plausible ranges and are shown in Table 1. The model was then applied to the more\ncomplex stimulation protocols by solving the differential equations semi-analytically, i.e. separately\nfor every spike and the following interspike interval. As measure for the prediction error of our\nmodel we used the normalized mean-square error E\n\nE =\n\n1\nN\n\nN(cid:88)i=1(cid:16) \u2206wexp\n\ni \u2212 \u2206wmod\n\ni\n\n\u03c3i\n\n,\n\n(10)\n\n(cid:17)2\n\ni\n\ni\n\nand \u2206wmod\n\nwhere \u2206wexp\nare the experimentally measured and the predicted modi\ufb01cations of synap-\ntic strength in the ith experiment; N is the number of data points (N = 18 for the visual cortex data\nset, N = 11 for the hippocampal data set). \u03c3i is the standard error of the mean of the experimental\ndata. Additionally we counted the number of correctly predicted signs S of synaptic modi\ufb01cation,\ni.e. induced depression or potentiation. The prediction error for both data sets is shown in Table 1.\n\n5\n\n\u2212150\u2212100\u221250050100\u22120.500.51\u2206 t (ms)\u2206 w(5,89,5)(5,20,5)(5,84,5)00.10.20.3(5,5)(10,10)(15,5)(5,15)00.20.30.4(5,5)(10,10)(15,5)(5,15)00.10.20.30.4 ACBDInterspike interval (ms)Interspike interval (ms)Interspike interval (ms)prepost\u2206 wExperimentModel\fFigure 4: Synaptic change depending on frequency f and phase shift \u2206\u03c6 of pre- and postsynaptic\nrate modulations for different baseline rates x0. The color codes are identical within each column\nand in arbitrary units. Note the strong suppression with increasing baseline rate for cortical synapses\nwhich is due to strong attenuation effects of pre- and postsynaptic contributions. It is weaker for\nhippocampal synapses because we found the postsynaptic attenuation to be bounded (u0 = 0.7).\n\n4 Phase, frequency and baseline rate dependence of STDP with contribution\n\ndynamics\n\nAs shown in the previous section our model can reproduce the experimental \ufb01ndings of synaptic\nweight changes in response to spike sequences surprisingly well and yields better \ufb01ts than former\nstudies (e.g. [22]). The proposed framework, however, is not restricted to spike sequences but al-\nlows to investigate synaptic changes depending on arbitrary pre- and postsynaptic activities. For\ninstance it could be used for investigations of the plasticity effects in simulations with inhomoge-\nneous Poisson processes. Taking x(t) to be \ufb01ring rates of Poissonian spike trains our account of\nSTDP represents a useful approximation for the expected changes of synaptic strength depending\non the time courses of xpre and xpost (compare e.g. [24]). Therefore our model can serve also as\nbuilding block in rate based network models for investigation of the joint dynamics of neuronal\nactivities and synaptic weights.\nHere, we demonstrate the bene\ufb01t of our approach for determining the \ufb01lter properties of STDP\nsubject to CD, i.e. we use the equations together with the parameters from the experiments for\ndetermining the dependency of weight changes on frequency, relative phase \u2206\u03c6 and baseline rates\nof modulated pre- and postsynaptic \ufb01ring rates. While for substantial modulations of \ufb01ring rates\nthe nonlinearities are dif\ufb01cult to be treated analytically, for small periodical modulations around a\nbaseline rate x0 the corresponding synaptic changes can be calculated analytically. This is done by\nconsidering\n\nxpre(t) = x0 + \u03b5 cos(2\u03c0f t)\n\n(11)\nwhich for small \u03b5 < x0 allows linearization of all equations from which one obtains \u2206W =\n\u2206w/(T \u03b5pre\u03b5post), where T = 1/f = 2\u03c0/\u03c9 is the period of the respective oscillations. Neglect-\n\nand xpost(t) = x0 + \u03b5 cos(2\u03c0f t \u2212 \u2206\u03c6) ,\n\n6\n\n137205010013720501001372050100137205010013720501001372050100Modulation frequency f [Hz]Phase shift \u2206\u03d5\ufffd0-\ufffd\ufffd/2-\ufffd/2\ufffd0-\ufffd\ufffd/2-\ufffd/2\ufffd0-\ufffd\ufffd/2-\ufffd/2\ufffd0-\ufffd\ufffd/2-\ufffd/2\ufffd0-\ufffd\ufffd/2-\ufffd/2\ufffd0-\ufffd\ufffd/2-\ufffd/2CortexHippocampusx0 = 1\ufffdHzx0 = 5\ufffdHzx0 = 10\ufffdHzx0 = 1\ufffdHzx0 = 5\ufffdHzx0 = 30\ufffdHz-11 \u2206W (a.u.)0 \fing transients this \ufb01nally yields the expected weight changes per unit time. Though lengthy the\ncalculations are straightforward and presented in the supplementary material. We here show only\nthe exact result for the case of constant u = 1 and z = 1:\n\n\u2206W =\n\n\u03c9\u03c4pre\u03c4post(cid:112)\u03c92(\u03c4post \u2212 \u03c4pre)2 + (1 + \u03c92\u03c4pre\u03c4post)2\n\npre\u03c92)(1 + \u03c4 2\n\n2(1 + \u03c4 2\n\npost\u03c92)\n\n\u00b7sin(cid:16)\u2206\u03c6+arctan\n\n\u03c9(\u03c4post \u2212 \u03c4pre)\n\n1 + \u03c92\u03c4pre\u03c4post(cid:17) (12)\n\nThe analytical results for the case with CD are shown graphically in Figure 4 using the parameters\nfrom cortex and hippocampus, respectively (see Tab. 1). These plots contain the main \ufb01ndings:\n(1) rate modulations in the theta frequency range ((cid:39) 7Hz) lead to strongest synaptic changes, (2)\nalso for phase-zero synchronous rate modulations weight changes are positive, (3) in hippocampus\nmaximal weight change magnitudes occur at baseline rates around 5 Hz, and (4) for high baseline\n0 for the visual\nrates weight changes become suppressed (\u223c 1/x0 for the hippocampus, \u223c 1/x2\ncortex). Numerical simulations with \ufb01nite rate modulations were found to con\ufb01rm these analytical\npredictions surprisingly well. Also for the nonlinear regime and Poissionian spike trains deviations\nremained moderate.\n\n5 Discussion\n\nSTDP has been proposed to represent a fundamental mechanism underlying learning and many\nmodels explored its computational role (examples are [25, 26, 27]). In contrast, research targeting\nthe computational roles of dynamical phenomena inherent in STDP are in the beginning (see [9]).\nHere, we here formulated a minimal, yet biologically plausible model including the dynamics of how\nneuronal activity contributes to STDP. We found that our model reproduces the synaptic changes in\nresponse to spike sequences in experiments in cortex and hippocampus with high accuracy.\nUsing the corresponding parameters our model predicts weight changes depending on temporal\nstructures in the pre- and postsynaptic activities including spike sequences and varying \ufb01ring rates.\nWhen applied to pre- and postsynaptic rate modulations our approach quanti\ufb01es synaptic changes\ndepending on frequency and phase shifts between pre- and postsynaptic activities. A rigorous per-\nturbation analysis of our model reveals that the dynamical \ufb01lter properties of STDP make weight\nchanges sensitively dependent on combinations of speci\ufb01c features of pre- and postsynaptic signals.\nIn particular, our analysis indicates that both cortical as well as hippocampal STDP is most suscep-\ntible for modulations in the theta frequency range. It predicts the dependency of synaptic changes\non pre- and postsynaptic phase relations of rate modulations. These results are in line with experi-\nmental results on the relation of theta rhythms and learning. For instance in hippocampus it is well\nestablished that theta oscillations are relevant for learning (for a recent paper see [28]). Furthermore,\nspike activities in hippocampus exhibit speci\ufb01c phase relations with the theta rhythm (for a review\nsee [29]). Also, it has been found that during learning cortex and hippocampus tend to synchronize\nwith particular phase relations that depend on the novelty of the item to be learned ([30]). The results\npresented here underline these \ufb01ndings and make testable predictions for the corresponding synaptic\nchanges.\nAlso, we \ufb01nd potentiation for zero phase differences and strong attenuation of weight changes at\nlarge baseline rates which is particularly strong for cortical synapses. This \ufb01nding suggests a mech-\nanism for restricting weight changes with high activity levels and that STDP is de facto switched off\nwhen large \ufb01ring rates are required for the execution of a function as opposed to learning phases;\nduring the latter baseline rates should be rather low, which is particularly relevant in cortex. While\nfor cortical synapses our analysis predicts that very low baseline activities are contributing most to\nweight changes, in hippocampus synaptic modi\ufb01cations peak at baseline \ufb01ring rates x0 around 5 Hz,\nwhich suggests that x0 can control learning.\nOur study suggests that the \ufb01lter properties of STDP originating from the dynamics of pre- and\npostsynaptic activity contributions are in fact exploited for learning in the brain. In particular, shifts\nin baseline rates, as well as the frequency and the respective phases of pre- and postsynaptic rate\nmodulations induced by theta oscillations could be tuned to match the values that make STDP most\nsusceptible for synaptic modi\ufb01cations. A fascinating possibility thereby is that these features could\nbe used to control the learning rate which would represent a novel mechanism in addition to other\ncontrol signals as e.g. neuromodulators.\n\n7\n\n\fReferences\n[1] W. Levy and O. Steward. 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