{"title": "A Criterion for the Convergence of Learning with Spike Timing Dependent Plasticity", "book": "Advances in Neural Information Processing Systems", "page_first": 763, "page_last": 770, "abstract": "", "full_text": "A Criterion for the Convergence of Learning\n\nwith Spike Timing Dependent Plasticity\n\nRobert Legenstein and Wolfgang Maass\nInstitute for Theoretical Computer Science\n\nTechnische Universitaet Graz\n\nA-8010 Graz, Austria\n\nflegi,maassg@igi.tugraz.at\n\nAbstract\n\nWe investigate under what conditions a neuron can learn by experimen-\ntally supported rules for spike timing dependent plasticity (STDP) to pre-\ndict the arrival times of strong \u201cteacher inputs\u201d to the same neuron. It\nturns out that in contrast to the famous Perceptron Convergence Theo-\nrem, which predicts convergence of the perceptron learning rule for a\nsimpli\ufb01ed neuron model whenever a stable solution exists, no equally\nstrong convergence guarantee can be given for spiking neurons with\nSTDP. But we derive a criterion on the statistical dependency structure of\ninput spike trains which characterizes exactly when learning with STDP\nwill converge on average for a simple model of a spiking neuron. This\ncriterion is reminiscent of the linear separability criterion of the Percep-\ntron Convergence Theorem, but it applies here to the rows of a correlation\nmatrix related to the spike inputs. In addition we show through computer\nsimulations for more realistic neuron models that the resulting analyti-\ncally predicted positive learning results not only hold for the common\ninterpretation of STDP where STDP changes the weights of synapses,\nbut also for a more realistic interpretation suggested by experimental\ndata where STDP modulates the initial release probability of dynamic\nsynapses.\n\n1\n\nIntroduction\n\nNumerous experimental data show that STDP changes the value wold of a synaptic weight\nafter pairing of the \ufb01ring of the presynaptic neuron at time tpre with a \ufb01ring of the postsy-\nnaptic neuron at time tpost = tpre + (cid:1)t to wnew = wold + (cid:1)w according to the rule\n\nwnew = (cid:26) minfwmax; wold + W+ (cid:1) e(cid:0)(cid:1)t=(cid:28)+ g\n\nmaxf0; wold (cid:0) W(cid:0) (cid:1) e(cid:1)t=(cid:28)(cid:0) g\n\n;\n;\n\nif (cid:1)t > 0\nif (cid:1)t (cid:20) 0 ;\n\n(1)\n\nwith some parameters W+; W(cid:0); (cid:28)+; (cid:28)(cid:0) > 0 (see [1]). If during training a teacher induces\n\ufb01ring of the postsynaptic neuron, this rule becomes somewhat analogous to the well-known\nperceptron learning rule for McCulloch-Pitts neurons (= \u201cperceptrons\u201d). The Perceptron\nConvergence Theorem states that this rule enables a perceptron to learn, starting from any\ninitial weights, after \ufb01nitely many errors any transformation that it could possibly imple-\nment. However, we have constructed examples of input spike trains and teacher spike trains\n\n\f(omitted in this abstract) such that although a weight vector exists which produces the de-\nsired \ufb01ring and which is stable under STDP, learning with STDP does not converge to a\nstable solution. On the other hand experiments in vivo have shown that neurons can be\ntaught by suitable teacher input to adopt a given \ufb01ring response [2, 3] (although the spike-\ntiming dependence is not exploited there). We show in section 2 that such convergence of\nlearning can be explained by STDP in the average case, provided that a certain criterion is\nmet for the statistical dependence among Poisson spike inputs. The validity of the proposed\ncriterion is tested in section 3 for more realistic models for neurons and synapses.\n\n2 An analytical criterion for the convergence of STDP\n\nThe average case analysis in this section is based on the linear Poisson neuron model (see\n[4, 5]). This neuron model outputs a spike train S post(t) which is a realization of a Poisson\nprocess with the underlying instantaneous \ufb01ring rate Rpost(t). We represent a spike train\n\nS(t) as a sum of Dirac-(cid:14) functions S(t) = Pk (cid:14)(t (cid:0) tk), where tk is the kth spike time of\n\nthe spike train. The effect of an input spike at input i at time t0 is modeled by an increase\nin the instantaneous \ufb01ring rate of an amount wi(t0)(cid:15)(t (cid:0) t0), where (cid:15) is a response kernel\nand wi(t0) is the synaptic ef\ufb01cacy of synapse i at time t0. We assume (cid:15)(s) = 0 for s < 0\n0 ds (cid:15)(s) = 1 (normalization of the response kernel), and (cid:15)(s) (cid:21) 0 for all s\nas well as wi (cid:21) 0 for all i (excitatory inputs). In the linear model, the contributions of all\ninputs are summed up linearly:\n\n(causality), R 1\n\nRpost(t) =\n\nn\n\nXj=1Z 1\n\n0\n\nds wj(t (cid:0) s) (cid:15)(s) Sj(t (cid:0) s) ,\n\n(2)\n\nwhere S1; : : : ; Sn are the n presynaptic spike trains. Note that in this spike generation\nprocess, the generation of an output spike is independent of previous output spikes.\nThe STDP-rule (1) avoids the growth of weights beyond bounds 0 and wmax by simple\nclipping. Alternatively one can make the weight update dependent on the actual weight\nvalue. In [5] a general rule is suggested where the weight dependence has the form of a\npower law with a non-negative exponent (cid:22). This weight update rule is de\ufb01ned by\n\nif (cid:1)t > 0\nif (cid:1)t (cid:20) 0\n\n(cid:0)W(cid:0) (cid:1) w(cid:22) (cid:1) e(cid:1)t=(cid:28)(cid:0)\n\n(cid:1)w = (cid:26) W+ (cid:1) (1 (cid:0) w)(cid:22) (cid:1) e(cid:0)(cid:1)t=(cid:28)+\n;\n;\nwhere we assumed for simplicity that wmax = 1.\nInstead of looking at speci\ufb01c input\nspike trains, we consider the average behavior of the weight vector for (possibly correlated)\nhomogeneous Poisson input spike trains. Hence, the change (cid:1)wi is a random variable with\na mean drift and \ufb02uctuations around it. We will in the following focus on the drift by\nassuming that individual weight changes are very small and only averaged quantities enter\nthe learning dynamics, see [6]. Let Si be the spike train of input i and let S (cid:3) be the output\nspike train of the neuron. The mean drift of synapse i at time t can be approximated as\n\n(3)\n\n,\n\n_wi(t) = W+(1 (cid:0) wi)(cid:22)Z 1\n\n0\n\nds e(cid:0)s=(cid:28) Ci(s; t) (cid:0) W(cid:0)w(cid:22)\n\ni Z 0\n\n(cid:0)1\n\nds es=(cid:28) Ci(s; t)\n\n,\n\n(4)\n\nwhere Ci(s; t) = hSi(t)S (cid:3)(t + s)iE is the ensemble averaged correlation function between\ninput i and the output of the neuron (see [5, 6]). For the linear Poisson neuron model, input-\noutput correlations can be described by means of correlations in the inputs. We de\ufb01ne the\nnormalized cross correlation between input spike trains Si and Sj with a common rate\nr > 0 as\n\n(cid:0) 1 ,\n\n(5)\n\nC 0\n\nij(s) =\n\nhSi(t) Sj(t + s)iE\n\nr2\n\nwhich assumes value 0 for uncorrelated Poisson spike trains. We assume in this article that\nij is constant over time. In our setup, the output of the neuron during learning is clamped\nC 0\n\n\fto the teacher spike train S (cid:3) which is the output of a neuron with the target weight vector\nw(cid:3). Therefore, the input-output correlations Ci(s; t) are also constant over time and we\ndenote them by Ci(s) in the following. In our neuron model, correlations are shaped by the\nresponse kernel (cid:15)(s) and they enter the learning equation (4) with respect to the learning\nwindow. This motivates the de\ufb01nition of window correlations c+\nij for the positive\nand negative learning window respectively:\n\nij and c(cid:0)\n\n0\n\n1\n\nc(cid:6)\nij = 1 +\n\n(cid:28) Z 1\n\nds e(cid:0)s=(cid:28) Z 1\nWe call the matrices C (cid:6) = fc(cid:6)\nijgi;j=1;:::;n the window correlation matrices. Note that\nwindow correlations are non-negative and that for homogeneous Poisson input spike trains\nand for a non-negative response kernel, they are positive. For soft weight bounds and\n(cid:22) > 0, a synaptic weight can converge to a value arbitrarily close to 0 or 1, but not to one\nof these values directly. This motivates the following de\ufb01nition of learnability.\n\nij((cid:6)s (cid:0) s0)\n\nds0 (cid:15)(s0)C 0\n\n(6)\n\n.\n\n0\n\nDe\ufb01nition 2.1 We say that a target weight vector w(cid:3) 2 f0; 1gn can approximately be\nlearned in a supervised paradigm by STDP with soft weight bounds on homogeneous Pois-\nson input spike trains (short: \u201cw(cid:3) can be learned\u201d) if and only if there exist W+; W(cid:0) > 0,\nsuch that for (cid:22) ! 0 the ensemble averaged weight vector hw(t)iE with learning dynamics\ngiven by Equation 4 converges to w(cid:3) for any initial weight vector w(0) 2 [0; 1]n.\n\nWe are now ready to formulate an analytical criterion for learnability:\n\nTheorem 2.1 A weight vector w(cid:3) can be learned (when being teached with S (cid:3)) for ho-\nmogeneous Poisson input spike trains with window correlation matrices C + and C (cid:0) to a\nlinear Poisson neuron with non-negative response kernel if and only if w(cid:3) 6= 0 and\n\nkc+\nik\nkc(cid:0)\ni = 1 and w(cid:3)\nfor all pairs hi; ji 2 f1; : : : ; ng2 with w(cid:3)\n\nPn\nk=1 w(cid:3)\nPn\nk=1 w(cid:3)\n\n> Pn\nk=1 w(cid:3)\nPn\nk=1 w(cid:3)\n\nkc+\njk\nkc(cid:0)\nj = 0.\n\njk\n\nik\n\nProof idea: The correlation between an input and the teacher induced output is (by Eq. 2):\n\nCi(s) = hSi(t) S (cid:3)(t + s)iE =\n\nn\n\nXj=1\n\nw(cid:3)\n\nj Z 1\n\n0\n\nds0 (cid:15)(s0) hSi(t) Sj(t + s (cid:0) s0)iE .\n\nSubstitution of this equation into Eq. 4 yields the synaptic drift\n\nW+(1 (cid:0) wi)(cid:22)\n\nn\n\nXj=1\n\nw(cid:3)\n\nj c+\n\nij (cid:0) W(cid:0)w(cid:22)\n\ni\n\nn\n\nXj=1\n\n_wi = (cid:28) r22\n4\ni (cid:19)(cid:0)1\n\n(cid:3)1=(cid:22)\n\nw(cid:3)\n\nij3\nj c(cid:0)\n5\n\n.\n\n(7)\n\n, where (cid:3)i denotes W+\nW(cid:0)\n\nWe \ufb01nd the equilibrium points w(cid:22)i of synapse i by setting _wi = 0 in Eq. 7. This\nyields w(cid:22)i = (cid:18)1 + 1\nzero if w(cid:3) = 0 which implies that w(cid:3) = 0 cannot be learned. For w(cid:3) 6= 0, one can\nshow that w(cid:22) = (w(cid:22)1; : : : ; w(cid:22)n) is the only equilibrium point of the system and that it\nis stable. Since the system decomposes into n independent one-dimensional systems,\nconvergence to w(cid:3) is guaranteed for all initial conditions. Furthermore, one sees that\nlim(cid:22)!0 w(cid:22)i = 1 if and only if (cid:3)i > 1, and lim(cid:22)!0 w(cid:22)i = 0 if and only if (cid:3)i < 1.\nTherefore, lim(cid:22)!0 w(cid:22) = w(cid:3) holds if and only if (cid:3)i > 1 for all i with w(cid:3)\ni = 1 and (cid:3)i < 1\nfor all i with w(cid:3)\n\ni = 0. The theorem follows from the de\ufb01nition of (cid:3)i.\n\n. Note that the drift is\n\nj=1 w(cid:3)\nj=1 w(cid:3)\n\nj c+\nij\nj c(cid:0)\nij\n\nPn\nPn\n\n\fFor a wide class of cross-correlation functions, one can establish a relationship between\nlearnability by STDP and the well-known concept of linear separability from linear alge-\nbra.1 Because of synaptic delays, the response of a spiking neuron to an input spike is\ndelayed by some time t0. One can model such a delay in the response kernel by the restric-\ntion (cid:15)(s) = 0 for all s (cid:20) t0. In the following Corollary we consider the case where input\ncorrelations C 0\n\nij(s) appear only in a time window smaller than the delay:\n\nCorollary 2.1 If there exists a t0 (cid:21) 0 such that (cid:15)(s) = 0 for all s (cid:20) t0 and C 0\nij(s) = 0 for\nall s < (cid:0)t0; i; j 2 f1; : : : ; ng, then the following holds for the case of homogeneous\nPoisson input spike trains to a linear Poisson neuron with positive response kernel (cid:15):\n\nA weight vector w(cid:3) can be learned if and only if w(cid:3) 6= 0 and w(cid:3) linearly separates the list\nL = hhc+\n\nn are the rows of C +.\n\nnii, where c+\n\n1i; : : : ; hc+\n\n1 ; : : : ; c+\n\n1 ; w(cid:3)\n\nn ; w(cid:3)\n\nProof idea: From the assumptions of the corollary it follows that c(cid:0)\nij = 1. In this case, the\ncondition in Theorem 2.1 is equivalent to the statement that w(cid:3) linearly separates the list\nL = hhc+\n\n1i; : : : ; hc+\n\n1 ; w(cid:3)\n\nn ; w(cid:3)\n\nnii.\n\nCorollary 2.1 can be viewed as an analogon of the Perceptron Convergence Theorem for the\naverage case analysis of STDP. Its formulation is tight in the sense that linear separability of\nthe list L alone (as opposed to linear separability by the target vector w(cid:3)) is not suf\ufb01cient\nto imply learnability. For uncorrelated input spike trains of rate r > 0, the normalized\ncross correlation functions are given by C 0\nr (cid:14)(s), where (cid:14)ij is the Kronecker\ndelta function. The positive window correlation matrix C + is therefore essentially a scaled\nversion of the identity matrix. The following corollary then follows from Corollary 2.1:\n\nij(s) = (cid:14)ij\n\nCorollary 2.2 A target weight vector w(cid:3) 2 f0; 1gn can be learned in the case of uncorre-\nlated Poisson input spike trains to a linear Poisson neuron with positive response kernel (cid:15)\nsuch that (cid:15)(s) = 0 for all s (cid:20) 0 if and only if w(cid:3) 6= 0.\n\n3 Computer simulations of supervised learning with STDP\n\nIn order to make a theoretical analysis feasible, we needed to make in section 2 a number of\nsimplifying assumptions on the neuron model and the synapse model. In addition a number\nof approximations had to be used in order to simplify the estimates. We consider in this\nsection the more realistic integrate-and-\ufb01re model2 for neurons and a model for synapses\nwhich are subject to paired-pulse depression and paired-pulse facilitation, in addition to the\nlong term plasticity induced by STDP [7]. This model describes synapses with parameters\nU (initial release probability), D (depression time constant), and F (facilitation time con-\nstant) in addition to the synaptic weight w. The parameters U, D, and F were randomly\n\n1Let c1; : : : ; cm 2 Rn and y1; : : : ; ym 2 f0; 1g. We say that a vector w 2 Rn linearly separates\nthe list hhc1; y1i; : : : ; hcm; ymii if there exists a threshold (cid:2) such that yi = sign(ci (cid:1) w (cid:0) (cid:2))\nfor i = 1; : : : ; m. We de\ufb01ne sign(z ) = 1 if z (cid:21) 0 and sign(z) = 0 otherwise.\n\n2The membrane potential Vm of the neuron is given by (cid:28)m\n\ndVm\ndt = (cid:0)(Vm (cid:0) Vresting) + Rm (cid:1)\n(Isyn(t) + Ibackground + Iinject(t)) where (cid:28)m = Cm (cid:1) Rm = 30ms is the membrane time constant,\nRm = 1M (cid:10) is the membrane resistance, Isyn(t) is the current supplied by the synapses, Ibackground\nis a constant background current, and Iinject(t) represents currents induced by a \u201cteacher\u201d. IfV m\nexceeds the threshold voltage Vthresh it is reset to Vreset = 14:2mV and held there for the length\nTref ract = 3ms of the absolute refractory period.Neuron parameters: Vresting = 0V , Ibackground\nrandomly chosen for each trial from the interval [13:5nA; 14:5nA]. Vthresh was set such that each\nneuron spiked at a rate of about 25 Hz. This resulted in a threshold voltage slightly above 15mV .\nSynaptic parameters: Synaptic currents were modeled as exponentially decaying currents with decay\ntime constants (cid:28)S = 3ms ((cid:28)S = 6ms) for excitatory (inhibitory) synapses.\n\n\fchosen from Gaussian distributions that were based on empirically found data for such con-\nnections. We also show that in some cases a less restrictive teacher forcing suf\ufb01ces, that\ntolerates undesired \ufb01ring of the neuron during training. The results of section 2 predict that\nthe temporal structure of correlations has a strong in\ufb02uence on the outcome of a learning\nexperiment. We used input spike trains with cross correlations that decay exponentially\nwith a correlation decay constant (cid:28)cc.3 In experiment 1 we consider temporal correlations\nwith (cid:28)cc=10ms. Since such \u201cbroader\u201d correlations are not problematic for STDP, sharper\ncorrelations ((cid:28)cc=6ms) are considered in experiment 2.\nExperiment 1 (correlated input with (cid:28)cc=10ms): In this experiment, a leaky integrate-\nand-\ufb01re neuron received inputs from 100 dynamic synapses. 90% of these synapses were\nexcitatory and 10% were inhibitory. For each excitatory synapse, the maximal ef\ufb01cacy\nwmax was chosen from a Gaussian distribution with mean 54 and SD 10:8, bounded by\n54 (cid:6) 3SD. The 90 excitatory inputs were divided into 9 groups of 10 synapses per group.\nSpike trains were correlated within groups with correlation coef\ufb01cients between 0 and 0:8,\nwhereas there were virtually no correlations between spike trains of different groups.4 Tar-\nget weight vectors w(cid:3) were chosen in the most adverse way: half of the weights of w(cid:3)\nwithin each group was set to 0, the other half to its maximal value wmax (see Fig. 1C).\n\nA\n\ntarget\n\ntrained\n\nB\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n\n0\n\n0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1\n\n1.2\n\n1.4\n\n1.6\n\n1.8\n\n2\n\ntime [sec]\n\nC\n\nw\n\n60\n\n40\n\n20\n\n0\n\n0\n\nD\n\nw\n\n60\n\n40\n\n20\n\n0\n\n0\n\ntarget\n\n20\n\n40\nSynapse\n\n60\n\n80\n\ntrained\n\n20\n\n40\nSynapse\n\n60\n\n80\n\nangular error [rad]\nspike correlation [s =5ms]\n\n500\n\n1000\n\n1500\n\n2000\n\n2500\n\ntime [sec]\n\nFigure 1: Learning a target weight vector w\n(cid:3) on correlated Poisson inputs. A) Output spike train on\ntest data after one hour of training (trained) compared to the target output (target). B) Evolution of\n(cid:3) that implements F in radiant (angular error,\nthe angle between weight vector w(t) and the vector w\nsolid line), and spike correlation (dashed line). C) Target weight vector w\n(cid:3) consisting of elements\nwith value 0 or the value wmax assigned to that synapse. D) Corresponding weights of the learned\nvector w(t) after 40 minutes of training. (All time data refer to simulated biological time)\n\nBefore training, the weights of all excitatory synapses were initialized by randomly chosen\nsmall values. Weights of inhibitory synapses remained \ufb01xed throughout the experiment.\nInformation about the target weight vector w(cid:3) was given to the neuron only in the form of\nshort current injections (1 (cid:22)A for 0.2 ms) at those times when the neuron with the weight\nvector w(cid:3) would have produced a spike. Learning was implemented as standard STDP\n(see rule 1) with parameters (cid:28)+ = (cid:28)(cid:0) = 20ms, W+ = 0:45, W(cid:0)=W+ = 1:05. Additional\ninhibitory input was given to the neuron during training that reduced the occurrence of non-\n\n3We constructed input spike trains with normalized cross correlations (see Equation 5) approxi-\n2(cid:28)ccr e(cid:0)jsj=(cid:28)cc between inputs i and j for a mean input rate of r = 20Hz,\n\nmately given by C 0\na correlation coef\ufb01cient cij, and a correlation decay constant of (cid:28)cc = 10ms.\n\nij(s) = ccij\n\n4The correlation coef\ufb01cient cij for spike trains within group k consisting of 10 spike trains was\n\nset to cij = cck = 0:1 (cid:3) (k (cid:0) 1) for k = 1; : : : ; 9.\n\n\fteacher-induced \ufb01ring of the neuron (see text below).5 Two different performance measures\nwere used for analyzing the learning progress. The \u201cspike correlation\u201d measures for test\ninputs that were not used for training (but had been generated by the same process) the\ndeviation between the output spike train produced by the target weight vector w(cid:3) for this\ninput, and the output spike train produced for the same input by the neuron with the current\nweight vector w(t)6. The angular error measures the angle between the current weight\nvector w(t) and the target weight vector w(cid:3). The results are shown in Fig. 1. One can\nsee that the deviation of the learned weight vector shown in panel D from the target weight\nvector w(cid:3) (panel C) is very small, even for highly correlated groups of synapses with het-\nerogeneous target weights. No signi\ufb01cant changes in the results were observed for longer\nsimulations (4 hours simulated biological time), showing stability of learning. On 20 trials\n(each with a new random distribution of maximal weights wmax, different initializations\nw(0) of the weight vector before learning, and new Poisson spike trains), a spike correla-\ntion of 0:83 (cid:6) 0:06 was achieved (angular error 6:8 (cid:6) 4:7 degrees). Note that learning is not\nonly based on teacher spikes but also on non teacher-induced \ufb01ring. Therefore, strongly\ncorrelated groups of inputs tend to cause autonomous (i.e., not teacher-induced) \ufb01ring of\nthe neuron which results in weight increases for all weights within the corresponding group\nof synapses according to well-known results for STDP [8, 5]. Obviously this effect makes\nit quite hard to learn a target weight vector w(cid:3) where half of the weights for each corre-\nlated group have value 0. The effect is reduced by the additional inhibitory input during\ntraining which reduces undesired \ufb01ring. However, without this input a spike correlation of\n0:79 (cid:6) 0:09 could still be achieved (angular error 14:1 (cid:6) 10 degrees).\n\nA\n\nl\n\nn\no\ni\nt\na\ne\nr\nr\no\nc\n \ne\nk\np\ns\n\ni\n\nB\n\n1\n\n0.9\n\n0.8\n\n0.7\n\n0.6\n\n0.5\n\n0.4\n\n0.3\n\n0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n\ninput correlation cc\n\nl\n\nn\no\ni\nt\na\ne\nr\nr\no\nc\n \ne\nk\np\ns\n\u2212\n1\n\ni\n\n0.35\n\n0.3\n\n0.25\n\n0.2\n\n0.15\n\n0.1\n\n0.05\n\n0\n\n0\n\npredicted to be learnable\npredicted to be not learnable\n\n5\n\n10\n\n15\n\nweight error [\u00b0]\n\nFigure 2: A) Spike correlation achieved for correlated inputs (solid line). Some inputs were cor-\nrelated with cc plotted on the x-axis. Also, as a control the spike correlation achieved by randomly\ndrawn weight vectors is shown (dashed line, where half of the weights were set to wmax and the other\nweights were set to 0). B) Comparison between theory and simulation results for a leaky integrate-\nand-\ufb01re neuron and input correlations between 0:1 and 0:5 ((cid:28)cc = 6ms). Each cross (open circle)\nmarks a trial where the target vector was learnable (not learnable) according to Theorem 2.1. The\nactual learning performance of STDP is plotted for each trial in terms of the weight error (x-axis) and\n1 minus the spike correlation (y-axis).\n\nExperiment 2 (testing the theoretical predictions for (cid:28)cc=6ms): In order to evaluate the\ndependence of correlation among inputs we proceeded in a setup similar to experiment\n1. 4 input groups consisting each of 10 input spike trains were constructed for which the\ncorrelations within each group had the same value cc while the input spike train to the\nother 50 excitatory synapses were uncorrelated. Again, half of the weights of w(cid:3) within\n\n5We added 30 inhibitory synapses with weights drawn from a gamma distribution with mean 25\n\nand standard deviation 7:5, that received additional 30 uncorrelated Poisson spike trains at 20 Hz.\n\n6For that purpose each spike in these two output spike trains was replaced by a Gaussian function\nwith an SD of 5 ms. The spike correlation between both output spike trains was de\ufb01ned as the\ncorrelation between the resulting smooth functions of time (for segments of length 100 s).\n\n\feach correlated group (and within the uncorrelated group) was set to 0, the other half to a\nrandomly chosen maximal value. The learning performance after 1 hour of training for 20\ntrials is plotted in Fig. 2A for 7 different values of the correlation cc ((cid:28)cc = 6ms) that is\napplied in 4 of the input groups (solid line).\n\nIn order to test the approximate validity of Theorem 2.1 for leaky integrate-and-\ufb01re neu-\nrons and dynamic synapses, we repeated the above experiment for input correlations\ncc = 0:1; 0:2; 0:3; 0:4, and 0:5. For each correlation value, 20 learning trials (with different\ntarget vectors) were simulated. For each trial we \ufb01rst checked whether the (randomly cho-\nsen) target vector w(cid:3) was learnable according to the condition given in Theorem 2:1 (65%\nof the 100 learning trials were classi\ufb01ed as being learnable).7 The actual performance of\nlearning with STDP was evaluated after 50 minutes of training.8 The result is shown in Fig.\n2B. It shows that the theoretical prediction of learnability or non-learnability for the case\nof simpler neuron models and synapses from Theorem 2.1 translates in a biologically more\nrealistic scenario into a quantitative grading of the learning performance that can ultimately\nbe achieved with STDP.\n\nA\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n0\n\nangular error [rad]\nweight deviation\nspike correlation\n\n500\n\n1000 1500 2000 2500\ntime [sec]\n\nB\n\nU\n\nC\n\nU\n\n0.2\n\n0.1\n\n0\n\n0.2\n\n0.1\n\n0\n\ntarget\n\n5\n\n5\n\n15\n\n20\n\n10\n\nSynapse\ntrained\n\n10\n\nSynapse\n\n15\n\n20\n\nFigure 3: Results of modulation of initial release probabilities U. A) Performance of U-learning for\na generic learning task (see text). B) Twenty values of the target U vector (each component assumes\nits maximal possible value or the value 0). C) Corresponding U values after 42 minutes of training.\n\nExperiment 3 (Modulation of initial release probabilities U by STDP): Experimental\ndata from [9] suggest that synaptic plasticity does not change the uniform scaling of the\namplitudes of EPSPs resulting from a presynaptic spike train (i.e., the parameter w), but\nrather redistributes the sum of their amplitudes. If one assumes that STDP changes the pa-\nrameter U that determines the synaptic release probability for the \ufb01rst spike in a spike train,\nwhereas the weight w remains unchanged, then the same experimental data that support the\nclassical rule for STDP, support the following rule for changing U:\n\nmaxf0; Uold (cid:0) U(cid:0) (cid:1) e(cid:1)t=(cid:28)(cid:0) g\n\nUnew = (cid:26) minfUmax; Uold + U+ (cid:1) e(cid:0)(cid:1)t=(cid:28)+ g\nwith suitable nonnegative parameters Umax; U+; U(cid:0); (cid:28)+; (cid:28)(cid:0).\nFig. 3 shows results of an experiment where U was modulated with rule (8) (similar to\nexperiment 1, but with uncorrelated inputs). 20 repetitions of this experiment yielded after\n42 minutes of training the following results: spike correlation 0:88 (cid:6) 0:036, angular error\n27:9 (cid:6) 3:7 degrees, for U+ = 0:0012, U(cid:0)=U+ = 1:055. Apparently the output spike train\nis less sensitive to changes in the values of U than to changes in w. Consequently, since\n\nif (cid:1)t > 0\nif (cid:1)t (cid:20) 0 ;\n\n(8)\n\n;\n;\n\n7We had chosen a response kernel of the form (cid:15)(s) = 1\n\n(e(cid:0)s=(cid:28)1 (cid:0) e(cid:0)s=(cid:28)2 ) with (cid:28)1 = 2ms\nand (cid:28)2 = 1ms (Least mean squares \ufb01t of the double exponential to the peri-stimulus-time histogram\n(PSTH) of the neuron, which re\ufb02ects the probability of spiking as a function of times since an input\nspike), and calculated the window correlations c+\n\n(cid:28)1(cid:0)(cid:28)2\n\nij and c(cid:0)\n\nij numerically.\n\n8To guarantee the best possible performance for each learning trial, training was performed on 27\n\ndifferent values for W(cid:0)=W+ between 1:02 and 1:15.\n\n\fonly the behavior of a neuron with vector U(cid:3) but not the vector U(cid:3) is made available to\nthe neuron during training, the resulting correlation between target- and actual output spike\ntrains is quite high, whereas angular error between U(cid:3) and U(t), as well as the average\ndeviation in U, remain rather large.\nWe also repeated experiment 1 (correlated Poisson inputs) with rule (8) for U-learning.\n20 repetitions with different target weights and different initial conditions yielded after 35\nminutes of training: spike correlation 0:75 (cid:6) 0:08, angular error 39:3 (cid:6) 4:8 degrees, for\nU+ = 8 (cid:1) 10(cid:0)4, U(cid:0)=U+ = 1:09.\n\n4 Discussion\n\nThe main conclusion of this article is that for many common distributions of input spikes\na spiking neuron can learn with STDP and teacher-induced input currents any map from\ninput spike trains to output spike trains that it could possibly implement in a stable manner.\n\nWe have shown in section 2 that a mathematical average case analysis can be carried out\nfor supervised learning with STDP. This theoretical analysis produces the \ufb01rst criterion\nthat allows us to predict whether supervised learning with STDP will succeed in spite of\ncorrelations among Poisson input spike trains. For the special case of \u201csharp correlations\u201d\n(i.e. when the cross correlations vanish for time shifts larger than the synaptic delay) this\ncriterion can be formulated in terms of linear separability of the rows of a correlation matrix\nrelated to the spike input, and its mathematical form is therefore reminiscent of the well-\nknown condition for learnability in the case of perceptron learning. In this sense Corollary\n2.1 can be viewed as an analogon of the Perceptron Convergence Theorem for spiking\nneurons with STDP.\n\nFurthermore we have shown that an alternative interpretation of STDP where one assumes\nthat it modulates the initial release probabilities U of dynamic synapses, rather than their\nscaling factors w, gives rise to very satisfactory convergence results for learning.\nAcknowledgment: We would like to thank Yves Fregnac, Wulfram Gerstner, and espe-\ncially Henry Markram for inspiring discussions.\n\nReferences\n[1] L. F. Abbott and S. B. Nelson. Synaptic plasticity: taming the beast. Nature Neurosci., 3:1178\u2013\n\n1183, 2000.\n\n[2] Y. Fregnac, D. Shulz, S. Thorpe, and E. Bienenstock. A cellular analogue of visual cortical\n\nplasticity. Nature, 333(6171):367\u2013370, 1988.\n\n[3] D. Debanne, D. E. Shulz, and Y. Fregnac. Activity dependent regulation of on- and off-responses\n\nin cat visual cortical receptive \ufb01elds. Journal of Physiology, 508:523\u2013548, 1998.\n\n[4] R. Kempter, W. Gerstner, and J. L. van Hemmen. Intrinsic stabilization of output rates by spike-\n\nbased hebbian learning. Neural Computation, 13:2709\u20132741, 2001.\n\n[5] R. G\u00a8utig, R. Aharonov, S. Rotter, and H. Sompolinsky. Learning input correlations through\nnon-linear temporally asymmetric hebbian plasticity. Journal of Neurosci., 23:3697\u20133714, 2003.\n[6] R. Kempter, W. Gerstner, and J. L. van Hemmen. Hebbian learning and spiking neurons. Phys.\n\nRev. E, 59(4):4498\u20134514, 1999.\n\n[7] H. Markram, Y. Wang, and M. Tsodyks. Differential signaling via the same axon of neocortical\n\npyramidal neurons. PNAS, 95:5323\u20135328, 1998.\n\n[8] S. Song, K. D. Miller, and L. F. Abbott. Competitive hebbian learning through spike-timing\n\ndependent synaptic plasticity. Nature Neuroscience, 3:919\u2013926, 2000.\n\n[9] H. Markram and M. Tsodyks. Redistribution of synaptic ef\ufb01cacy between neocortical pyramidal\n\nneurons. Nature, 382:807\u2013810, 1996.\n\n\f", "award": [], "sourceid": 2824, "authors": [{"given_name": "Robert", "family_name": "Legenstein", "institution": null}, {"given_name": "Wolfgang", "family_name": "Maass", "institution": null}]}