{"title": "Perfect Associative Learning with Spike-Timing-Dependent Plasticity", "book": "Advances in Neural Information Processing Systems", "page_first": 1709, "page_last": 1717, "abstract": "Recent extensions of the Perceptron, as e.g. the Tempotron, suggest that this theoretical concept is highly relevant also for understanding networks of spiking neurons in the brain. It is not known, however, how the computational power of the Perceptron and of its variants might be accomplished by the plasticity mechanisms of real synapses. Here we prove that spike-timing-dependent plasticity having an anti-Hebbian form for excitatory synapses as well as a spike-timing-dependent plasticity of Hebbian shape for inhibitory synapses are sufficient for realizing the original Perceptron Learning Rule if the respective plasticity mechanisms act in concert with the hyperpolarisation of the post-synaptic neurons. We also show that with these simple yet biologically realistic dynamics Tempotrons are efficiently learned. The proposed mechanism might underly the acquisition of mappings of spatio-temporal activity patterns in one area of the brain onto other spatio-temporal spike patterns in another region and of long term memories in cortex. Our results underline that learning processes in realistic networks of spiking neurons depend crucially on the interactions of synaptic plasticity mechanisms with the dynamics of participating neurons.", "full_text": "Perfect Associative Learning with\nSpike-Timing-Dependent Plasticity\n\nChristian Albers\n\nInstitute of Theoretical Physics\n\nUniversity of Bremen\n\n28359 Bremen, Germany\n\nMaren Westkott\n\nInstitute of Theoretical Physics\n\nUniversity of Bremen\n\n28359 Bremen, Germany\n\ncalbers@neuro.uni-bremen.de\n\nmaren@neuro.uni-bremen.de\n\nKlaus Pawelzik\n\nInstitute of Theoretical Physics\n\nUniversity of Bremen\n\n28359 Bremen, Germany\n\npawelzik@neuro.uni-bremen.de\n\nAbstract\n\nRecent extensions of the Perceptron as the Tempotron and the Chronotron sug-\ngest that this theoretical concept is highly relevant for understanding networks of\nspiking neurons in the brain. It is not known, however, how the computational\npower of the Perceptron might be accomplished by the plasticity mechanisms of\nreal synapses. Here we prove that spike-timing-dependent plasticity having an\nanti-Hebbian form for excitatory synapses as well as a spike-timing-dependent\nplasticity of Hebbian shape for inhibitory synapses are suf\ufb01cient for realizing the\noriginal Perceptron Learning Rule if these respective plasticity mechanisms act in\nconcert with the hyperpolarisation of the post-synaptic neurons. We also show that\nwith these simple yet biologically realistic dynamics Tempotrons and Chronotrons\nare learned. The proposed mechanism enables incremental associative learning\nfrom a continuous stream of patterns and might therefore underly the acquisition\nof long term memories in cortex. Our results underline that learning processes\nin realistic networks of spiking neurons depend crucially on the interactions of\nsynaptic plasticity mechanisms with the dynamics of participating neurons.\n\n1 Introduction\n\nPerceptrons are paradigmatic building blocks of neural networks [1]. The original Perceptron Learn-\ning Rule (PLR) is a supervised learning rule that employs a threshold to control weight changes,\nwhich also serves as a margin to enhance robustness [2, 3]. If the learning set is separable, the PLR\nalgorithm is guaranteed to converge in a \ufb01nite number of steps [1], which justi\ufb01es the term \u2019perfect\nlearning\u2019.\n\nAssociative learning can be considered a special case of supervised learning where the activity of the\noutput neuron is used as a teacher signal such that after learning missing activities are \ufb01lled in. For\nthis reason the PLR is very useful for building associative memories in recurrent networks where\nit can serve to learn arbitrary patterns in a \u2019quasi-unsupervised\u2019 way. Here it turned out to be far\nmore ef\ufb01cient than the simple Hebb rule, leading to a superior memory capacity and non-symmetric\nweights [4]. Note also that over-learning from repetitions of training examples is not possible with\nthe PLR because weight changes vanish as soon as the accumulated inputs are suf\ufb01cient, a property\n\n1\n\n\fwhich in contrast to the na\u00a8\u0131ve Hebb rule makes it suitable also for incremental learning of associative\nmemories from sequential presentation of patterns.\n\nOn the other hand, it is not known if and how real synaptic mechanisms might realize the success-\ndependent self-regulation of the PLR in networks of spiking neurons in the brain. For example in\nthe Tempotron [5], a generalization of the perceptron to spatio-temporal patterns, learning was con-\nceived even somewhat less biological than the PLR, since here it not only depends on the potential\nclassi\ufb01cation success, but also on a process that is not local in time, namely the localization of the\nabsolute maximum of the (virtual) postsynaptic membrane potential of the post-synaptic neuron.\nThe simpli\ufb01ed tempotron learning rule, while biologically more plausible, still relies on a reward\nsignal which tells each neuron speci\ufb01cally that it should have spiked or not. Taken together, while\nhighly desirable, the feature of self regulation in the PLR still poses a challenge for biologically\nrealistic synaptic mechanisms.\n\nThe classical form of spike-timing-dependent plasticity (STDP) for excitatory synapses (here de-\nnoted CSTDP) states that the causal temporal order of \ufb01rst pre-synaptic activity and then postsy-\nnaptic activity leads to long-term potentiation of the synapse (LTP) while the reverse order leads to\nlong-term depression (LTD)[6, 7, 8]. More recently, however, it became clear that STDP can exhibit\ndifferent dependencies on the temporal order of spikes. In particular, it was found that the reversed\ntemporal order (\ufb01rst post- then presynaptic spiking) could lead to LTP (and vice versa; RSTDP),\ndepending on the location on the dendrite [9, 10]. For inhibitory synapses some experiments were\nperformed which indicate that here STDP exists as well and has the form of CSTDP [11]. Note that\nCSTDP of inhibitory synapses in its effect on the postsynaptic neuron is equivalent to RSTDP of\nexcitatory synapses. Additionally it has been shown that CSTDP does not always rely on spikes, but\nthat strong subthreshold depolarization can replace the postsynaptic spike for LTD while keeping\nthe usual timing dependence [12]. We therefore assume that there exists a second threshold for the\ninduction of timing dependent LTD. For simplicity and without loss of generality, we restrict the\nstudy to RSTDP for synapses that in contradiction to Dale\u2019s law can change their sign.\n\nIt is very likely that plasticity rules and dynamical properties of neurons co-evolved to take advan-\ntage of each other. Combining them could reveal new and desirable effects. A modeling example\nfor a bene\ufb01cial effect of such an interplay was investigated in [13], where CSTDP interacted with\nspike-frequency adaptation of the postsynaptic neuron to perform a gradient descent on a square\nerror. Several other studies investigate the effect of STDP on network function, however mostly\nwith a focus on stability issues (e.g. [14, 15, 16]). In contrast, we here focus on the construc-\ntive role of STDP for associative learning. First we prove that RSTDP of excitatory synapses (or\nCSTDP on inhibitory synapses) when acting in concert with neuronal after-hyperpolarisation and\ndepolarization-dependent LTD is suf\ufb01cient for realizing the classical Perceptron learning rule, and\nthen show that this plasticity dynamics realizes a learning rule suited for the Tempotron and the\nChronotron [17].\n\n2 Ingredients\n\n2.1 Neuron model and network structure\n\nWe assume a feed-forward network of N presynaptic neurons and one postsynaptic integrate-and-\n\ufb01re neuron with a membrane potential U governed by\n\n\u03c4U \u02d9U = \u2212U + Isyn + Iext,\n\n(1)\n\nwhere Isyn denotes the input from the presynaptic neurons, and Iext is an input which can be used\nto drive the postsynaptic neuron to spike at certain times. When the neuron reaches a threshold\npotential Uthr, it is reset to a reset potential Ureset < 0, from where it decays back to the resting\npotential U\u221e = 0 with time constant \u03c4U . Spikes and other signals (depolarization) take \ufb01nite times\nto travel down the axon (\u03c4a) and the dendrite (\u03c4d). Synaptic transmission takes the form of delta\npulses, which reach the soma of the postsynaptic neuron after time \u03c4a + \u03c4d, and are modulated by\npre:\nthe synaptic weight w. We denote the presynaptic spike train of neuron i as xi with spike times ti\n\nxi(t) = Xti\n\npre\n\n\u03b4(t \u2212 ti\n\npre).\n\n2\n\n(2)\n\n\fA\n\nUthr\n\nUst\n\nU\u00a5\n\nz(t)\n\nw(t)\n\nx(t)\n\nB\n\npostsynaptic trace y\n\npresynaptic spikes x\n\nx\n\nsubthreshold events z(t)\n\nFigure 1: Illustration of STDP mechanism. A: Upper trace (red) is the membrane potential of the\npostsynaptic neuron. Shown are the \ufb01ring threshold Uthr and the threshold for LTD Ust. Middle\ntrace (black) is the variable z(t), the train of LTD threshold crossing events. Please note that the \ufb01rst\nspike in z(t) occurs at a different time than the neuronal spike. Bottom traces show w(t) (yellow)\nand \u00afx (blue) of a selected synapse. The second event in z reads out the trace of the presynaptic\nspike \u00afx, leading to LTD. B: Learning rule (4) is equivalent to RSTDP. A postsynaptic spike leads\nto an instantaneous jump in the trace \u00afy (top left, red line), which decays exponentially. Subsequent\npresynaptic spikes (dark blue bars and corresponding thin gray bars in the STDP window) \u201cread\u201d out\nthe state of the trace for the respective \u2206t = tpre \u2212 tpost. Similarly, z(t) reads out the presynaptic\ntrace \u00afx (lower left, blue line). Sampling for all possible times results in the STDP window (right).\n\nA postsynaptic neuron receives the input Isyn(t) = Pi wixi(t \u2212 \u03c4a \u2212 \u03c4d). The postsynaptic spike\ntrain is similarly denoted by y(t) = Ptpost\n\n\u03b4(t \u2212 tpost).\n\n2.2 The plasticity rule\n\nThe plasticity rule we employ mimics reverse STDP: A postsynaptic spike which arrives at the\nsynapse shortly before a presynaptic spike leads to synaptic potentiation. For synaptic depression\nthe relevant signal is not the spike, but the point in time where U (t) crosses an additional threshold\nUst from below, with U\u221e < Ust < Uthr (\u201csubthreshold threshold\u201d). These events are modelled as\n\u03b4-pulses in the function z(t) = Pk \u03b4(t\u2212tk), where tk are the times of the aforementioned threshold\n\ncrossing events (see Fig. 1 A for an illustration of the principle). The temporal characteristic of\n(reverse) STDP is preserved: If a presynaptic spike occurs shortly before the membrane potential\ncrosses this threshold, the synapse depresses. Timing dependent LTD without postsynaptic spiking\nhas been observed, although with classical timing requirements [12].\n\nWe formalize this by letting pre- and postsynaptic spikes each drive a synaptic trace:\n\n\u03c4pre \u02d9\u00afx = \u2212\u00afx + x(t \u2212 \u03c4a)\n\u03c4post \u02d9\u00afy = \u2212\u00afy + y(t \u2212 \u03c4d).\n\n(3)\n\nThe learning rule is a read\u2013out of the traces by spiking and threshold crossing events, respectively:\n\n\u02d9w \u221d \u00afyx(t \u2212 \u03c4a) \u2212 \u03b3 \u00afxz(t \u2212 \u03c4d),\n\n(4)\n\nwhere \u03b3 is a factor which scales depression and potentiation relative to each other. Fig. 1 B shows\nhow this plasticity rule creates RSTDP.\n\n3 Equivalence to Perceptron Learning Rule\n\nThe Perceptron Learning Rule (PLR) for positive binary inputs and outputs is given by\n\n\u2206w\u00b5\n\ni \u221d xi,\u00b5\n\n0 (2y\u00b5\n\n0 \u2212 1)\u0398 [\u03ba \u2212 (2y\u00b5\n\n0 \u2212 1)(h\u00b5 \u2212 \u03d1)] ,\n\n(5)\n\n3\n\n\frobustness against noise after convergence, h\u00b5 = Pi wixi,\u00b5\n\nwhere xi,\u00b5\n0 \u2208 {0, 1} denotes the activity of presynaptic neuron i in pattern \u00b5 \u2208 {1, . . . , P },\ny\u00b5\n0 \u2208 {0, 1} signals the desired response to pattern \u00b5, \u03ba > 0 is a margin which ensures a certain\nis the input to a postsynaptic neuron,\n\u03d1 denotes the \ufb01ring threshold, and \u0398(x) denotes the Heaviside step function. If the P patterns are\nlinearly separable, the perceptron will converge to a correct solution of the weights in a \ufb01nite number\nof steps. For random patterns this is generally the case for P < 2N . A \ufb01nite margin \u03ba reduces the\ncapacity.\n\n0\n\nInterestingly, for the case of temporally well separated synchronous spike patterns the combination\nof RSTDP-like synaptic plasticity dynamics with depolarization-dependent LTD and neuronal hy-\nperpolarization leads to a plasticity rule which can be mapped to the Perceptron Learning Rule. To\ncut down unnecessary notation in the derivation, we drop the indices i and \u00b5 except where necessary\nand consider only times 0 \u2264 t \u2264 \u03c4a + 2\u03c4d.\nWe consider a single postsynaptic neuron with N presynaptic neurons, with the condition \u03c4d < \u03c4a.\nDuring learning, presynaptic spike patterns consisting of synchronous spikes at time t = 0 are\ninduced, concurrent with a possibly occuring postsynaptic spike which signals the class the presy-\nnaptic pattern belongs to. This is equivalent to the setting of a single layered perceptron with bi-\nnary neurons. With x0 and y0 used as above we can write the pre- and postsynaptic activity as\nx(t) = x0\u03b4(t) and y(t) = y0\u03b4(t). The membrane potential of the postsynaptic neuron depends on\ny0:\n\nU (t) = y0Ureset exp(\u2212t/\u03c4U )\n\nU (\u03c4a + \u03c4d) = y0Ureset exp(\u2212(\u03c4a + \u03c4d)/\u03c4U ) = y0Uad.\n\nSimilarly, the synaptic current is\n\nIsyn(t) = Xi\nIsyn(\u03c4a + \u03c4d) = Xi\n\nwixi\n\n0\u03b4(t \u2212 \u03c4a \u2212 \u03c4d)\n\nwixi\n\n0 = Iad.\n\nThe activity traces at the synapses are\n\n\u00afx(t) = x0\u0398(t \u2212 \u03c4a)\n\n\u00afy(t) = y0\u0398(t \u2212 \u03c4d)\n\nexp(\u2212(t \u2212 \u03c4a)/\u03c4pre)\n\n\u03c4pre\n\nexp(\u2212(t \u2212 \u03c4d)/\u03c4post)\n\n\u03c4post\n\n.\n\n(6)\n\n(7)\n\n(8)\n\nThe variable of threshold crossing z(t) depends on the history of the postsynaptic neurons, which\nagain can be written with the aid of y0 as:\n\nz(t) = \u0398(Iad + y0Uad \u2212 Ust)\u03b4(t \u2212 \u03c4a \u2212 \u03c4d).\n\n(9)\nHere, \u0398 re\ufb02ects the condition for induction of LTD. Only when the postsynaptic input at time\nt = \u03c4a + \u03c4d is greater than the residual hyperpolarization (Uad < 0!) plus the threshold Ust, a\npotential LTD event gets enregistered. These are the ingredients for the plasticity rule (4):\n\n\u2206w \u221dZ [\u00afyx(t \u2212 \u03c4a) \u2212 \u03b3 \u00afxz(t \u2212 \u03c4d)] dt\n\n=x0y0\n\nexp(\u2212(\u03c4a + \u03c4d)/\u03c4post)\n\n\u03c4post\n\n\u2212 \u03b3x0\n\nexp(\u22122\u03c4d/\u03c4pre)\n\n\u03c4pre\n\n\u0398(Iad + y0Uad \u2212 Ust).\n\n(10)\n\nWe shorten this expression by choosing \u03b3 such that the factors of both terms are equal, which we\ncan drop subsequently:\n\nWe expand the equation by adding and substracting y0\u0398(Iad + y0Uad \u2212 Ust):\n\n\u2206w \u221d x0 (y0 \u2212 \u0398(Iad + y0Uad \u2212 Ust)) .\n\n(11)\n\n\u2206w \u221dx0 [y0(1 \u2212 \u0398(Iad + y0Uad \u2212 Ust)) \u2212 (1 \u2212 y0)\u0398(Iad + y0Uad \u2212 Ust)]\n\n=x0 [y0\u0398(\u2212Iad \u2212 Uad + Ust) \u2212 (1 \u2212 y0)\u0398(Iad \u2212 Ust)] .\n\n(12)\n\nWe used 1 \u2212 \u0398(x) = \u0398(\u2212x) in the last transformation, and dropped y0 from the argument of the\nHeaviside functions, as the two terms are seperated into the two cases y0 = 0 and y0 = 1. We do a\n\n4\n\n\fsimilar transformation to construct an expression G that turns either into the argument of the left or\nright Heaviside function depending on y0. That expression is\n\nG = Iad \u2212 Ust + y0(\u22122Iad \u2212 Uad + 2Ust),\n\nwith which we replace the arguments:\n\n\u2206w \u221d x0y0\u0398(G) \u2212 x0(1 \u2212 y0)\u0398(G) = x0(2y0 \u2212 1)\u0398(G).\n\n(13)\n\n(14)\n\nThe last task is to show that G and the argument of the Heaviside function in equation (5) are\nequivalent. For this we choose Iad = h, Uad = \u22122\u03ba and Ust = \u03d1 \u2212 \u03ba and keep in mind, that\n\u03d1 = Uthr is the \ufb01ring threshold. If we put this into G we get\n\nG =Iad \u2212 Ust + y0(\u22122Iad \u2212 Uad + 2Ust)\n\n=h \u2212 \u03d1 + \u03ba + 2y0h + 2y0\u03ba + 2y0\u03d1 \u2212 2y0\u03ba\n=\u03ba \u2212 (2y0 \u2212 1)(h \u2212 \u03d1),\n\n(15)\n\nwhich is the same as the argument of the Heaviside function in equation (5). Therefore, we have\nshown the equivalence of both learning rules.\n\n4 Associative learning of spatio-temporal spike patterns\n\n4.1 Tempotron learning with RSTDP\n\nThe condition of exact spike synchrony used for the above equivalence proof can be relaxed to\ninclude the association of spatio-temporal spike patterns with a desired postsynaptic activity. In the\nfollowing we take the perspective of the postsynaptic neuron which during learning is externally\nactivated (or not) to signal the respective class by spiking at time t = 0 (or not). During learning in\neach trial presynaptic spatio-temporal spike patterns are presented in the time span 0 < t < T , and\nplasticity is ruled by (4). For these conditions the resulting synaptic weights realize a Tempotron\nwith substantial memory capacity.\n\nA Tempotron is an integrate-and-\ufb01re neuron with input weights adjusted to perform arbitrary clas-\nsi\ufb01cations of (sparse) spike patterns [5, 18]. To implement a Tempotron, we make two changes\nto the model. First, we separate the time scales of membrane potential and hyperpolarization by\nintroducing a variable \u03bd:\n\n\u03c4\u03bd \u02d9\u03bd = \u2212\u03bd .\n\n(16)\nImmediately after a postsynaptic spike, \u03bd is reset to \u03bdspike < 0. The reason is that the length\nof hyperpolarization determines the time window where signi\ufb01cant learning can take place. To\nimprove comparability with the Tempotron as presented originally in [5], we set T = 0.5s and\n\u03c4\u03bd = \u03c4post = 0.2s, so that the postsynaptic neuron can learn to spike almost anywhere over the time\nwindow, and we introduce postsynaptic potentials (PSP) with a \ufb01nite rise time:\n\n\u03c4s \u02d9Isyn = \u2212Isyn +Xi\n\nwixi(t \u2212 \u03c4a),\n\n(17)\n\nwhere wi denotes the synaptic weight of presynaptic neuron i. With \u03c4s = 3ms and \u03c4U = 15ms the\nPSPs match the ones used in the original Tempotron study. This second change has little impact on\nthe capacity or otherwise. With these changes, the membrane potential is governed by\n\n\u03c4U \u02d9U = (\u03bd \u2212 U ) + Isyn(t \u2212 \u03c4d).\n\n(18)\n\nA postsynaptic spike resets U to \u03bdspike = Ureset < 0. Ureset is the initial hyperpolarization which\nis induced after a spike, which relaxes back to zero with the time constant \u03c4\u03bd \u226b \u03c4U . Presynaptic\nspikes add up linearly, and for simplicity we assume that both the axonal and the dendritic delay are\nnegligibly small: \u03c4a = \u03c4d = 0.\nIt is a natural choice to set \u03c4U = \u03c4pre and \u03c4\u03bd = \u03c4post. \u03c4U sets the time scale for the summation\nof EPSP contributing to spurious spikes, \u03c4\u03bd sets the time window where the desired spikes can lie.\nThey therefore should coincide with LTD and LTP, respectivly.\n\n5\n\n\fFigure 2: Illustration of Perceptron learning with RSTDP with subthreshold LTD and postsynaptic\nhyperpolarization. Shown are the traces \u00afx, \u00afy and U . Pre- and postsynaptic spikes are displayed as\nblack bars at t = 0. A: Learning in the case of y0 = 1, i.e. a postsynaptic spike as the desired\noutput. Initially the weights are too low and the synaptic current (summed PSPs) is smaller than\nUst. Weight change is LTP only until during pattern presentation the membrane potential hits Ust.\nAt this point LTP and LTD cancel exactly, and learning stops. B: Pattern completion for y0 = 1.\nShown are the same traces as in A at the absence of an inital postsynaptic spike. The membrane\npotential after learning is drawn as a dashed line to highlight the amplitude. Without the initial hy-\nperpolarization, the synaptic current after learning is large enough to cross the spiking threshold, the\npostsynaptic neuron \ufb01res the desired spike. Learning until Ust is reached ensures a minimum height\nof synaptic currents and therefore robustness against noise. C: Pattern presentation and completion\nfor y0 = 0. Initially, the synaptic current during pattern presentation causes a spike and conse-\nquently LTD. Learning stops when the membrane potential stays below Ust. Again, this ensures a\ncertain robustness against noise, analogous to the margin in the PLR.\n\n6\n\n\fA\n\nB\n\nFigure 3: Performance of Tempotron and Chronotron after convergence. A: Classi\ufb01cation perfor-\nmance of the Tempotron. Shown is the fraction of pattern which elicit the desired postsynaptic activ-\nity upon presentation. Perfect recall for all N is achieved up to \u03b1 = 0.18. Beyond that mark, some\nof the patterns become incorrectly classi\ufb01ed. The inset shows the learning curves for \u03b1 = 7/16. The\n\ufb01nal fraction of correctly classi\ufb01ed pattern is the average fraction of the last 500 blocks of each run.\nB: Performance of the Chronotron. Shown is the fraction of pattern which during recall succeed in\nproducing the correct postsynaptic spike time in a window of length 30 ms after the teacher spike.\nSee supplemental material for a detailed description. Please note that the scale of the load axis is\ndifferent in A and B.\n\nTable 1: Parameters for Tempotron learning\n\n\u03c4U , \u03c4pre\n15 ms\n\n\u03c4\u03bd, \u03c4post\n200 ms\n\n\u03c4s\n3 ms\n\nUthr\n20 mV 19 mV -20 mV 10\u22125\n\n\u03bdspike\n\nUst\n\n\u03b7\n\n\u03b3\n2\n\n4.1.1 Learning performance\n\nWe test the performance of networks of N input neurons at classifying spatio-temporal spike patterns\nby generating P = \u03b1N patterns, which we repeatedly present to the network. In each pattern,\neach presynaptic neuron spikes exactly once at a \ufb01xed time in each presentation, with spike times\nuniformly distributed over the trial. Learning is organized in learning blocks. In each block all P\npatterns are presented in randomized order. Synaptic weights are initialized as zero, and are updated\nafter each pattern presentation. After each block, we test if the postsynaptic output matches the\ndesired activity for each pattern. If during training a postsynaptic spike at t = 0 was induced, the\noutput can lie anytime in the testing trial for a positive outcome. To test scaling of the capacity,\nwe generate networks of 100 to 600 neurons and present the patterns until the classi\ufb01cation error\nreaches a plateau. Examples of learning curves (Classi\ufb01cation error over time) are shown in Fig. 3.\nFor each combination of \u03b1 and N , we run 40 simulations. The \ufb01nal classi\ufb01cation error is the mean\nover the last 500 blocks, averaged over all runs. The parameters we use in the simulations are shown\nin Tab. 1. Fig. 3 shows the \ufb01nal classi\ufb01cation performance as a function of the memory load \u03b1, for\nall network sizes we use. Up to a load of 0.18, the networks learns to perfectly classify each pattern.\nHigher loads leave a residual error which increases with load. The drop in performance is steeper\nfor larger networks. In comparison, the simpli\ufb01ed Tempotron learning rule proposed in [5] achieves\nperfect classi\ufb01cation up to \u03b1 \u2248 1.5, i.e. one order of magnitude higher.\n\n4.2 Chronotron learning with RSTDP\n\nIn the Chronotron [17] input spike patterns become associated with desired spike trains. There are\ndifferent learning rules which can achieve this mapping, including E\u2013learning, I\u2013learning, ReSuMe\nand PBSNLR [17, 19, 20]. The plasticity mechanism presented here has the tendency to generate\npostsynaptic spikes as close in time as possible to the teacher spike during recall. The presented\nlearning principle is therefore a candidate for Chronotron learning. The average distance of these\n\n7\n\n\fspikes depends on the time constants of hyperpolarization and the learning window, especially \u03c4post.\nThe modi\ufb01cations of the model necessary to implement Chronotron learning are described in the\nsupplement. The resulting capacity, i.e. the ability to generate the desired spike times within a short\nwindow in time, is shown in Fig. 3 B. Up to a load of \u03b1 = 0.01, the recall is perfect within the limits\nof the learning window \u03c4lw = 30ms. Inspection of the spike times reveals that the average distance\nof output spikes to the respective teacher spike is much shorter than the learning window (\u2248 2ms\nfor \u03b1 = 0.01, see supplemental Fig. 1).\n\n5 Discussion\n\nWe present a new and biologically highly plausible approach to learning in neuronal networks.\nRSTDP with subthreshold LTD in concert with hyperpolarisation is shown to be mathematically\nequivalent to the Perceptron learning rule for activity patterns consisting of synchronous spikes,\nthereby inheriting the highly desirable properties of the PLR (convergence in \ufb01nite time, stop condi-\ntion if performance is suf\ufb01cient and robustness against noise). This provides a biologically plausible\nmechanism to build associative memories with a capacity close to the theoretical maximum. Equiv-\nalence of STDP with the PRL was shown before in [21], but this equivalence only holds on average.\nWe would like to stress that we here present a novel approach that ensures exact mathematical eqi-\nvalence to the PRL.\n\nThe mechanism proposed here is complementary to a previous approach [13] which uses CSTDP\nin combination with spike frequency adaptation to perform gradient descent learning on a squared\nerror. However, that approach relies on an explicit teacher signal, and is not applicable to auto-\nassociative memories in recurrent networks. Most importantly, the approach presented here inherits\nthe important feature of selfregulation and fast convergence from the original Perceptron which is\nabsent in [13].\n\nFor sparse spatio-temporal spike patterns extensive simulations show that the same mechanism is\nable to learn Tempotrons and Chronotrons with substantial memory capacity. In the case of the\nTempotron, the capacity achieved with this mechanism is lower than with a comparably plausible\nlearning rule. However, in the case of the Chronotron the capacity comes close to the one obtained\nwith a commonly employed, supervised spike time learning rule. Moreover, these rules are biolog-\nically quite unrealistic. A prototypical example for such a supervised learning rule is the Temptron\nrule proposed by G\u00a8utig and Sompolinski [5]. Essentially, after a pattern presentation the complete\ntime course of the membrane potential during the presentation is examined, and if classi\ufb01cation was\nerroneous, the synaptic weights which contributed most to the absolute maximum of the potential\nare changed. In other words, the neurons would have to able to retrospectivly disentangle contri-\nbutions to their membrane potential at a certain time in the past. As we showed here, RSTDP with\nsubthreshold LTD together with postsynaptic hyperpolarization for the \ufb01rst time provides a realistic\nmechanism for Tempotron and Chronotron learning.\n\nSpike after-hyperpolarization is often neglected in theoretical studies or assumed to only play a role\nin network stabilization by providing refractoriness. Depolarization dependent STDP receives little\nattention in modeling studies (but see [22]), possibly because there are only few studies which show\nthat such a mechanism exists [12, 23]. The novelty of the learning mechanism presented here lies\nin the constructive roles both play in concert. After-hyperpolarization allows synaptic potentiation\nfor presynaptic inputs immediately after the teacher spike without causing additional non-teacher\nspikes, which would be detrimental for learning. During recall, the absence of the hyperpolarization\nensures the then desired threshold crossing of the membrane potential (see Fig. 2 B). Subthreshold\nLTD guarantees convergence of learning. It counteracts synaptic potentiation when the membrane\npotential becomes suf\ufb01ciently high after the teacher spike. The combination of both provides the\nlearning margin, which makes the resulting network robust against noise in the input. Taken together,\nour results show that the interplay of neuronal dynamics and synaptic plasticity rules can give rise\nto powerful learning dynamics.\n\nAcknowledgments\n\nThis work was in part funded by the German ministry for Science and Education (BMBF), grant\nnumber 01GQ0964. We are grateful to the anonymus reviewers who pointed out an error in \ufb01rst\nversion of the proof.\n\n8\n\n\fReferences\n[1] Hertz J, Krogh A, Palmer RG (1991) Introduction to the Theory of Neural Computation., Addison-Wesley.\n[2] Rosenblatt F (1957) The Perceptron\u2013a perceiving and recognizing automaton. Report 85-460-1.\n[3] Minsky ML, Papert SA (1969) Perceptrons. 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