{"title": "Synchrony Detection by Analogue VLSI Neurons with Bimodal STDP Synapses", "book": "Advances in Neural Information Processing Systems", "page_first": 1027, "page_last": 1034, "abstract": "", "full_text": "Synchrony Detection by Analogue VLSI\nNeurons with Bimodal STDP Synapses\n\nAdria Bo(cid:12)ll-i-Petit\n\nAlan F. Murray\n\nThe University of Edinburgh\n\nThe University of Edinburgh\n\nEdinburgh, EH9 3JL\n\nScotland\n\nEdinburgh, EH9 3JL\n\nScotland\n\nadria.bofill@ee.ed.ac.uk\n\nalan.murray@ee.ed.ac.uk\n\nAbstract\n\nWe present test results from spike-timing correlation learning ex-\nperiments carried out with silicon neurons with STDP (Spike Tim-\ning Dependent Plasticity) synapses. The weight change scheme\nof the STDP synapses can be set to either weight-independent or\nweight-dependent mode. We present results that characterise the\nlearning window implemented for both modes of operation. When\npresented with spike trains with di(cid:11)erent types of synchronisation\nthe neurons develop bimodal weight distributions. We also show\nthat a 2-layered network of silicon spiking neurons with STDP\nsynapses can perform hierarchical synchrony detection.\n\n1\n\nIntroduction\n\nTraditionally, Hebbian learning algorithms have interpreted Hebb\u2019s postulate in\nterms of coincidence detection. They are based on mean spike (cid:12)ring rates correla-\ntions between presynaptic and postsynaptic spikes rather than upon precise timing\ndi(cid:11)erences between presynaptic and postsynaptic spikes.\n\nIn recent years, new forms of synaptic plasticity that rely on precise spike-timing\ndi(cid:11)erences between presynaptic and postsynaptic spikes have been discovered in\nseveral biological systems[1][2][3]. These forms of plasticity, generally termed Spike\nTiming Dependent Plasticity (STDP), increase the synaptic e(cid:14)cacy of a synapse\nwhen a presynaptic spike reaches the neuron a few milliseconds before the postsy-\nnaptic action potential. In contrast, when the postsynaptic neuron (cid:12)res immediately\nbefore the presynaptic neuron the strength of the synapse diminishes.\n\nMuch debate has taken place regarding the precise characteristics of the learning\nrules underlying STDP [4]. The presence of weight dependence in the learning rule\nhas been identi(cid:12)ed as having a dramatic e(cid:11)ect on the computational properties of\nSTDP. When weight modi(cid:12)cations are independent of the weight value, a strong\ncompetition takes places between the synapses. Hence, even when no spike-timing\ncorrelation is present in the input, synapses develop maximum or minimum strength\nso that a bimodal weight distribution emerges from learning[5]. Conversely, if the\nlearning rule is strongly weight-dependent, such that strong synapses receive less po-\ntentiation than weaker ones while depression is independent of the synaptic strength,\n\n\fa smooth unimodal weight distribution emerges from the learning process[6].\n\nIn this paper we present circuits to support STDP on silicon. Bimodal weight\ndistributions are e(cid:11)ectively binary. Hence, they are suited to analog VLSI imple-\nmentation, as the main barrier to the implementation of on-chip learning, the long\nterm storage of precise analog weight values, can be rendered unimportant. How-\never, weight-independent STDP creates a highly unstable learning process that may\nhinder learning when only low levels of spike-timing correlations exist and neurons\nhave few synapses. The circuits proposed here introduce a tunable weight depen-\ndence mechanism which stabilises the learning process. This allows (cid:12)ner correlations\nto be detected than does a weight-independent scheme. In the weight-dependent\nlearning experiments reported here the weight-dependence is set at moderate levels\nsuch that bimodal weight distributions still result from learning.\n\nThe analogue VLSI implementation of spike-based learning was (cid:12)rst investi-\ngated in [7]. The authors used a weight-dependent scheme and concentrated on the\nweight normalisation properties of the learning rule. In [8], we proposed circuits\nto implement asymmetric STDP which lacked the weight-dependent mechanism.\nMore recently, others have also investigated asymmetric STDP learning using VLSI\nsystems[9][10]. STDP synapses that contain an explicit bistable mechanism have\nbeen proposed in [10]. Long-term bistable synapses are a good technological so-\nlution for weight storage. However, the maximum and minimum weight limits in\nbimodal STDP already act as natural attractors. An explicit bistable mechanism\nmay increase the instability of the learning process and may hinder, in consequence,\nthe detection of subtle correlations. In contrast, the circuits that we propose here\nintroduce a mechanism that tends to stabilise learning.\n\n2 STDP circuits\n\nThe circuits in Figure 1 implement the asymmetric decaying learning window with\nthe abrupt transition at the origin that is so characteristic of STDP. The weight of\neach synapse is represented by the charge stored on its weight capacitor Cw. The\nstrength of the weight is inversely proportional to Vw. The closer the value of Vw\nis to GND , the stronger is the synapse.\n\nOur silicon spiking neurons signal their (cid:12)ring events with the sequence of pulses\nseen in Figure 1c. Signal post bp is back-propagated to the a(cid:11)erent synapses of the\nneuron. Long is a longer pulse (a few (cid:22)s) used in the current neuron (termed as\nsignal postLong in Figure 1b). Long is also sent to input synapses of following neu-\nrons in the activity path (see preLong in 1a). Finally, spikeOut is the presynaptic\nspike for the next receiving neuron (termed pre in Figure 1a). More details on the\nimplementation of the silicon neuron can be found in [11]\n\nIn Figure 1a, if preLong is long enough (a few (cid:22)s) the voltage created by Ibpot on\nthe diode connected transistor N5 is copied to the gate of N2. This voltage across\nCpot decays with time from its peak value due to a leakage current set by Vbpot.\nWhen the postsynaptic neuron (cid:12)res, a back propagation pulse post bp switches N3\non. Therefore, the weight is potentiated (Vw decreased) by an amount which re(cid:13)ects\nthe time elapsed since the last presynaptic event.\n\nA weight dependence mechanism is introduced by the simple linearised V-I con-\n(cid:12)guration P5-P6 and current mirror N7-N6 (see Figure 1a). P5 is a low gain tran-\nsistor operated in strong inversion whereas P6 is a wide transistor made to operate\nin weak inversion such that it has even higher gain. When the value of Vw decreases\n(weight increase) the current through P5-P6 increases, but P5 is maintained in the\nlinear region by the high gain transistor. Thus, a current proportional to the value\nof the weight is subtracted from Ibpot. The resulting smaller current injected into\nN5 will cause a drop in the peak of potentiation for large weight values.\n\n\fpre\n\nP1\n\nP4\n\nVw\n\nCw\n\nP2\n\nP3\n\npost_bp\n\nLong\n\nspikeOut\n\nP5\n\nP6\n\nN7\n\nIbpot\n\nVr\n\npreLong\n\nN3\n\npost_bp\n\nN6\n\nN5\n\nN8\n\nVbpot\n\nN2\n\nN1\n\nCpot\n\nIdep\n\n( c )\n\nIbdep\n\npostLong\n\nIdep_1\n\nIdep_N\n\nN1\n\nN2\nVbdep\n\nN3\n\nCdep\n\nN4\n\n( a )\n\n( b )\n\nFigure 1: Weight change circuits. (a) The strength of the synapse is inversely proportional\nto the value of Vw. The lower Vw, the smaller the weight of the synapse. This section of\nthe weight change circuit detects causal spike correlations. (b) A single depression circuit\npresent in the soma of the neuron creates the decaying shape of the depression side of the\nlearning window. (c) Waveforms of pulses that signal an action potential event. They are\nused to stimulate the weight change circuits.\n\nIn a similar manner to potentiation, the weight is weakened by the circuit of\nFigure 1b when it detects a non-causal interaction between a presynaptic and a\npostsynaptic spike. When a postsynaptic spike event is generated a postLong pulse\ncharges Cdep. The charge accumulated leaks linearly through N3 at a rate set by\nVbdep. A set of non-linear decaying currents (IdepX ) is sent to the weight change\ncircuits placed in the input synapse (see Idep in Figure 1a). When a presynaptic\nspike reaches a synapse P1 is switched on.\nIf this occurs soon enough after the\npostLong pulse was generated, Vw is brought closer to Vdd (weight strength de-\ncreased). Only one depression circuit per neuron is required since the depression\npart of the learning rule is independent of the weight value.\n\nA chip including 5 spiking neurons with STDP synapses has been fabricated us-\ning a standard 0.6(cid:22)m CMOS process. Each neuron has 6 learning synapses, a single\nexcitatory non-learning synapse and a single inhibitory one. Along with the silicon\nneuron circuits, the chip contains several voltage bu(cid:11)ers that allow us to monitor\nthe behaviour of the neuron. The testing setup uses a networked logic analysis sys-\ntem to stimulate the silicon neuron and to capture the results of on-chip learning.\nAn externally addressable circuit creates preLong and pre pulses to stimulate the\nsynapses.\n\n3 Weight-independent learning rule\n\n3.1 Characterisation\n\nA weight-independent weight change regime is obtained by setting Vr to Vdd in\nthe weight change circuit presented in Figure 1 . The resulting learning window\non silicon can be seen in Figure 2. Each point in the curve was obtained from the\nstimulation of the (cid:12)x synapse and a learning synapse with a varying delay between\nthem. As can be seen in the (cid:12)gure, the circuit is highly tunable. Figure 2a shows\nthat the peaks for potentiation and depression can be set independently. Also, as\nshown in Figure 2b the decay of the learning window for both sides of the curve can\nbe set independently of the maximum weight change with Vbdep and Vbpot. Since the\nweight-dependent mechanism is switched o(cid:11), the curve of the learning window is\nthe same for a wide range of Vw. Obviously, when the weight voltage Vw approaches\n\n\f)\n \n\nV\n\n \n(\n \n \n \n\nw\n\nV\n\n0.4\n\n0.3\n\n0.2\n\n0.1\n\n0\n\n\u22120.1\n\n\u22120.2\n\n\u22120.3\n\n\u221230\n\n\u221220\n\n0.3\n\n0.25\n\n0.2\n\n0.15\n\n0.1\n\n0.05\n\n0\n\n)\n \n\nV\n\n \n(\n \n \n \n\nw\n\nV\n\n\u22120.05\n\n\u22120.1\n\n20 \n\n30 \n\n40 \n\n\u22120.15\n\n\u221240\n\n\u221230\n\n\u221220\n\n20 \n\n30 \n\n40 \n\n\u221210\nt\npre\n\n0    \n\n \u2212 t\n\npost\n\n10 \n   ( ms )\n\n( b )\n\n\u221210\nt\npre\n\n10 \n\n    ( ms )\n\n0    \n\npost\n\n \u2212 t\n( a )\n\nFigure 2: Experimental learning window for weight-independent STDP. The curves show\nthe weight modi(cid:12)cation induced in the weight of a learning synapse for di(cid:11)erent time in-\ntervals between the presynaptic and the postsynaptic spike. For the results shown, the\nsynapses were operated in a weight-independent mode. (a) The peaks of the learning win-\ndow is shown for 4 di(cid:11)erent settings. The peak for potentiation and depression are tuned\nindependently with Ibpot and Ibdep (b) The rate of decay of the learning window for po-\ntentiation and depression can be set independently without a(cid:11)ecting the maximum weight\nchange.\n\nany of the power supply rails a saturation e(cid:11)ect occurs as the transistors injecting\ncurrent in the weight capacitor leave saturation. For the learning experiment with\nweight-independent weight change the area under the potentiation curve should be\napproximately 50% smaller than the area under the depression region.\n\n3.2 Learning spike-timing correlations with weight-independent\n\nlearning\n\nWe stimulated a 6-synapse silicon neuron with 6 independent Poisson-distributed\nspike trains with a rate of 30Hz. An absolute refractory period of 10ms was enforced\nbetween consecutive spikes of each train. Refractoriness helps break the temporal\naxis into disjoint segments so that presynaptic spikes can make less noisy \"predic-\ntions\" of the postsynaptic time of (cid:12)ring. We introduced spike-timing correlations\nbetween the inputs for synapses 1 and 2. Synapses 3 to 6 were uncorrelated.\n\nThe evolution of the 6 weights for one of such experiments is show in Figure 3.\nThe correlated inputs shared 35% of the spike-timings. They were constructed by\nmerging two independent 19.5Hz Poisson-distributed spike trains with a common\n10.5Hz spike train. As can be seen in Figure 3 the weights of synapses that receive\ncorrelated activity reach maximum strength (Vw close to GND) whereas the rest\ndecay towards Vdd. Clearly, the bimodal weight distribution re(cid:13)ects the correlation\npattern of the input signals.\n\n3.3 Hierarchical synchrony detection\n\nTo experiment with hierarchical synchrony detection we included in the chip a\nsmall 2-layered network of STDP silicon neurons with the con(cid:12)guration shown\nin Figure 4. Neurons in the (cid:12)rst layer were stimulated with independent sets of\nPoisson-distributed spike trains with a mean spiking rate of 30Hz. As with the\nexperiments presented in the preceding section, a 10ms refractory period was\nforced between consecutive spikes. A primary level of correlation was introduced\nfor each neuron in the (cid:12)rst layer as signalled by the arrowed bridge between the\n\nD\nD\n\f)\n \n\nV\n\n \n(\n \n \n \n\nV\n\nw\n\n4.5\n\n4\n\n3.5\n\n3\n\n2.5\n\n2\n\n1.5\n\n1\n\n0.5\n\n0\n0\n\nV\n\n \nw5\n\nV\n\n \nw6\n\nV\n\n \nw4\n\nV\n\n \nw3\n\nV\n\n, V\n\n \nw2\n\nw1\n\n1\n\n2\n\n3\n\n4\n\ntime   ( s )\n\n5\n\n6\n\n7\n\n8\n\nFigure 3: Learning experiment with weight-independent STDP.\n\n0.25\n\n0.5\n\n0.5\n\n0.5\n\n0.5\n\nN1\n\nN2\n\nN3\n\nN4\n\nN5\n\nFigure 4: Final weight values for a 2-layered network of STDP silicon neurons.\n\ninputs of synapses 1 and 2 of each neuron. For the results shown here these 2\ninputs of each neuron shared 50% of the spike-timings (indicated with 0.5 on top\nof the double-arrowed bridge of Figure 4). A secondary level of correlation was\nintroduced between the inputs of synapses 1 and 2 of both N1 and N2, as signalled\nby the arrow linking the (cid:12)rst level of correlations of N1 and N2. This second level\nof correlations is weaker, with only 25% of shared spikes (indicated with 0.25 in\nFigure 4). The two direct inputs of N5, in the second layer, were also Poisson\ndistributed but had a rate of 15Hz.\n\nThe evolution of the weights recorded for the experiment just described is\npresented in Figure 5. On the left, we see the weight evolution for N1. The weights\ncorresponding to synapses 1 and 2 evolve towards the maximum value (i.e. GND).\nThe weights of the remaining synapses, which receive random activity, decrease\n(i.e. Vw close to Vdd). The other neurons in the 1st layer have weight evolutions\nsimilar to that of N1. Synapses with synchronised activity corresponding to the\n1st level of correlations win the competition imposed by STDP. The Vw traces\non the right-hand side of Figure 5 show how N5 in the second layer captures the\nsecondary level of correlation. Weights of the synapses receiving input from N1\nand N2 are reinforced while the rest are decreased towards the minimum possible\nweight value (Vw = Vdd). Clearly, the second layer only captures features from\nsignals which have already a basic level of interesting features (primary level of\ncorrelations) detected by the (cid:12)rst layer.\n\nIn Figure 4, we have represented graphically the (cid:12)nal weight distribution for\nall synapses. As marked by (cid:12)lled circles, only synapses in the path of hierarchical\n\n\f4.5\n\n4\n\n3.5\n\n3\n\n2.5\n\n2\n\n1.5\n\n1\n\n0.5\n\n)\n \n\nV\n\n \n(\n \n \n \n\nV\n\nw\n\nN5\n\nV\n\n \nw4\n\nV\n\n    \nw3\n\nV\n\n \nw5\n\nV\n\n \nw6\n\nV\n\nw1\n\n   ,   V\n\n \nw2\n\n5\n\n10\n\n15\n\n20\n\n25\n\n30\n\n35\n\n40\n\ntime   ( s )\n\n \n(\n \n \n \n\nV\n\nw\n\n2\n\n1.5\n\n1\n\n0.5\n\n0\n0\n\nV\n\n \nw3\n\nV\n\n \nw4\n\nN1\n\n4.5\n\n4\n\n3.5\n\n3\n\n)\n \n\nV\n\n2.5\n\nV\n\n  ,  V\n\n \nw6\n\nw5\n\nV\n\n  ,   V\n\n \nw2\n\nw1\n\n0\n0\n\n2\n\n4\n\n6\n\ntime   ( s )\n\n8\n\n10\n\nFigure 5: Hierarchical synchrony detection. (a) Weight evolution of neuron in (cid:12)rst layer.\n(b) Weight evolution of output neuron in 2nd layer.\n\nsynchrony activity develop maximum weight strength.\nIn contrast, weights with\n(cid:12)nal minimum strength are indicated by empty circles. These correspond to\nsynapses of (cid:12)rst layer neurons which received uncorrelated inputs or synapses of\nN5 which received inputs from neurons stimulated without a secondary level of\ncorrelations (N3-N4).\n\n4 Weight-dependent learning rule\n\n4.1 Characterisation\n\nThe STDP synapses presented can also be operated in weight-dependent mode.\nThe weight dependent learning window implemented is similar to that which seems\nto underly some STDP recordings from biological neurons [6]. Figure 6a shows\nchip results of the weight-dependent learning rule. The weight change curve for\npotentiation is given for 3 di(cid:11)erent weight values. The larger the weight value (low\nVw), the smaller the degree of potentiation induced in the synapse. The depression\nside of the learning window is una(cid:11)ected by the weight value since the depression\ncircuit shown in Figure 1b does not have an explicit weight-dependent mechanism.\n\n4.2 Learning spike-timing correlations with weight-dependent learning\n\nFigure 6b shows the weight evolution for an experiment where the correlated ac-\ntivity between synapses 1 and 2 consisted of only 20% of common spike-timings.\nAs in the weight-independent experiments, the mean (cid:12)ring rate was 30Hz and a\nrefractory period of 10ms was enforced.\n\nFinally, we stimulated a neuron in weight-dependent mode with a form of syn-\nchrony where spike-timings coincided in a time window (window of correlation)\ninstead of being perfectly matched (syn0-1). The uncorrelated inputs (syn2-5) were\nPoisson-distributed spike trains. The synchrony data was an inhomogeneous Pois-\nson spike train with a rate modulated by a binary signal with random transition\npoints. Figure 7 shows a normalised histogram of spike intervals between the corre-\nlated inputs for synapses 0 and 1 (Figure 7a) and the histogram of the uncorrelated\ninputs for synapses 2 and 3 (Figure 7b). Again, as can be seen in Figure 7c the\nneuron with weight-dependent STDP can detect this low-level of synchrony with\nnon-coincident spikes. Clearly, the bimodal weight distribution identi(cid:12)es the syn-\n\n\f0.25\n\n0.2\n\n0.15\n\n0.1\n\n0.05\n\n0\n\n)\n \n\nV\n\n \n(\n \n \n \n\nw\n\nV\n\n\u22120.05\n\n\u22120.1\n\n\u22120.15\n\n\u221225 \u221220  \u221215 \u221210 \u22125 \nt\npre\n\n 0 \n \u2212 t\n\n   ( ms )\n\nWinit = 0.75V \n\nWinit = 2V \n\nWinit = 3.25 \n\n5 \n\n10   15  20  \n\n25\n\n30 \n\npost\n(a)\n\n4.5\n\n4\n\n3.5\n\n V\n\nw6\n\n V\n\nw5\n\n)\n \n\nV\n\n \n(\n \n \n \n\nw\n\nV\n\n3\n\n2.5\n\n2\n\n1.5\n\n1\n\n0.5\n\n0\n0\n\n V\n\nw4\n\nV\n\n \nw3\n\n V\n\nw1\n\n V\n\nw2\n\n2\n\n4\n\n6\n\n8\n\n10\n\n12\n\n14\n\n16\n\ntime   ( s )\n\n(b)\n\nFigure 6: (a) Experimental learning window for weight-dependent STDP (b) Learning\nexperiment with weight-dependent STDP. Synapses 1 and 2 share 20% of spike-timings.\nThe other synapses receive completely uncorrelated activity. Correlated activity causes\nsynapses to develop strong weights (Vw close to GND).\n\nchrony pattern of the inputs.\n\n5 Conclusions\n\nThe circuits presented can be used to study both weight-dependent and weight-\nindependent learning rules. The in(cid:13)uence of weight-dependence on the (cid:12)nal weight\ndistribution has been studied extensively[5][6].\nIn this paper, we have concen-\ntrated on the stabilising e(cid:11)ect that moderate weight-dependence can have on learn-\ning processes that develop bimodal weight distributions. By introducing weight-\ndependence subtle spike-timing correlations can be detected.\n\nWe have also shown experimentally that a small feed-forward network of silicon\nneurons with STDP synapses can detect a hierarchical synchrony structure embed-\nded in noisy spike trains.\n\nWe are currently investigating the synchrony ampli(cid:12)cation properties of silicon\nneurons with bimodal STDP. We are also working on a new chip that uses lateral-\ninhibitory connections between neurons to classify data with complex synchrony\npatterns.\n\nReferences\n\n[1] G-Q. Bi and M m Poo. Synaptic modi(cid:12)cations in cultured hippocampal neurons;\ndependence on spike timing, synaptic strength and postsynaptic cell type. Journal of\nNeuroscience, 18:10464{10472, 1998.\n\n[2] L.I. Zhang, H.W. Tao, C.E. Holt, W.A. Harris, and M m. Poo. A critical window\nfor cooperation and competition among developing retinotectal synapses. Nature,\n395:37{44, 1998.\n\n[3] H. Markram, J. Lubke, M. Frotscher, and B. Sakmann. Regulation of synaptic e(cid:14)cacy\n\nby coincidence of postsynaptic APs and EPSPs. Science, 275:213{215, 1997.\n\n[4] A. Kepecs, M.C.W van Rossum, S. Song, and J. Tegner. Spike-timing-dependent\nplasticity: common themes and divergent vistas. Biological Cybernetics, 87:446{458,\n2002.\n\n[5] S. Song, K.D. Miller, and L.F. Abbott. Competitive Hebbian learning through spike-\n\ntiming dependent synaptic plasticity. Nature Neuroscience, 3:919{926, 2000.\n\nD\n\f0.05\n\n0.045\n\n0.04\n\n0.035\n\n0.03\n\n0.025\n\n0.02\n\n0.015\n\n0.01\n\n0.005\n\nn\no\n\ni\nt\n\nl\n\na\ne\nr\nr\no\nC\n\n0.02\n\n0.04\n\n0.06\n\n0\n\n\u22120.06\n\n\u22120.04\n\n\u22120.02\n\n0.02\n\n0.04\n\n0.06\n\n0\n\nt  ( s )\n(b)\n\nV\n\n , V\n\nw13\n\nw14\n\n0.05\n\n0.045\n\n0.04\n\n0.035\n\n0.03\n\n0.025\n\n0.02\n\n0.015\n\n0.01\n\n0.005\n\nn\no\n\ni\nt\n\nl\n\na\ne\nr\nr\no\nC\n\n0\n\n\u22120.06\n\n\u22120.04\n\n\u22120.02\n\n0\nt ( s )\n(a)\n\n4.5\n\n4\n\n3.5\n\n3\n\n2.5\n\n)\n \n\nV\n\nV\n\n \nw15\n\nV\n\n \nw12\n\nV\n\n \nw11\nV\n\n \nw10\n\n10\n\n20\n\n30\n\n40\n\n50\n\n60\n\ntime   ( s )\n\n(c)\n\n \n(\n \n \n \n\nV\n\nw\n\n2\n\n1.5\n\n1\n\n0.5\n\n0\n0\n\nFigure 7: Detection of non-coincident spike-timing synchrony with weight-dependent\nSTDP.(a) Normalised spike interval histogram of the 2 correlated inputs (synapses 0 and\n1). (b) Normalised spike interval histogram between 2 uncorrelated inputs (synapses 2-5)\n(c) Synapses 0 and 1 win the learning competition.\n\n[6] M. van Rossum and G.G. Turrigiano. Corrrelation based learning from spike timing\n\ndependent plasticity. Neurocomputing, 38-40:409{415, 2001.\n\n[7] P. Hal(cid:12)ger, M. Mahowald, and L. Watts. A spike based learning neuron in analog\nIn M.C. Mozer, M.I. Jordan, and T. Petsche, editors, Advances in Neural\n\nVLSI.\nInformation Processing Systems 9, pages 692{698. MIT Press, 1996.\n\n[8] A. Bo(cid:12)ll, A. F. Murray, and D. P. Thompson. Circuits for VLSI implementation\nof temporally asymmetric Hebbian learning.\nIn T. G. Dietterich, S. Becker, and\nZ. Ghahramani, editors, Advances in Neural Information Processing Systems 14. MIT\nPress, 2002.\n\n[9] R. J. Vogelstein, F. Tenore, R. Philipp, M. S. Adlerstein, D. H. Goldberg, and\nIn\nG. Cauwenberghs. Spike timing-dependent plasticity in the address domain.\nS. Becker, S. Thrun, and Klaus Obermayer, editors, Advances in Neural Informa-\ntion Processing Systems 15. MIT Press, 2003.\n\n[10] G. Indiveri. Circuits for bistable spike-timing-dependent plasticity neuromorphic vlsi\nsynapses. In S. Becker, S. Thrun, and Klaus Obermayer, editors, Advances in Neural\nInformation Processing Systems 15. MIT Press, 2003.\n\n[11] A. Bo(cid:12)ll i Petit and A.F. Murray. Learning temporal correlations in biologically-\ninspired aVLSI. In IEEE Internation Symposium on Circuits and Systems, volume 5,\npages 817{820, 2003.\n\nD\nD\n\f", "award": [], "sourceid": 2375, "authors": [{"given_name": "Adria", "family_name": "Bofill-i-petit", "institution": null}, {"given_name": "Alan", "family_name": "Murray", "institution": null}]}