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246 nips-2013-Perfect Associative Learning with Spike-Timing-Dependent Plasticity

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Author: Christian Albers, Maren Westkott, Klaus Pawelzik

Abstract: Recent extensions of the Perceptron as the Tempotron and the Chronotron suggest that this theoretical concept is highly relevant for understanding networks of spiking neurons in the brain. It is not known, however, how the computational power of the Perceptron 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 these 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 and Chronotrons are learned. The proposed mechanism enables incremental associative learning from a continuous stream of patterns and might therefore underly the acquisition 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.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 de Abstract Recent extensions of the Perceptron as the Tempotron and the Chronotron suggest that this theoretical concept is highly relevant for understanding networks of spiking neurons in the brain. [sent-7, score-0.149]

2 It is not known, however, how the computational power of the Perceptron might be accomplished by the plasticity mechanisms of real synapses. [sent-8, score-0.258]

3 The proposed mechanism enables incremental associative learning from a continuous stream of patterns and might therefore underly the acquisition of long term memories in cortex. [sent-11, score-0.233]

4 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. [sent-12, score-0.638]

5 The original Perceptron Learning Rule (PLR) is a supervised learning rule that employs a threshold to control weight changes, which also serves as a margin to enhance robustness [2, 3]. [sent-14, score-0.187]

6 Associative learning can be considered a special case of supervised learning where the activity of the output neuron is used as a teacher signal such that after learning missing activities are ﬁlled in. [sent-16, score-0.28]

7 For this reason the PLR is very useful for building associative memories in recurrent networks where it can serve to learn arbitrary patterns in a ’quasi-unsupervised’ way. [sent-17, score-0.216]

8 On the other hand, it is not known if and how real synaptic mechanisms might realize the successdependent self-regulation of the PLR in networks of spiking neurons in the brain. [sent-20, score-0.361]

9 The simpliﬁed tempotron learning rule, while biologically more plausible, still relies on a reward signal which tells each neuron speciﬁcally that it should have spiked or not. [sent-22, score-0.408]

10 Taken together, while highly desirable, the feature of self regulation in the PLR still poses a challenge for biologically realistic synaptic mechanisms. [sent-23, score-0.253]

11 In particular, it was found that the reversed temporal order (ﬁrst post- then presynaptic spiking) could lead to LTP (and vice versa; RSTDP), depending on the location on the dendrite [9, 10]. [sent-26, score-0.228]

12 For inhibitory synapses some experiments were performed which indicate that here STDP exists as well and has the form of CSTDP [11]. [sent-27, score-0.147]

13 Note that CSTDP of inhibitory synapses in its effect on the postsynaptic neuron is equivalent to RSTDP of excitatory synapses. [sent-28, score-0.736]

14 Additionally it has been shown that CSTDP does not always rely on spikes, but that strong subthreshold depolarization can replace the postsynaptic spike for LTD while keeping the usual timing dependence [12]. [sent-29, score-0.901]

15 It is very likely that plasticity rules and dynamical properties of neurons co-evolved to take advantage of each other. [sent-32, score-0.311]

16 A modeling example for a beneﬁcial effect of such an interplay was investigated in [13], where CSTDP interacted with spike-frequency adaptation of the postsynaptic neuron to perform a gradient descent on a square error. [sent-34, score-0.556]

17 When the neuron reaches a threshold potential Uthr , it is reset to a reset potential Ureset < 0, from where it decays back to the resting potential U∞ = 0 with time constant τU . [sent-42, score-0.412]

18 Synaptic transmission takes the form of delta pulses, which reach the soma of the postsynaptic neuron after time τa + τd , and are modulated by the synaptic weight w. [sent-44, score-0.745]

19 We denote the presynaptic spike train of neuron i as xi with spike times ti : pre δ(t − ti ). [sent-45, score-1.049]

20 pre xi (t) = ti pre 2 (2) A B Uthr postsynaptic trace y Ust U¥ presynaptic spikes x x z(t) w(t) x(t) subthreshold events z(t) Figure 1: Illustration of STDP mechanism. [sent-46, score-1.079]

21 A: Upper trace (red) is the membrane potential of the postsynaptic neuron. [sent-47, score-0.668]

22 Middle trace (black) is the variable z(t), the train of LTD threshold crossing events. [sent-49, score-0.14]

23 Please note that the ﬁrst spike in z(t) occurs at a different time than the neuronal spike. [sent-50, score-0.339]

24 The second event in z reads out the trace of the presynaptic ¯ spike x, leading to LTD. [sent-52, score-0.551]

25 A postsynaptic spike leads ¯ to an instantaneous jump in the trace y (top left, red line), which decays exponentially. [sent-54, score-0.764]

26 Subsequent ¯ presynaptic spikes (dark blue bars and corresponding thin gray bars in the STDP window) “read” out the state of the trace for the respective ∆t = tpre − tpost . [sent-55, score-0.402]

27 Similarly, z(t) reads out the presynaptic trace x (lower left, blue line). [sent-56, score-0.242]

28 ¯ A postsynaptic neuron receives the input Isyn (t) = i wi xi (t − τa − τd ). [sent-58, score-0.585]

29 The postsynaptic spike train is similarly denoted by y(t) = tpost δ(t − tpost ). [sent-59, score-0.809]

30 2 The plasticity rule The plasticity rule we employ mimics reverse STDP: A postsynaptic spike which arrives at the synapse shortly before a presynaptic spike leads to synaptic potentiation. [sent-61, score-2.071]

31 For synaptic depression the relevant signal is not the spike, but the point in time where U (t) crosses an additional threshold Ust from below, with U∞ < Ust < Uthr (“subthreshold threshold”). [sent-62, score-0.273]

32 These events are modelled as δ-pulses in the function z(t) = k δ(t−tk ), where tk are the times of the aforementioned threshold crossing events (see Fig. [sent-63, score-0.157]

33 The temporal characteristic of (reverse) STDP is preserved: If a presynaptic spike occurs shortly before the membrane potential crosses this threshold, the synapse depresses. [sent-65, score-0.766]

34 Timing dependent LTD without postsynaptic spiking has been observed, although with classical timing requirements [12]. [sent-66, score-0.52]

35 We formalize this by letting pre- and postsynaptic spikes each drive a synaptic trace: ˙ ¯ x τpre x = −¯ + x(t − τa ) ˙ ¯ y τpost y = −¯ + y(t − τd ). [sent-67, score-0.723]

36 (3) The learning rule is a read–out of the traces by spiking and threshold crossing events, respectively: w ∝ y x(t − τa ) − γ xz(t − τd ), ˙ ¯ ¯ (4) where γ is a factor which scales depression and potentiation relative to each other. [sent-68, score-0.356]

37 1 B shows how this plasticity rule creates RSTDP. [sent-70, score-0.302]

38 3 Equivalence to Perceptron Learning Rule The Perceptron Learning Rule (PLR) for positive binary inputs and outputs is given by µ µ µ ∆wi ∝ xi,µ (2y0 − 1)Θ [κ − (2y0 − 1)(hµ − ϑ)] , 0 3 (5) where xi,µ ∈ {0, 1} denotes the activity of presynaptic neuron i in pattern µ ∈ {1, . [sent-71, score-0.418]

39 , P }, 0 µ y0 ∈ {0, 1} signals the desired response to pattern µ, κ > 0 is a margin which ensures a certain robustness against noise after convergence, hµ = i wi xi,µ is the input to a postsynaptic neuron, 0 ϑ denotes the ﬁring threshold, and Θ(x) denotes the Heaviside step function. [sent-74, score-0.578]

40 If the P patterns are linearly separable, the perceptron will converge to a correct solution of the weights in a ﬁnite number of steps. [sent-75, score-0.186]

41 Interestingly, for the case of temporally well separated synchronous spike patterns the combination of RSTDP-like synaptic plasticity dynamics with depolarization-dependent LTD and neuronal hyperpolarization leads to a plasticity rule which can be mapped to the Perceptron Learning Rule. [sent-78, score-1.323]

42 We consider a single postsynaptic neuron with N presynaptic neurons, with the condition τd < τa . [sent-80, score-0.759]

43 During learning, presynaptic spike patterns consisting of synchronous spikes at time t = 0 are induced, concurrent with a possibly occuring postsynaptic spike which signals the class the presynaptic pattern belongs to. [sent-81, score-1.698]

44 With x0 and y0 used as above we can write the pre- and postsynaptic activity as x(t) = x0 δ(t) and y(t) = y0 δ(t). [sent-83, score-0.461]

45 The membrane potential of the postsynaptic neuron depends on y0 : U (t) = y0 Ureset exp(−t/τU ) U (τa + τd ) = y0 Ureset exp(−(τa + τd )/τU ) = y0 Uad . [sent-84, score-0.769]

46 (6) Similarly, the synaptic current is wi xi δ(t − τa − τd ) 0 Isyn (t) = i wi xi = Iad . [sent-85, score-0.247]

47 0 Isyn (τa + τd ) = (7) i The activity traces at the synapses are exp(−(t − τa )/τpre ) τpre exp(−(t − τd )/τpost ) y (t) = y0 Θ(t − τd ) ¯ . [sent-86, score-0.183]

48 τpost x(t) = x0 Θ(t − τa ) ¯ (8) The variable of threshold crossing z(t) depends on the history of the postsynaptic neurons, which again can be written with the aid of y0 as: z(t) = Θ(Iad + y0 Uad − Ust )δ(t − τa − τd ). [sent-87, score-0.517]

49 Only when the postsynaptic input at time t = τa + τd is greater than the residual hyperpolarization (Uad < 0! [sent-89, score-0.542]

50 These are the ingredients for the plasticity rule (4): ∆w ∝ [¯x(t − τa ) − γ xz(t − τd )] dt y ¯ =x0 y0 exp(−(τa + τd )/τpost ) exp(−2τd /τpre ) − γx0 Θ(Iad + y0 Uad − Ust ). [sent-91, score-0.302]

51 4 Associative learning of spatio-temporal spike patterns 4. [sent-101, score-0.387]

52 1 Tempotron learning with RSTDP The condition of exact spike synchrony used for the above equivalence proof can be relaxed to include the association of spatio-temporal spike patterns with a desired postsynaptic activity. [sent-102, score-1.176]

53 In the following we take the perspective of the postsynaptic neuron which during learning is externally activated (or not) to signal the respective class by spiking at time t = 0 (or not). [sent-103, score-0.625]

54 During learning in each trial presynaptic spatio-temporal spike patterns are presented in the time span 0 < t < T , and plasticity is ruled by (4). [sent-104, score-0.825]

55 For these conditions the resulting synaptic weights realize a Tempotron with substantial memory capacity. [sent-105, score-0.189]

56 A Tempotron is an integrate-and-ﬁre neuron with input weights adjusted to perform arbitrary classiﬁcations of (sparse) spike patterns [5, 18]. [sent-106, score-0.527]

57 First, we separate the time scales of membrane potential and hyperpolarization by introducing a variable ν: τν ν = −ν . [sent-108, score-0.339]

58 ˙ (16) Immediately after a postsynaptic spike, ν is reset to νspike < 0. [sent-109, score-0.445]

59 The reason is that the length of hyperpolarization determines the time window where signiﬁcant learning can take place. [sent-110, score-0.181]

60 2s, so that the postsynaptic neuron can learn to spike almost anywhere over the time window, and we introduce postsynaptic potentials (PSP) with a ﬁnite rise time: ˙ τs Isyn = −Isyn + wi xi (t − τa ), (17) i where wi denotes the synaptic weight of presynaptic neuron i. [sent-113, score-1.871]

61 With these changes, the membrane potential is governed by ˙ τU U = (ν − U ) + Isyn (t − τd ). [sent-116, score-0.213]

62 (18) A postsynaptic spike resets U to νspike = Ureset < 0. [sent-117, score-0.725]

63 τU sets the time scale for the summation of EPSP contributing to spurious spikes, τν sets the time window where the desired spikes can lie. [sent-121, score-0.209]

64 5 Figure 2: Illustration of Perceptron learning with RSTDP with subthreshold LTD and postsynaptic hyperpolarization. [sent-123, score-0.515]

65 Pre- and postsynaptic spikes are displayed as ¯ ¯ black bars at t = 0. [sent-125, score-0.534]

66 Initially the weights are too low and the synaptic current (summed PSPs) is smaller than Ust . [sent-129, score-0.189]

67 Weight change is LTP only until during pattern presentation the membrane potential hits Ust . [sent-130, score-0.285]

68 Shown are the same traces as in A at the absence of an inital postsynaptic spike. [sent-133, score-0.46]

69 The membrane potential after learning is drawn as a dashed line to highlight the amplitude. [sent-134, score-0.213]

70 Without the initial hyperpolarization, the synaptic current after learning is large enough to cross the spiking threshold, the postsynaptic neuron ﬁres the desired spike. [sent-135, score-0.85]

71 Learning until Ust is reached ensures a minimum height of synaptic currents and therefore robustness against noise. [sent-136, score-0.233]

72 Initially, the synaptic current during pattern presentation causes a spike and consequently LTD. [sent-138, score-0.57]

73 Learning stops when the membrane potential stays below Ust . [sent-139, score-0.213]

74 Shown is the fraction of pattern which elicit the desired postsynaptic activity upon presentation. [sent-143, score-0.527]

75 Shown is the fraction of pattern which during recall succeed in producing the correct postsynaptic spike time in a window of length 30 ms after the teacher spike. [sent-150, score-0.906]

76 1 Learning performance We test the performance of networks of N input neurons at classifying spatio-temporal spike patterns by generating P = αN patterns, which we repeatedly present to the network. [sent-155, score-0.467]

77 In each pattern, each presynaptic neuron spikes exactly once at a ﬁxed time in each presentation, with spike times uniformly distributed over the trial. [sent-156, score-0.77]

78 After each block, we test if the postsynaptic output matches the desired activity for each pattern. [sent-160, score-0.497]

79 If during training a postsynaptic spike at t = 0 was induced, the output can lie anytime in the testing trial for a positive outcome. [sent-161, score-0.725]

80 To test scaling of the capacity, we generate networks of 100 to 600 neurons and present the patterns until the classiﬁcation error reaches a plateau. [sent-162, score-0.158]

81 2 Chronotron learning with RSTDP In the Chronotron [17] input spike patterns become associated with desired spike trains. [sent-180, score-0.732]

82 The plasticity mechanism presented here has the tendency to generate postsynaptic spikes as close in time as possible to the teacher spike during recall. [sent-182, score-1.19]

83 The average distance of these 7 spikes depends on the time constants of hyperpolarization and the learning window, especially τpost . [sent-184, score-0.244]

84 the ability to generate the desired spike times within a short window in time, is shown in Fig. [sent-188, score-0.4]

85 Inspection of the spike times reveals that the average distance of output spikes to the respective teacher spike is much shorter than the learning window (≈ 2ms for α = 0. [sent-192, score-0.862]

86 This provides a biologically plausible mechanism to build associative memories with a capacity close to the theoretical maximum. [sent-197, score-0.267]

87 The mechanism proposed here is complementary to a previous approach [13] which uses CSTDP in combination with spike frequency adaptation to perform gradient descent learning on a squared error. [sent-200, score-0.35]

88 For sparse spatio-temporal spike patterns extensive simulations show that the same mechanism is able to learn Tempotrons and Chronotrons with substantial memory capacity. [sent-203, score-0.428]

89 However, in the case of the Chronotron the capacity comes close to the one obtained with a commonly employed, supervised spike time learning rule. [sent-205, score-0.376]

90 A prototypical example for such a supervised learning rule is the Temptron rule proposed by G¨ tig and Sompolinski [5]. [sent-207, score-0.186]

91 Essentially, after a pattern presentation the complete u time course of the membrane potential during the presentation is examined, and if classiﬁcation was erroneous, the synaptic weights which contributed most to the absolute maximum of the potential are changed. [sent-208, score-0.57]

92 In other words, the neurons would have to able to retrospectivly disentangle contributions to their membrane potential at a certain time in the past. [sent-209, score-0.269]

93 As we showed here, RSTDP with subthreshold LTD together with postsynaptic hyperpolarization for the ﬁrst time provides a realistic mechanism for Tempotron and Chronotron learning. [sent-210, score-0.702]

94 After-hyperpolarization allows synaptic potentiation for presynaptic inputs immediately after the teacher spike without causing additional non-teacher spikes, which would be detrimental for learning. [sent-214, score-0.815]

95 During recall, the absence of the hyperpolarization ensures the then desired threshold crossing of the membrane potential (see Fig. [sent-215, score-0.499]

96 It counteracts synaptic potentiation when the membrane potential becomes sufﬁciently high after the teacher spike. [sent-218, score-0.516]

97 Taken together, our results show that the interplay of neuronal dynamics and synaptic plasticity rules can give rise to powerful learning dynamics. [sent-220, score-0.496]

98 [5] G¨ tig R, Sompolinsky H (2006) The Tempotron: a neuron that learns spike timing-based decisions. [sent-231, score-0.477]

99 [9] Froemke RC, Poo MM, Dan Y (2005) Spike-timing-dependent synaptic plasticity depends on dendritic location. [sent-239, score-0.424]

100 [14] Song S, Miller KD, Abbott LF (2000) Competitive Hebbian learning through spike-timing-dependent synaptic plasticity. [sent-248, score-0.189]

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