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166 nips-2001-Self-regulation Mechanism of Temporally Asymmetric Hebbian Plasticity

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Author: N. Matsumoto, M. Okada

Abstract: Recent biological experimental ﬁndings have shown that the synaptic plasticity depends on the relative timing of the pre- and postsynaptic spikes which determines whether Long Term Potentiation (LTP) occurs or Long Term Depression (LTD) does. The synaptic plasticity has been called “Temporally Asymmetric Hebbian plasticity (TAH)”. Many authors have numerically shown that spatiotemporal patterns can be stored in neural networks. However, the mathematical mechanism for storage of the spatio-temporal patterns is still unknown, especially the eﬀects of LTD. In this paper, we employ a simple neural network model and show that interference of LTP and LTD disappears in a sparse coding scheme. On the other hand, it is known that the covariance learning is indispensable for storing sparse patterns. We also show that TAH qualitatively has the same eﬀect as the covariance learning when spatio-temporal patterns are embedded in the network. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 jp Abstract Recent biological experimental ﬁndings have shown that the synaptic plasticity depends on the relative timing of the pre- and postsynaptic spikes which determines whether Long Term Potentiation (LTP) occurs or Long Term Depression (LTD) does. [sent-7, score-0.411]

2 The synaptic plasticity has been called “Temporally Asymmetric Hebbian plasticity (TAH)”. [sent-8, score-0.302]

3 Many authors have numerically shown that spatiotemporal patterns can be stored in neural networks. [sent-9, score-0.209]

4 However, the mathematical mechanism for storage of the spatio-temporal patterns is still unknown, especially the eﬀects of LTD. [sent-10, score-0.331]

5 In this paper, we employ a simple neural network model and show that interference of LTP and LTD disappears in a sparse coding scheme. [sent-11, score-0.344]

6 On the other hand, it is known that the covariance learning is indispensable for storing sparse patterns. [sent-12, score-0.297]

7 We also show that TAH qualitatively has the same eﬀect as the covariance learning when spatio-temporal patterns are embedded in the network. [sent-13, score-0.239]

8 1 Introduction Recent biological experimental ﬁndings have indicated that the synaptic plasticity depends on the relative timing of the pre- and post- synaptic spikes which determines whether Long Term Potentiation (LTP) occurs or Long Term Depression (LTD) does [1, 2, 3]. [sent-14, score-0.474]

9 Many authors have numerically shown that spatio-temporal patterns can be stored in neural networks [6, 7, 8, 9, 10, 11]. [sent-19, score-0.209]

10 discussed the variablity of spike generation about the network consisting of spiking neurons using TAH [6]. [sent-21, score-0.307]

11 Yoshioka also discussed the associative memory network consisting of spiking neurons using TAH [11]. [sent-24, score-0.383]

12 Munro and Hernandez numerically showed that a network can retrieve spatio-temporal patterns even in a noisy environment owing to LTD [9]. [sent-26, score-0.328]

13 However, they did not discuss the reason why TAH was eﬀective in terms of the storage and retrieval of the spatio-temporal patterns. [sent-27, score-0.298]

14 Since TAH has not only the eﬀect of LTP but that of LTD, the interference of LTP and LTD may prevent retrieval of the patterns. [sent-28, score-0.188]

15 To investigate this unknown mathematical mechanism for retrieval, we employ an associative memory network consisting of binary neurons. [sent-29, score-0.367]

16 We show the mechanism that the spatio-temporal patterns can be retrieved in this network. [sent-32, score-0.223]

17 There are many works concerned with associative memory networks that store spatio-temporal patterns by the covariance learning [12, 13]. [sent-33, score-0.414]

18 It is wellknown that the covariance learning is indispensable when the sparse patterns are embedded in a network as attractors [15, 16]. [sent-35, score-0.544]

19 The information on the ﬁring rate for the stored patterns is not indispensable for TAH, although it is indispensable for the covariance learning. [sent-36, score-0.619]

20 We theoretically show that TAH qualitatively has the same eﬀect as the covariance learning when the spatio-temporal patterns are embedded in the network. [sent-37, score-0.239]

21 This means that the diﬀerence in spike times induces LTP or LTD, and the eﬀect of the ﬁring rate information can be canceled out by this spike time diﬀerence. [sent-38, score-0.381]

22 We also use discrete time steps and the following synchronous updating rule, N ui (t) = Jij xj (t), (1) j=1 xi (t + 1) = Θ(ui (t) − θ), Θ(u) = 1, u ≥ 0 0, u < 0, (2) (3) where xi (t) is the state of the i-th neuron at time t, ui (t) its internal potential, and θ a uniform threshold. [sent-43, score-0.519]

23 If the i-th neuron ﬁres at time t, its state is xi (t) = 1; otherwise, xi (t) = 0. [sent-44, score-0.218]

24 Jij is µ the synaptic weight from the j-th neuron to the i-th neuron. [sent-46, score-0.196]

25 Each element ξi of µ µ µ µ the µ-th memory pattern ξ = (ξ1 , ξ2 , · · ·, ξ N ) is generated independently by, µ µ µ Prob[ξi = 1] = 1 − Prob[ξi = 0] = f. [sent-47, score-0.192]

26 µ E[ξi ] (4) The expectation of ξ is = f, and thus, f can be considered as the mean ﬁring rate of the memory pattern. [sent-48, score-0.226]

27 The memory pattern is “sparse” when f → 0, and this coding scheme is called “sparse coding”. [sent-49, score-0.235]

28 The synaptic weight Jij follows the synaptic plasticity that depends on the diﬀerence in spike times between the i-th (post-) and j-th (pre-) neurons. [sent-50, score-0.437]

29 Figure 1(b) shows that LTP occurs when the j-th neuron ﬁres one time step before the i-th neuron µ+1 µ does, ξi = ξj = 1, and that LTD occurs when the j-th neuron ﬁres one time step µ−1 µ after the i-th neuron does, ξi = ξj = 1. [sent-55, score-0.402]

30 Jij = N f(1 − f) µ=1 i (5) The number of memory patterns is p = αN where α is deﬁned as the “loading rate”. [sent-64, score-0.211]

31 If the loading rate is larger than αC , the pattern sequence becomes unstable. [sent-66, score-0.325]

32 We show that p memory patterns are retrieved periodically like ξ1 → ξ 2 → · · · → ξ p → ξ 1 → · · ·. [sent-69, score-0.283]

33 One candidate algorithm for controlling the threshold value is to maintain the mean ﬁring rate of the network at that of memory pattern, f, as follows, N N 1 1 xi (t) = Θ(ui (t) − θ(t)). [sent-73, score-0.508]

34 (6) f = N N i=1 i=1 It is known that the obtained threshold value is nearly optimal, since it approximately gives a maximal storage capacity value [16]. [sent-74, score-0.458]

35 3 Theory Many neural network models that store and retrieve sequential patterns by TAH have been discussed by many authors [7, 8, 9, 10]. [sent-75, score-0.38]

36 For example, Munro and Hernandez showed that their model could retrieve a stored pattern sequence even in a noisy environment [9]. [sent-77, score-0.278]

37 Here, we discuss the mechanism that the network learned by TAH can store and retrieve sequential patterns. [sent-80, score-0.343]

38 Before providing details of the retrieval process, we discuss a simple situation where the number of memory patterns is very small relative to the number of neurons, i. [sent-81, score-0.329]

39 Then, the internal potential u i (t) of the equation (1) is given by, t+1 t−1 ui (t) = ξi − ξi . [sent-85, score-0.196]

40 The ﬁrst term ξi of the equation (7) is a signal term for the recall of the pattern ξt+1 , which is designed to be retrieved at time t+1, and the second term t−1 ξi can interfere in retrieval of ξt+1 . [sent-87, score-0.263]

41 If the t+1 threshold θ(t) is set between 0 and +1, ξi = 0 isn’t inﬂuenced by the interference t−1 t+1 t−1 of ξi = 1. [sent-90, score-0.215]

42 We consider the probability distribution of the internal potential ui (t) to examine how the interference of LTD inﬂuences the retrieval of ξt+1 . [sent-92, score-0.383]

43 (8) Since the threshold θ(t) is set between 0 and +1, the state xi (t + 1) is 1 with probability f − f 2 and 0 with 1 − f + f 2 . [sent-95, score-0.193]

44 The overlap between the state x(t + 1) and the memory pattern ξ t+1 is given by, mt+1 (t + 1) = 1 N f(1 − f) N i=1 t+1 (ξi − f)xi (t + 1) = 1 − f. [sent-96, score-0.333]

45 This means that the interference of LTD disappears in a sparse limit, and the model can retrieve the next pattern ξt+1 . [sent-98, score-0.4]

46 Next, we discuss whether the information on the ﬁring rate is indispensable for TAH or not. [sent-100, score-0.277]

47 To investigate this, we consider the case that the number of memory patterns is extensively large, i. [sent-101, score-0.238]

48 Using the equation (9), the internal potential ui (t) of the i-th neuron at time t is represented as, t+1 t−1 ui (t) = (ξi − ξi )mt (t) + zi (t), (10) p zi (t) = µ=t µ+1 µ−1 (ξi − ξi )mµ (t). [sent-104, score-0.47]

49 (11) zi (t) is called the “cross-talk noise”, which represents contributions from non-target patterns excluding ξt−1 and prevents the target pattern ξt+1 from being retrieved. [sent-105, score-0.212]

50 It is well-known that the covariance learning is indispensable when the sparse patterns are embedded in a network as attractors [15, 16]. [sent-107, score-0.544]

51 Under sparse coding schemes, unless the covariance learning is employed, the cross-talk noise does diverge in the large N limit. [sent-108, score-0.271]

52 The information on the ﬁring rate for the stored patterns is not indispensable for TAH, although it is indispensable for the covariance learning. [sent-110, score-0.619]

53 If a pattern sequence can be stored, the cross-talk noise is obeyed by a Gaussian distribution with mean 0 and time-dependent variance σ 2 (t). [sent-112, score-0.189]

54 According to the statistical neurodynamics, we obtain the recursive equations for the overlap mt (t) between the network state x(t) and the target pattern ξt and the variance σ2 (t). [sent-115, score-0.502]

55 2σ(t − 1) 2σ(t − 1) 2σ(t − 1) These equations reveal that the variance σ2 (t) of cross-talk noise does not diverge as long as a pattern sequence can be retrieved. [sent-122, score-0.241]

56 Therefore, the variance of cross-talk noise doesn’t diverge, and this is another factor for the network learned by TAH to store and retrieve a pattern sequence. [sent-127, score-0.388]

57 We conclude that the diﬀerence in spike times induces LTP or LTD, and the eﬀect of the ﬁring rate information can be canceled out by this spike times diﬀerence. [sent-128, score-0.381]

58 4 Results We investigate the property of our model and examine the following two conditions: a ﬁxed threshold and a time-dependent threshold, using the statistical neurodynamics and computer simulations. [sent-129, score-0.231]

59 overlap (solid), activity/f (dashed) Figure 2 shows how the overlap mt (t) and the mean ﬁring rate of the network, 1 x(t) = N i xi (t), depend on the loading rate α when the mean ﬁring rate of ¯ the memory pattern is f = 0. [sent-130, score-1.055]

60 52, where the storage capacity is maximum with respect to the threshold θ. [sent-132, score-0.458]

61 The stored pattern sequence can be retrieved when the initial overlap m1 (1) is greater than the critical value mC . [sent-133, score-0.398]

62 The lower line indicates how the critical initial overlap m C depends on the loading rate α. [sent-134, score-0.403]

63 In other words, the lower line represents the basin of attraction for the retrieved sequence. [sent-135, score-0.235]

64 The upper line denotes a steady value of overlap mt (t) when the pattern sequence is retrieved. [sent-136, score-0.399]

65 mt (t) is obtained by setting the initial state to the ﬁrst memory pattern: x(1) = ξ1 . [sent-137, score-0.287]

66 The dashed line shows a steady value of the normalized mean ﬁring rate of network, x(t)/f, for the pattern sequence. [sent-140, score-0.263]

67 The critical overlap (the lower line) and the overlap at the stationary state (the upper line). [sent-156, score-0.291]

68 The dashed line shows the mean ﬁring rate of the network divided ﬁring rate which is 0. [sent-157, score-0.355]

69 overlap (solid), activity/f (dashed) Next, we examine the threshold control scheme in the equation (6), where the threshold is controlled to maintain the mean ﬁring rate of the network at f. [sent-165, score-0.618]

70 q(t) 1 in equation (15) is equal to the mean ﬁring rate because q(t) = N N (xi (t))2 = i=1 N 1 i=1 xi (t) under the condition xi (t) = {0, 1}. [sent-166, score-0.256]

71 (17) 2 Figure 3 shows the overlap mt (t) as a function of loading rate α with f = 0. [sent-168, score-0.469]

72 The basin of attraction becomes larger than that of the ﬁxed threshold condition, θ = 0. [sent-172, score-0.232]

73 This means that even if the initial state x(1) is diﬀerent from the ﬁrst memory pattern ξ1 , that is, the state includes a lot of noise, the pattern sequence can be retrieved. [sent-175, score-0.365]

74 The critical overlap (the lower line) and the overlap at the stationary state (the upper line) when the threshold is changing over time to maintain mean ﬁring rate of the network at f. [sent-189, score-0.633]

75 The dashed line shows the mean ﬁring rates of the network divided ﬁring rate which is 0. [sent-190, score-0.266]

76 The basin of attraction become larger than that of the ﬁxed threshold condition: Figure 2. [sent-192, score-0.232]

77 Finally, we discuss how the storage capacity depends on the ﬁring rate f of the 1 memory pattern. [sent-193, score-0.591]

78 It is known that the storage capacity diverges as f | log f | in a sparse limit, f → 0 [19, 20]. [sent-194, score-0.489]

79 Therefore, we investigate the asymptotic property of the storage capacity in a sparse limit. [sent-195, score-0.462]

80 Figure 4 shows how the storage capacity depends on the ﬁring rate where the threshold is controlled to maintain the network 1 activity at f (symbol ◦). [sent-196, score-0.668]

81 The storage capacity diverges as f | log f | in a sparse limit. [sent-197, score-0.489]

82 The storage capacity as a function of f in the case of maintaining activity at f (symbol ◦). [sent-202, score-0.352]

83 Ths 1 storage capacity diverges as f | log f | in a sparse limit. [sent-203, score-0.489]

84 45 1/|log f| 5 Discussion Using a simple neural network model, we have discussed the mechanism that TAH enables the network to store and retrieve a pattern sequence. [sent-217, score-0.496]

85 First, we showed that the interference of LTP and LTD disappeared in a sparse coding scheme. [sent-218, score-0.269]

86 This is a factor to enable the network to store and retrieve a pattern sequence. [sent-219, score-0.334]

87 Next, we showed the mechanism that TAH qualitatively had the same eﬀect as the covariance learning by analyzing the stability of the stored pattern sequence and the retrieval process by means of the statistical neurodynamics. [sent-220, score-0.407]

88 Consequently, the variance of cross-talk noise didn’t diverge, and this is another factor for the network learned by TAH to store and retrieve a pattern sequence. [sent-221, score-0.388]

89 We conclude that the diﬀerence in spike times induces LTP or LTD, and the eﬀect of the ﬁring rate information can be canceled out by this spike times diﬀerence. [sent-222, score-0.381]

90 To improve the retrieval property of the basin of attraction, we introduced a threshold control algorithm where a threshold value was adjusted to maintain the mean ﬁring rate of the network at that of a memory pattern. [sent-224, score-0.701]

91 We also found that the loading rate diverged as f | log f | in a sparse limit, f → 0. [sent-226, score-0.299]

92 Here, we compare the storage capacity of our model with that of the model using the covariance learning (Figure 5). [sent-227, score-0.417]

93 We calculate the storage capacity αCOV from their dynamical equations and compare these of our model, C αT AH , by the ratio of α T AH /αCOV . [sent-229, score-0.379]

94 The contribution of LTD reduces the storage capacity of our model to half. [sent-233, score-0.352]

95 Therefore, in terms of the storage capacity, the covariance learning is better than TAH. [sent-234, score-0.245]

96 But, as we discussed previously, the information of the ﬁring rate is indispensable in TAH. [sent-235, score-0.265]

97 The comparison of the storage capacity of our model with that of the model using the covariance learning. [sent-242, score-0.417]

98 As f decreases, the ratio of storage capacity approaches 0. [sent-243, score-0.352]

99 Synaptic modiﬁcations in cultured hippocampal neurons: Dependence on spike timing, synaptic strength, and postsynaptic cell type. [sent-257, score-0.283]

100 Temporally asymmetric hebbian learning, spike timing and neuronal response variability. [sent-283, score-0.349]

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