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249 nips-2011-Sequence learning with hidden units in spiking neural networks

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Author: Johanni Brea, Walter Senn, Jean-pascal Pfister

Abstract: We consider a statistical framework in which recurrent networks of spiking neurons learn to generate spatio-temporal spike patterns. Given biologically realistic stochastic neuronal dynamics we derive a tractable learning rule for the synaptic weights towards hidden and visible neurons that leads to optimal recall of the training sequences. We show that learning synaptic weights towards hidden neurons signiﬁcantly improves the storing capacity of the network. Furthermore, we derive an approximate online learning rule and show that our learning rule is consistent with Spike-Timing Dependent Plasticity in that if a presynaptic spike shortly precedes a postynaptic spike, potentiation is induced and otherwise depression is elicited.

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sentIndex sentText sentNum sentScore

1 Sequence learning with hidden units in spiking neural networks Johanni Brea, Walter Senn and Jean-Pascal Pﬁster Department of Physiology University of Bern B¨ hlplatz 5 u CH-3012 Bern, Switzerland {brea, senn, pfister}@pyl. [sent-1, score-0.521]

2 ch Abstract We consider a statistical framework in which recurrent networks of spiking neurons learn to generate spatio-temporal spike patterns. [sent-3, score-0.788]

3 Given biologically realistic stochastic neuronal dynamics we derive a tractable learning rule for the synaptic weights towards hidden and visible neurons that leads to optimal recall of the training sequences. [sent-4, score-1.92]

4 We show that learning synaptic weights towards hidden neurons signiﬁcantly improves the storing capacity of the network. [sent-5, score-1.06]

5 Furthermore, we derive an approximate online learning rule and show that our learning rule is consistent with Spike-Timing Dependent Plasticity in that if a presynaptic spike shortly precedes a postynaptic spike, potentiation is induced and otherwise depression is elicited. [sent-6, score-0.705]

6 1 Introduction Learning to produce temporal sequences is a general problem that the brain needs to solve. [sent-7, score-0.185]

7 Early attempts to model sequence learning used a simple asymmetric Hebbian learning rule [10, 20, 6] and succeeded to store sequences of random patterns, but perform poorly as soon as there are temporal correlations between the patterns [3]. [sent-9, score-0.511]

8 Other studies [14] included a reservoir of hidden neurons but assumed weights towards the hidden neurons to be ﬁxed. [sent-11, score-1.957]

9 Here we start by deﬁning a stochastic neuronal dynamics - that can be arbitrarily complicated (e. [sent-13, score-0.159]

10 This stochastic dynamics deﬁnes the overall probability distribution which is parametrized by the synaptic weights. [sent-16, score-0.181]

11 The goal of learning is to adapt synaptic weights such that the model distribution approximates as good as possible the target distribution of temporal sequences. [sent-17, score-0.274]

12 This can be seen as the extension of the maximum likelihood approach of Barber [2] where we add stochastic hidden neurons with plastic weights. [sent-18, score-0.849]

13 1 A B ht−1 ht stochastic hidden neurons ht−1 ht vt−1 vt stochastic visible neurons vt−1 vt Figure 1: Graphical representation of the conditional dependencies of the joint distribution over visible and hidden sequences. [sent-20, score-2.984]

14 A Graphical model used for the derivation of the learning rule in section 2 and the example in section 4. [sent-21, score-0.173]

15 B Markovian model used in the example with binary neurons in section 3. [sent-22, score-0.49]

16 The resulting learning rule is local (but modulated by a global factor), causal and biologically relevant in the sense that it shares important features with Spike-Timing Dependent Plasticity (STDP). [sent-23, score-0.252]

17 We also derive an online version of the learning rule and show numerically that it performs almost equally well as the exact batch learning rule. [sent-24, score-0.277]

18 2 Learning a distribution of sequences Let us consider temporal sequences v = {vt,i |t = 0 . [sent-25, score-0.361]

19 Nv } of Nv visible neurons over the interval [0, T ]. [sent-31, score-0.865]

20 from a target distribution P ∗ (v) that must be learned by a model which consists of Nv visible neurons and Nh hidden neurons. [sent-46, score-1.261]

21 The model distribution over those visible sequences is denoted by Pθ (v) = h Pθ (v, h) where θ denotes the model parameters, h = {ht,i |t = 0 . [sent-47, score-0.551]

22 Nh } the hidden temporal sequence and Pθ (v, h) the joint distribution over the visible and the hidden sequences. [sent-53, score-1.153]

23 (2), it is possible to approximate it by sampling the visible sequences v from the target distribution P ∗ (v) and the hidden sequences from the posterior distribution Pθ (h|v) given the visible ones. [sent-60, score-1.493]

24 Indeed, at a time t the posterior distribution over ht does not only depend on the past visible activity but also on the future visible activity, since it is conditioned on the whole visible activity v0:T from time step 0 to T . [sent-62, score-1.633]

25 We exploit that in all neuronal network models of interest, neuronal ﬁring at any time point is conditionally independent given the past activity of the network. [sent-69, score-0.287]

26 Using the chain rule this means that we can write the joint distribution Pθ (v, h) (see Fig. [sent-70, score-0.207]

27 The sampling can be accomplished by clamping the visible neurons to a target sequence v and let the hidden dynamics run, i. [sent-72, score-1.374]

28 at time t, ht is sampled from Pθ (ht |v0:t−1 h0:t−1 ). [sent-74, score-0.218]

30 Note that in the absence of hidden neurons, this factor γθ (v, h) is equal to one and the maximum likelihood learning rule [2, 18] is recovered. [sent-82, score-0.498]

31 N do h ∼ Qθ (h|v) α(v) ← α(v) + Rθ (v|h) ∂ log Pθ (v,h) ∂θ Pθ (v) ← Pθ (v) + N −1 Rθ (v|h) end for α(v) θ ← θ + η Pθ (v) end while return θ 3 B C G 10 20 30 10 20 30 time step D 10 20 30 time step E 10 20 30 time step perform ance unit number A 1. [sent-89, score-0.24]

32 A The target distribution contained only this training pattern for 30 visible neurons and 30 time steps. [sent-96, score-1.09]

33 B-F, H-J Overlay of 20 recalls after learning with 15 000 training pattern presentations, B with only visible neurons and a simple asymmetric Hebb rule (see main text) C only visible neurons and learning rule Eq. [sent-97, score-2.251]

34 (5) D static weights towards 30 hidden neurons (Reservoir Computing) E learning rule Eq. [sent-98, score-1.239]

35 G Learning curves for the training pattern in A for only visible neurons (black line), static weights towards hidden (blue line), online learning approximation (purple line) exact learning rule (red line). [sent-101, score-1.799]

36 The performance was measured in one minus average Hamming distance per neuron per time step (see main text). [sent-102, score-0.191]

37 I Recall with a network of 30 visible and 10 hidden neurons without learning the weights towards hidden neurons. [sent-104, score-1.706]

38 J Recall after training the same network with learning rule Eq. [sent-105, score-0.252]

39 3 Binary neurons In order to illustrate the learning rule given by Eq. [sent-107, score-0.663]

40 Let x denote the activity of the visible and hidden neurons, i. [sent-109, score-0.776]

41 For the distribution over the initial conditions Pθ (v0 ) and Pθ (h0 ) we choose delta distributions such that v0 is equal to the ﬁrst state of the training sequence and h0 is an arbitrary but ﬁxed vector of binary values. [sent-123, score-0.185]

42 If we assume that the weights wij are the only adaptable parameters in this model, 4 A B 1. [sent-124, score-0.294]

43 5 20 40 60 80 100 20 40 60 80 100 seq u en ce len gth number of hidden units Figure 3: Adding trainable hidden neurons leads to much better recall performance than having static hidden neurons or no hidden neurons at all. [sent-136, score-2.985]

44 A Comparison of the performance after 20000 learning cycles between static (blue curve) and dynamic weights (red curve) towards hidden neurons for a network with 30 visible and different numbers of hidden neurons in a training task with a uncorrelated random pattern of length 60 time steps. [sent-137, score-2.497]

45 For B we generated random, uncorrelated sequences of different length and compared the performance after 20000 learning cycles for only visible neurons (black curve), static weights towards hidden (blue curve) and dynamic weights towards hidden (red curve). [sent-138, score-2.134]

46 (3) and (6) we ﬁnd ∂ log Pw (x) β = ∂wij 2 T (xt,i − xt,i Pθ (xt,i |xt−1 ) )xt−1,j , (8) t=1 where xt,i Pθ (xt,i |xt−1 ) = g(ut,i )δt − (1 − g(ut,i )δt) and the indices i and j run over all visible and hidden neurons. [sent-141, score-0.733]

47 2) where the distribution over sequences is a delta distribution P ∗ (v) = δ(v − v ∗ ) around a single pattern v ∗ (Fig. [sent-144, score-0.324]

48 a non-Markovian pattern, thus making it a dift=0 T ∗ ∗ ﬁcult pattern to learn with a simple asymmetric Hebb rule ∆wij ∝ t=0 vt+1,i vt,j (Fig. [sent-147, score-0.295]

49 The performance was measured ∗ by one minus the Hamming distance per visible neuron and time step 1−(T Nv )−1 t,i |vt,i −vt,i |/2 between target pattern and recall pattern averaged over 100 runs. [sent-150, score-0.792]

50 Adding hidden neurons without learning the weights towards hidden neurons is similar to the idea used in the framework of Reservoir Computing (for a review see [13]): the visible states feed a ﬁxed reservoir of neurons that returns a non-linear transformation of the input. [sent-151, score-2.847]

51 Only the readout from hidden to visible neurons and in our case the recurrent connections in the visible layer are trained. [sent-152, score-1.614]

52 To assure a sensible distribution of weights towards hidden units, we used the weights that were obtained after learning with Eq. [sent-153, score-0.604]

53 Obviously, without training the reservoir the performance is always worse compared to a system with an equal number of hidden neurons but dynamic weights (Fig. [sent-155, score-1.11]

54 With only a few hidden neurons our rule is also capable to learn patterns where the visible neurons are silent during a few time-steps. [sent-157, score-1.9]

55 After learning the weights towards 10 hidden neurons with learning rule Eq. [sent-160, score-1.153]

56 2) or static weights towards hidden neurons the time gap was not learned (see Fig. [sent-164, score-1.124]

57 5 w arbitrary units 0 40 20 0 20 t post t pre ms 40 Figure 4: The learning rule Eq. [sent-166, score-0.206]

58 (11) is compatible with Spike-Timing Dependent Plasticity (STDP): the weight gets potentiated if a presynaptic spike is followed by a postsynaptic spike and depressed otherwise. [sent-167, score-0.406]

59 3 we used again delta target distributions P ∗ (v) = δ(v − v ∗ ) with random uncorrelated patterns v ∗ of different length. [sent-170, score-0.168]

60 For a pattern of length 2Nv = 60 only Nv /2 = 15 trainable hidden neurons are sufﬁcient to reach perfect recall (see Fig. [sent-172, score-0.978]

61 This is in clear contrast to the case of static hidden weights. [sent-174, score-0.411]

62 Again the static weights were obtained by reshufﬂing those that we obtained after learning with Eq. [sent-175, score-0.166]

63 3B compares the capacity of our learning rule with Nh = Nv = 30 hidden neurons to the case of no hidden neuron or static weights towards hidden neurons. [sent-178, score-2.017]

64 Without learning the weights towards hidden neurons the performance drops to almost chance level for sequences of 45 or more time steps, whereas with our learning rule this decrease of performance occurs only at sequences of 100 or more time steps. [sent-179, score-1.505]

65 4 Limit to Continuous Time Starting from the neurons in the last section we show that in the limit to continuous time we can implement the sequence learning task with stochastic spiking neurons [7]. [sent-180, score-1.239]

66 First note that the state of a neuron at time t in the model described in the previous section is fully deﬁned by ut,i := j wij xt−1,j (see Eq. [sent-181, score-0.342]

67 The weighted sum j wij xt−1,j is the response of neuron i to the spikes of its presynaptic neurons and its own spikes. [sent-183, score-0.985]

68 In a more realistic model the postsynaptic neuron feels the inﬂuence of presynaptic spikes through a perturbation of the membrane potential on the order of a few milliseconds, which in the limit to continuous time clearly cannot be modeled by a one-time step response. [sent-185, score-0.486]

69 (6) by ∞ ut,i = ∞ κs xt−s,i + s=1 j=i =:xκ t,i ǫs xt−s,j , wij (10) s=1 =:xǫ t,j where xt−s,i ∈ {0, 1}. [sent-187, score-0.214]

70 The kernel ǫ models the time-course of the response to a presynaptic spike and κ the refractoriness. [sent-188, score-0.262]

71 (9) we note that we can scale Rw (v|h) without changing the learning rule Eq. [sent-191, score-0.173]

72 We use the scaling Rw (v|h) → Rw (v|h) := (g0 δt)−Sv Rw (v|h), where Sv denotes the total number of spikes T Nv in the visible sequence v, i. [sent-193, score-0.47]

73 With neuron i’s response to past spiking activity ui (t) = xκ (t) + j=i wij xǫ (t) and the escape rate function ρi (t) = g0 exp (βui (t)) we j i recovered the deﬁning equations of a simpliﬁed stochastic spike response model [7]. [sent-200, score-0.827]

74 4 we display the weight change after forcing two neurons to ﬁre with a ﬁxed time lag. [sent-202, score-0.524]

75 Our learning rule is consistent with STDP in the sense that a presynaptic spike followed by a postsynaptic spike leads to potentiation and to depression otherwise. [sent-204, score-0.647]

76 5 Approximate online version Without hidden neurons the learning rule found by using Eq. [sent-206, score-1.053]

77 (11) is straightforward to implement in an online way where the parameters are updated at every moment in time according to wij ∝ ˙ (xi (t) − ρi (t))xǫ (t) instead of waiting with the update until a training batch ﬁnished. [sent-207, score-0.405]

78 Finding j an online version of the learning algorithm for networks with hidden neurons turns out to be a challenge, since we need to know the whole sequences v and h in order to evaluate the importance factor Rθ (v|h)/ Rθ (v|h′ ) Qθ (h′ |v) . [sent-208, score-1.095]

79 Here we propose to use in each time step an approximation of the importance factor based on the network dynamics during the preceding period of typical sequence length and multiply it by the low-pass ﬁltered change of parameters. [sent-209, score-0.259]

80 To ﬁnd an online estimate of Rθ (v, h′ ) Qθ (h′ |v) we assume that a training pattern v ∼ P ∗ (v) is presented a few times in a row and after time N T , with N ∈ N, N ≫ 1, a new training pattern is picked from the training distribution. [sent-216, score-0.392]

81 Under this assumption we can replace the average over 7 hidden sequences by a low-pass ﬁlter of r with time constant N T , see Eq. [sent-217, score-0.501]

82 during the time interval [0, τ ), with τ on the order of the kernel time constant τm - the hidden activity h(s) is drawn from a given distribution P (h(s)). [sent-221, score-0.503]

83 2A) in section 3, the performance of the online rule is close to the one of the batch rule (Fig. [sent-225, score-0.45]

84 6 Discussion Learning long and temporally correlated sequences with neural networks is a difﬁcult task. [sent-227, score-0.239]

85 In this paper we suggested a statistical model with hidden neurons and derived a learning rule that leads to optimal recall of the learned sequences given the neuronal dynamics. [sent-228, score-1.243]

86 The learning rule is derived by minimizing the Kullback-Leibler divergence from training distribution to model distribution with a variant of the EM-algorithm, where we use importance sampling to draw hidden sequences given the visible training sequence. [sent-229, score-1.296]

87 Choosing an appropriate distribution in the importance sampling step we are able to circumvent inference which usually makes the training of non-Markovian models hard. [sent-230, score-0.197]

88 We showed that it is ready to be implemented with biologically realistic neurons and that an approximate online version exists. [sent-232, score-0.634]

89 Our approach follows the ideas outlined in [2], where sequence learning was considered with visible neurons. [sent-233, score-0.426]

90 Here we extended this model by adding stochastic hidden neurons that help to perform well with sequences of linearly depend states - including non-Markovian sequences - or long sequences. [sent-234, score-1.158]

91 As in [18] we look at the limit of continuous time and ﬁnd that the learning rule is consistent with Spike-Timing Dependent Plasticity. [sent-235, score-0.244]

92 In contrast to Reservoir Computing [13] we train the weights towards hidden neurons which clearly helps to improve performance. [sent-236, score-0.98]

93 Our learning rule does not need a “wake” and a “sleep” phase as we know it from Boltzmann machines [1, 8]. [sent-237, score-0.198]

94 Viewed in a different light our learning algorithm has a nice interpretation: as in reinforcement learning, the hidden neurons explore different sequences, where each trial leads to a global reward signal that modulates the weight change. [sent-238, score-0.815]

95 However, in contrast to common reinforcement learning the reward is not provided by an external teacher but depends solely on the internal dynamics and the visible neurons do not explore but are clamped to the training sequence. [sent-239, score-0.985]

96 To make our model even more biologically relevant, future work should aim for a biological implementation of the global importance factor that depends on the spike timing and the membrane potential of all the visible neurons (see Eq. [sent-240, score-1.123]

97 Phase diagram and storage capacity of sequence processing u neural networks. [sent-279, score-0.188]

98 Matching storage and recall: hippocampal spike timingdependent plasticity and phase response curves. [sent-313, score-0.319]

99 Efﬁcient methods for sampling spike trains in networks of coupled neurons. [sent-340, score-0.175]

100 A tutorial on hidden Markov models and selected applications in speech recognition. [sent-352, score-0.325]

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