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73 nips-2012-Coding efficiency and detectability of rate fluctuations with non-Poisson neuronal firing

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Author: Shinsuke Koyama

Abstract: Statistical features of neuronal spike trains are known to be non-Poisson. Here, we investigate the extent to which the non-Poissonian feature affects the efﬁciency of transmitting information on ﬂuctuating ﬁring rates. For this purpose, we introduce the Kullback-Leibler (KL) divergence as a measure of the efﬁciency of information encoding, and assume that spike trains are generated by time-rescaled renewal processes. We show that the KL divergence determines the lower bound of the degree of rate ﬂuctuations below which the temporal variation of the ﬁring rates is undetectable from sparse data. We also show that the KL divergence, as well as the lower bound, depends not only on the variability of spikes in terms of the coefﬁcient of variation, but also signiﬁcantly on the higher-order moments of interspike interval (ISI) distributions. We examine three speciﬁc models that are commonly used for describing the stochastic nature of spikes (the gamma, inverse Gaussian (IG) and lognormal ISI distributions), and ﬁnd that the time-rescaled renewal process with the IG distribution achieves the largest KL divergence, followed by the lognormal and gamma distributions.

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sentIndex sentText sentNum sentScore

1 Coding efﬁciency and detectability of rate ﬂuctuations with non-Poisson neuronal ﬁring Shinsuke Koyama∗ Department of Statistical Modeling The Institute of Statistical Mathematics 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan skoyama@ism. [sent-1, score-0.198]

2 jp Abstract Statistical features of neuronal spike trains are known to be non-Poisson. [sent-3, score-0.272]

3 For this purpose, we introduce the Kullback-Leibler (KL) divergence as a measure of the efﬁciency of information encoding, and assume that spike trains are generated by time-rescaled renewal processes. [sent-5, score-0.644]

4 We show that the KL divergence determines the lower bound of the degree of rate ﬂuctuations below which the temporal variation of the ﬁring rates is undetectable from sparse data. [sent-6, score-0.464]

5 We also show that the KL divergence, as well as the lower bound, depends not only on the variability of spikes in terms of the coefﬁcient of variation, but also signiﬁcantly on the higher-order moments of interspike interval (ISI) distributions. [sent-7, score-0.181]

6 1 Introduction Characterizing the statistical features of spike time sequences in the brain is important for understanding how the brain represents information about stimuli or actions in the sequences of spikes. [sent-9, score-0.221]

7 In the problem of estimating ﬁring rates, there is a minimum degree of rate ﬂuctuation below which a rate estimator cannot detect the temporal variation of the ﬁring rate [23]. [sent-15, score-0.48]

8 If, for instance, the degree of temporal variation of the rate is on the same order as that of the noise, a constant rate might be chosen as the most likely estimate for a given spike train. [sent-16, score-0.501]

9 It is, therefore, interesting to see how the minimum degree of rate ﬂuctuation depends on the non-Poissonian feature of spike trains. [sent-17, score-0.33]

10 In this study, we investigate the extent to which the non-Poissonian feature of spike trains affects the encoding efﬁciency of rate ﬂuctuations. [sent-18, score-0.423]

11 jp 1 tectability of rate ﬂuctuations depends on the encoding efﬁciency. [sent-21, score-0.169]

12 For this purpose, we introduce the Kullback-Leibler (KL) divergence to measure the encoding efﬁciency, and assume that spike sequences are generated by time-rescaled renewal processes. [sent-22, score-0.633]

13 With the aid of analytical and numerical studies, we suggest that the lower bound of detectable rate ﬂuctuations, below which the empirical Bayes decoder cannot detect the rate ﬂuctuations, is uniquely determined by the KL divergence. [sent-23, score-0.465]

14 We also ﬁnd that among the three ISI distributions, the IG distribution achieves the highest efﬁciency of coding information on rate ﬂuctuations. [sent-25, score-0.149]

15 2 Encoding rate ﬂuctuations using time-rescaled renewal processes Deﬁnitions of time-rescaled renewal processes and KL divergence We introduce time-rescaled renewal processes for a model of neuronal spike trains constructed in the ∞ following way. [sent-26, score-1.401]

16 , yn } independently from f κ (y), and arrange the ISIs sequentially to form a spike train of the unit rate; ith spike is given by summing the previous ISIs as s i = i yj . [sent-36, score-0.336]

17 This transformation ensures that the instantaneous ﬁring rate of {t i } corresponds to λ(t), 0 while the shape of the ISI distribution f κ (y), which characterizes the ﬁring irregularity, is unchanged in time. [sent-44, score-0.152]

18 This is in agreement with the empirical fact that the degree of irregularity in neuronal ﬁring is generally maintained in cortical processing [21, 22], while the ﬁring rate λ(t) changes in time. [sent-45, score-0.373]

19 We next introduce the KL divergence for measuring the encoding efﬁciency of ﬂuctuating rates. [sent-48, score-0.201]

20 (2) 0 Consider a probability density of a renewal process that has the same ISI density f κ (x) and the constant rate µ: n pκ ({ti }|µ) = n=1 µfκ (µ(ti − ti−1 )). [sent-50, score-0.465]

21 (3) The KL divergence between p κ ({ti }|{λ(t)}) and pκ ({ti }|µ) is, then, deﬁned as Dκ (λ(t)||µ) := ∞ 1 T →∞ T n=0 lim × log T 0 T t1 T ··· tn−1 pκ ({ti }|{λ(t)}) pκ ({ti }|{λ(t)}) dt1 dt2 · · · dtn . [sent-51, score-0.256]

22 pκ ({ti }|µ) (4) Since it is deﬁned as the entropy of a renewal process with the ﬂuctuating rate λ(t) relative to that with the constant rate µ, D κ (λ(t)||µ) can be interpreted as the amount of information on the rate ﬂuctuations encoded into spike trains. [sent-52, score-0.849]

23 (4) and further assuming ergodicity of spike trains, the KL divergence can be expressed as Dκ (λ(t)||µ) = lim n→∞ tn 1 pκ ({ti }|{λ(t)}) log − t0 pκ ({ti }|µ) 1 n→∞ tn − t0 = n log λ(ti ) + log fκ (Λ(ti ) − Λ(ti−1 )) lim i=1 − log µ − log fκ (µ(ti − ti−1 )) . [sent-56, score-0.852]

24 (5) This expression can be used for computing the KL divergence numerically by simulating a large number of spikes n 1. [sent-57, score-0.221]

25 Three ISI distributions and their KL divergence In order to examine the behavior of the KL divergence, we use the three speciﬁc ISI distributions for fκ (y) (the gamma, inverse Gaussian (IG) and lognormal distributions), which have been used to describe the stochastic nature of ISIs [9, 10, 14]. [sent-58, score-0.59]

26 The KL divergence for the three models is analytically solvable when the rate ﬂuctuation has a long time scale relative to the mean ISI. [sent-61, score-0.328]

27 Here, we show the derivation for the gamma distribution. [sent-62, score-0.191]

28 (The derivations for the IG and lognormal distributions are essentially the same. [sent-63, score-0.347]

29 (5) leads to Dκ (λ(t)||µ) = 1 n→∞ tn − t0 n log λ(ti ) + (κ − 1) log[Λ(ti ) − Λ(ti−1 )] lim i=1 − (κ − 1) log(ti − ti−1 ) − κµ log µ, tn t0 n λ(t)dt → µ and tn −t0 → µ as n → ∞. [sent-66, score-0.441]

30 By introducing the “averaged” ¯ ¯ ﬁring rate in the ith ISI: λi := Λ(ti )−Λ(ti−1 ) , we obtain log[Λ(ti ) − Λ(ti−1 )] = log λi + log(ti − ti −ti−1 ¯ ti−1 ). [sent-67, score-0.608]

31 Assuming that the time scale of the rate ﬂuctuation is longer than the mean ISI so that λi is approximated to λ(t i ), Eq. [sent-68, score-0.147]

32 (9) becomes where we used 1 tn −t0 (9) Dκ (λ(t)||µ) 1 n→∞ tn − t0 n = κ lim = κ lim T →∞ 1 T i=1 T 0 i log λ(ti ) − κµ log µ δ(t − ti ) log λ(t)dt − µ log µ . [sent-69, score-0.912]

33 (10) The ﬂuctuation in the apparent spike count is given by the variance to mean ratio as represented by the Fano factor [8]. [sent-70, score-0.168]

34 For the renewal process in which ISIs are drawn from a given distribution 2 function, it is proven that the Fano factor is related to the ISI variability with F ≈ C V [4]. [sent-71, score-0.291]

35 Thus, for a long range time scale in which a serial correlation of spikes is negligible, the spike train in Eq. [sent-72, score-0.227]

36 (10) can be approximated to n i=1 δ(t − ti ) ≈ λ(t) + 3 λ(t)/κξ(t), (11) where ξ(t) is a ﬂuctuating process such that ξ(t) = 0 and ξ(t)ξ(t ) = δ(t − t ). [sent-73, score-0.496]

37 Using this, the ﬁrst term on the rhs of (10) can be evaluated as 1 T →∞ T T lim 1 T →∞ T T λ(t) log λ(t)dt + lim 0 λ(t)/κ log λ(t)ξ(t)dt = λ log λ λ , (12) 0 where the second term on the lhs has vanished due to a property of stochastic integrals. [sent-74, score-0.232]

38 Therefore, the KL divergence of the gamma distribution is obtained as Dκ (λ(t)||µ) = κ λ log λ λ − µ log µ . [sent-75, score-0.441]

39 (13) In the same way, the KL divergence for the IG and lognormal distributions are, respectively, derived as 1 µ κ+1 Dκ (λ(t)||µ) = log µ − λ log λ λ + (λ − µ)2 λ , (14) 2 2 2µ and µ log µ 1 (log µ)2 − λ log λ λ + λ(log λ)2 λ . [sent-76, score-0.704]

40 Dκ (λ(t)||µ) = (15) Results We compute the KL divergence for the three models, in which the rate ﬂuctuates according to the Ornstein-Uhlenbeck process. [sent-78, score-0.311]

41 Formally, the rate process is given by λ(t) = [x(t)] + , where [·]+ is the rectiﬁcation function: [x]+ = x, 0, x>0 otherwise (16) and x(t) is derived from the Ornstein-Uhlenbeck process: dx(t) x(t) − µ =− +σ dt τ 2 ξ(t), τ (17) where ξ(t) is the Gaussian white noise. [sent-79, score-0.24]

42 Figure 1b depicts the KL divergence as a function of σ for C V =0. [sent-80, score-0.162]

43 The KL divergence for the three models increases as σ is increased and as C V is decreased, which is rather obvious since larger σ and smaller C V imply lower noise entropy of spike trains. [sent-84, score-0.372]

44 One nontrivial result is that, even if the three models share the same values of σ and C V , the KL divergence of each model signiﬁcantly differs from that of the others: the IG distribution achieves the largest KL divergence, followed by the lognormal and gamma distributions. [sent-85, score-0.681]

45 The difference in the KL divergence among the three models becomes larger as C V grows larger. [sent-86, score-0.181]

46 Since the three models share the same ﬁring rate λ(t) and CV , it can be concluded that the higher-order (more than second-order) moments of ISI distributions strongly affect the KL divergence. [sent-87, score-0.229]

47 In order to conﬁrm this result for another rate process, we examine a sinusoidal rate process, λ(t) = µ + σ sin t/τ , and observe the same behavior as the Ornstein-Uhlenbeck rate process (Figure 1c). [sent-88, score-0.492]

48 3 Decoding ﬂuctuating rates using the empirical Bayes method In this section, we show that the KL divergence (4) determines the lower bound of the degree of rate ﬂuctuation below which the empirical Bayes estimator cannot detect rate ﬂuctuations. [sent-89, score-0.594]

49 The empirical Bayes method We consider decoding a ﬂuctuation rate λ(t) from a given spike train {t i } := {t1 . [sent-90, score-0.364]

50 , tn } in an observation interval [0, T ] by the empirical Bayes method. [sent-93, score-0.147]

51 6 KL divergence KL divergence gamma IG lognormal (c) (b) 0. [sent-98, score-0.824]

52 8 Figure 1: (a) The gamma (blue), IG (green) and lognormal (red) ISI distribution functions for CV =0. [sent-114, score-0.5]

53 (b) The KL divergence as a function of σ for C V =0. [sent-117, score-0.162]

54 5, when the rate ﬂuctuates according to the Ornstein-Uhlenbeck process (17) with µ = 1 and τ = 10. [sent-119, score-0.175]

55 The blue, green and red indicate the KL divergence for the gamma, IG and lognormal distribution, respectively. [sent-120, score-0.471]

56 (c) The KL divergence for the sinusoidally modulated rate, λ(t) = µ + σ sin t/τ , with µ = 1 and τ = 10. [sent-124, score-0.241]

57 Detectability of rate ﬂuctuations We ﬁrst examine the gamma distribution (6). [sent-131, score-0.345]

58 For synthetic spike trains (n = 1, 000) generated by the time-rescaled renewal process with the gamma ISI distribution, in which the rate ﬂuctuates according to the Ornstein-Uhlenbeck process (17) with µ = 1 and τ = 10, we attempt to decode λ(t) using the empirical Bayes decoder. [sent-132, score-0.921]

59 When σ is increased or C V is decreased, the γ 5 empirical Bayes estimator exhibits a phase transition corresponding to the switch of the most likely rate estimation from (II) to (I) (Figure 2b). [sent-134, score-0.188]

60 Note that below the critical point of this phase transition, the empirical Bayes method provides a constant rate as the most likely estimation even if the true rate process ﬂuctuates. [sent-135, score-0.363]

61 The critical point, thus, gives the lower bound for the degree of detectable rate ﬂuctuations. [sent-136, score-0.233]

62 It is also conﬁrmed, using numerical simulations, that the phase transition occurs not only with the gamma distribution, but also with the IG and lognormal distributions (Figure 2c,d). [sent-137, score-0.589]

63 For the time-rescaled renewal process with the gamma ISI distribution, we could analytically derive the formula that the lower bound satisﬁes as: Dκ (λ(t)||µ) = 4 maxη φ(0) , φ(u)e−ηu du ∞ 0 (21) where φ(u) is the correlation function of λ(t). [sent-138, score-0.575]

64 (21) itself does not depend on the gamma distribution. [sent-143, score-0.191]

65 We investigated if this formula is also applicable to the IG and lognormal distributions, and found that the theoretical lower bounds (the solid lines in Figure 2c,d) indeed do correspond to those obtained by the numerical simulations; this result implies that Eq. [sent-144, score-0.443]

66 (21) is applicable to more general time-rescaled renewal processes. [sent-145, score-0.246]

67 The lower bound of the IG distribution is the lowest, followed by the lognormal and gamma distributions, which is expected from the result in Figure 1b, as the lower bound is identically determined by the KL divergence via Eq. [sent-147, score-0.748]

68 4 Discussion In this study, we ﬁrst examined the extent to which spike trains derived from time-rescaled renewal processes encode information on ﬂuctuating rates. [sent-150, score-0.569]

69 The encoding efﬁciency is measured by the KL divergence between two renewal processes with ﬂuctuating and constant rates. [sent-151, score-0.48]

70 We showed that the KL divergence signiﬁcantly differs among the gamma, IG and lognormal ISI distributions, even if these three processes share the same rate ﬂuctuation λ(t) and C V (Figure 1b). [sent-152, score-0.653]

71 Among the three distributions, the IG distribution achieves the largest KL divergence, followed by the lognormal and gamma distributions. [sent-154, score-0.519]

72 A similar result has been reported for stationary renewal processes [12]. [sent-155, score-0.279]

73 Since the KL divergence gives the distance between two probability distributions, Eq. [sent-156, score-0.162]

74 (4) is naturally related to the ability to discriminate between a ﬂuctuating rate and a constant rate. [sent-157, score-0.147]

75 In fact, the lower bound of the degree of rate ﬂuctuation, below which the empirical Bayes decoder cannot discriminate the underlying ﬂuctuating rate from a constant rate, satisﬁes the formula (21). [sent-158, score-0.459]

76 There commonly exists a lower bound below which the underlying rate ﬂuctuations are undetectable, not only in the empirical Bayes method with the above prior distribution (18), but also with other prior distributions, and in other rate estimators such as a time-histogram. [sent-159, score-0.331]

77 The lower bound in these methods has been derived for inhomogeneous Poisson processes as τ σ 2 /µ ∼ O(1), where τ , σ and µ are the time scale, amplitude and mean of the rate ﬂuctuation, respectively [23]. [sent-160, score-0.276]

78 Here, the crucial step for this generalization is incorporating the KL divergence into the formula. [sent-163, score-0.162]

79 Note that the formula (21) was derived analytically under the assumption of the gamma ISI distribution, and then was shown to hold for the IG and lognormal ISI distributions with numerical simulations. [sent-164, score-0.628]

80 The analytical tractability of the gamma family lies in the fact that it is the only scale family that admits the mean as a sufﬁcient statistic. [sent-165, score-0.213]

81 (21) is applicable to more general time-rescaled renewal processes (even to “non-renewal” processes), which is open to future research. [sent-167, score-0.279]

82 5 Figure 2: (a) Left: the phase diagram for sequences generated by the time-rescaled renewal process with the gamma ISI distribution. [sent-208, score-0.53]

83 The ordinate represents the amplitude of rate ﬂuctuation σ, and abscissa represents CV of the gamma ISI distribution. [sent-209, score-0.38]

84 The dots represent the result of numerical simulations in which the empirical Bayes decoder provides a ﬂuctuating rate estimation (ˆ > 0). [sent-210, score-0.242]

85 The solid line represents ˆ the theoretical lower bound obtained by the formula (21). [sent-212, score-0.129]

86 Right: raster plots of sample spike trains and the estimated rates. [sent-213, score-0.236]

87 (c, ˆ d) The phase diagrams for the IG and lognormal ISI distributions. [sent-223, score-0.361]

88 (f-h) The phase diagrams for the gamma, IG and lognormal ISI distributions, when the rate process is given by λ(t) = µ + σ sin t/τ with µ = 1 and τ = 10. [sent-225, score-0.569]

89 Appendix: Implementation of the empirical Bayes method Discretization To construct a practical algorithm for performing empirical Bayes decoding, we ﬁrst divide the time axis into a set of intervals (t i−1 , ti ] (i = 1, . [sent-230, score-0.49]

90 We assume that the ﬁring rate within each interval (ti−1 , ti ] does not change drastically (which is a reasonable assumption in practice), so that it can be approximated to a constant value λ i . [sent-234, score-0.599]

91 Letting Ti = ti − ti−1 be the ith ISI, the probability density of {T i } ≡ {T1 , T2 , . [sent-235, score-0.456]

92 , Tn }, given the rate process {λ i } ≡ {λ1 , λ2 . [sent-238, score-0.175]

93 The rate process is linked with the latent process via xi = log λi . [sent-243, score-0.328]

94 (18), is derived as p γ ({xi }) = n p(x1 ) i=2 pγ (xi |xi−1 ), where (xi − xi−1 )2 1 exp − 2 , γ (Ti + Ti−1 ) πγ 2 (Ti + Ti−1 ) pγ (xi |xi−1 ) = (22) and p(x1 ) is the probability density function of the initial latent rate variable. [sent-248, score-0.202]

95 We note that this provides a good approximation to the original continuous-time model if the timescale of the rate ﬂuctuation is larger than the mean ISI. [sent-250, score-0.13]

96 The gamma (6), IG (7) and lognormal (8) distributions are included in this family. [sent-256, score-0.538]

97 The MAP estimate of the latent process is obtained by maximizing the log posterior distribution: n l(x) = log p(x1 ) + i=2 n log pγ (xi |xi−1 ) + i=1 log pκ (Ti |xi ) + const. [sent-269, score-0.274]

98 Inferring neural ﬁring rates from spike trains using Gaussian processes. [sent-315, score-0.278]

99 Random walk models for the spike activity of a single neuron. [sent-342, score-0.168]

100 Detection limit for rate ﬂuctuations in inhomogeneous poisson processes. [sent-454, score-0.174]

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Information in a network may ∗ Authors contributed equally 1 originate from several possible sources: external sensory input, external neural network input, or alternatively, it may originate within the network itself as a memory, or as a computation. Whatever the source, this initial assumption leads directly to the conclusion that a network of integrate-and-ﬁre neurons can optimally represent a signal while exhibiting properties (a)–(c). A major problem with this framework is that network connectivity must be completely speciﬁed a priori, and requires the tuning of N 2 parameters, where N is the number of neurons in the network. Although this is feasible mathematically, it is unclear how a real network could tune itself into this optimal regime. In this work, we solve this problem using a simple synaptic learning rule. The key insight is that the plasticity rule can be derived from the same basic principle as the spiking rule in the earlier work—namely, that any change should improve information representation. Surprisingly, this can be achieved with a local, Hebbian learning rule, where synaptic plasticity is proportional to the product of presynaptic ﬁring rates with post-synaptic membrane potentials. Spiking and synaptic plasticity then work hand in hand towards the same goal: the spiking of a neuron decreases the representation error on a fast time scale, thereby giving rise to the actual population representation; synaptic plasticity decreases the representation error on a slower time scale, thereby improving or maintaining the population representation. For a large set of initial connectivities and spiking dynamics, neural networks are driven into a balanced regime, where excitation and inhibition cancel each other and where spike trains are asynchronous and irregular. Furthermore, the learning rule that we derive reproduces the main features of STDP (property (d) above). In this way, a network can learn to represent information optimally, with synaptic, neural and network dynamics consistent with those observed experimentally. 1 Derivation of the learning rule for a single neuron We begin by deriving a learning rule for a single neuron with an autapse (a self-connection) (Fig. 1A). Our approach is to derive synaptic dynamics for the autapse and spiking dynamics for the neuron such that the neuron learns to optimally represent a time-varying input signal. We will derive a learning rule for networks of neurons later, after we have developed the fundamental concepts for the single neuron case. Our ﬁrst step is to derive optimal spiking dynamics for the neuron, so that we have a target for our learning rule. We do this by making two simple assumptions [11]. First, we assume that the neuron can provide an estimate or read-out x(t) of a time-dependent signal x(t) by ﬁltering its spike train ˆ o(t) as follows: ˙ x(t) = −ˆ(t) + Γo(t), ˆ x (1) where Γ is a ﬁxed read-out weight, which we will refer to as the neuron’s “output kernel” and the spike train can be written as o(t) = i δ(t − ti ), where {ti } are the spike times. Next, we assume that the neuron only produces a spike if that spike improves the read-out, where we measure the read-out performance through a simple squared-error loss function: 2 L(t) = x(t) − x(t) . ˆ (2) With these two assumptions, we can now derive optimal spiking dynamics. First, we observe that if the neuron produces an additional spike at time t, the read-out increases by Γ, and the loss function becomes L(t|spike) = (x(t) − (x(t) + Γ))2 . This allows us to restate our spiking rule as follows: ˆ the neuron should only produce a spike if L(t|no spike) > L(t|spike), or (x(t) − x(t))2 > (x(t) − ˆ (x(t) + Γ))2 . Now, squaring both sides of this inequality, deﬁning V (t) ≡ Γ(x(t) − x(t)) and ˆ ˆ deﬁning T ≡ Γ2 /2 we ﬁnd that the neuron should only spike if: V (t) > T. (3) We interpret V (t) to be the membrane potential of the neuron, and we interpret T as the spike threshold. This interpretation allows us to understand the membrane potential functionally: the voltage is proportional to a prediction error—the difference between the read-out x(t) and the actual ˆ signal x(t). A spike is an error reduction mechanism—the neuron only spikes if the error exceeds the spike threshold. This is a greedy minimisation, in that the neuron ﬁres a spike whenever that action decreases L(t) without considering the future impact of that spike. Importantly, the neuron does not require direct access to the loss function L(t). 2 To determine the membrane potential dynamics, we take the derivative of the voltage, which gives ˙ ˙ us V = Γ(x − x). (Here, and in the following, we will drop the time index for notational brevity.) ˙ ˆ ˙ Now, using Eqn. (1) we obtain V = Γx − Γ(−x + Γo) = −Γ(x − x) + Γ(x + x) − Γ2 o, so that: ˙ ˆ ˆ ˙ ˙ V = −V + Γc − Γ2 o, (4) where c = x + x is the neural input. This corresponds exactly to the dynamics of a leaky integrate˙ and-ﬁre neuron with an inhibitory autapse1 of strength Γ2 , and a feedforward connection strength Γ. The dynamics and connectivity guarantee that a neuron spikes at just the right times to optimise the loss function (Fig. 1B). In addition, it is especially robust to noise of different forms, because of its error-correcting nature. If x is constant in time, the voltage will rise up to the threshold T at which point a spike is ﬁred, adding a delta function to the spike train o at time t, thereby producing a read-out x that is closer to x and causing an instantaneous drop in the voltage through the autapse, ˆ by an amount Γ2 = 2T , effectively resetting the voltage to V = −T . We now have a target for learning—we know the connection strength that a neuron must have at the end of learning if it is to represent information optimally, for a linear read-out. We can use this target to derive synaptic dynamics that can learn an optimal representation from experience. Speciﬁcally, we consider an integrate-and-ﬁre neuron with some arbitrary autapse strength ω. The dynamics of this neuron are given by ˙ V = −V + Γc − ωo. (5) This neuron will not produce the correct spike train for representing x through a linear read-out (Eqn. (1)) unless ω = Γ2 . Our goal is to derive a dynamical equation for the synapse ω so that the spike train becomes optimal. We do this by quantifying the loss that we are incurring by using the suboptimal strength, and then deriving a learning rule that minimises this loss with respect to ω. The loss function underlying the spiking dynamics determined by Eqn. (5) can be found by reversing the previous membrane potential analysis. First, we integrate the differential equation for V , assuming that ω changes on time scales much slower than the membrane potential. We obtain the following (formal) solution: V = Γx − ω¯, o (6) ˙ where o is determined by o = −¯ + o. The solution to this latter equation is o = h ∗ o, a convolution ¯ ¯ o ¯ of the spike train with the exponential kernel h(τ ) = θ(τ ) exp(−τ ). As such, it is analogous to the instantaneous ﬁring rate of the neuron. Now, using Eqn. (6), and rewriting the read-out as x = Γ¯, we obtain the loss incurred by the ˆ o sub-optimal neuron, L = (x − x)2 = ˆ 1 V 2 + 2(ω − Γ2 )¯ + (ω − Γ2 )2 o2 . o ¯ Γ2 (7) We observe that the last two terms of Eqn. (7) will vanish whenever ω = Γ2 , i.e., when the optimal reset has been found. We can therefore simplify the problem by deﬁning an alternative loss function, 1 2 V , (8) 2 which has the same minimum as the original loss (V = 0 or x = x, compare Eqn. (2)), but yields a ˆ simpler learning algorithm. We can now calculate how changes to ω affect LV : LV = ∂LV ∂V ∂o ¯ =V = −V o − V ω ¯ . (9) ∂ω ∂ω ∂ω We can ignore the last term in this equation (as we will show below). Finally, using simple gradient descent, we obtain a simple Hebbian-like synaptic plasticity rule: τω = − ˙ ∂LV = V o, ¯ ∂ω (10) where τ is the learning time constant. 1 This contribution of the autapse can also be interpreted as the reset of an integrate-and-ﬁre neuron. Later, when we generalise to networks of neurons, we shall employ this interpretation. 3 This synaptic learning rule is capable of learning the synaptic weight ω that minimises the difference between x and x (Fig. 1B). During learning, the synaptic weight changes in proportion to the postˆ synaptic voltage V and the pre-synaptic ﬁring rate o (Fig. 1C). As such, this is a Hebbian learning ¯ rule. Of course, in this single neuron case, the pre-synaptic neuron and post-synaptic neuron are the same neuron. The synaptic weight gradually approaches its optimal value Γ2 . However, it never completely stabilises, because learning never stops as long as neurons are spiking. Instead, the synapse oscillates closely about the optimal value (Fig. 1D). This is also a “greedy” learning rule, similar to the spiking rule, in that it seeks to minimise the error at each instant in time, without regard for the future impact of those changes. To demonstrate that the second term in Eqn. (5) can be neglected we note that the equations for V , o, and ω deﬁne a system ¯ of coupled differential equations that can be solved analytically by integrating between spikes. This results in a simple recurrence relation for changes in ω from the ith to the (i + 1)th spike, ωi+1 = ωi + ωi (ωi − 2T ) . τ (T − Γc − ωi ) (11) This iterative equation has a single stable ﬁxed point at ω = 2T = Γ2 , proving that the neuron’s autaptic weight or reset will approach the optimal solution. 2 Learning in a homogeneous network We now generalise our learning rule derivation to a network of N identical, homogeneously connected neurons. This generalisation is reasonably straightforward because many characteristics of the single neuron case are shared by a network of identical neurons. We will return to the more general case of heterogeneously connected neurons in the next section. We begin by deriving optimal spiking dynamics, as in the single neuron case. This provides a target for learning, which we can then use to derive synaptic dynamics. As before, we want our network to produce spikes that optimally represent a variable x for a linear read-out. We assume that the read-out x is provided by summing and ﬁltering the spike trains of all the neurons in the network: ˆ ˙ x = −ˆ + Γo, ˆ x (12) 2 where the row vector Γ = (Γ, . . . , Γ) contains the read-out weights of the neurons and the column vector o = (o1 , . . . , oN ) their spike trains. Here, we have used identical read-out weights for each neuron, because this indirectly leads to homogeneous connectivity, as we will demonstrate. Next, we assume that a neuron only spikes if that spike reduces a loss-function. This spiking rule is similar to the single neuron spiking rule except that this time there is some ambiguity about which neuron should spike to represent a signal. Indeed, there are many different spike patterns that provide exactly the same estimate x. For example, one neuron could ﬁre regularly at a high rate (exactly like ˆ our previous single neuron example) while all others are silent. To avoid this ﬁring rate ambiguity, we use a modiﬁed loss function, that selects amongst all equivalent solutions, those with the smallest neural ﬁring rates. We do this by adding a ‘metabolic cost’ term to our loss function, so that high ﬁring rates are penalised: ¯ L = (x − x)2 + µ o 2 , ˆ (13) where µ is a small positive constant that controls the cost-accuracy trade-off, akin to a regularisation parameter. Each neuron in the optimal network will seek to reduce this loss function by ﬁring a spike. Speciﬁcally, the ith neuron will spike whenever L(no spike in i) > L(spike in i). This leads to the following spiking rule for the ith neuron: Vi > Ti (14) where Vi ≡ Γ(x − x) − µoi and Ti ≡ Γ2 /2 + µ/2. We can naturally interpret Vi as the membrane ˆ potential of the ith neuron and Ti as the spiking threshold of that neuron. As before, we can now derive membrane potential dynamics: ˙ V = −V + ΓT c − (ΓT Γ + µI)o, 2 (15) The read-out weights must scale as Γ ∼ 1/N so that ﬁring rates are not unrealistically small in large networks. We can see this by calculating the average ﬁring rate N oi /N ≈ x/(ΓN ) ∼ O(N/N ) ∼ O(1). i=1 ¯ 4 where I is the identity matrix and ΓT Γ + µI is the network connectivity. We can interpret the selfconnection terms {Γ2 +µ} as voltage resets that decrease the voltage of any neuron that spikes. This optimal network is equivalent to a network of identical integrate-and-ﬁre neurons with homogeneous inhibitory connectivity. The network has some interesting dynamical properties. The voltages of all the neurons are largely synchronous, all increasing to the spiking threshold at about the same time3 (Fig. 1F). Nonetheless, neural spiking is asynchronous. The ﬁrst neuron to spike will reset itself by Γ2 + µ, and it will inhibit all the other neurons in the network by Γ2 . This mechanism prevents neurons from spik- x 3 The ﬁrst neuron to spike will be random if there is some membrane potential noise. V (A) (B) x x ˆ x 10 1 0.1 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 1 D 0.5 V V 0 ˆ x V ˆ x (C) 1 0 1 2 0 0.625 25 25.625 (D) start of learning 1 V 50 200.625 400 400.625 1 2.4 O 1.78 ω 1.77 25 neuron$ 0 1 2 !me$ 3 4 25 1 5 V 400.625 !me$ (F) 25 1 2.35 1.05 1.049 400 25.625 !me$ (E) neuron$ 100.625 200 end of learning 1.4 1.35 ω 100 !me$ 1 V 1 O 50.625 0 1 2 !me$ 3 4 5 V !me$ !me$ Figure 1: Learning in a single neuron and a homogeneous network. (A) A single neuron represents an input signal x by producing an output x. (B) During learning, the single neuron output x (solid red ˆ ˆ line, top panel) converges towards the input x (blue). Similarly, for a homogeneous network the output x (dashed red line, top panel) converges towards x. Connectivity also converges towards optimal ˆ connectivity in both the single neuron case (solid black line, middle panel) and the homogeneous net2 2 work case (dashed black line, middle panel), as quantiﬁed by D = maxi,j ( Ωij − Ωopt / Ωopt ) ij ij at each point in time. Consequently, the membrane potential reset (bottom panel) converges towards the optimal reset (green line, bottom panel). Spikes are indicated by blue vertical marks, and are produced when the membrane potential reaches threshold (bottom panel). Here, we have rescaled time, as indicated, for clarity. (C) Our learning rule dictates that the autapse ω in our single neuron (bottom panel) changes in proportion to the membrane potential (top panel) and the ﬁring rate (middle panel). (D) At the end of learning, the reset ω ﬂuctuates weakly about the optimal value. (E) For a homogeneous network, neurons spike regularly at the start of learning, as shown in this raster plot. Membrane potentials of different neurons are weakly correlated. (F) At the end of learning, spiking is very irregular and membrane potentials become more synchronous. 5 ing synchronously. The population as a whole acts similarly to the single neuron in our previous example. Each neuron ﬁres regularly, even if a different neuron ﬁres in every integration cycle. The design of this optimal network requires the tuning of N (N − 1) synaptic parameters. How can an arbitrary network of integrate-and-ﬁre neurons learn this optimum? As before, we address this question by using the optimal network as a target for learning. We start with an arbitrarily connected network of integrate-and-ﬁre neurons: ˙ V = −V + ΓT c − Ωo, (16) where Ω is a matrix of connectivity weights, which includes the resets of the individual neurons. Assuming that learning occurs on a slow time scale, we can rewrite this equation as V = ΓT x − Ω¯ . o (17) Now, repeating the arguments from the single neuron derivation, we modify the loss function to obtain an online learning rule. Speciﬁcally, we set LV = V 2 /2, and calculate the gradient: ∂LV = ∂Ωij Vk k ∂Vk =− ∂Ωij Vk δki oj − ¯ k Vk Ωkl kl ∂ ol ¯ . ∂Ωij (18) We can simplify this equation considerably by observing that the contribution of the second summation is largely averaged out under a wide variety of realistic conditions4 . Therefore, it can be neglected, and we obtain the following local learning rule: ∂LV ˙ = V i oj . ¯ τ Ωij = − ∂Ωij (19) This is a Hebbian plasticity rule, whereby connectivity changes in proportion to the presynaptic ﬁring rate oj and post-synaptic membrane potential Vi . We assume that the neural thresholds are set ¯ to a constant T and that the neural resets are set to their optimal values −T . In the previous section we demonstrated that these resets can be obtained by a Hebbian plasticity rule (Eqn. (10)). This learning rule minimises the difference between the read-out and the signal, by approaching the optimal recurrent connection strengths for the network (Fig. 1B). As in the single neuron case, learning does not stop, so the connection strengths ﬂuctuate close to their optimal value. During learning, network activity becomes progressively more asynchronous as it progresses towards optimal connectivity (Fig. 1E, F). 3 Learning in the general case Now that we have developed the fundamental concepts underlying our learning rule, we can derive a learning rule for the more general case of a network of N arbitrarily connected leaky integrateand-ﬁre neurons. Our goal is to understand how such networks can learn to optimally represent a ˙ J-dimensional signal x = (x1 , . . . , xJ ), using the read-out equation x = −x + Γo. We consider a network with the following membrane potential dynamics: ˙ V = −V + ΓT c − Ωo, (20) where c is a J-dimensional input. We assume that this input is related to the signal according to ˙ c = x + x. This assumption can be relaxed by treating the input as the control for an arbitrary linear dynamical system, in which case the signal represented by the network is the output of such a computation [11]. However, this further generalisation is beyond the scope of this work. As before, we need to identify the optimal recurrent connectivity so that we have a target for learning. Most generally, the optimal recurrent connectivity is Ωopt ≡ ΓT Γ + µI. The output kernels of the individual neurons, Γi , are given by the rows of Γ, and their spiking thresholds by Ti ≡ Γi 2 /2 + 4 From the deﬁnition of the membrane potential we can see that Vk ∼ O(1/N ) because Γ ∼ 1/N . Therefore, the size of the ﬁrst term in Eqn. (18) is k Vk δki oj = Vi oj ∼ O(1/N ). Therefore, the second term can ¯ ¯ be ignored if kl Vk Ωkl ∂ ol /∂Ωij ¯ O(1/N ). This happens if Ωkl O(1/N 2 ) as at the start of learning. It also happens towards the end of learning if the terms {Ωkl ∂ ol /∂Ωij } are weakly correlated with zero mean, ¯ or if the membrane potentials {Vi } are weakly correlated with zero mean. 6 µ/2. With these connections and thresholds, we ﬁnd that a network of integrate-and-ﬁre neurons ˆ ¯ will produce spike trains in such a way that the loss function L = x − x 2 + µ o 2 is minimised, ˆ where the read-out is given by x = Γ¯ . We can show this by prescribing a greedy5 spike rule: o a spike is ﬁred by neuron i whenever L(no spike in i) > L(spike in i) [11]. The resulting spike generation rule is Vi > Ti , (21) ˆ where Vi ≡ ΓT (x − x) − µ¯i is interpreted as the membrane potential. o i 5 Despite being greedy, this spiking rule can generate ﬁring rates that are practically identical to the optimal solutions: we checked this numerically in a large ensemble of networks with randomly chosen kernels. (A) x1 … x … 1 1 (B) xJJ x 10 L 10 T T 10 4 6 8 1 Viii V D ˆˆ ˆˆ x11 xJJ x x F 0.5 0 0.4 … … 0.2 0 0 2000 4000 !me (C) x V V 1 x 10 x 3 ˆ x 8 0 x 10 1 2 3 !me 4 5 4 0 1 4 0 1 8 V (F) Ρ(Δt) E-‐I input 0.4 ˆ x 0 3 0 1 x 10 1.3 0.95 x 10 ˆ x 4 V (E) 1 x 0 end of learning 50 neuron neuron 50 !me 2 0 ˆ x 0 0.5 ISI Δt 1 2 !me 4 5 4 1.5 1.32 3 2 0.1 Ρ(Δt) x E-‐I input (D) start of learning 0 2 !me 0 0 0.5 ISI Δt 1 Figure 2: Learning in a heterogeneous network. (A) A network of neurons represents an input ˆ signal x by producing an output x. (B) During learning, the loss L decreases (top panel). The difference between the connection strengths and the optimal strengths also decreases (middle panel), as 2 2 quantiﬁed by the mean difference (solid line), given by D = Ω − Ωopt / Ωopt and the maxi2 2 mum difference (dashed line), given by maxi,j ( Ωij − Ωopt / Ωopt ). The mean population ﬁring ij ij rate (solid line, bottom panel) also converges towards the optimal ﬁring rate (dashed line, bottom panel). (C, E) Before learning, a raster plot of population spiking shows that neurons produce bursts ˆ of spikes (upper panel). The network output x (red line, middle panel) fails to represent x (blue line, middle panel). The excitatory input (red, bottom left panel) and inhibitory input (green, bottom left panel) to a randomly selected neuron is not tightly balanced. Furthermore, a histogram of interspike intervals shows that spiking activity is not Poisson, as indicated by the red line that represents a best-ﬁt exponential distribution. (D, F) At the end of learning, spiking activity is irregular and ˆ Poisson-like, excitatory and inhibitory input is tightly balanced and x matches x. 7 How can we learn this optimal connection matrix? As before, we can derive a learning rule by minimising the cost function LV = V 2 /2. This leads to a Hebbian learning rule with the same form as before: ˙ τ Ωij = Vi oj . ¯ (22) Again, we assume that the neural resets are given by −Ti . Furthermore, in order for this learning rule to work, we must assume that the network input explores all possible directions in the J-dimensional input space (since the kernels Γi can point in any of these directions). The learning performance does not critically depend on how the input variable space is sampled as long as the exploration is extensive. In our simulations, we randomly sample the input c from a Gaussian white noise distribution at every time step for the entire duration of the learning. We ﬁnd that this learning rule decreases the loss function L, thereby approaching optimal network connectivity and producing optimal ﬁring rates for our linear decoder (Fig. 2B). In this example, we have chosen connectivity that is initially much too weak at the start of learning. Consequently, the initial network behaviour is similar to a collection of unconnected single neurons that ignore each other. Spike trains are not Poisson-like, ﬁring rates are excessively large, excitatory and inhibitory ˆ input is unbalanced and the decoded variable x is highly unreliable (Fig. 2C, E). As a result of learning, the network becomes tightly balanced and the spike trains become asynchronous, irregular and Poisson-like with much lower rates (Fig. 2D, F). However, despite this apparent variability, the population representation is extremely precise, only limited by the the metabolic cost and the discrete nature of a spike. This learnt representation is far more precise than a rate code with independent Poisson spike trains [11]. In particular, shufﬂing the spike trains in response to identical inputs drastically degrades this precision. 4 Conclusions and Discussion In population coding, large trial-to-trial spike train variability is usually interpreted as noise [2]. We show here that a deterministic network of leaky integrate-and-ﬁre neurons with a simple Hebbian plasticity rule can self-organise into a regime where information is represented far more precisely than in noisy rate codes, while appearing to have noisy Poisson-like spiking dynamics. Our learning rule (Eqn. (22)) has the basic properties of STDP. Speciﬁcally, a presynaptic spike occurring immediately before a post-synaptic spike will potentiate a synapse, because membrane potentials are positive immediately before a postsynaptic spike. Furthermore, a presynaptic spike occurring immediately after a post-synaptic spike will depress a synapse, because membrane potentials are always negative immediately after a postsynaptic spike. This is similar in spirit to the STDP rule proposed in [12], but different to classical STDP, which depends on post-synaptic spike times [9]. This learning rule can also be understood as a mechanism for generating a tight balance between excitatory and inhibitory input. We can see this by observing that membrane potentials after learning can be interpreted as representation errors (projected onto the read-out kernels). Therefore, learning acts to minimise the magnitude of membrane potentials. Excitatory and inhibitory input must be balanced if membrane potentials are small, so we can equate balance with optimal information representation. Previous work has shown that the balanced regime produces (quasi-)chaotic network dynamics, thereby accounting for much observed cortical spike train variability [13, 14, 4]. Moreover, the STDP rule has been known to produce a balanced regime [16, 17]. Additionally, recent theoretical studies have suggested that the balanced regime plays an integral role in network computation [15, 13]. In this work, we have connected these mechanisms and functions, to conclude that learning this balance is equivalent to the development of an optimal spike-based population code, and that this learning can be achieved using a simple Hebbian learning rule. Acknowledgements We are grateful for generous funding from the Emmy-Noether grant of the Deutsche Forschungsgemeinschaft (CKM) and the Chaire d’excellence of the Agence National de la Recherche (CKM, DB), as well as a James Mcdonnell Foundation Award (SD) and EU grants BACS FP6-IST-027140, BIND MECT-CT-20095-024831, and ERC FP7-PREDSPIKE (SD). 8 References [1] Tolhurst D, Movshon J, and Dean A (1982) The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vision Res 23: 775–785. [2] Shadlen MN, Newsome WT (1998) The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. J Neurosci 18(10): 3870–3896. [3] Zohary E, Newsome WT (1994) Correlated neuronal discharge rate and its implication for psychophysical performance. Nature 370: 140–143. [4] Renart A, de la Rocha J, Bartho P, Hollender L, Parga N, Reyes A, & Harris, KD (2010) The asynchronous state in cortical circuits. Science 327, 587–590. [5] Ecker AS, Berens P, Keliris GA, Bethge M, Logothetis NK, Tolias AS (2010) Decorrelated neuronal ﬁring in cortical microcircuits. Science 327: 584–587. [6] Okun M, Lampl I (2008) Instantaneous correlation of excitation and inhibition during ongoing and sensory-evoked activities. Nat Neurosci 11, 535–537. [7] Shu Y, Hasenstaub A, McCormick DA (2003) Turning on and off recurrent balanced cortical activity. Nature 423, 288–293. [8] Gentet LJ, Avermann M, Matyas F, Staiger JF, Petersen CCH (2010) Membrane potential dynamics of GABAergic neurons in the barrel cortex of behaving mice. Neuron 65: 422–435. [9] Caporale N, Dan Y (2008) Spike-timing-dependent plasticity: a Hebbian learning rule. Annu Rev Neurosci 31: 25–46. [10] Boerlin M, Deneve S (2011) Spike-based population coding and working memory. PLoS Comput Biol 7, e1001080. [11] Boerlin M, Machens CK, Deneve S (2012) Predictive coding of dynamic variables in balanced spiking networks. under review. [12] Clopath C, B¨ sing L, Vasilaki E, Gerstner W (2010) Connectivity reﬂects coding: a model of u voltage-based STDP with homeostasis. Nat Neurosci 13(3): 344–352. [13] van Vreeswijk C, Sompolinsky H (1998) Chaotic balanced state in a model of cortical circuits. Neural Comput 10(6): 1321–1371. [14] Brunel N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory neurons. J Comput Neurosci 8, 183–208. [15] Vogels TP, Rajan K, Abbott LF (2005) Neural network dynamics. Annu Rev Neurosci 28: 357–376. [16] Vogels TP, Sprekeler H, Zenke F, Clopath C, Gerstner W. (2011) Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks. Science 334(6062):1569– 73. [17] Song S, Miller KD, Abbott LF (2000) Competitive Hebbian learning through spike-timingdependent synaptic plasticity. Nat Neurosci 3(9): 919–926. 9

4 0.11686244 239 nips-2012-Neuronal Spike Generation Mechanism as an Oversampling, Noise-shaping A-to-D converter

Author: Dmitri B. Chklovskii, Daniel Soudry

Abstract: We test the hypothesis that the neuronal spike generation mechanism is an analog-to-digital (AD) converter encoding rectified low-pass filtered summed synaptic currents into a spike train linearly decodable in postsynaptic neurons. Faithful encoding of an analog waveform by a binary signal requires that the spike generation mechanism has a sampling rate exceeding the Nyquist rate of the analog signal. Such oversampling is consistent with the experimental observation that the precision of the spikegeneration mechanism is an order of magnitude greater than the cut -off frequency of low-pass filtering in dendrites. Additional improvement in the coding accuracy may be achieved by noise-shaping, a technique used in signal processing. If noise-shaping were used in neurons, it would reduce coding error relative to Poisson spike generator for frequencies below Nyquist by introducing correlations into spike times. By using experimental data from three different classes of neurons, we demonstrate that biological neurons utilize noise-shaping. Therefore, the spike-generation mechanism can be viewed as an oversampling and noise-shaping AD converter. The nature of the neural spike code remains a central problem in neuroscience [1-3]. In particular, no consensus exists on whether information is encoded in firing rates [4, 5] or individual spike timing [6, 7]. On the single-neuron level, evidence exists to support both points of view. On the one hand, post-synaptic currents are low-pass-filtered by dendrites with the cut-off frequency of approximately 30Hz [8], Figure 1B, providing ammunition for the firing rate camp: if the signal reaching the soma is slowly varying, why would precise spike timing be necessary? On the other hand, the ability of the spike-generation mechanism to encode harmonics of the injected current up to about 300Hz [9, 10], Figure 1B, points at its exquisite temporal precision [11]. Yet, in view of the slow variation of the somatic current, such precision may seem gratuitous and puzzling. The timescale mismatch between gradual variation of the somatic current and high precision of spike generation has been addressed previously. Existing explanations often rely on the population nature of the neural code [10, 12]. Although this is a distinct possibility, the question remains whether invoking population coding is necessary. Other possible explanations for the timescale mismatch include the possibility that some synaptic currents (for example, GABAergic) may be generated by synapses proximal to the soma and therefore not subject to low-pass filtering or that the high frequency harmonics are so strong in the pre-synaptic spike that despite attenuation, their trace is still present. Although in some cases, these explanations could apply, for the majority of synaptic inputs to typical neurons there is a glaring mismatch. The perceived mismatch between the time scales of somatic currents and the spike-generation mechanism can be resolved naturally if one views spike trains as digitally encoding analog somatic currents [13-15], Figure 1A. Although somatic currents vary slowly, information that could be communicated by their analog amplitude far exceeds that of binary signals, such as all- or-none spikes, of the same sampling rate. Therefore, faithful digital encoding requires sampling rate of the digital signal to be much higher than the cut-off frequency of the analog signal, socalled over-sampling. Although the spike generation mechanism operates in continuous time, the high temporal precision of the spikegeneration mechanism may be viewed as a manifestation of oversampling, which is needed for the digital encoding of the analog signal. Therefore, the extra order of magnitude in temporal precision available to the spike-generation mechanism relative to somatic current, Figure 1B, is necessary to faithfully encode the amplitude of the analog signal, thus potentially reconciling the firing rate and the spike timing points of view [13-15]. Figure 1. Hybrid digital-analog operation of neuronal circuits. A. Post-synaptic currents are low-pass filtered and summed in dendrites (black) to produce a somatic current (blue). This analog signal is converted by the spike generation mechanism into a sequence of all-or-none spikes (green), a digital signal. Spikes propagate along an axon and are chemically transduced across synapses (gray) into post-synatpic currents (black), whose amplitude reflects synaptic weights, thus converting digital signal back to analog. B. Frequency response function for dendrites (blue, adapted from [8]) and for the spike generation mechanism (green, adapted from [9]). Note one order of magnitude gap between the cut off frequencies. C. Amplitude of the summed postsynaptic currents depends strongly on spike timing. If the blue spike arrives just 5ms later, as shown in red, the EPSCs sum to a value already 20% less. Therefore, the extra precision of the digital signal may be used to communicate the amplitude of the analog signal. In signal processing, efficient AD conversion combines the principle of oversampling with that of noise-shaping, which utilizes correlations in the digital signal to allow more accurate encoding of the analog amplitude. This is exemplified by a family of AD converters called modulators [16], of which the basic one is analogous to an integrate-and-fire (IF) neuron [13-15]. The analogy between the basic modulator and the IF neuron led to the suggestion that neurons also use noise-shaping to encode incoming analog current waveform in the digital spike train [13]. However, the hypothesis of noise-shaping AD conversion has never been tested experimentally in biological neurons. In this paper, by analyzing existing experimental datasets, we demonstrate that noise-shaping is present in three different classes of neurons from vertebrates and invertebrates. This lends support to the view that neurons act as oversampling and noise-shaping AD converters and accounts for the mismatch between the slowly varying somatic currents and precise spike timing. Moreover, we show that the degree of noise-shaping in biological neurons exceeds that used by basic  modulators or IF neurons and propose viewing more complicated models in the noise-shaping framework. This paper is organized as follows: We review the principles of oversampling and noise-shaping in Section 2. In Section 3, we present experimental evidence for noise-shaping AD conversion in neurons. In Section 4 we argue that rectification of somatic currents may improve energy efficiency and/or implement de-noising. 2 . Oversampling and noise-shaping in AD converters To understand how oversampling can lead to more accurate encoding of the analog signal amplitude in a digital form, we first consider a Poisson spike encoder, whose rate of spiking is modulated by the signal amplitude, Figure 2A. Such an AD converter samples an analog signal at discrete time points and generates a spike with a probability given by the (normalized) signal amplitude. Because of the binary nature of spike trains, the resulting spike train encodes the signal with a large error even when the sampling is done at Nyquist rate, i.e. the lowest rate for alias-free sampling. To reduce the encoding error a Poisson encoder can sample at frequencies, fs , higher than Nyquist, fN – hence, the term oversampling, Figure 2B. When combined with decoding by lowpass filtering (down to Nyquist) on the receiving end, this leads to a reduction of the error, which can be estimated as follows. The number of samples over a Nyquist half-period (1/2fN) is given by the oversampling ratio: . As the normalized signal amplitude, , stays roughly constant over the Nyquist half-period, it can be encoded by spikes generated with a fixed probability, x. For a Poisson process the variance in the number of spikes is equal to the mean, . Therefore, the mean relative error of the signal decoded by averaging over the Nyquist half-period: , (1) indicating that oversampling reduces transmission error. However, the weak dependence of the error on the oversampling frequency indicates diminishing returns on the investment in oversampling and motivates one to search for other ways to lower the error. Figure 2. Oversampling and noise-shaping in AD conversion. A. Analog somatic current (blue) and its digital code (green). The difference between the green and the blue curves is encoding error. B. Digital output of oversampling Poisson encoder over one Nyquist half-period. C. Error power spectrum of a Nyquist (dark green) and oversampled (light green) Poisson encoder. Although the total error power is the same, the fraction surviving low-pass filtering during decoding (solid green) is smaller in oversampled case. D. Basic  modulator. E. Signal at the output of the integrator. F. Digital output of the  modulator over one Nyquist period. G. Error power spectrum of the  modulator (brown) is shifted to higher frequencies and low-pass filtered during decoding. The remaining error power (solid brown) is smaller than for Poisson encoder. To reduce encoding error beyond the ½ power of the oversampling ratio, the principle of noiseshaping was put forward [17]. To illustrate noise-shaping consider a basic AD converter called  [18], Figure 2D. In the basic  modulator, the previous quantized signal is fed back and subtracted from the incoming signal and then the difference is integrated in time. Rather than quantizing the input signal, as would be done in the Poisson encoder,  modulator quantizes the integral of the difference between the incoming analog signal and the previous quantized signal, Figure 2F. One can see that, in the oversampling regime, the quantization error of the basic  modulator is significantly less than that of the Poisson encoder. As the variance in the number of spikes over the Nyquist period is less than one, the mean relative error of the signal is at most, , which is better than the Poisson encoder. To gain additional insight and understand the origin of the term noise-shaping, we repeat the above analysis in the Fourier domain. First, the Poisson encoder has a flat power spectrum up to the sampling frequency, Figure 2C. Oversampling preserves the total error power but extends the frequency range resulting in the lower error power below Nyquist. Second, a more detailed analysis of the basic  modulator, where the dynamics is linearized by replacing the quantization device with a random noise injection [19], shows that the quantization noise is effectively differentiated. Taking the derivative in time is equivalent to multiplying the power spectrum of the quantization noise by frequency squared. Such reduction of noise power at low frequencies is an example of noise shaping, Figure 2G. Under the additional assumption of the white quantization noise, such analysis yields: , (2) which for R >> 1 is significantly better performance than for the Poisson encoder, Eq.(1). As mentioned previously, the basic  modulator, Figure 2D, in the continuous-time regime is nothing other than an IF neuron [13, 20, 21]. In the IF neuron, quantization is implemented by the spike generation mechanism and the negative feedback corresponds to the after-spike reset. Note that resetting the integrator to zero is strictly equivalent to subtraction only for continuous-time operation. In discrete-time computer simulations, the integrator value may exceed the threshold, and, therefore, subtraction of the threshold value rather than reset must be used. Next, motivated by the -IF analogy, we look for the signs of noise-shaping AD conversion in real neurons. 3 . Experimental evidence of noise-shaping AD conversion in real neurons In order to determine whether noise-shaping AD conversion takes place in biological neurons, we analyzed three experimental datasets, where spike trains were generated by time-varying somatic currents: 1) rat somatosensory cortex L5 pyramidal neurons [9], 2) mouse olfactory mitral cells [22, 23], and 3) fruit fly olfactory receptor neurons [24]. In the first two datasets, the current was injected through an electrode in whole-cell patch clamp mode, while in the third, the recording was extracellular and the intrinsic somatic current could be measured because the glial compartment included only one active neuron. Testing the noise-shaping AD conversion hypothesis is complicated by the fact that encoded and decoded signals are hard to measure accurately. First, as somatic current is rectified by the spikegeneration mechanism, only its super-threshold component can be encoded faithfully making it hard to know exactly what is being encoded. Second, decoding in the dendrites is not accessible in these single-neuron recordings. In view of these difficulties, we start by simply computing the power spectrum of the reconstruction error obtained by subtracting a scaled and shifted, but otherwise unaltered, spike train from the somatic current. The scaling factor was determined by the total weight of the decoding linear filter and the shift was optimized to maximize information capacity, see below. At the frequencies below 20Hz the error contains significantly lower power than the input signal, Figure 3, indicating that the spike generation mechanism may be viewed as an AD converter. Furthermore, the error power spectrum of the biological neuron is below that of the Poisson encoder, thus indicating the presence of noise-shaping. For dataset 3 we also plot the error power spectrum of the IF neuron, the threshold of which is chosen to generate the same number of spikes as the biological neuron. 4 somatic current biological neuron error Poisson encoder error I&F; neuron error 10 1 10 0 Spectral power, a.u. Spectral power, a.u. 10 3 10 -1 10 -2 10 -3 10 2 10 -4 10 0 10 20 30 40 50 60 Frequency [Hz] 70 80 90 0 10 20 30 40 50 60 70 80 90 100 Frequency [Hz] Figure 3. Evidence of noise-shaping. Power spectra of the somatic current (blue), difference between the somatic current and the digital spike train of the biological neuron (black), of the Poisson encoder (green) and of the IF neuron (red). Left: datset 1, right: dataset 3. Although the simple analysis presented above indicates noise-shaping, subtracting the spike train from the input signal, Figure 3, does not accurately quantify the error when decoding involves additional filtering. An example of such additional encoding/decoding is predictive coding, which will be discussed below [25]. To take such decoding filter into account, we computed a decoded waveform by convolving the spike train with the optimal linear filter, which predicts the somatic current from the spike train with the least mean squared error. Our linear decoding analysis lends additional support to the noise-shaping AD conversion hypothesis [13-15]. First, the optimal linear filter shape is similar to unitary post-synaptic currents, Figure 4B, thus supporting the view that dendrites reconstruct the somatic current of the presynaptic neuron by low-pass filtering the spike train in accordance with the noise-shaping principle [13]. Second, we found that linear decoding using an optimal filter accounts for 60-80% of the somatic current variance. Naturally, such prediction works better for neurons in suprathreshold regime, i.e. with high firing rates, an issue to which we return in Section 4. To avoid complications associated with rectification for now we focused on neurons which were in suprathreshold regime by monitoring that the relationship between predicted and actual current is close to linear. 2 10 C D 1 10 somatic current biological neuron error Poisson encoder error Spectral power, a.u. Spectral power, a.u. I&F; neuron error 3 10 0 10 -1 10 -2 10 -3 10 2 10 -4 0 10 20 30 40 50 60 Frequency [Hz] 70 80 90 10 0 10 20 30 40 50 60 70 80 90 100 Frequency [Hz] Figure 4. Linear decoding of experimentally recorded spike trains. A. Waveform of somatic current (blue), resulting spike train (black), and the linearly decoded waveform (red) from dataset 1. B. Top: Optimal linear filter for the trace in A, is representative of other datasets as well. Bottom: Typical EPSPs have a shape similar to the decoding filter (adapted from [26]). C-D. Power spectra of the somatic current (blue), the decdoding error of the biological neuron (black), the Poisson encoder (green), and IF neuron (red) for dataset 1 (C) dataset 3 (D). Next, we analyzed the spectral distribution of the reconstruction error calculated by subtracting the decoded spike train, i.e. convolved with the computed optimal linear filter, from the somatic current. We found that at low frequencies the error power is significantly lower than in the input signal, Figure 4C,D. This observation confirms that signals below the dendritic cut-off frequency of 20-30Hz can be efficiently communicated using spike trains. To quantify the effect of noise-shaping we computed information capacity of different encoders: where S(f) and N(f) are the power spectra of the somatic current and encoding error correspondingly and the sum is computed only over the frequencies for which S(f) > N(f). Because the plots in Figure 4C,D use semi-logrithmic scale, the information capacity can be estimated from the area between a somatic current (blue) power spectrum and an error power spectrum. We find that the biological spike generation mechanism has higher information capacity than the Poisson encoder and IF neurons. Therefore, neurons act as AD converters with stronger noise-shaping than IF neurons. We now return to the predictive nature of the spike generation mechanism. Given the causal nature of the spike generation mechanism it is surprising that the optimal filters for all three datasets carry most of their weight following a spike, Figure 4B. This indicates that the spike generation mechanism is capable of making predictions, which are possible in these experiments because somatic currents are temporally correlated. We note that these observations make delay-free reconstruction of the signal possible, thus allowing fast operation of neural circuits [27]. The predictive nature of the encoder can be captured by a  modulator embedded in a predictive coding feedback loop [28], Figure 5A. We verified by simulation that such a nested architecture generates a similar optimal linear filter with most of its weight in the time following a spike, Figure 5A right. Of course such prediction is only possible for correlated inputs implying that the shape of the optimal linear filter depends on the statistics of the inputs. The role of predictive coding is to reduce the dynamic range of the signal that enters , thus avoiding overloading. A possible biological implementation for such integrating feedback could be Ca2+ 2+ concentration and Ca dependent potassium channels [25, 29]. Figure 5. Enhanced  modulators. A.  modulator combined with predictive coder. In such device, the optimal decoding filter computed for correlated inputs has most of its weight following a spike, similar to experimental measurements, Figure 4B. B. Second-order  modulator possesses stronger noise-shaping properties. Because such circuit contains an internal state variable it generates a non-periodic spike train in response to a constant input. Bottom trace shows a typical result of a simulation. Black – spikes, blue – input current. 4 . Possible reasons for current rectification: energy efficiency and de-noising We have shown that at high firing rates biological neurons encode somatic current into a linearly decodable spike train. However, at low firing rates linear decoding cannot faithfully reproduce the somatic current because of rectification in the spike generation mechanism. If the objective of spike generation is faithful AD conversion, why would such rectification exist? We see two potential reasons: energy efficiency and de-noising. It is widely believed that minimizing metabolic costs is an important consideration in brain design and operation [30, 31]. Moreover, spikes are known to consume a significant fraction of the metabolic budget [30, 32] placing a premium on their total number. Thus, we can postulate that neuronal spike trains find a trade-off between the mean squared error in the decoded spike train relative to the input signal and the total number of spikes, as expressed by the following cost function over a time interval T: , (3) where x is the analog input signal, s is the binary spike sequence composed of zeros and ones, and is the linear filter. To demonstrate how solving Eq.(3) would lead to thresholding, let us consider a simplified version taken over a Nyquist period, during which the input signal stays constant: (4) where and normalized by w. Minimizing such a cost function reduces to choosing the lowest lying parabola for a given , Figure 6A. Therefore, thresholding is a natural outcome of minimizing a cost function combining the decoding error and the energy cost, Eq.(3). In addition to energy efficiency, there may be a computational reason for thresholding somatic current in neurons. To illustrate this point, we note that the cost function in Eq. (3) for continuous variables, st, may be viewed as a non-negative version of the L1-norm regularized linear regression called LASSO [33], which is commonly used for de-noising of sparse and Laplacian signals [34]. Such cost function can be minimized by iteratively applying a gradient descent and a shrinkage steps [35], which is equivalent to thresholding (one-sided in case of non-negative variables), Figure 6B,C. Therefore, neurons may be encoding a de-noised input signal. Figure 6. Possible reasons for rectification in neurons. A. Cost function combining encoding error squared with metabolic expense vs. input signal for different values of the spike number N, Eq.(4). Note that the optimal number of spikes jumps from zero to one as a function of input. B. Estimating most probable “clean” signal value for continuous non-negative Laplacian signal and Gaussian noise, Eq.(3) (while setting w = 1). The parabolas (red) illustrate the quadratic loglikelihood term in (3) for different values of the measurement, s, while the linear function (blue) reflects the linear log-prior term in (3). C. The minimum of the combined cost function in B is at zero if s , and grows linearly with s, if s >. 5 . Di scu ssi on In this paper, we demonstrated that the neuronal spike-generation mechanism can be viewed as an oversampling and noise-shaping AD converter, which encodes a rectified low-pass filtered somatic current as a digital spike train. Rectification by the spike generation mechanism may subserve both energy efficiency and de-noising. As the degree of noise-shaping in biological neurons exceeds that in IF neurons, or basic , we suggest that neurons should be modeled by more advanced  modulators, e.g. Figure 5B. Interestingly,  modulators can be also viewed as coders with error prediction feedback [19]. Many publications studied various aspects of spike generation in neurons yet we believe that the framework [13-15] we adopt is different and discuss its relationship to some of the studies. Our framework is different from previous proposals to cast neurons as predictors [36, 37] because a different quantity is being predicted. The possibility of perfect decoding from a spike train with infinite temporal precision has been proven in [38]. Here, we are concerned with a more practical issue of how reconstruction error scales with the over-sampling ratio. Also, we consider linear decoding which sets our work apart from [39]. Finally, previous experiments addressing noiseshaping [40] studied the power spectrum of the spike train rather than that of the encoding error. Our work is aimed at understanding biological and computational principles of spike-generation and decoding and is not meant as a substitute for the existing phenomenological spike-generation models [41], which allow efficient fitting of parameters and prediction of spike trains [42]. Yet, the theoretical framework [13-15] we adopt may assist in building better models of spike generation for a given somatic current waveform. First, having interpreted spike generation as AD conversion, we can draw on the rich experience in signal processing to attack the problem. Second, this framework suggests a natural metric to compare the performance of different spike generation models in the high firing rate regime: a mean squared error between the injected current waveform and the filtered version of the spike train produced by a model provided the total number of spikes is the same as in the experimental data. The AD conversion framework adds justification to the previously proposed spike distance obtained by subtracting low-pass filtered spike trains [43]. As the framework [13-15] we adopt relies on viewing neuronal computation as an analog-digital hybrid, which requires AD and DA conversion at every step, one may wonder about the reason for such a hybrid scheme. Starting with the early days of computers, the analog mode is known to be advantageous for computation. For example, performing addition of many variables in one step is possible in the analog mode simply by Kirchhoff law, but would require hundreds of logical gates in the digital mode [44]. However, the analog mode is vulnerable to noise build-up over many stages of computation and is inferior in precisely communicating information over long distances under limited energy budget [30, 31]. 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