nips nips2006 nips2006-17 knowledge-graph by maker-knowledge-mining

17 nips-2006-A recipe for optimizing a time-histogram

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Author: Hideaki Shimazaki, Shigeru Shinomoto

Abstract: The time-histogram method is a handy tool for capturing the instantaneous rate of spike occurrence. In most of the neurophysiological literature, the bin size that critically determines the goodness of the ﬁt of the time-histogram to the underlying rate has been selected by individual researchers in an unsystematic manner. We propose an objective method for selecting the bin size of a time-histogram from the spike data, so that the time-histogram best approximates the unknown underlying rate. The resolution of the histogram increases, or the optimal bin size decreases, with the number of spike sequences sampled. It is notable that the optimal bin size diverges if only a small number of experimental trials are available from a moderately ﬂuctuating rate process. In this case, any attempt to characterize the underlying spike rate will lead to spurious results. Given a paucity of data, our method can also suggest how many more trials are needed until the set of data can be analyzed with the required resolution. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 jp Abstract The time-histogram method is a handy tool for capturing the instantaneous rate of spike occurrence. [sent-8, score-0.512]

2 In most of the neurophysiological literature, the bin size that critically determines the goodness of the ﬁt of the time-histogram to the underlying rate has been selected by individual researchers in an unsystematic manner. [sent-9, score-0.901]

3 We propose an objective method for selecting the bin size of a time-histogram from the spike data, so that the time-histogram best approximates the unknown underlying rate. [sent-10, score-1.127]

4 The resolution of the histogram increases, or the optimal bin size decreases, with the number of spike sequences sampled. [sent-11, score-1.237]

5 It is notable that the optimal bin size diverges if only a small number of experimental trials are available from a moderately ﬂuctuating rate process. [sent-12, score-0.929]

6 In this case, any attempt to characterize the underlying spike rate will lead to spurious results. [sent-13, score-0.56]

7 The shape of a PSTH depends on the choice of the bin size. [sent-16, score-0.569]

8 With too large a bin size, one cannot represent the detailed time-dependent rate, while with too small a bin size, the time-histogram ﬂuctuates greatly and one cannot discern the underlying spike rate. [sent-17, score-1.543]

9 There exists an ideal bin size for estimating the spike rate for each set of experimental data. [sent-18, score-1.132]

10 We devised a method of selecting the bin size objectively so that a PSTH best approximates the underlying rate, which is unknown. [sent-20, score-0.788]

11 In the course of our study, we found an interesting paper that proposed an empirical method of choosing the histogram bin size for a probability density function (Rudemo M, (1982) Scandinavian Journal of Statistics 9: 65-78 [3]). [sent-21, score-0.734]

12 Given a set of experimental data, we wish to not only determine the optimal bin size, but also estimate how many more experimental trials should be performed in order to obtain a resolution we deem sufﬁcient. [sent-24, score-0.687]

13 It was revealed by a theoretical analysis that the optimal bin size may diverge for a small number of spike sequences derived from a moderately ﬂuctuating rate [4]. [sent-25, score-1.469]

14 This implies that any attempt to characterize the underlying rate will lead to spurious results. [sent-26, score-0.221]

15 The present method can indicate the divergence of the optimal bin size only from the spike data. [sent-27, score-1.072]

16 2 Methods We consider sequences of spikes repeatedly recorded from identical experimental trials. [sent-29, score-0.386]

17 A recent analysis revealed that in vivo spike trains are not simply random, but possess inter-spike-interval distributions intrinsic and speciﬁc to individual neurons [5, 6]. [sent-30, score-0.467]

18 However, spikes accumulated from a large number of spike trains recorded from a single neuron are, in the majority, mutually independent. [sent-31, score-0.588]

19 Being free from the intrinsic inter-spike-interval distributions of individual spike trains, the accumulated spikes can be regarded as being derived repeatedly from Poisson processes of an identical time-dependent rate [7, 8]. [sent-32, score-0.728]

20 We suggest a method for minimizing the MISE with respect to the bin size ∆. [sent-34, score-0.672]

21 The difﬁculty of the present problem comes from the fact that the underlying spike rate λt is not known. [sent-35, score-0.531]

22 1 Selection of the bin size ˆ We choose the (bar-graph) PSTH as a way to estimate the rate λt , and explore a method to select the bin size of a PSTH that minimizes MISE in Eq. [sent-37, score-1.443]

23 A PSTH is constructed simply by counting the number of spikes that belong to each bin. [sent-39, score-0.165]

24 The number of spikes accumulated from all n sequences in the ith interval is counted as ki . [sent-41, score-0.524]

25 The bar height at the ith bin is given by ki /n∆. [sent-42, score-0.73]

26 Given a bin of width ∆, the expected height of a bar graph for t ∈ [0, ∆] is the time-averaged rate, 1 ∆ λt dt. [sent-43, score-0.687]

27 (2) ∆ 0 The total number of spikes k from n spike sequences that enter a bin of width ∆ obeys a Poisson distribution with the expected number n∆θ, θ= k p(k | n∆θ) = (n∆θ) −n∆θ e . [sent-44, score-1.366]

28 (3) ˆ The unbiased estimator for θ is given as θ = k/(n∆), which is the empirical height of the bar graph for t ∈ [0, ∆]. [sent-46, score-0.225]

29 By segmenting the total observation period T into N intervals of size ∆, the MISE deﬁned in Eq. [sent-47, score-0.181]

30 Hereafter we denote the average over those segmented rate λt+(i−1)∆ as an average over an ensemble of (segmented) rate functions {λt } deﬁned in an interval of t ∈ [0, ∆]: MISE = 1 ∆ ∆ 0 ˆ E ( θ − λt )2 dt. [sent-49, score-0.303]

31 (5) Table 1: A method for bin size selection for a PSTH (i) Divide the observation period T into N bins of width ∆, and count the number of spikes ki from all n sequences that enter the ith bin. [sent-50, score-1.345]

32 (ii) Construct the mean and variance of the number of spikes {ki } as, N N 1 1 ¯ ¯ k≡ ki , and v ≡ (ki − k)2 . [sent-51, score-0.25]

33 N i=1 N i=1 (iii) Compute the cost function, Cn (∆) = ¯ 2k − v . [sent-52, score-0.158]

34 (n∆)2 Repeat i through iii while changing the bin size ∆ to search for ∆∗ that minimizes Cn (∆). [sent-53, score-0.645]

35 (iv) ˆ The expectation E now refers to the average over the spike count, or θ = k/(n∆), given a rate function λt , or its mean value, θ. [sent-54, score-0.517]

36 (6) 0 ˆ The ﬁrst and second terms are respectively the stochastic ﬂuctuation of the estimator θ around the expected mean rate θ, and the temporal ﬂuctuation of λt around its mean θ over an interval of length ∆, averaged over the segments. [sent-56, score-0.319]

37 (6) can further be decomposed into two parts, 1 ∆ ∆ (λt − θ + θ − θ)2 dt = 0 1 ∆ ∆ (λt − θ ) 2 dt − (θ − θ ) 2 . [sent-58, score-0.247]

38 (7) represents a mean squared ﬂuctuation of the underlying rate λt from the mean rate θ , and is independent of the bin size ∆, because 1 ∆ ∆ 2 (λt − θ ) dt = 0 1 T T 2 (λt − θ ) dt. [sent-60, score-1.139]

39 (8) 0 We deﬁne a cost function by subtracting this term from the original MISE, Cn (∆) ≡ MISE − = 1 ∆ ∆ (λt − θ ) 2 dt 0 ˆ E(θ − θ)2 − (θ − θ ) 2 . [sent-61, score-0.27]

40 (9) This cost function corresponds to the “risk function” in the report by Rudemo, (Eq. [sent-62, score-0.158]

41 (9) represents the temporal ﬂuctuation of the expected mean rate θ for individual intervals of period ∆. [sent-66, score-0.255]

42 As the expected mean rate is not an observable quantity, we must replace the ﬂuctuation of the expected mean rate with that of the ˆ ˆ observable estimator θ. [sent-67, score-0.484]

43 Using the decomposition rule for an unbiased estimator (E θ = θ), ˆ ˆ ˆ ˆ E(θ − E θ )2 = E(θ − θ + θ − θ )2 = E(θ − θ)2 + (θ − θ ) 2 , (10) the cost function is transformed into ˆ ˆ ˆ Cn (∆) = 2 E(θ − θ)2 − E(θ − E θ )2 . [sent-68, score-0.286]

44 (11) Due to the assumed Poisson nature of the point process, the number of spikes k counted in each bin obeys a Poisson distribution: the variance of k is equal to the mean. [sent-69, score-0.771]

45 For the estimated rate deﬁned ˆ as θ = k/(n∆), this variance-mean relation corresponds to ˆ E(θ − θ)2 = 1 ˆ E θ. [sent-70, score-0.176]

46 (11), the cost function is given as a function of the estimator θ, 2 ˆ ˆ ˆ E θ − E(θ − E θ )2 . [sent-73, score-0.242]

47 n∆ Cn (∆) = (13) The optimal bin size is obtained by minimizing the cost function Cn (∆): ∆∗ ≡ arg min Cn (∆). [sent-74, score-0.838]

48 (13) with the sample spike counts, the method is converted into a user-friendly recipe summarized in Table 1. [sent-76, score-0.432]

49 2 Extrapolation of the cost function With the method developed in the preceding subsection, we can determine the optimal bin size for a given set of experimental data. [sent-78, score-0.887]

50 In this section, we develop a method to estimate how the optimal bin size decreases when more experimental trials are added to the data set. [sent-79, score-0.768]

51 Assume that we are in possession of n spike sequences. [sent-80, score-0.359]

52 The ﬂuctuation of the expected mean rate ˆ (θ − θ )2 in Eq. [sent-81, score-0.18]

53 (10) is replaced with the empirical ﬂuctuation of the time-histogram θn using the ˆ ˆ decomposition rule for the unbiased estimator θn satisfying E θn = θ, ˆ ˆ ˆ ˆ E(θn − E θn )2 = E(θn − θ + θ − θ )2 = E(θn − θ)2 + (θ − θ )2 . [sent-82, score-0.161]

54 (15) The expected cost function for m sequences can be obtained by substituting the above equation into Eq. [sent-83, score-0.369]

55 (12), and ˆ E(θm − θ)2 = we obtain Cm (∆|n) = 1 1 ˆ ˆ E θm = E θn , m∆ m∆ 1 1 − m n 1 ˆ E θn + Cn (∆) , ∆ (17) (18) ˆ where Cn (∆) is the original cost function, Eq. [sent-86, score-0.158]

56 By replacing the expectation with sample spike count averages, the cost function for m sequences can be extrap¯ olated as Cm (∆|n) with this formula, using the sample mean k and variance v of the numbers of spikes, given n sequences and the bin size ∆. [sent-88, score-1.614]

57 It may come to pass that the original cost function Cn (∆) computed for n spike sequences does not have a minimum, or have a minimum at a bin size comparable to the observation period T . [sent-90, score-1.457]

58 In such a case, with the method summarized in Table 2, one may estimate the critical number of sequences nc above which the cost function has a ﬁnite bin size ∆∗ , and consider carrying out more experiments to obtain a reasonable rate estimation. [sent-91, score-1.487]

59 In the case that the optimal bin size exhibits continuous divergence, the cost function can be expanded as Cn (∆) ∼ µ 1 1 − n nc 1 1 + u 2, ∆ ∆ (19) where we have introduced nc and u, which are independent of n. [sent-92, score-1.324]

60 The optimal bin size undergoes a phase transition from the vanishing 1/∆∗ for n < nc to a ﬁnite 1/∆∗ for n > nc . [sent-93, score-1.146]

61 In this case, the inverse optimal bin size is expanded in the vicinity of nc as 1/∆∗ ∝ (1/n − 1/nc ). [sent-94, score-0.917]

62 We can Table 2: A method for extrapolating the cost function for a PSTH (A) Construct the extrapolated cost function, ¯ 1 1 k Cm (∆|n) = − + Cn (∆), m n n∆2 ¯ using the sample mean k and variance v of the number of spikes obtained from n sequences of spikes. [sent-95, score-0.843]

63 m (C) Repeat A and B while changing m, and plot 1/∆∗ vs 1/m to search for m the critical value 1/m = 1/ˆ c above which 1/∆∗ practically vanishes. [sent-97, score-0.187]

64 n m ˆm estimate the critical value nc by applying this asymptotic relation to the set of ∆∗ estimated from ˆ Cm (∆|n) for various values of m: 1 1 − m nc ˆ 1 ∝ ∆∗ m . [sent-98, score-0.573]

65 (20) It should be noted that there are cases that the optimal bin size exhibits discontinuous divergence from a ﬁnite value. [sent-99, score-0.772]

66 Even in such cases, the plot of {1/m, 1/∆∗ } could be useful in exploring a discontinuous transition from nonvanishing values of 1/∆∗ to practically vanishing values. [sent-100, score-0.16]

67 3 Theoretical cost function In this section, we obtain a “theoretical” cost function directly from a process with a known underlying rate, λt , and compare it with the “empirical” cost function which can be evaluated without knowing the rate process. [sent-102, score-0.666]

68 Note that this theoretical cost function is not available in real experimental conditions in which the underlying rate is not known. [sent-103, score-0.416]

69 ˆ The present estimator θ ≡ k/(n∆) is a uniformly minimum variance unbiased estimator (UMVUE) of θ, which achieves the lower bound of the Cram´ r-Rao inequality [9, 10], e ∞ ˆ E(θ − θ) = − 2 k=0 ∂ 2 log p (k|θ) p (k|θ) ∂θ2 −1 = θ . [sent-104, score-0.212]

70 (9), the cost function is represented as Cn (∆) = = θ 2 − (θ − θ ) n∆ ∆ ∆ µ 1 − 2 φ (t1 − t2 ) dt1 dt2 , n∆ ∆ 0 0 (22) where µ is the mean rate, and φ(t) is the autocorrelation function of the rate ﬂuctuation, λt − µ. [sent-106, score-0.316]

71 Based on the symmetry φ(t) = φ(−t), the cost function can be rewritten as Cn (∆) = ≈ µ 1 − 2 n∆ ∆ µ 1 − n∆ ∆ ∆ (∆ − |t|)φ(t) dt −∆ ∞ φ(t) dt + −∞ 1 ∆2 ∞ |t|φ(t) dt, (23) −∞ which can be identiﬁed with Eq. [sent-107, score-0.382]

72 (19) with parameters given by ∞ nc = µ φ(t) dt , (24) −∞ ∞ u = |t|φ(t) dt. [sent-108, score-0.325]

73 −∞ (25) B Underlying rate, t 60 30 0 0 A Cost function, 100 2 Spike Sequences C 200 1 Empirical cost function 0 1 2 Theoretical cost function ^ Histograms, D 0 t 120 60 0 -100 60 30 0 0. [sent-109, score-0.316]

74 5 0 60 30 0 0 1 2 Time, t Figure 1: A: (Dots): The empirical cost function, Cn (∆), computed from spike data according to the method in Table 1. [sent-114, score-0.578]

75 (Solid line): The “theoretical” cost function computed directly from the underlying ﬂuctuating rate, with Eq. [sent-115, score-0.245]

76 (Below): Time-histograms made using three types of bin sizes: too small, optimal, and too large. [sent-119, score-0.569]

77 Model parameters: the number of sequences n = 30; total observation period T = 30 [sec]; the mean rate µ = 30 [1/s]; the amplitude of rate ﬂuctuation σ = 10 [1/s]; time scale of rate ﬂuctuation τ = 0. [sent-120, score-0.728]

78 3 Results Our ﬁrst objective was to develop a method for selecting the ideal bin size using spike sequences derived repeatedly from Poisson processes, all with a given identical rate λt . [sent-122, score-1.401]

79 The MISE of the PSTH from the underlying rate is minimized by minimizing the cost function Cn (∆). [sent-123, score-0.35]

80 Figure 1A displays the cost function computed with the method summarized in Table 1. [sent-124, score-0.238]

81 This “empirical” cost function is compared with the “theoretical” cost function Eq. [sent-125, score-0.316]

82 (22) that is computed directly from the underlying rate λt . [sent-126, score-0.213]

83 The ﬁgure exhibits that the “empirical” cost function is consistent with the “theoretical” cost function. [sent-127, score-0.352]

84 The time-histogram constructed using the optimal bin size is compared with those constructed using non-optimal bin sizes in Figs. [sent-128, score-1.295]

85 1B, demonstrating the effectiveness of the present method of bin size selection. [sent-129, score-0.672]

86 We also tested a method for extrapolating the cost function. [sent-130, score-0.231]

87 Figures 2A and B demonstrate the extrapolated cost functions for several sequences with differing values of m and the plot of {1/m, 1/∆∗ } for estimating the critical value 1/m = 1/ˆ c , above which 1/∆∗ practically vann ishes. [sent-131, score-0.625]

88 Figure 2C depicts the critical number nc estimated from the smaller or larger numbers of ˆ spike sequences n. [sent-132, score-0.867]

89 The empirically estimated critical number nc approximates the theoretically ˆ predicted critical number nc computed using Eq. [sent-133, score-0.699]

90 Note that the critical number is correctly estimated from the small number of sequences, with which the optimal bin size practically diverges (n < nc ). [sent-135, score-1.122]

91 4 Summary We have developed a method for optimizing the bin size, so that the PSTH best represents the (unknown) underlying spike rate. [sent-136, score-1.001]

92 For a small number of spike sequences derived from a modestly Extrapolated cost function Estimated optimal bin size B A 10 12 m=10 5 6 m=20 0 m=30 -5 0 0 2 4 6 8 0. [sent-137, score-1.387]

93 1 1/m ^ Estimated critical number, nc C 40 30 20 10 0 0 10 20 30 40 Number of sequences, n Figure 2: A: Extrapolated cost functions Cm (∆|n) plotted against 1/∆ for several numbers of sequences m = 10, 20 and 30 computed from n = 10 sample sequences. [sent-140, score-0.678]

94 B: The plot of {1/m, 1/∆∗ } used for estimating the critical value 1/m = 1/ˆ c , above which 1/∆∗ practically vanishes. [sent-141, score-0.187]

95 C: The n number of spike sequences n used to obtain the extrapolated cost function Cm (∆|n) and an estimated critical number nc . [sent-142, score-1.116]

96 Model parameters: the number of sequences n = 10; total observation ˆ period T = 30 [sec]; the mean rate µ = 30 [1/s]; the amplitude of rate ﬂuctuation σ = 4 [1/s]; time scale of rate ﬂuctuation τ = 0. [sent-143, score-0.728]

97 This theoretical nc is depicted as the horizontal dashed line. [sent-148, score-0.257]

98 ﬂuctuating rate, the cost function does not have a minimum, implying the uselessness of the rate estimation. [sent-149, score-0.284]

99 Our method can nevertheless extrapolate the cost function for any number of spike sequences, and suggest how many trials are needed in order to obtain a meaningful time-histogram with the required accuracy. [sent-150, score-0.563]

100 The suitability of the present method was demonstrated by application to spike sequences generated by time-dependent Poisson processes. [sent-151, score-0.555]

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