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104 nips-2007-Inferring Neural Firing Rates from Spike Trains Using Gaussian Processes

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Author: Maneesh Sahani, Byron M. Yu, John P. Cunningham, Krishna V. Shenoy

Abstract: Neural spike trains present challenges to analytical efforts due to their noisy, spiking nature. Many studies of neuroscientiﬁc and neural prosthetic importance rely on a smoothed, denoised estimate of the spike train’s underlying ﬁring rate. Current techniques to ﬁnd time-varying ﬁring rates require ad hoc choices of parameters, offer no conﬁdence intervals on their estimates, and can obscure potentially important single trial variability. We present a new method, based on a Gaussian Process prior, for inferring probabilistically optimal estimates of ﬁring rate functions underlying single or multiple neural spike trains. We test the performance of the method on simulated data and experimentally gathered neural spike trains, and we demonstrate improvements over conventional estimators. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk 3 Abstract Neural spike trains present challenges to analytical efforts due to their noisy, spiking nature. [sent-8, score-0.821]

2 Many studies of neuroscientiﬁc and neural prosthetic importance rely on a smoothed, denoised estimate of the spike train’s underlying ﬁring rate. [sent-9, score-0.622]

3 Current techniques to ﬁnd time-varying ﬁring rates require ad hoc choices of parameters, offer no conﬁdence intervals on their estimates, and can obscure potentially important single trial variability. [sent-10, score-0.207]

4 We present a new method, based on a Gaussian Process prior, for inferring probabilistically optimal estimates of ﬁring rate functions underlying single or multiple neural spike trains. [sent-11, score-0.713]

5 We test the performance of the method on simulated data and experimentally gathered neural spike trains, and we demonstrate improvements over conventional estimators. [sent-12, score-0.677]

6 Even when experimental conditions are repeated closely, the same neuron may produce quite different spike trains from trial to trial. [sent-14, score-0.812]

7 This variability may be due to both randomness in the spiking process and to differences in cognitive processing on different experimental trials. [sent-15, score-0.089]

8 One common view is that a spike train is generated from a smooth underlying function of time (the ﬁring rate) and that this function carries a signiﬁcant portion of the neural information. [sent-16, score-0.66]

9 If this is the case, questions of neuroscientiﬁc and neural prosthetic importance may require an accurate estimate of the ﬁring rate. [sent-17, score-0.074]

10 Unfortunately, these estimates are complicated by the fact that spike data gives only a sparse observation of its underlying rate. [sent-18, score-0.571]

11 Typically, researchers average across many trials to ﬁnd a smooth estimate (averaging out spiking noise). [sent-19, score-0.118]

12 However, averaging across many roughly similar trials can obscure important temporal features [1]. [sent-20, score-0.094]

13 Thus, estimating the underlying rate from only one spike train (or a small number of spike trains believed to be generated from the same underlying rate) is an important but challenging problem. [sent-21, score-1.509]

14 The most common approach to the problem has been to collect spikes from multiple trials in a peristimulus-time histogram (PSTH), which is then sometimes smoothed by convolution or splines [2], [3]. [sent-22, score-0.164]

15 Bin sizes and smoothness parameters are typically chosen ad hoc (but see [4], [5]) and the result is fundamentally a multi-trial analysis. [sent-23, score-0.077]

16 An alternative is to convolve a single spike train with a kernel. [sent-24, score-0.584]

17 Again, the kernel shape and time scale are frequently ad hoc. [sent-25, score-0.099]

18 1 More recently, point process likelihood methods have been adapted to spike data [6]–[8]. [sent-28, score-0.542]

19 These methods optimize (implicitly or explicitly) the conditional intensity function λ(t|x(t), H(t)) — which gives the probability of a spike in [t, t + dt), given an underlying rate function x(t) and the history of previous spikes H(t) — with respect to x(t). [sent-29, score-0.713]

20 In a regression setting, this rate x(t) may be learned as a function of an observed covariate, such as a sensory stimulus or limb movement. [sent-30, score-0.082]

21 In other ﬁelds, a theory of generalized Cox processes has been developed, where the point process is conditionally Poisson, and x(t) is obtained by applying a link function to a draw from a random process, often a Gaussian process (GP) (e. [sent-39, score-0.105]

22 However, the link function, which ensures a nonnegative intensity, introduces possibly undesirable artifacts. [sent-43, score-0.09]

23 For instance, an exponential link leads to a process that grows less smooth as the intensity increases. [sent-44, score-0.12]

24 First, we adapt the theory of GP-driven point processes to incorporate a history-dependent conditional likelihood, suitable for spike trains. [sent-46, score-0.509]

25 We show that GP methods employing evidence optimization outperform both kernel smoothing and maximum-likelihood point process models. [sent-49, score-0.104]

26 2 Gaussian Process Model For Spike Trains Spike trains can often be well modelled by gamma-interval point processes [6], [10]. [sent-50, score-0.256]

27 We assume the underlying nonnegative ﬁring rate x(t) : t ∈ [0, T ] is a draw from a GP, and then we assume that our spike train is a conditionally inhomogeneous gamma-interval process (IGIP), given x(t). [sent-51, score-0.894]

28 The spike train is represented by a list of spike times y = {y0 , . [sent-52, score-1.093]

29 [6]) can be written as: p(yi | yi−1 , x(t)) = γx(yi ) γ Γ(γ) γ−1 yi x(u)du yi−1 yi exp −γ x(u)du . [sent-58, score-0.15]

30 (2) yi−1 The true p0 (·) and pT (·) under this gamma-interval spiking model are not closed form, so we simplify these distributions as intervals of an inhomogeneous Poisson process (IP). [sent-59, score-0.242]

31 The discretized IGIP output process is now (ignoring terms that scale with ∆): 1 The IGIP is one of a class of renewal models that works well for spike data (much better than inhomogeneous Poisson; see [6], [10]). [sent-67, score-0.738]

32 Other log-concave renewal models such as the inhomogeneous inverse-Gaussian interval can be chosen, and the implementation details remain unchanged. [sent-68, score-0.196]

33 2 N p(y | x) = i=1 γxyi γ Γ(γ) yi −1 yi −1 γ−1 exp −γ xk ∆ k=yi−1 y0 −1 · xy0 exp − xk ∆ k=yi−1 n−1 xk ∆ · exp − k=0 (3) xk ∆ , k=yN where the ﬁnal two terms are p0 (·) and pT (·), respectively [11]. [sent-69, score-0.354]

34 Our goal is to estimate a smoothly varying ﬁring rate function from spike times. [sent-70, score-0.591]

35 Since our intensity function is nonnegative, however, it is sensible to treat µ instead as a hyperparameter and let it be optimized to a positive value. [sent-79, score-0.105]

36 As written, the model assumes only one observed spike train; it may be that we have m trials believed to be generated from the same ﬁring rate proﬁle. [sent-82, score-0.653]

37 Our method naturally incorporates this case: deﬁne m p(y(i) | x), where y(i) denotes the ith spike train observed. [sent-83, score-0.584]

38 1 Finding an Optimal Firing Rate Estimate Algorithmic Approach Ideally, we would calculate the posterior on ﬁring rate p(x | y) = θ p(x | y, θ)p(θ)dθ (integrating over the hyperparameters θ), but this problem is intractable. [sent-86, score-0.146]

39 We consider two approximations: replacing the integral by evaluation at the modal θ, and replacing the integral with a sum over a discrete grid of θ values. [sent-87, score-0.099]

40 We ﬁrst consider choosing a modal hyperparameter set (ML-II model selection, see [12]), i. [sent-88, score-0.13]

42 Below we show this integrand is log concave in x. [sent-98, score-0.071]

43 Further, since we are modelling a non-zero mean GP, most of the Laplace approximated probability mass lies in the nonnegative orthant (as is the case with the true posterior). [sent-100, score-0.117]

44 Accordingly, we write: 2 Another reasonable approach would consider each trial as having a different rate function x that is a draw from a GP with a nonstationary mean function µ(t). [sent-101, score-0.156]

45 Instead of inferring a mean rate function x ∗ , we would learn a distribution of means. [sent-102, score-0.105]

46 The Laplace approximation also naturally provides conﬁdence intervals from the approximated posterior covariance (Σ −1 + Λ∗ )−1 . [sent-109, score-0.072]

48 This approximate integration consistently yields better results than a modal hyperparameter set, so we will only consider approximate integration for the remainder of this report. [sent-115, score-0.18]

49 For the Laplace approximation at any value of θ, we require the modal estimate of ﬁring rate x ∗ , which is simply the MAP estimator: x∗ = argmax p(x | y) = argmax p(y | x)p(x). [sent-116, score-0.245]

50 Typically a link or squashing function would be included to enforce nonnegativity in x, but this can distort the intensity space in unintended ways. [sent-118, score-0.116]

51 Accordingly, we can use a log barrier Newton method to efﬁciently solve for the global MAP estimator of ﬁring rate x∗ [15]. [sent-145, score-0.112]

52 In the case of multiple spike train observations, we need only add extra terms of negative log likelihood from the observation model. [sent-146, score-0.614]

53 With 1 ms time resolution (or similar), this method would be restricted to spike trains lasting less than a second, and even this problem would be burdensome. [sent-158, score-0.837]

54 Since we have evenly spaced resolution of our data x in time indices ti , Σ is Toeplitz; thus multiplication by Σ can be done using Fast Fourier Transform (FFT) methods [16]. [sent-165, score-0.084]

55 For the modal hyperparameter scheme (as opposed to approximately integrating over the hyperparameters), gradients of Eq. [sent-168, score-0.153]

56 These data sizes translate to seconds of spike data at 1 ms resolution, long enough for most electrophysiological trials. [sent-172, score-0.551]

57 4 Results We tested the methods developed here using both simulated neural data, where the true ﬁring rate was known by construction, and in real neural spike trains, where the true ﬁring rate was estimated by a PSTH that averaged many similar trials. [sent-174, score-0.828]

58 We compared the IGIP-likelihood GP method (hereafter, GP IGIP) to other rate estimators (kernel smoothers, Bayesian Adaptive Regressions Splines or BARS [3], and variants of the GP method) using root mean squared difference (RMS) to the true ﬁring rate. [sent-177, score-0.082]

59 PSTH and kernel methods approximate the mean conditional intensity λ(t) = EH(t) [λ(t|x(t), H(t))]. [sent-178, score-0.119]

60 For a renewal process, we know (by the time rescaling theorem [7], [11]) that λ(t) = x(t), and thus we can compare the GP IGIP (which ﬁnds x(t)) directly to the kernel methods. [sent-179, score-0.162]

61 To conﬁrm that hyperparameter optimization improves performance, we also compared GP IGIP results to maximum likelihood (ML) estimates of x(t) using ﬁxed hyperparameters θ. [sent-180, score-0.12]

62 To evaluate the importance of an observation model with spike history dependence (the IGIP of Eq. [sent-182, score-0.509]

63 3), we also compared GP IGIP to an inhomogeneous Poisson (GP IP) observation model (again with a GP prior on x(t); simply γ = 1 in Eq. [sent-183, score-0.105]

64 For the approximate integration, we chose a grid consisting of the empirical mean rate for µ (that is, total spike count N divided by total time T ) 2 and (γ, log(σf ), log(κ)) ∈ [1, 2, 4] × [4, . [sent-193, score-0.617]

65 1; 4 spike trains 50 16 14 40 Firing Rate (spikes/sec) Firing Rate (spikes/sec) 45 35 30 25 20 15 10 10 8 6 4 2 5 0 12 0 0. [sent-219, score-0.765]

66 3; 1 spike train Figure 1: Sample GP ﬁring rate estimate. [sent-237, score-0.666]

67 1 represent experimentally gathered ﬁring rate proﬁles (according to the methods in [17]). [sent-240, score-0.129]

68 In each of the plots, the empirical average ﬁring rate of the spike trains is shown in bold red. [sent-241, score-0.887]

69 For simulated spike trains, the spike trains were generated from each of these empirical average ﬁring rates using an IGIP (γ = 4, comparable to ﬁts to real neural data). [sent-242, score-1.42]

70 For real neural data, the spike train(s) were selected as a subset of the roughly 200 experimentally recorded spike trains that were used to construct the ﬁring rate proﬁle. [sent-243, score-1.447]

71 These spike trains are shown as a train of black dots, each dot indicating a spike event time (the y-axis position is not meaningful). [sent-244, score-1.349]

72 This spike train or group of spike trains is the only input given to each of the ﬁtting models. [sent-245, score-1.349]

73 In thin green and magenta, we have two kernel smoothed estimates of ﬁring rates; each represents the spike trains convolved with a normal distribution of a speciﬁed standard deviation (50 and 100 ms). [sent-246, score-0.894]

74 We also smoothed these spike trains with adaptive kernel [18], ﬁxed ML (as described above), BARS [3], and 150 ms kernel smoothers. [sent-247, score-1.015]

75 The light blue envelopes around the bold blue GP ﬁring rate estimate represent the 95% conﬁdence intervals. [sent-252, score-0.172]

76 We then ran all methods 100 times on each ﬁring rate proﬁle, using (separately) simulated and real neural spike trains. [sent-256, score-0.709]

77 The four panels correspond to the same rate proﬁles shown in Fig. [sent-262, score-0.108]

78 In each panel, the top, middle, and bottom bar graphs correspond to the method on 1, 4, and 8 spike trains, respectively. [sent-264, score-0.509]

79 6 50 GP Methods GP IP Fixed ML short Kernel Smoothers medium long adaptive BARS 50 % 0 50 % % 0 BARS % 0 50 Kernel Smoothers medium long adaptive 0 50 % short % 0 50 GP Methods GP IP Fixed ML 0 (a) L20061107. [sent-271, score-0.208]

80 1; 1,4,8 spike trains 50 GP Methods GP IP Fixed ML short Kernel Smoothers medium long adaptive BARS (b) L20061107. [sent-273, score-0.869]

81 1; 1,4,8 spike trains 50 % 0 50 % % 0 BARS % 0 50 Kernel Smoothers medium long adaptive 0 50 % short % 0 50 GP Methods GP IP Fixed ML 0 (c) L20061107. [sent-275, score-0.869]

82 3; 1,4,8 spike trains Figure 2: Average percent RMS improvement of GP IGIP method (with model selection) vs. [sent-279, score-0.765]

83 Blue improvement bars are for simulated spike trains; red improvement bars are for real neural spike trains. [sent-282, score-1.252]

84 the long bandwidth kernel in panel (b), real spike trains, 4 and 8 spike trains), outperforming the GP IGIP method requires an optimal kernel choice, which can not be judged from the data alone. [sent-286, score-1.218]

85 In particular, the adaptive kernel method generally performed more poorly than GP IGIP. [sent-287, score-0.102]

86 5 Discussion We have demonstrated a new method that accurately estimates underlying neural ﬁring rate functions and provides conﬁdence intervals, given one or a few spike trains as input. [sent-291, score-0.946]

87 Estimating underlying ﬁring rates is especially challenging due to the inherent noise in spike trains. [sent-293, score-0.576]

88 Having only a few spike trains deprives the method of many trials to reduce spiking noise. [sent-294, score-0.883]

89 It is important here to remember why we care about single trial or small number of trial estimates, since we believe that in general the neural processing on repeated trials is not identical. [sent-295, score-0.193]

90 In this study we show both simulated and real neural spike trains. [sent-297, score-0.627]

91 Simulated data provides a good test environment for this method, since the underlying ﬁring rate is known, but it lacks the experimental proof of real neural spike trains (where spiking does not exactly follow a gamma-interval process). [sent-298, score-1.009]

92 Taken together, however, we believe the real and simulated data give good evidence of the general improvements offered by this method. [sent-300, score-0.114]

93 In simulation, GP IGIP generally outperforms the other smoothers (though, by considerably less than in other panels). [sent-304, score-0.087]

94 This may indicate that, in the low ﬁring rate regime, the IGIP is a poor model for real neural spiking. [sent-306, score-0.149]

95 Furthermore, some neural spike trains may be inherently ill-suited to analysis. [sent-309, score-0.802]

96 With spike trains of only a few spikes/sec, it will be impossible for any method to ﬁnd interesting structure in the ﬁring rate. [sent-311, score-0.765]

97 Several studies have investigated the inhomogeneous gamma and other more general models (e. [sent-313, score-0.105]

98 [6], [19]), including the inhomogeneous inverse gaussian (IIG) interval and inhomogeneous Markov interval (IMI) processes. [sent-315, score-0.21]

99 The methods of this paper apply immediately to any log-concave inhomogeneous renewal process in which inhomogeneity is generated by time-rescaling (this includes the IIG and several others). [sent-316, score-0.229]

100 Another direction for this work is to consider signiﬁcant nonstationarity in the spike data. [sent-318, score-0.509]

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