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305 nips-2013-Spectral methods for neural characterization using generalized quadratic models

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Author: Il M. Park, Evan W. Archer, Nicholas Priebe, Jonathan W. Pillow

Abstract: We describe a set of fast, tractable methods for characterizing neural responses to high-dimensional sensory stimuli using a model we refer to as the generalized quadratic model (GQM). The GQM consists of a low-rank quadratic function followed by a point nonlinearity and exponential-family noise. The quadratic function characterizes the neuron’s stimulus selectivity in terms of a set linear receptive ﬁelds followed by a quadratic combination rule, and the invertible nonlinearity maps this output to the desired response range. Special cases of the GQM include the 2nd-order Volterra model [1, 2] and the elliptical Linear-Nonlinear-Poisson model [3]. Here we show that for “canonical form” GQMs, spectral decomposition of the ﬁrst two response-weighted moments yields approximate maximumlikelihood estimators via a quantity called the expected log-likelihood. The resulting theory generalizes moment-based estimators such as the spike-triggered covariance, and, in the Gaussian noise case, provides closed-form estimators under a large class of non-Gaussian stimulus distributions. We show that these estimators are fast and provide highly accurate estimates with far lower computational cost than full maximum likelihood. Moreover, the GQM provides a natural framework for combining multi-dimensional stimulus sensitivity and spike-history dependencies within a single model. We show applications to both analog and spiking data using intracellular recordings of V1 membrane potential and extracellular recordings of retinal spike trains. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Spectral methods for neural characterization using generalized quadratic models Il Memming Park∗123 , Evan Archer∗13 , Nicholas Priebe14 , & Jonathan W. [sent-1, score-0.149]

2 edu Abstract We describe a set of fast, tractable methods for characterizing neural responses to high-dimensional sensory stimuli using a model we refer to as the generalized quadratic model (GQM). [sent-10, score-0.337]

3 The GQM consists of a low-rank quadratic function followed by a point nonlinearity and exponential-family noise. [sent-11, score-0.167]

4 The quadratic function characterizes the neuron’s stimulus selectivity in terms of a set linear receptive ﬁelds followed by a quadratic combination rule, and the invertible nonlinearity maps this output to the desired response range. [sent-12, score-0.677]

5 Here we show that for “canonical form” GQMs, spectral decomposition of the ﬁrst two response-weighted moments yields approximate maximumlikelihood estimators via a quantity called the expected log-likelihood. [sent-14, score-0.188]

6 The resulting theory generalizes moment-based estimators such as the spike-triggered covariance, and, in the Gaussian noise case, provides closed-form estimators under a large class of non-Gaussian stimulus distributions. [sent-15, score-0.544]

7 We show that these estimators are fast and provide highly accurate estimates with far lower computational cost than full maximum likelihood. [sent-16, score-0.112]

8 Moreover, the GQM provides a natural framework for combining multi-dimensional stimulus sensitivity and spike-history dependencies within a single model. [sent-17, score-0.373]

9 We show applications to both analog and spiking data using intracellular recordings of V1 membrane potential and extracellular recordings of retinal spike trains. [sent-18, score-0.522]

10 1 Introduction Although sensory stimuli are high-dimensional, sensory neurons are typically sensitive to only a small number of stimulus features. [sent-19, score-0.485]

11 These ﬁlters, which describe how the stimulus is integrated over space and time, can be considered the ﬁrst stage in a “cascade” model of neural responses. [sent-21, score-0.368]

12 In the well-known linear-nonlinear-Poisson (LNP) cascade model, ﬁlter outputs are combined via a nonlinear function to produce an instantaneous spike rate, which generates spikes via an inhomogeneous Poisson process [4, 5]. [sent-22, score-0.277]

13 The most popular methods for dimensionality reduction with spike train data involve the ﬁrst two moments of the spike-triggered stimulus distribution: (1) the spike-triggered average (STA) [7–9]; and (2) major and minor eigenvectors of spike-triggered covariance (STC) matrix [10, 11]1 . [sent-23, score-0.679]

14 Related moment-based estimators have also appeared in the statistics literature under the names “inverse regression” and “sufﬁcient dimensionality reduction”, although the connection to STA and STC analysis does not appear to have been noted previously [12, 13]. [sent-25, score-0.108]

15 1 1 Generalized Quadratic Model linear filters nonlinear quadratic function noise analog or recurrent filters stimulus spikes . [sent-26, score-0.638]

16 response Figure 1: Schematic of generalized quadratic model (GQM) for analog or spike train data. [sent-29, score-0.403]

17 Recently, Park and Pillow [3] described a connection between STA/STC analysis and maximum likelihood estimators based on a quantity called the expected log-likelihood (EL). [sent-32, score-0.107]

18 The EL results from replacing the nonlinear term in the log-likelihood and with its expectation over the stimulus distribution. [sent-33, score-0.374]

19 When the stimulus is Gaussian, the EL depends only on moments (mean spike rate, STA, STC, and stimulus mean and covariance) and leads to a closed-form spectral estimate for LNP ﬁlters, which has STC analysis as a special case. [sent-34, score-0.917]

20 More recently, Ramirez and Paninski derived ELbased estimators for the linear Gaussian model and proposed fast EL-based inference methods for generalized linear models (GLMs) [14]. [sent-35, score-0.106]

21 Here, we show that the EL framework can be extended to a more general class that we refer to as the generalized quadratic model (GQM). [sent-36, score-0.12]

22 The GQM represents a straightforward extension of the generalized linear model GLM [15, 16] wherein the linear predictor is replaced by a quadratic function (Fig. [sent-37, score-0.12]

23 For Gaussian and Poisson GQMs, we derive computationally efﬁcient EL-based estimators that apply to a variety of non-Gaussian stimulus distributions; this substantially extends previous work on the conditions of validity for moment-based estimators [7,17–19]. [sent-39, score-0.497]

24 In the Gaussian case, the EL-based estimator has a closed form solution that relies only on the ﬁrst two responseweighted moments and the ﬁrst four stimulus moments. [sent-40, score-0.379]

25 , where the response depends on multiple projections of the stimulus) and dependencies on spike history. [sent-43, score-0.27]

26 We show that spectral estimates of a low-dimensional feature space are nearly as accurate as maximum likelihood estimates (for GQMs without spike-history), and demonstrate the applicability of GQMs for both analog and spiking data. [sent-44, score-0.216]

27 A GLM has three basic components: a linear stimulus ﬁlter, an invertible nonlinearity (or “inverse link” function), and an exponential-family noise model. [sent-46, score-0.46]

28 The GLM describes the conditional response y to a vector stimulus x as: y|x ∼ P(f (w x)), (1) where w is the ﬁlter, f is the nonlinearity, and P(λ) denotes a noise distribution function with mean λ. [sent-47, score-0.464]

29 From the standpoint of dimensionality reduction, the GLM makes the strong modeling assumption that response y depends upon x only via its one-dimensional projection onto w. [sent-48, score-0.129]

30 (2) Spike-triggered covariance analysis and related methods provide low-cost estimates of the ﬁlters {wi } under Poisson or Bernoulli noise models, but only under restrictive conditions on the stimulus 2 distribution (e. [sent-53, score-0.455]

31 Semi-parametric estimators like “maximally informative dimensions” (MID) eliminate these restrictions [20], but do not practically scale beyond two or three ﬁlters without additional modeling assumptions [21]. [sent-56, score-0.079]

32 The generalized quadratic model (GQM) provides a tractable middle ground between the GLM and general multi-ﬁlter LN models. [sent-57, score-0.12]

33 The GQM allows for multi-dimensional stimulus dependence, yet restricts the nonlinearity to be a transformed quadratic function [22–25]. [sent-58, score-0.506]

34 The GQM can be written: y|x ∼ P(f (Q(x))), (3) where Q(x) = x Cx + b x + a denotes a quadratic function of x, governed by a (possibly lowrank) symmetric matrix C, a vector b, and a scalar a. [sent-59, score-0.093]

35 Note that the GQM may be regarded as a GLM in the space of quadratically transformed stimuli [6], although this approach does not allow Q(x) to be parametrized directly in terms of a projection onto a small number of linear ﬁlters. [sent-60, score-0.122]

36 Finally, we show that the GQM provides a natural framework for combining multi-dimensional stimulus sensitivity with dependencies on spike train history or other response covariates. [sent-63, score-0.672]

37 The expected log-likelihood (EL) results from replacing the nonlinear term with its expectation over the stimulus distribution P (x), which in neurophysiology settings is often known a priori to the experimenter. [sent-67, score-0.402]

38 Maximizing the EL results in maximum expected log-likelihood (MEL) estimators that have very low computational cost while achieving nearly the accuracy of full maximum likelihood (ML) estimators. [sent-68, score-0.107]

39 Spectral decompositions derived from the EL provide estimators that generalize STA/STC analysis. [sent-69, score-0.079]

40 In the following, we derive MEL estimators for three special cases—two for the Gaussian noise model, and one for the Poisson noise model. [sent-70, score-0.173]

41 1 Gaussian GQMs Gaussian noise provides a natural model for analog neural response variables like membrane potential or ﬂuorescence. [sent-72, score-0.278]

42 The canonical nonlinearity for Gaussian noise is the identity function, f (x) = x. [sent-73, score-0.153]

43 2 (5) i=1 1 The expected log-likelihood results from replacing the troublesome nonlinear term N i Q(xi )2 by its expectation over the stimulus distribution. [sent-76, score-0.402]

44 This is justiﬁed by the law of large numbers, which 2 When responses yi are spike counts, these correspond to the STA and STC. [sent-77, score-0.253]

45 y (6) 2σ Gaussian stimuli If the stimuli are drawn from a Gaussian distribution, x ∼ N (0, Σ), then we have (from [26]): E[Q(x)2 ] = 2 Tr (CΣ)2 + Tr(bT Σb) + (Tr(CΣ) + a)2 . [sent-80, score-0.2]

46 Axis-symmetric stimuli More generally, we can derive the MEL estimator for stimuli with arbitrary axis-symmetric distributions with ﬁnite 4th-order moments. [sent-82, score-0.2]

47 a, b, and C, respectively, we obtain conditions for the MEL estimates: y = E [Q(x)] = a + b E[x] + Tr(CE[xx ]) ¯ µ = E [Q(x)x] = aE[x] + b E[xx ] + Cij E[xi xj x] i,j Λ = E Q(x)xx = aE[xx ] + bi E[xi xx ] + i Cij E[xi xj xx ] i,j where the subindices within the sums are for components. [sent-98, score-0.23]

48 If we further assume that the stimulus is whitened so that E[xx ] = I, sufﬁcient stimulus statistics are the 4th order even moments, which we represent with the matrix Mij = E x2 x2 . [sent-101, score-0.678]

49 i j In general, when the marginals are not identical but the joint distribution is axis-symmetric, Cii diag(x2 x2 , · · · , x2 x2 ) + i 1 i d Cij E[xi xj xx ] = ij i Cij Mij ei ej (11) i=j = diag(1 (I ◦ C)M ) + C ◦ M ◦ (11 − I). [sent-102, score-0.136]

50 We can solve these sets of linear equations for the diagonal terms and off-diagonal terms separately obtaining, [Cmel ]ij = Λij 2Mij , −1 Ω(M − 11 ) 4 i=j , i=j (12) 2D stimulus distribution neural response assumed stimulus distribution Gaussian [r2 = 0. [sent-104, score-0.785]

51 99] time Figure 2: Maximum expected log-likelihood (MEL) estimators for a Gaussian GQM under different assumptions about the stimulus distribution. [sent-107, score-0.446]

52 Performance is evaluated on a cross-validation test set with no noise for each C, and we see a huge loss in performance as a result of incorrect assumption about the stimulus distribution. [sent-115, score-0.386]

53 This gives the simpliﬁed formula (also 1 2 1 given in [27]): [Cmel ]ij = Λij 2µ22 , Λii −¯ y µ4 −µ22 , i=j i=j (14) When the stimulus is not Gaussian or the marginals not identical, the estimates obtained from (eq. [sent-118, score-0.372]

54 2 Poisson GQM Poisson noise provides a natural model for discrete events like spike counts, and extends easily to point process models for spike trains. [sent-125, score-0.363]

55 The canonical nonlinearity for Poisson noise is exponential, f (x) = exp(x), so the canonical-form Poisson GQM is: y|x ∼ Poiss(exp(Q(x))). [sent-126, score-0.153]

56 Under a zero-mean Gaussian stimulus distribution with covariance Σ, the closed-form MEL estimates are (from [3]): 1 bmel = Λ + y 2µµ ¯ −1 µ, Cmel = 1 2 1 Σ−1 − y Λ + y 2µµ ¯ ¯ −1 , (16) 1 where we assume that Λ + y 2µµ is invertible. [sent-130, score-0.44]

57 5 filter estimation error optimization time MELE (1st eigenvector) rank−1 MELE rank−1 ML 2 0 seconds L1 error Figure 3: Rank-1 quadratic ﬁlter reconstruction performance. [sent-133, score-0.148]

58 10 -1 10 4 5 10 10 # of samples 1 0 4 5 10 10 # of samples Mixture-of-Gaussians stimuli Results for Gaussian stimuli extend naturally to mixtures of Gaussians, which can be used to approximate arbitrary stimulus distributions. [sent-137, score-0.539]

59 The EL for mixture-of-Gaussian stimuli can be computed simply via the linearity of expectation. [sent-138, score-0.1]

60 1 Spectral estimation for low-dimensional models Low-rank parameterization We have so far focused upon MEL estimators for the parameters a, b, and C. [sent-142, score-0.079]

61 Under the GQM, a low-dimensional stimulus dependence is equivalent to having a low-rank C. [sent-144, score-0.339]

62 We can obtain spectral estimates of a low-dimensional GQM by performing an eigenvector decomposition of Cmel and selecting the eigenvectors corresponding to the largest p eigenvalues. [sent-146, score-0.153]

63 2 Consistency of subspace estimates If the conditional probability y|x = y|β x for a matrix β, the neural feature space is spanned by the columns of β. [sent-157, score-0.109]

64 As a generalization of STC, we introduce moment-based dimensionality reduction 6 A quadratic filter w 1 B quadratic filter w 2 0 −2 space (0. [sent-158, score-0.352]

65 8 1 Figure 4: GQM ﬁt and prediction for intracellular recording in cat V1 with a trinary noise stimulus. [sent-169, score-0.153]

66 (A) On top, estimated linear (b) and quadratic (w1 and w2 ) ﬁlters for the GQM, lagged by 20ms. [sent-170, score-0.093]

67 (B) Cross-validated model prediction (red) and n = 94 recordings with repeats of identical stimulus (light grey) along with their mean (black). [sent-172, score-0.392]

68 techniques that recover (portions of) β and show the relationship of these techniques to the MEL estimators of GQM. [sent-175, score-0.079]

69 When the response is binary, this coincides with the traditional STA/STC analysis, which is provably consistent only in the case of stimuli drawn from a spherically symmetric (for STA) or independent Gaussian distribution (for STC) [5]. [sent-177, score-0.178]

70 Below, we argue that this procedure can identify the subspace when y has mean f (β x) with ﬁnite variance, f is some function, and the stimulus distribution is zero-mean with white covariance, i. [sent-178, score-0.365]

71 Effective estimation of the subspace depends critically on both the stimulus distribution and the form of f . [sent-185, score-0.365]

72 Under the GQM, the eigenvectors of E yxx are closely related to the expected log-likelihood estimators we derived earlier. [sent-186, score-0.213]

73 1 Results Intracellular membrane potential We ﬁt a Gaussian GQM to intracellular recordings of membrane potential from a neuron in cat V1, using a 2D spatiotemporal “ﬂickering bars” stimulus aligned with the cell’s preferred orientation (Fig. [sent-192, score-0.634]

74 6 minute recording of responses to non-repeating trinary noise stimulus . [sent-197, score-0.483]

75 We validated the model using responses to 94 repeats of a 1 second frozen noise stimulus. [sent-198, score-0.112]

76 Although the cell was classiﬁed as “simple”, meaning that its response is predominately linear, the GQM ﬁt reveals two quadratic ﬁlters that also inﬂuence the membrane potential response. [sent-201, score-0.272]

77 7 stimulus filter gain linear rate prediction (test data) spike history 2 1 GLM ( GQM ( 0. [sent-204, score-0.637]

78 2 0 10 20 30 time (stimulus frames) 40 0 100 time (ms) 200 1 sec Figure 5: (left) GLM and GQM ﬁlters ﬁt to spike responses of a retinal ganglion cell stimulated with a 120 Hz binary full ﬁeld noise stimulus [28]. [sent-210, score-0.739]

79 The GLM has only linear stimulus and spike history ﬁlters (top left) while the GQM contains all four ﬁlters. [sent-211, score-0.56]

80 Quadratic ﬁlter outputs are squared and then subtracted from other inputs, giving them a suppressive effect on spiking (although quadratic excitation is also possible). [sent-213, score-0.187]

81 2 Retinal ganglion spike train The Poisson GLM provides a popular model for neural spike trains due to its ability to incorporate dependencies on spike history (e. [sent-216, score-0.642]

82 The GLM achieves this by incorporating a one-dimensional linear projection of spike history as an input to the model. [sent-220, score-0.243]

83 In general, however, a spike train may exhibit dependencies on more than one linear projection of spike history. [sent-221, score-0.372]

84 The GQM extends the GLM by allowing multiple stimulus ﬁlters and multiple spike-history ﬁlters. [sent-222, score-0.339]

85 , as found in complex cells) and produce dynamic spike patterns unachievable by GLMs. [sent-225, score-0.158]

86 We ﬁt a Poisson GQM with a quadratic history ﬁlter to data recorded from a retinal ganglion cell driven by a full-ﬁeld white noise stimulus [28]. [sent-226, score-0.672]

87 For ease of comparison, we ﬁt a Poisson GLM, then added quadratic stimulus and history ﬁlters, initialized using a spectral decomposition of the MEL estimate (eq. [sent-227, score-0.536]

88 Both quadratic ﬁlters (which enter with negative sign), have a suppressive effect on spiking (Fig. [sent-229, score-0.187]

89 The quadratic stimulus ﬁlter induces strong suppression at a delay of 5 frames, while the quadratic spike history ﬁlter induces strong suppression during a 50 ms window after a spike. [sent-231, score-0.811]

90 Unlike the GLM, the GQM allows multiple stimulus and history ﬁlters and yet remains tractable for likelihood-based inference. [sent-233, score-0.402]

91 We have derived expected log-likelihood estimators in a general form that reveals a deep connection between likelihood-based and moment-based inference methods. [sent-234, score-0.107]

92 We have shown that GQM performs well on neural data, both for discrete (spiking) and analog (voltage) data. [sent-235, score-0.076]

93 Second-order volterra ﬁltering and its application to nonlinear system identiﬁcation. [sent-253, score-0.098]

94 Bayesian inference for spiking neuron models with a sparsity prior. [sent-279, score-0.093]

95 Real-time performance of a movement-senstivive neuron in the blowﬂy visual system: coding and information transmission in short spike sequences. [sent-302, score-0.189]

96 A point process framework for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects. [sent-342, score-0.182]

97 Dimensionality reduction in neural models: An information-theoretic generalization of spike-triggered average and covariance analysis. [sent-368, score-0.092]

98 Analyzing neural responses to natural signals: maximally informative dimensions. [sent-376, score-0.119]

99 Maximally informative ”stimulus energies” in the analysis of neural responses to natural signals. [sent-402, score-0.094]

100 Precision of spike trains in primate retinal ganglion cells. [sent-446, score-0.264]

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