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45 nips-2008-Characterizing neural dependencies with copula models

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Author: Pietro Berkes, Frank Wood, Jonathan W. Pillow

Abstract: The coding of information by neural populations depends critically on the statistical dependencies between neuronal responses. However, there is no simple model that can simultaneously account for (1) marginal distributions over single-neuron spike counts that are discrete and non-negative; and (2) joint distributions over the responses of multiple neurons that are often strongly dependent. Here, we show that both marginal and joint properties of neural responses can be captured using copula models. Copulas are joint distributions that allow random variables with arbitrary marginals to be combined while incorporating arbitrary dependencies between them. Different copulas capture different kinds of dependencies, allowing for a richer and more detailed description of dependencies than traditional summary statistics, such as correlation coefﬁcients. We explore a variety of copula models for joint neural response distributions, and derive an efﬁcient maximum likelihood procedure for estimating them. We apply these models to neuronal data collected in macaque pre-motor cortex, and quantify the improvement in coding accuracy afforded by incorporating the dependency structure between pairs of neurons. We ﬁnd that more than one third of neuron pairs shows dependency concentrated in the lower or upper tails for their ﬁring rate distribution. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Characterizing neural dependencies with copula models Pietro Berkes Volen Center for Complex Systems Brandeis University, Waltham, MA 02454 berkes@brandeis. [sent-1, score-1.042]

2 uk Abstract The coding of information by neural populations depends critically on the statistical dependencies between neuronal responses. [sent-5, score-0.225]

3 However, there is no simple model that can simultaneously account for (1) marginal distributions over single-neuron spike counts that are discrete and non-negative; and (2) joint distributions over the responses of multiple neurons that are often strongly dependent. [sent-6, score-0.448]

4 Here, we show that both marginal and joint properties of neural responses can be captured using copula models. [sent-7, score-1.104]

5 Copulas are joint distributions that allow random variables with arbitrary marginals to be combined while incorporating arbitrary dependencies between them. [sent-8, score-0.347]

6 Different copulas capture different kinds of dependencies, allowing for a richer and more detailed description of dependencies than traditional summary statistics, such as correlation coefﬁcients. [sent-9, score-0.369]

7 We explore a variety of copula models for joint neural response distributions, and derive an efﬁcient maximum likelihood procedure for estimating them. [sent-10, score-1.019]

8 We apply these models to neuronal data collected in macaque pre-motor cortex, and quantify the improvement in coding accuracy afforded by incorporating the dependency structure between pairs of neurons. [sent-11, score-0.281]

9 We ﬁnd that more than one third of neuron pairs shows dependency concentrated in the lower or upper tails for their ﬁring rate distribution. [sent-12, score-0.278]

10 The stochastic spiking activity of individual neurons in cortex is often well described by a Poisson distribution. [sent-14, score-0.173]

11 Responses from multiple neurons also exhibit strong dependencies (i. [sent-15, score-0.228]

12 Recent work has focused on the construction of large parametric models that capture inter-neuronal dependencies using generalized linear point-process models [5, 6, 7, 8, 9] and binary second-order maximum-entropy models [10, 11, 12]. [sent-22, score-0.228]

13 Although these approaches are quite powerful, they model spike trains only in very ﬁne time bins, and thus describe the dependencies in neural spike count distributions only implicitly. [sent-23, score-0.289]

14 Modeling the joint distribution of neural activities is therefore an important open problem. [sent-24, score-0.122]

15 Here we show how to construct non-independent joint distributions over ﬁring rates using copulas. [sent-25, score-0.108]

16 Top row: The marginal distributions (the leftmost marginal is uniform, by deﬁnition of copula). [sent-28, score-0.196]

17 cortex; ﬁnally, in Section 6 we review the insights provided by neural copula models and discuss several extensions and future directions. [sent-31, score-0.913]

18 , un ) : [0, 1]n → [0, 1] is a multivariate distribution function on the unit cube with uniform marginals [13, 14]. [sent-35, score-0.265]

19 The basic idea behind copulas is quite simple, and is closely related to that of histogram equalization: for a random variable yi with continuous cumulative distribution function (cdf) Fi , the random variable ui := Fi (yi ) is uniformly distributed on the interval [0, 1]. [sent-36, score-0.351]

20 , Fn and joint distribution F , there exist a unique copula C such that for all ui : −1 −1 C(u1 , . [sent-44, score-0.965]

21 , Fn (yn )) (2) is a n-variate distribution function with marginal distribution functions F1 , . [sent-59, score-0.134]

22 This result gives a way to derive a copula given the joint and marginal distributions (using Eq. [sent-63, score-1.041]

23 1), and also, more importantly here, to construct a joint distribution by specifying the marginal distributions and the dependency structure separately (Eq. [sent-64, score-0.296]

24 For example, one can keep the dependency structure ﬁxed and vary the marginals (Fig. [sent-66, score-0.207]

25 1), or vice versa given ﬁxed marginal distributions deﬁne new joint distributions using parametrized copula families (Fig. [sent-67, score-1.219]

26 Since copulas do not depend on the marginals, one can deﬁne in this way dependency measures that are insensitive to non-linear transformations of the individual variables [14] and generalize correlation coefﬁcients, which are only appropriate for elliptic distributions. [sent-71, score-0.305]

27 The copula representation has also been used to estimate the conditional entropy of neural latencies by separating the contribution of the individual latencies from that coming from their correlations [16]. [sent-72, score-1.007]

28 One notable example is the Gaussian copula, which generalizes the dependency structure of the multivariate Gaussian distribution to arbitrary marginal distribution (Fig. [sent-74, score-0.28]

29 1), and is deﬁned as C(u1 , u2 ; Σ) = ΦΣ φ−1 (u1 ), φ−1 (u2 ) , 2 (3) Figure 2: Samples drawn from a joint distribution with ﬁxed Gaussian marginals and dependency structure deﬁned by parametric copula families, as indicated by the labels. [sent-75, score-1.176]

30 where φ(u) is the cdf of the univariate Gaussian with mean 0 and variance 1, and ΦΣ is the cdf of a standard multivariate Gaussian with mean 0 and covariance matrix Σ. [sent-80, score-0.166]

31 Other families derive from the economics literature, and are typically one-parameter families that capture various possible dependencies, for example dependencies only in one of the tails of the distribution. [sent-81, score-0.466]

32 Table 1 shows the deﬁnition of the copula distributions used in this paper (see [14], for an overview of known copulas and copula construction methods). [sent-82, score-1.956]

33 3 Maximum Likelihood estimation for discrete marginal distributions In the case where the random variables have discrete distribution functions, as in the case of neural ﬁring rates, only a weaker version of Theorem 1 is valid: there always exists a copula that satisﬁes Eq. [sent-83, score-1.113]

34 With discrete data, the probability of a particular outcome is determined by an integral over the region of [0, 1]n corresponding to that outcome; any two copulas that integrate to the same values on all such regions produce the same joint distribution. [sent-85, score-0.291]

35 These marginals can be given by the empirical cumulative distribution of ﬁring rates (as in this paper) or by any parametrized family of univariate distributions (such as Poisson). [sent-88, score-0.296]

36 , un ) du , ··· = argmax F1 (y1 −1) 3 Fn (yn −1) (5) θ u p(u|θ) = cθ (u1 , . [sent-92, score-0.109]

37 3 Frank Figure 3: Graphical representation of the copula model with discrete marginals. [sent-101, score-0.884]

38 Uniform marginals u are drawn from the copula density function cθ (u1 , . [sent-102, score-0.965]

39 The discrete marginals are then generated deterministically using the inverse cdf of the marginals, which are parametrized by λ. [sent-106, score-0.25]

40 5 2 4 1 3 5 −5 6 0 4 7 8 5 5 9 10 10 Figure 4: Distribution of the maximum likelihood estimation of the parameters of four copula families, for various setting of their parameter (x-axis). [sent-111, score-0.891]

41 The last equation is the copula probability mass inside the volume deﬁned by the vertices Fi (yi ) and Fi (yi − 1), and can be readily computed using the copula distribution Cθ (u1 , . [sent-115, score-1.759]

42 For example, in the bivariate case one obtains argmax p(y1 , y2 |θ) = argmax Cθ (u1 , u2 ) + Cθ (u− , u− ) − Cθ (u− , u2 ) − Cθ (u1 , u− ) , 1 2 1 2 θ (6) θ where ui = Fi (yi ) and u− = Fi (yi − 1). [sent-119, score-0.145]

43 Given neural data in the form of ﬁring rates y1 , y2 from a pair of neurons, we collect the empirical cumulative histogram of responses, Fi (k) = P (yi ≤ k). [sent-122, score-0.15]

44 The data is then transformed through the cdfs ui = Fi (yi ), and the copula model is ﬁt according to Eq. [sent-123, score-0.925]

45 If a parametric distribution family is used for the marginals, the parameters of the copula θ and those of the marginals λ can be estimated simultaneously, or alternatively λ can be ﬁtted ﬁrst, followed by θ. [sent-125, score-1.04]

46 We checked for biases in ML estimation due to a limited amount of data and low ﬁring rate by generating data from the discrete copula model (Fig. [sent-127, score-0.902]

47 3), for a number of copula families and Poisson marginals with parameters λ1 = 2, λ2 = 3. [sent-128, score-1.076]

48 Inaccuracy in the estimation becomes larger as the copulas approach functional dependency (i. [sent-132, score-0.308]

49 , u2 = f (u1 ) for a deterministic function f ), as it is the case for the Gaussian copula when ρ tends to 1, and for the Gumbel copula as θ goes to inﬁnity. [sent-134, score-1.712]

50 4                   Figure 5: Empirical joint distribution and copula ﬁt for two neuron pairs. [sent-138, score-0.985]

51 The top row shows two neurons that have dependencies mainly in the upper tails of their marginal distribution. [sent-139, score-0.442]

52 4 Results To demonstrate the ability of copula models to ﬁt joint ﬁring rate distribution, we model neural data recorded using a multi-electrode array implanted in the pre-motor cortex (PMd) area of a macaque monkey [18, 19]. [sent-147, score-1.109]

53 We ﬁt the copula model using the marginal distribution of neural activity over the entire recording session, including data recorded between trials (i. [sent-151, score-1.041]

54 , the stimulus-conditional response distribution), the marginal response distribution is an important statistical object in its own right, and has been the focus of recent much literature [10, 11]. [sent-156, score-0.147]

55 For example, the joint activity across neurons, averaged over stimuli, is the only distribution the brain has access to, and must be sufﬁcient for learning to construct representations of the external world. [sent-157, score-0.099]

56 We collected spike responses in 100ms bins, and selected at random, without repetition, a training set of 4000 bins and a test set of 2000 bins. [sent-158, score-0.138]

57 Out of a total of 194 neurons we select a subset of 33 neurons that ﬁred a minimum of 2500 spikes over the whole data set. [sent-159, score-0.198]

58 For every pair of neurons in this subset (528 pairs), we ﬁt the parameters of several copula families to the joint ﬁring rate. [sent-160, score-1.136]

59 Figure 5 shows two examples of the kind of the dependencies present in the data set and how they are ﬁt by different copula families. [sent-161, score-0.985]

60 This is conﬁrmed by the empirical copula, which shows the probability mass in the regions deﬁned by the cdfs of the marginal distribution. [sent-163, score-0.156]

61 Since the marginal cdfs are discrete, the data is projected on a discrete set of points on the unit cube; the colors in the empirical copula plots represent the probability mass in the region where the marginal cdfs are constant. [sent-164, score-1.182]

62 The axis in the empirical copula should be interpreted as the quantiles of the marginal distributions – for example, 0. [sent-165, score-0.986]

63 The higher probability mass in the upper right corner of the plot thus means that the two neurons tend to be in the upper tails of the distributions simultaneously, and thus to have higher ﬁring rates together. [sent-167, score-0.302]

64 On the right, one can see that this dependency structure is well captured by the Gumbel copula ﬁt. [sent-168, score-0.969]

65 Although this is not readily visible in the joint histogram, the dependency becomes clear in the empirical copula plot. [sent-170, score-1.007]

66 This structure is captured by the Frank copula ﬁt. [sent-171, score-0.887]

67 The goodness-of-ﬁt of the copula families is evaluated by cross-validation: We ﬁt different models on training data, and compute the log-likelihood of test data under the ﬁtted model. [sent-175, score-0.984]

68 The models are scored according to the difference between the log-likelihood of a model that assumes independent neurons and the log-likelihood of the copula model. [sent-176, score-0.972]

69 This measure (appropriately renormalized) can be interpreted as the number of bits per second that can be saved when coding the ﬁring rate by taking into account the dependencies encoded by the copula family. [sent-177, score-1.012]

70 (7) (8) (9) We took particular care in selecting a small set of copula families that would be able to capture the dependencies occurring in the data. [sent-179, score-1.113]

71 Some of the families that we considered at ﬁrst capture similar kind of dependencies, and their scores are highly correlated. [sent-180, score-0.128]

72 For example, the Frank and Gaussian copulas are able to represent both positive and negative dependencies in the data, and simultaneously in lower and upper tails, although the dependencies in the tails are less strong for the Frank family (compare the copula densities in Figs. [sent-181, score-1.46]

73 An advantage of the Frank copula is that it is much more efﬁcient to ﬁt, since the Gaussian copula requires multiple evaluations of the bivariate Gaussian cdf, which requires expensive numerical calculations. [sent-185, score-1.764]

74 In addition, The Gaussian copula was also found to be more prone to overﬁtting on this data set (Fig. [sent-186, score-0.856]

75 With similar procedures we shortlisted a total 3 families that cover the vast majority of dependencies in our data set: Frank, Clayton, and Gumbel copulas. [sent-189, score-0.24]

76 Examples of the copula density of these families can be found in Figs. [sent-190, score-0.967]

77 The Clayton and Gumbel copulas describe dependencies in the lower and upper tails of the distributions, respectively. [sent-192, score-0.444]

78 We didn’t ﬁnd any example of neuron pairs where the dependency would be in the upper tail of the distribution for one and in the lower tail for the other distribution, or more complicated dependencies. [sent-193, score-0.236]

79 Dependencies in the data set seem thus to be widespread, despite the fact that individual neurons are recorded from electrodes that are up to 4. [sent-196, score-0.139]

80 The most common dependencies structures over all neuron pairs are given by the Gaussian-like dependencies of the Frank copula (54% of the pairs). [sent-200, score-1.188]

81 Interestingly, a large proportion of the neurons showed dependencies concentrated in the upper tails (Gumbel copula, 22%) or lower tails (Clayton copula, 16%) of the distributions (Fig. [sent-201, score-0.47]

82 1 We computed the signiﬁcance level by generating an artiﬁcial data set using independent neurons with the same empirical pdf as the monkey data. [sent-203, score-0.146]

83 Right: Pie chart of the copula families that best ﬁt the neuron pairs. [sent-210, score-1.014]

84 5 Discussion The results presented here show that it is possible to represent neuronal spike responses using a model that preserves discrete, non-negative marginals while incorporating various types of dependencies between neurons. [sent-211, score-0.385]

85 However, many copula families have only one or two parameters, regardless of the copula dimensionality. [sent-215, score-1.823]

86 If the dependency structure across a neural population is relatively homogeneous, then these copulas may be useful in that they can be estimated using far less data than required, e. [sent-216, score-0.384]

87 On the other hand, if the dependencies within a population vary markedly for different pairs of neurons (as in the data set examined here), such copulas will lack the ﬂexibility to capture the complicated dependencies within a full population. [sent-219, score-0.647]

88 In such cases, we can still apply the Gaussian copula (and other copulas derived from elliptically symmetric distributions), since it is parametrized by the same covariance matrix as a n-dimensional Gaussian. [sent-220, score-1.109]

89 However, the Gaussian copula becomes prohibitively expensive to ﬁt in high dimensions, since evaluating the likelihood requires an exponential number of evaluations of the multivariate Gaussian cdf, which itself must be computed numerically. [sent-221, score-0.921]

90 One challenge for future work will therefore be to design new parametric families of copulas whose parameters grow with the number of neurons, but remain tractable enough for maximum-likelihood estimation. [sent-222, score-0.35]

91 Recently, Kirshner [20] proposed a copula-based representation for multivariate distributions using a model that averages over tree-structured copula distributions. [sent-223, score-0.94]

92 The basic idea is that pairwise copulas can be easily combined to produce a tree-structured representation of a multivariate distribution, and that averaging over such trees gives an even more ﬂexible class of multivariate distributions. [sent-224, score-0.304]

93 Another future challenge is to combine explicit models of the stimulus-dependence underlying neural responses with models capable of capturing their joint response dependencies. [sent-226, score-0.207]

94 The data set analyzed here concerned the distribution over spike responses during all all stimulus conditions (i. [sent-227, score-0.127]

95 , the marginal distribution over responses, as opposed to the the conditional response distribution given a stimulus). [sent-229, score-0.154]

96 Although this marginal response distribution is interesting in its own right, for many applications one is interested in separating correlations that are induced by external stimuli from internal correlations due to the network interactions. [sent-230, score-0.241]

97 One obvious approach is to consider a hybrid model with a Linear-Nonlinear-Poisson model [21] capturing stimulus-induced correlation, adjoined to a copula distribution that models the residual dependencies between neurons (Fig. [sent-231, score-1.142]

98 The LNP part of the model removes stimulus-induced correlations from the neural data, so that the copula model can take into account residual network-related dependencies. [sent-237, score-0.943]

99 A point process framework for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects. [sent-283, score-0.167]

100 Bayesian inference for spiking neuron models with a sparsity prior. [sent-298, score-0.099]

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