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286 nips-2013-Robust learning of low-dimensional dynamics from large neural ensembles

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Author: David Pfau, Eftychios A. Pnevmatikakis, Liam Paninski

Abstract: Recordings from large populations of neurons make it possible to search for hypothesized low-dimensional dynamics. Finding these dynamics requires models that take into account biophysical constraints and can be ﬁt efﬁciently and robustly. Here, we present an approach to dimensionality reduction for neural data that is convex, does not make strong assumptions about dynamics, does not require averaging over many trials and is extensible to more complex statistical models that combine local and global inﬂuences. The results can be combined with spectral methods to learn dynamical systems models. The basic method extends PCA to the exponential family using nuclear norm minimization. We evaluate the effectiveness of this method using an exact decomposition of the Bregman divergence that is analogous to variance explained for PCA. We show on model data that the parameters of latent linear dynamical systems can be recovered, and that even if the dynamics are not stationary we can still recover the true latent subspace. We also demonstrate an extension of nuclear norm minimization that can separate sparse local connections from global latent dynamics. Finally, we demonstrate improved prediction on real neural data from monkey motor cortex compared to ﬁtting linear dynamical models without nuclear norm smoothing. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Robust learning of low-dimensional dynamics from large neural ensembles David Pfau Eftychios A. [sent-1, score-0.274]

2 edu Abstract Recordings from large populations of neurons make it possible to search for hypothesized low-dimensional dynamics. [sent-6, score-0.235]

3 Finding these dynamics requires models that take into account biophysical constraints and can be ﬁt efﬁciently and robustly. [sent-7, score-0.252]

4 The basic method extends PCA to the exponential family using nuclear norm minimization. [sent-10, score-0.501]

5 We evaluate the effectiveness of this method using an exact decomposition of the Bregman divergence that is analogous to variance explained for PCA. [sent-11, score-0.208]

6 We show on model data that the parameters of latent linear dynamical systems can be recovered, and that even if the dynamics are not stationary we can still recover the true latent subspace. [sent-12, score-0.638]

7 We also demonstrate an extension of nuclear norm minimization that can separate sparse local connections from global latent dynamics. [sent-13, score-0.659]

8 Finally, we demonstrate improved prediction on real neural data from monkey motor cortex compared to ﬁtting linear dynamical models without nuclear norm smoothing. [sent-14, score-0.823]

9 1 Introduction Progress in neural recording technology has made it possible to record spikes from ever larger populations of neurons [1]. [sent-15, score-0.41]

10 Analysis of these large populations suggests that much of the activity can be explained by simple population-level dynamics [2]. [sent-16, score-0.498]

11 Typically, this low-dimensional activity is extracted by principal component analysis (PCA) [3, 4, 5], but in recent years a number of extensions have been introduced in the neuroscience literature, including jPCA [6] and demixed principal component analysis (dPCA) [7]. [sent-17, score-0.542]

12 A downside of these methods is that they do not treat either the discrete nature of spike data or the positivity of ﬁring rates in a statistically principled way. [sent-18, score-0.231]

13 One alternative is to ﬁt a more complex statistical model directly from spike data, where temporal dependencies are attributed to latent low dimensional dynamics [8, 9]. [sent-20, score-0.477]

14 State space models can include complex interactions such as switching linear dynamics [10] and direct coupling between neurons [11]. [sent-22, score-0.394]

15 These methods have drawbacks too: they are typically ﬁt by approximate EM [12] or other methods that are prone to local minima, the number of latent dimensions is typically chosen ahead of time, and a certain class of possible dynamics must be chosen before doing dimensionality reduction. [sent-23, score-0.363]

16 Our approach is convex and based on recent advances in system identiﬁcation using nuclear norm minimization [13, 14, 15], a convex relaxation of matrix rank minimization. [sent-25, score-0.761]

17 Unlike the standard GLM, where the inputs driving the neurons are observed, we assume that the driving activity is unobserved, but lies on some low dimensional subspace. [sent-36, score-0.24]

18 Thus, instead of ﬁtting a linear receptive ﬁeld, the goal of learning in low-dimensional GLMs is to accurately recover the latent subspace of activity. [sent-38, score-0.339]

19 Let xt ∈ Rm be the value of the dynamics at time t. [sent-39, score-0.217]

20 To turn this into spiking activity, we project this into the space of neurons: yt = Cxt + b is a vector in Rn , n m, where each dimension of yt corresponds to one neuron. [sent-40, score-0.594]

21 C ∈ Rn×m denotes the subspace of the neural population and b ∈ Rn the bias vector for all the neurons. [sent-41, score-0.334]

22 As yt can take on negative values, we cannot use this directly as a ﬁring rate, and so we pass each element of yt through some convex and log-concave increasing point-wise nonlinearity f : R → R+ . [sent-42, score-0.671]

23 To account for biophysical effects such as refractory periods, bursting, and direct synaptic connections, we include a linear dependence on spike history before the nonlinearity. [sent-44, score-0.51]

24 We can extend this simple model by adding dynamics to the low-dimensional latent state, including input-driven dynamics. [sent-47, score-0.311]

25 In this case the model is closely related to the common input model used in neuroscience [11], the difference being that the observed input is added to xt rather than being directly mapped to yt . [sent-48, score-0.362]

26 The case without history terms and with linear Gaussian dynamics is a wellstudied state space model for neural data, usually ﬁt by EM [19, 12, 20], though a consistent spectral method has been derived [16] for the case f (·) = exp(·). [sent-49, score-0.419]

27 Unlike these methods, our approach largely decouples the problem of dimensionality reduction and learning dynamics: even in the case of nonstationary, non-Gaussian dynamics where A, B and Cov[ ] change over time, we can still robustly recover the latent subspace spanned by xt . [sent-50, score-0.613]

28 1 Learning Nuclear norm minimization In the case that the spike history terms D1:k are zero, the natural rate at time t is yt = Cxt + b, so all yt are elements of some m-dimensional afﬁne space given by the span of the columns of C offset by b. [sent-52, score-1.131]

29 Ideally we would minimize λnT rank(A(Y )) − t=1 log p(st |yt ), where λ controls how much we trade off between a simple solution and the likelihood of the data, however general rank minimization is a hard non convex problem. [sent-59, score-0.248]

30 Instead we replace the matrix rank with its convex envelope: the sum of singular values or nuclear norm · ∗ [13], which can be seen as the analogue of the 1 norm for vector sparsity. [sent-60, score-0.831]

31 Our problem then becomes: √ min λ nT ||A(Y )||∗ − Y T log p(st |yt ) (3) t=1 Since the log likelihood scales linearly with the size of the data, and the singular values scale with √ the square root of the size, we also add a factor of nT in front of the nuclear norm term. [sent-61, score-0.701]

32 In the examples in this paper, we assume spikes are drawn from a Poisson distribution: N log p(st |yt ) = sit log f (yit ) − f (yit ) − log sit ! [sent-62, score-0.293]

33 2 Stable principal component pursuit The model above is appropriate for cases where the spike history terms Dτ are zero, that is the observed data can entirely be described by some low-dimensional global dynamics. [sent-66, score-0.502]

34 In real data neurons exhibit history-dependent behavior like bursting and refractory periods. [sent-67, score-0.229]

35 Moreover if the recorded neurons are close to each other some may have direct synaptic connections. [sent-68, score-0.311]

36 1 it is clear that yt is no longer restricted to a lowdimensional afﬁne space. [sent-70, score-0.297]

37 This is an extension of stable principal component pursuit [23], which separates sparse and low-rank components of a noise-corrupted matrix. [sent-78, score-0.288]

38 One can also consider the use of a group sparsity penalty where each group collects a speciﬁc synaptic weight across all the k time lags. [sent-81, score-0.242]

39 As we recover a subspace spanned by the columns of Y rather than a single parameter, this presents a challenge. [sent-84, score-0.302]

40 For the case of PCA, we can project the held out data onto a subspace spanned by principal components and compute what fraction of total variance is explained by this subspace. [sent-86, score-0.59]

41 For any exponential family with natural parameters θ, link function g, function F such that F = g −1 and sufﬁcient statistic T , the log likelihood can be written as DF [θ||g(T (x))] − h(x), where D· [·||·] is a Bregman divergence [24]: DF [x||y] = F (x) − F (y) − (x − y)T F (y). [sent-88, score-0.242]

42 Given a matrix of natural rates recovered from training data, we compute the fraction of Bregman divergence explained by a sequence of subspaces as follows. [sent-92, score-0.545]

43 Let ui be the ith singular vector of (q) the recovered natural rates. [sent-93, score-0.284]

44 Let b be the mean natural rate, and let yt be the maximum likelihood natural rates restricted to the space spanned by u1 , . [sent-94, score-0.578]

45 , uq : q (q) yt k (q) = Dτ st−τ + b ui vit + τ =1 i=1 q (q) vt (q) = arg max log p st k ui vit + v Dτ st−τ + b (6) τ =1 i=1 (q) Here vt is the projection of yt onto the singular vectors. [sent-97, score-0.943]

46 Then the divergence from the mean explained by the qth dimension is given by (q−1) t DF yt (0) t DF yt (q) yt (7) g(st ) (0) where yt is the bias b plus the spike history terms. [sent-98, score-1.654]

47 For Gaussian noise g(x) = x and F (x) = 1 ||x||2 and this is exactly the variance explained by each principal component, while for 2 Poisson noise g(x) = log(x) and F (x) = i exp(xi ). [sent-100, score-0.246]

48 5 is difﬁcult, because the nuclear and 1 norm are not differentiable everywhere. [sent-106, score-0.501]

49 1 Nuclear norm minimization To ﬁnd the optimal Y we alternate between minimizing an augmented Lagrangian with respect to Y , minimizing with respect to an auxiliary variable Z, and performing gradient ascent on a Lagrange multiplier Λ. [sent-111, score-0.229]

50 2 Stable principal component pursuit To extend ADMM to the problem in Eq. [sent-119, score-0.244]

51 , Dk ), and stack the different time-shifted matrices of spike histories on top of one another to form a single spike history matrix H. [sent-124, score-0.461]

52 First, we show in the absence of spike history terms that the true low dimensional subspace can be recovered in the limit of large data, even when the dynamics are nonstationary. [sent-129, score-0.906]

53 Second, we show that spectral methods can accurately recover the transition matrix when dynamics are linear. [sent-130, score-0.403]

54 Lastly, we show that nuclear-norm penalized subspace recovery leads to improved prediction on real neural data recorded from macaque motor cortex. [sent-132, score-0.322]

55 For linear dynamical systems, the transition matrix was sampled from a Gaussian distribution, and the 5 0 −50 1600 1700 1800 1900 2000 2100 2200 2300 2400 Subspace Angle 1. [sent-134, score-0.227]

56 While the subspace remains the same, the dynamics switch between 5 different linear systems. [sent-138, score-0.416]

57 Left top: one dimension of the latent trajectory, switching from one set of dynamics to another (red line). [sent-139, score-0.349]

58 Left middle: ﬁring rates of a subset of neurons during the same switch. [sent-140, score-0.204]

59 Left bottom: covariance between spike counts for different neurons during each epoch of linear dynamics. [sent-141, score-0.305]

60 Right top: Angle between the true subspace and top principal components directly from spike data, from natural rates recovered by nuclear norm minimization, and from the true natural rates. [sent-142, score-1.435]

61 Right bottom: fraction of Bregman divergence explained by the top 1, 5 or 10 dimensions from nuclear norm minimization. [sent-143, score-0.761]

62 Dotted lines are variance explained by the same number of principal components. [sent-144, score-0.246]

63 1 the divergence explained by a given number of dimensions exceeds the variance explained by the same number of PCs. [sent-146, score-0.381]

64 We ﬁrst sought to show that we could accurately recover the subspace in which the dynamics take place even when those dynamics are not stationary. [sent-152, score-0.679]

65 We performed nuclear norm minimization on data generated from this model, varying the smoothing parameter λ from 10−3 to 10, and compared the subspace angle between the top 8 principal components and the true matrix C. [sent-154, score-1.029]

66 We found that when smoothing was optimized the recovered subspace was signiﬁcantly closer to the true subspace than the top principal components taken directly from spike data. [sent-156, score-0.921]

67 Increasing the amount of data from 1000 to 10000 time bins signiﬁcantly reduced the average subspace angle at the optimal λ. [sent-157, score-0.255]

68 The top PCs of the true natural rates Y , while not spanning exactly the same space as C due to differences between the mean column and true bias b, was still closer to the true subspace than the result of nuclear norm minimization. [sent-158, score-0.956]

69 We also computed the fraction of Bregman divergence explained by the sequence of spaces spanned by successive principal components, solving Eq. [sent-159, score-0.39]

70 We did not ﬁnd a clear drop at the true dimensionality of the subspace, but we did ﬁnd that a larger share of the divergence could be explained by the top dimensions than by PCA directly on spikes. [sent-161, score-0.307]

71 To show that the parameters of a latent dynamical system can be recovered, we investigated the performance of spectral methods on model data with linear Gaussian latent dynamics. [sent-164, score-0.424]

72 After estimating natural rates by nuclear norm minimization with λ = 0. [sent-166, score-0.68]

73 01 on 10 trials of 10000 time bins with unit-variance innovations t , we ﬁt the transition matrix A by subspace identiﬁcation (SSID) [26]. [sent-167, score-0.341]

74 2 Imaginary Component (b) (c) Figure 2: Recovered eigenvalues for the transition matrix of a linear dynamical system from model neural data. [sent-202, score-0.381]

75 (2b) Eigenvalues recovered from subspace identiﬁcation directly on spike counts. [sent-206, score-0.55]

76 (2c) Eigenvalues recovered from subspace identiﬁcation on the natural rates estimated by nuclear norm minimization. [sent-207, score-1.0]

77 nuclear norm minimization had little bias, and seemed to perform almost as well as SSID directly on the true natural rates. [sent-208, score-0.662]

78 We found that other methods for ﬁtting linear dynamical systems from the estimated natural rates were biased, as was SSID on the result of nuclear norm minimization without mean-centering (see the supplementary material for more details). [sent-209, score-0.82]

79 We incorporated spike history terms into our model data to see whether local connectivity and global dynamics could be separated. [sent-210, score-0.559]

80 We found that we could recover synaptic weights with an r2 up to . [sent-214, score-0.218]

81 4 on this data by combining both a nuclear norm and 1 penalty, compared to at most . [sent-215, score-0.501]

82 Somewhat surprisingly, at the extreme of either no nuclear norm penalty or a dominant nuclear norm penalty, increasing the 1 penalty never improved estimation. [sent-218, score-1.142]

83 Left: r2 between true and recovered synaptic weights across a range of parameters. [sent-255, score-0.404]

84 Right: scatter plot of true versus recovered synaptic weights, illustrating the effect of the nuclear norm term. [sent-258, score-0.905]

85 We ﬁt linear dynamical systems by subspace identiﬁcation as in Fig. [sent-264, score-0.339]

86 2, but as we did not have access to a “true” linear dynamical system for comparison, we evaluated our model ﬁts by approximating the held out log likelihood by Laplace-Gaussian ﬁltering [28]. [sent-265, score-0.376]

87 We found that a strong nuclear norm penalty improved prediction by several hundred bits per second, and that fewer dimensions were needed for optimal prediction as the nuclear norm penalty was increased. [sent-267, score-1.194]

88 The best ﬁt models predicted held out data nearly as well as models trained via EM, even though nuclear norm minimization is not directly maximizing the likelihood of a linear dynamical system. [sent-268, score-0.852]

89 16e−02 EM −2000 0 5 10 15 20 25 30 35 40 45 50 Number of Latent Dimensions 6 Discussion Figure 4: Log likelihood of held out motor cortex The method presented here has a number of straight- data versus number of latent dimensions for difforward extensions. [sent-273, score-0.418]

90 If the dimensionality of the la- ferent latent linear dynamical systems. [sent-274, score-0.234]

91 If there are also observed inputs ut then the term inside the nuclear norm should also include a projection orthogonal to the row space of the inputs. [sent-279, score-0.538]

92 This could enable joint learning of dynamics and receptive ﬁelds for small populations of neurons with high dimensional inputs. [sent-280, score-0.452]

93 It remains an open question what kinds of dynamics can be learned from the recovered natural parameters. [sent-284, score-0.452]

94 In this paper we have focused on linear systems, but nuclear norm minimization could just as easily be combined with spectral methods for switching linear systems and general nonlinear systems. [sent-285, score-0.656]

95 Harris, “Population rate dynamics and multineuron ﬁring patterns in sensory cortex,” The Journal of Neuroscience, vol. [sent-309, score-0.217]

96 Sahani, “Dynamical segmentation of single trials from population neural data,” Advances in neural information processing systems, vol. [sent-400, score-0.247]

97 Vandenberghe, “Interior-point method for nuclear norm approximation with application to system identiﬁcation,” SIAM Journal on Matrix Analysis and Applications, vol. [sent-425, score-0.544]

98 Vandenberghe, “Nuclear norm system identiﬁcation with missing inputs and outputs,” Systems & Control Letters, vol. [sent-431, score-0.245]

99 Sahani, “Spectral learning of linear dynamics from generalised-linear observations with application to neural population data,” Advances in neural information processing systems, vol. [sent-438, score-0.409]

100 Schapire, “A generalization of principal component analysis to the exponential family,” Advances in neural information processing systems, vol. [sent-465, score-0.241]

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