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266 nips-2013-Recurrent linear models of simultaneously-recorded neural populations

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Author: Marius Pachitariu, Biljana Petreska, Maneesh Sahani

Abstract: Population neural recordings with long-range temporal structure are often best understood in terms of a common underlying low-dimensional dynamical process. Advances in recording technology provide access to an ever-larger fraction of the population, but the standard computational approaches available to identify the collective dynamics scale poorly with the size of the dataset. We describe a new, scalable approach to discovering low-dimensional dynamics that underlie simultaneously recorded spike trains from a neural population. We formulate the Recurrent Linear Model (RLM) by generalising the Kalman-ﬁlter-based likelihood calculation for latent linear dynamical systems to incorporate a generalised-linear observation process. We show that RLMs describe motor-cortical population data better than either directly-coupled generalised-linear models or latent linear dynamical system models with generalised-linear observations. We also introduce the cascaded generalised-linear model (CGLM) to capture low-dimensional instantaneous correlations in neural populations. The CGLM describes the cortical recordings better than either Ising or Gaussian models and, like the RLM, can be ﬁt exactly and quickly. The CGLM can also be seen as a generalisation of a lowrank Gaussian model, in this case factor analysis. The computational tractability of the RLM and CGLM allow both to scale to very high-dimensional neural data. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Recurrent linear models of simultaneously-recorded neural populations Marius Pachitariu, Biljana Petreska, Maneesh Sahani Gatsby Computational Neuroscience Unit University College London, UK {marius,biljana,maneesh}@gatsby. [sent-1, score-0.062]

2 uk Abstract Population neural recordings with long-range temporal structure are often best understood in terms of a common underlying low-dimensional dynamical process. [sent-4, score-0.172]

3 Advances in recording technology provide access to an ever-larger fraction of the population, but the standard computational approaches available to identify the collective dynamics scale poorly with the size of the dataset. [sent-5, score-0.079]

4 We describe a new, scalable approach to discovering low-dimensional dynamics that underlie simultaneously recorded spike trains from a neural population. [sent-6, score-0.152]

5 We formulate the Recurrent Linear Model (RLM) by generalising the Kalman-ﬁlter-based likelihood calculation for latent linear dynamical systems to incorporate a generalised-linear observation process. [sent-7, score-0.26]

6 We show that RLMs describe motor-cortical population data better than either directly-coupled generalised-linear models or latent linear dynamical system models with generalised-linear observations. [sent-8, score-0.263]

7 We also introduce the cascaded generalised-linear model (CGLM) to capture low-dimensional instantaneous correlations in neural populations. [sent-9, score-0.205]

8 The CGLM describes the cortical recordings better than either Ising or Gaussian models and, like the RLM, can be ﬁt exactly and quickly. [sent-10, score-0.038]

9 The CGLM can also be seen as a generalisation of a lowrank Gaussian model, in this case factor analysis. [sent-11, score-0.061]

10 The computational tractability of the RLM and CGLM allow both to scale to very high-dimensional neural data. [sent-12, score-0.033]

11 1 Introduction Many essential neural computations are implemented by large populations of neurons working in concert, and recent studies have sought both to monitor increasingly large groups of neurons [1, 2] and to characterise their collective behaviour [3, 4]. [sent-13, score-0.278]

12 In this paper we introduce a new computational tool to model coordinated behaviour in very large neural data sets. [sent-14, score-0.11]

13 While we explicitly discuss only multi-electrode extracellular recordings, the same model can be readily used to characterise 2-photon calcium-marker image data, EEG, fMRI or even large-scale biologically-faithful simulations. [sent-15, score-0.081]

14 Populational neural data may be represented at each time point by a vector yt with as many dimensions as neurons, and as many indices t as time points in the experiment. [sent-16, score-0.634]

15 For spiking neurons, yt will have positive integer elements corresponding to the number of spikes ﬁred by each neuron in the time interval corresponding to the t-th bin. [sent-17, score-0.627]

16 As others have before [5, 6], we assume that the coordinated activity reﬂected in the measurement yt arises from a low-dimensional set of processes, collected into a vector xt , which is not directly observed. [sent-18, score-0.869]

17 However, unlike the previous studies, we construct a recurrent model in which the hidden processes xt are driven directly and explicitly by the measured neural signals y1 . [sent-19, score-0.361]

18 We assume for simplicity that xt evolves with linear dynamics and affects the future state of the neural signal yt in a generalised-linear manner, although both assumptions may be relaxed. [sent-24, score-0.907]

19 As in the latent dynamical system, the resulting model enforces a “bottleneck”, whereby predictions of yt based on y1 . [sent-25, score-0.795]

20 1 State prediction in the RLM is related to the Kalman ﬁlter [7] and we show in the next section a formal equivalence between the likelihoods of the RLM and the latent dynamical model when observation noise is Gaussian distributed. [sent-29, score-0.217]

21 However, spiking data is not well modelled as Gaussian, and the generalisation of our approach to Poisson noise leads to a departure from the latent dynamical approach. [sent-30, score-0.336]

22 Unlike latent linear models with conditionally Poisson observations, the parameters of our model can be estimated efﬁciently and without approximation. [sent-31, score-0.093]

23 We show that, perhaps in consequence, the RLM can provide superior descriptions of neural population data. [sent-32, score-0.082]

24 2 From the Kalman ﬁlter to the recurrent linear model (RLM) Consider a latent linear dynamical system (LDS) model with linear-Gaussian observations. [sent-33, score-0.34]

25 The latent process is parametrised by a dynamics matrix A and innovations covariance Q that describe the evolution of the latent state xt : P (xt |xt−1 ) = N (xt |Axt−1 , Q) , where N (x|µ, Σ) represents a normal distribution on x with mean µ and (co)variance Σ. [sent-36, score-0.661]

26 The output distribution is determined by an observation loading matrix C and a noise covariance R often taken to be diagonal so that all covariance is modelled by the latent process: P (yt |xt ) = N (yt |Cxt , R) . [sent-38, score-0.363]

27 In the LDS, the joint likelihood of the observations {yt } can be written as the product: T P (yt |y1 . [sent-39, score-0.041]

28 yT ) = P (y1 ) t=2 and in the Gaussian case can be computed using the usual Kalman ﬁlter approach to ﬁnd the conditional distributon at time t iteratively: P (yt+1 |y1 . [sent-45, score-0.023]

29 yt ) ˆ dxt+1 N (yt+1 |Cxt+1 , R) N (xt+1 |Axt , Vt+1 ) ˆ = N (yt+1 |CAxt , CVt+1 C + R) , ˆ where we have introduced the (ﬁltered) state estimate xt = E [xt |y1 . [sent-51, score-0.824]

30 yt ] and (predictive) unˆ certainty Vt+1 = E (xt+1 − Axt )2 |y1 . [sent-54, score-0.601]

31 Both quantities are computed recursively using the Kalman gain Kt = Vt C (CVt C + R)−1 , giving the following recursive recipe to calculate the conditional likelihood of yt+1 : ˆ ˆ ˆ xt = Axt−1 + Kt (yt − yt ) Vt+1 = A(I − Kt C)Vt A + Q ˆ ˆ yt+1 = CAxt ˆ P (yt+1 |y1 . [sent-58, score-0.865]

32 yt ) = N (yt+1 |yt+1 , CVt+1 C + R) For the Gaussian LDS, the Kalman gain Kt and state uncertainty Vt+1 (and thus the output covariance CVt+1 C + R) depend on the model parameters (A, C, R, Q) and on the time step—although as time grows they both converge to stationary values. [sent-61, score-0.767]

33 Thus, we might consider a relaxation of the Gaussian LDS model in which these matrices are taken to be stationary from the outset, and are parametrised independently so that they are no longer constrained to take on the “correct” values as computed for Kalman inference. [sent-63, score-0.071]

34 Let us call this parametric form of the Kalman gain W and the parametric form of the output covariance S. [sent-64, score-0.121]

35 Then the conditional likelihood iteration becomes ˆ ˆ ˆ xt = Axt−1 + W (yt − yt ) ˆ ˆ yt+1 = CAxt ˆ P (yt+1 |y1 . [sent-65, score-0.824]

36 Shaded LDS variables are observed, unshaded    circles are latent random variables     and squares are variables that depend deterministically on their parents. [sent-70, score-0.121]

37 In B the LDS is redrawn in terms of the random innovations ηt = xt − Axt−1 , facilitating the transition towards the RLM. [sent-71, score-0.307]

38 The RLM RLM is then obtained by replacing ηt with a deterministically derived ˆ estimate W (yt − yt ). [sent-72, score-0.65]

39 This is a relaxation of the Gaussian latent LDS model because W has more degrees of freedom than Q, as does S than R (at least if R is constrained to be diagonal). [sent-74, score-0.093]

40 The new model has a recurrent linear structure in that the random ˆ observation yt is fed back linearly to perturb the otherwise deterministic evolution of the state xt . [sent-75, score-1.054]

41 The RLM can be viewed as replacing the random innovation variables ηt = xt − Axt−1 ˆ with data-derived estimates W (yt − yt ); estimates which are made possible by the fact that ηt ˆ contributes to the variability of yt around yt . [sent-78, score-2.026]

42 3 Recurrent linear models with Poisson observations The discussion above has transformed a stochastic-latent LDS model with Gaussian output to an RLM with deterministic latent, but still with Gaussian output. [sent-79, score-0.088]

43 Our goal, however, is to ﬁt a model with an output distribution better suited to the binned point-processes that characterise neural spiking. [sent-80, score-0.143]

44 Both linear Kalman-ﬁltering steps above and the eventual stationarity of the inference parameters depend on the joint Gaussian structure of the assumed LDS model. [sent-81, score-0.039]

45 They would not apply if we were to begin a similar derivation from an LDS with Poisson output. [sent-82, score-0.026]

46 However, a tractable approach to modelling point-process data with low-dimensional temporal structure may be provided by introducing a generalised-linear output stage directly to the RLM. [sent-83, score-0.076]

47 This model is given by: ˆ ˆ ˆ xt = Axt−1 + W (yt − yt ) ˆ ˆ g(yt+1 ) = CAxt ˆ P (yt+1 |y1 . [sent-84, score-0.803]

48 yt ) = ExpFam(yt+1 |yt+1 ) (1) where ExpFam is an exponential-family distribution such as Poisson, and the element-wise link ˆ function g allows for a nonlinear mapping from xt to the predicted mean yt+1 . [sent-87, score-0.849]

49 In the following, we ˆ ˆ will write f for the inverse-link as is more common for neural models, so that yt+1 = f(CAxt ). [sent-88, score-0.033]

50 The simplest Poisson-based generalised-linear RLM might take as its output distribution ˆ P (yt |yt ) = Poisson(yti |ˆti ); y ˆ ˆ yt = f(CAxt−1 )) , i where yti is the spike count of the ith cell in bin t and the function f is non-negative. [sent-89, score-0.813]

51 However, comparison with the output distribution derived for the Gaussian RLM suggests that this choice would fail to capture the instantaneous covariance that the LDS formulation transfers to the output distribution (and which appears in the low-rank structure of S above). [sent-90, score-0.204]

52 One option is to bin the data more ﬁnely, thus diminishing the inﬂuence of the instantaneous covariance. [sent-92, score-0.082]

53 The alternative is to replace the independent Poissons with a correlated output distribution on spike counts. [sent-93, score-0.118]

54 The cascaded generalised-linear model introduced below is a natural choice, and we will show that it captures instantaneous correlations faithfully with very few hidden dimensions. [sent-94, score-0.172]

55 3 In practice, we also sometimes add a ﬁxed input µt to equation 1 that varies in time and determines the average behavior of the population or the peri-stimulus time histogram (PSTH). [sent-95, score-0.049]

56 The matrix A controls the evolution of the dynamical process xt . [sent-97, score-0.364]

57 The phenomenology of its dynamics is determined by the complex eigenvalues of A. [sent-98, score-0.111]

58 Eigenvalues with moduli close to 1 correspond to long timescales of ﬂuctuation around the PSTH. [sent-99, score-0.042]

59 Eigenvalues with non-zero imaginary part correspond to oscillatory components. [sent-100, score-0.02]

60 Finally, the dynamics will be stable iff all the eigenvalues lie within the unit disc. [sent-101, score-0.088]

61 The matrix C describes the dependence of the high-dimensional neural signals on the lowdimensional latent processes xt . [sent-102, score-0.347]

62 This generalised-linear stage ensures that the ﬁring rates are positive through the link function f, and the observation process is Poisson. [sent-104, score-0.096]

63 For other types of data, the generalised-linear stage might be replaced by other appropriate link functions and output distributions. [sent-105, score-0.122]

64 1 Relationship to other models RLMs are related to recurrent neural networks [8]. [sent-107, score-0.159]

65 The differences lie in the state evolution, which in the neural network is nonlinear: xt = h (Axt−1 + W yt−1 ); and in the recurrent term which depends on the observation rather than the prediction error. [sent-108, score-0.405]

66 On the data considered here, we found that using sigmoidal or threshold-linear functions h resulted in models comparable in likelihood to the RLM, and so we restricted our attention to simple linear dynamics. [sent-109, score-0.045]

67 We also found that ˆ using the prediction error term W (yt−1 − yt ) resulted in better models than the simple neural-net formulation, and we attribute this difference to the link between the RLM and Kalman inference. [sent-110, score-0.671]

68 It is also possible to work within the stochatic latent LDS framework, replacing the Gaussian output distribution with a generalised-linear Poisson output (e. [sent-111, score-0.212]

69 For an unobserved latent process xt , an inference procedure needs to be devised to estimate the posterior distribution on the entire sequence x1 . [sent-115, score-0.295]

70 However, with generalised-linear observations, inference becomes intractable and the necessary approximations [6] are computationally intense and can jeopardize the quality of the ﬁtted models. [sent-120, score-0.022]

71 By contrast, in the RLM xt is a deterministic function of data. [sent-121, score-0.221]

72 In effect, the Kalman ﬁlter has been built into the model as the accurate estimation procedure, and efﬁcient ﬁtting is possible by direct gradient ascent on the log-likelihood. [sent-122, score-0.02]

73 Note that to estimate the matrices A and W the gradient must be backpropagated through successive iterations of equation 1. [sent-125, score-0.022]

74 This technique, known as backpropagation-through-time, was ﬁrst described by [10] as a technique to ﬁt recurrent neural network models. [sent-126, score-0.159]

75 Backpropagation-through-time is thought to be inherently unstable when propagated past many timesteps and often the gradient is truncated prematurely [11]. [sent-128, score-0.023]

76 We found that using large values of momentum in the gradient ascent alleviated these instabilities and allowed us to use backpropagation without the truncation. [sent-129, score-0.043]

77 4 The cascaded generalised-linear model (CGLM) The link between the RLM and the LDS raises the possibility that a model for simultaneouslyrecorded correlated spike counts might be derived in a similar way, starting from a non-dynamical, but low-dimensional, Gaussian model. [sent-130, score-0.24]

78 Stationary models of population activity have attracted recent interest for their own sake (e. [sent-131, score-0.068]

79 [1]), and would also provide a way model correlations introduced by common innovations that were neglected by the simple Poisson form of the RLM. [sent-133, score-0.101]

80 Thus, we consider vectors y of spike counts from N neurons, without explicit reference to the time at which they were collected. [sent-134, score-0.09]

81 A Gaussian model for y can certainly describe correlations between the cells, but is ill-matched to discrete count observations. [sent-135, score-0.064]

82 Thus, as with the derivation of the RLM from the Kalman ﬁlter, we derive here a new generalisation of a low-dimensional, structured Gaussian model to spike count data. [sent-136, score-0.185]

83 4 The distribution of any multivariate variable y can be factorized into a “cascaded” product of multiple one-dimensional distributions: N P (yn |y < n and ≤ n restrict the matrix to the ﬁrst (n − 1) and n rows and/or columns respectively. [sent-137, score-0.019]

84 Thus, we might construct suitably structured linear weights for the CGLM by applying this result to the covariance matrix induced by the low-dimensional Gaussian model known as factor analysis [12]. [sent-138, score-0.069]

85 Factor analysis assumes that data are generated from a K-dimensional latent process x ∼ N (0, I), where I is the K×K identity matrix, and y has the conditional distribution P (y|x) = N (Λx, Ψ) with Ψ a diagonal matrix and Λ an N × K loading matrix. [sent-139, score-0.155]

86 This leads to a covariance of y given by Σ = Ψ + ΛΛT . [sent-140, score-0.05]

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