{"title": "Recurrent linear models of simultaneously-recorded neural   populations", "book": "Advances in Neural Information Processing Systems", "page_first": 3138, "page_last": 3146, "abstract": "Population neural recordings with long-range temporal structure are often best understood in terms of a shared 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. Here we describe a new, scalable approach to discovering the low-dimensional dynamics that underlie simultaneously recorded spike trains from a neural population. Our method is based on recurrent linear models (RLMs), and relates closely to timeseries models based on recurrent neural networks. We formulate RLMs for neural data by generalising the Kalman-filter-based likelihood calculation for latent linear dynamical systems (LDS) models 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 linear model (CLM) to capture low-dimensional instantaneous correlations in neural populations. The CLM describes the cortical recordings better than either Ising or Gaussian models and, like the RLM, can be fit exactly and quickly. The CLM can also be seen as a generalization of a low-rank Gaussian model, in this case factor analysis. The computational tractability of the RLM and CLM allow both to scale to very high-dimensional neural data.", "full_text": "Recurrent linear models of\n\nsimultaneously-recorded neural populations\n\nMarius Pachitariu, Biljana Petreska, Maneesh Sahani\n\nGatsby Computational Neuroscience Unit\n\n{marius,biljana,maneesh}@gatsby.ucl.ac.uk\n\nUniversity College London, UK\n\nAbstract\n\nPopulation neural recordings with long-range temporal structure are often best un-\nderstood in terms of a common underlying low-dimensional dynamical process.\nAdvances in recording technology provide access to an ever-larger fraction of the\npopulation, but the standard computational approaches available to identify the\ncollective dynamics scale poorly with the size of the dataset. We describe a new,\nscalable approach to discovering low-dimensional dynamics that underlie simul-\ntaneously recorded spike trains from a neural population. We formulate the Re-\ncurrent Linear Model (RLM) by generalising the Kalman-\ufb01lter-based likelihood\ncalculation for latent linear dynamical systems to incorporate a generalised-linear\nobservation process. We show that RLMs describe motor-cortical population data\nbetter than either directly-coupled generalised-linear models or latent linear dy-\nnamical system models with generalised-linear observations. We also introduce\nthe cascaded generalised-linear model (CGLM) to capture low-dimensional in-\nstantaneous correlations in neural populations. The CGLM describes the cortical\nrecordings better than either Ising or Gaussian models and, like the RLM, can be\n\ufb01t exactly and quickly. The CGLM can also be seen as a generalisation of a low-\nrank Gaussian model, in this case factor analysis. The computational tractability\nof the RLM and CGLM allow both to scale to very high-dimensional neural data.\n\n1\n\nIntroduction\n\nMany essential neural computations are implemented by large populations of neurons working in\nconcert, and recent studies have sought both to monitor increasingly large groups of neurons [1, 2]\nand to characterise their collective behaviour [3, 4]. In this paper we introduce a new computational\ntool to model coordinated behaviour in very large neural data sets. While we explicitly discuss only\nmulti-electrode extracellular recordings, the same model can be readily used to characterise 2-photon\ncalcium-marker image data, EEG, fMRI or even large-scale biologically-faithful simulations.\nPopulational neural data may be represented at each time point by a vector yt with as many dimen-\nsions as neurons, and as many indices t as time points in the experiment. For spiking neurons, yt\nwill have positive integer elements corresponding to the number of spikes \ufb01red by each neuron in\nthe time interval corresponding to the t-th bin. As others have before [5, 6], we assume that the\ncoordinated activity re\ufb02ected in the measurement yt arises from a low-dimensional set of processes,\ncollected into a vector xt, which is not directly observed. However, unlike the previous studies,\nwe construct a recurrent model in which the hidden processes xt are driven directly and explicitly\nby the measured neural signals y1 . . . yt\u22121. This assumption simpli\ufb01es the estimation process. We\nassume for simplicity that xt evolves with linear dynamics and affects the future state of the neural\nsignal yt in a generalised-linear manner, although both assumptions may be relaxed. As in the latent\ndynamical system, the resulting model enforces a \u201cbottleneck\u201d, whereby predictions of yt based on\ny1 . . . yt\u22121 must be carried by the low-dimensional xt.\n\n1\n\n\fState prediction in the RLM is related to the Kalman \ufb01lter [7] and we show in the next section a\nformal equivalence between the likelihoods of the RLM and the latent dynamical model when ob-\nservation noise is Gaussian distributed. However, spiking data is not well modelled as Gaussian,\nand the generalisation of our approach to Poisson noise leads to a departure from the latent dynam-\nical approach. Unlike latent linear models with conditionally Poisson observations, the parameters\nof our model can be estimated ef\ufb01ciently and without approximation. We show that, perhaps in\nconsequence, the RLM can provide superior descriptions of neural population data.\n\n2 From the Kalman \ufb01lter to the recurrent linear model (RLM)\n\nConsider a latent linear dynamical system (LDS) model with linear-Gaussian observations.\nIts\ngraphical model is shown in Fig. 1A. The latent process is parametrised by a dynamics matrix\nA and innovations covariance Q that describe the evolution of the latent state xt:\n\nP (xt|xt\u22121) = N (xt|Axt\u22121, Q) ,\n\nwhere N (x|\u00b5, \u03a3) represents a normal distribution on x with mean \u00b5 and (co)variance \u03a3. For brevity,\nwe omit here and below the special case of the \ufb01rst time-step, in which x1 is drawn from a multivari-\nate Gaussian. The output distribution is determined by an observation loading matrix C and a noise\ncovariance R often taken to be diagonal so that all covariance is modelled by the latent process:\n\nP (yt|xt) = N (yt|Cxt, R) .\n\nIn the LDS, the joint likelihood of the observations {yt} can be written as the product:\n\nP (y1 . . . yT ) = P (y1)\n\nP (yt|y1 . . . yt\u22121)\n\nT(cid:89)\n\nt=2\n\n(cid:90)\n(cid:90)\n\nand in the Gaussian case can be computed using the usual Kalman \ufb01lter approach to \ufb01nd the condi-\ntional distributon at time t iteratively:\n\nP (yt+1|y1 . . . yt) =\n\ndxt+1 P (yt+1|xt+1)P (xt+1|y1 . . . yt)\ndxt+1 N (yt+1|Cxt+1, R) N (xt+1|A\u02c6xt, Vt+1)\n\n=\n= N (yt+1|CA\u02c6xt, CVt+1C(cid:62) + R) ,\n\ncertainty Vt+1 = E(cid:2)(xt+1 \u2212 A\u02c6xt)2|y1 . . . yt\n\nwhere we have introduced the (\ufb01ltered) state estimate \u02c6xt = E [xt|y1 . . . yt] and (predictive) un-\nKalman gain Kt = VtC(cid:62)(CVtC(cid:62) + R)\u22121, giving the following recursive recipe to calculate the\nconditional likelihood of yt+1:\n\n(cid:3). Both quantities are computed recursively using the\n\n\u02c6xt = A\u02c6xt\u22121 + Kt(yt \u2212 \u02c6yt)\nVt+1 = A(I \u2212 KtC)VtA(cid:62) + Q\n\u02c6yt+1 = CA\u02c6xt\n\nP (yt+1|y1 . . . yt) = N (yt+1| \u02c6yt+1, CVt+1C(cid:62) + R)\n\nFor the Gaussian LDS, the Kalman gain Kt and state uncertainty Vt+1 (and thus the output covari-\nance CVt+1C(cid:62) + R) depend on the model parameters (A, C, R, Q) and on the time step\u2014although\nas time grows they both converge to stationary values. Neither depends on the observations.\nThus, we might consider a relaxation of the Gaussian LDS model in which these matrices are taken\nto be stationary from the outset, and are parametrised independently so that they are no longer\nconstrained to take on the \u201ccorrect\u201d values as computed for Kalman inference. Let us call this\nparametric form of the Kalman gain W and the parametric form of the output covariance S. Then\nthe conditional likelihood iteration becomes\n\n\u02c6xt = A\u02c6xt\u22121 + W (yt \u2212 \u02c6yt)\n\n\u02c6yt+1 = CA\u02c6xt\n\nP (yt+1|y1 . . . yt) = N (yt+1| \u02c6yt+1, S) .\n\n2\n\n\fA x1\n\nA\n\nC\n\ny1\n\nB x0\n\n\u03b71\n\ny1\n\nC x0\n\nA\n\nA\n\n\u03b72\n\nA\n\nx2\n\nC\n\ny2\n\nx1\n\ny2\n\nx1\n\nx3\n\nC\n\ny3\n\nx2\n\ny3\n\nx2\n\nCA\n\nW\n\nCA\n\nW\n\nCA\n\nA\n\n\u2022 \u2022 \u2022\n\nA\n\n\u2022 \u2022 \u2022\n\n\u03b73\n\n\u03b7T -1\n\n\uf8fc\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8fe LDS\n\nxT\n\nC\n\nyT\n\nxT -1\n\nyT\n\nA\n\nW\n\n\u2022 \u2022 \u2022\n\nA\n\nxT -1\n\nW\n\nCA\n\nRLM\n\ny1\n\ny2\n\ny3\n\nyT\n\nFigure 1: Graphical representa-\ntions of the latent linear dynamical\nsystem (LDS: A, B) and recurrent\nlinear model (RLM: C). Shaded\nvariables are observed, unshaded\ncircles are latent random variables\nand squares are variables that de-\npend deterministically on their par-\nents. In B the LDS is redrawn in\nterms of the random innovations\n\u03b7t = xt \u2212 Axt\u22121, facilitating the\ntransition towards the RLM. The\nRLM is then obtained by replacing\n\u03b7t with a deterministically derived\nestimate W (yt \u2212 \u02c6yt).\n\nThe parameters of this new model are A, C, W and S. This is a relaxation of the Gaussian latent\nLDS model because W has more degrees of freedom than Q, as does S than R (at least if R is\nconstrained to be diagonal). The new model has a recurrent linear structure in that the random\nobservation yt is fed back linearly to perturb the otherwise deterministic evolution of the state \u02c6xt.\nA graphical representation of this model is shown in Fig. 1C, along with a redrawn graph of the LDS\nmodel. The RLM can be viewed as replacing the random innovation variables \u03b7t = xt \u2212 Axt\u22121\nwith data-derived estimates W (yt \u2212 \u02c6yt); estimates which are made possible by the fact that \u03b7t\ncontributes to the variability of yt around \u02c6yt.\n\n3 Recurrent linear models with Poisson observations\n\nThe discussion above has transformed a stochastic-latent LDS model with Gaussian output to an\nRLM with deterministic latent, but still with Gaussian output. Our goal, however, is to \ufb01t a model\nwith an output distribution better suited to the binned point-processes that characterise neural spik-\ning. Both linear Kalman-\ufb01ltering steps above and the eventual stationarity of the inference param-\neters depend on the joint Gaussian structure of the assumed LDS model. They would not apply\nif we were to begin a similar derivation from an LDS with Poisson output. However, a tractable\napproach to modelling point-process data with low-dimensional temporal structure may be provided\nby introducing a generalised-linear output stage directly to the RLM. This model is given by:\n\n\u02c6xt = A\u02c6xt\u22121 + W (yt \u2212 \u02c6yt)\n\ng( \u02c6yt+1) = CA\u02c6xt\n\nP (yt+1|y1 . . . yt) = ExpFam(yt+1| \u02c6yt+1)\n\n(1)\n\nwhere ExpFam is an exponential-family distribution such as Poisson, and the element-wise link\nfunction g allows for a nonlinear mapping from xt to the predicted mean \u02c6yt+1. In the following, we\nwill write f for the inverse-link as is more common for neural models, so that \u02c6yt+1 = f(CA\u02c6xt).\nThe simplest Poisson-based generalised-linear RLM might take as its output distribution\n\nP (yt| \u02c6yt) =(cid:89)\n\nPoisson(yti|\u02c6yti);\n\n\u02c6yt = f(CA\u02c6xt\u22121)) ,\n\ni\n\nwhere yti is the spike count of the ith cell in bin t and the function f is non-negative. However,\ncomparison with the output distribution derived for the Gaussian RLM suggests that this choice\nwould fail to capture the instantaneous covariance that the LDS formulation transfers to the output\ndistribution (and which appears in the low-rank structure of S above). We can address this concern\nin two ways. One option is to bin the data more \ufb01nely, thus diminishing the in\ufb02uence of the instan-\ntaneous covariance. The alternative is to replace the independent Poissons with a correlated output\ndistribution on spike counts. The cascaded generalised-linear model introduced below is a natural\nchoice, and we will show that it captures instantaneous correlations faithfully with very few hidden\ndimensions.\n\n3\n\n\fIn practice, we also sometimes add a \ufb01xed input \u00b5t to equation 1 that varies in time and determines\nthe average behavior of the population or the peri-stimulus time histogram (PSTH).\n\n\u02c6yt+1 = f (\u00b5t + CAxt)\n\nNote that the matrices A and C retain their interpretation from the LDS models. The matrix A\ncontrols the evolution of the dynamical process xt. The phenomenology of its dynamics is deter-\nmined by the complex eigenvalues of A. Eigenvalues with moduli close to 1 correspond to long\ntimescales of \ufb02uctuation around the PSTH. Eigenvalues with non-zero imaginary part correspond\nto oscillatory components. Finally, the dynamics will be stable iff all the eigenvalues lie within the\nunit disc. The matrix C describes the dependence of the high-dimensional neural signals on the low-\ndimensional latent processes xt. In particular, equation 2 determines the \ufb01ring rate of the neurons.\nThis generalised-linear stage ensures that the \ufb01ring rates are positive through the link function f, and\nthe observation process is Poisson. For other types of data, the generalised-linear stage might be\nreplaced by other appropriate link functions and output distributions.\n\n3.1 Relationship to other models\n\nRLMs are related to recurrent neural networks [8]. The differences lie in the state evolution, which\nin the neural network is nonlinear: xt = h (Axt\u22121 + W yt\u22121); and in the recurrent term which\ndepends on the observation rather than the prediction error. On the data considered here, we found\nthat using sigmoidal or threshold-linear functions h resulted in models comparable in likelihood\nto the RLM, and so we restricted our attention to simple linear dynamics. We also found that\nusing the prediction error term W (yt\u22121 \u2212 \u02c6yt) resulted in better models than the simple neural-net\nformulation, and we attribute this difference to the link between the RLM and Kalman inference.\nIt is also possible to work within the stochatic latent LDS framework, replacing the Gaussian out-\nput distribution with a generalised-linear Poisson output (e.g. [6]). The main dif\ufb01culty here is the\nintractability of the estimation procedure. For an unobserved latent process xt, an inference pro-\ncedure needs to be devised to estimate the posterior distribution on the entire sequence x1 . . . xt.\nFor linear-Gaussian observations, this inference is tractable and is provided by Kalman smoothing.\nHowever, with generalised-linear observations, inference becomes intractable and the necessary ap-\nproximations [6] are computationally intense and can jeopardize the quality of the \ufb01tted models. By\ncontrast, in the RLM xt is a deterministic function of data. In effect, the Kalman \ufb01lter has been built\ninto the model as the accurate estimation procedure, and ef\ufb01cient \ufb01tting is possible by direct gradient\nascent on the log-likelihood. Empirically we did not encounter dif\ufb01culties with local minima during\noptimization, as has been reported for LDS models \ufb01t by approximate EM [9]. Multiple restarts\nfrom different random values of the parameters always led to models with similar likelihoods.\nNote that to estimate the matrices A and W the gradient must be backpropagated through succes-\nsive iterations of equation 1. This technique, known as backpropagation-through-time, was \ufb01rst\ndescribed by [10] as a technique to \ufb01t recurrent neural network models. Recent implementations\nhave demonstrated state-of-the-art language models [11]. Backpropagation-through-time is thought\nto be inherently unstable when propagated past many timesteps and often the gradient is truncated\nprematurely [11]. We found that using large values of momentum in the gradient ascent alleviated\nthese instabilities and allowed us to use backpropagation without the truncation.\n\n4 The cascaded generalised-linear model (CGLM)\n\nThe link between the RLM and the LDS raises the possibility that a model for simultaneously-\nrecorded correlated spike counts might be derived in a similar way, starting from a non-dynamical,\nbut low-dimensional, Gaussian model. Stationary models of population activity have attracted recent\ninterest for their own sake (e.g. [1]), and would also provide a way model correlations introduced\nby common innovations that were neglected by the simple Poisson form of the RLM. Thus, we\nconsider vectors y of spike counts from N neurons, without explicit reference to the time at which\nthey were collected. A Gaussian model for y can certainly describe correlations between the cells,\nbut is ill-matched to discrete count observations. Thus, as with the derivation of the RLM from the\nKalman \ufb01lter, we derive here a new generalisation of a low-dimensional, structured Gaussian model\nto spike count data.\n\n4\n\n\fThe distribution of any multivariate variable y can be factorized into a \u201ccascaded\u201d product of multi-\nple one-dimensional distributions:\n\nP (y) =\n\nP (yn|y<n) .\n\n(2)\n\nN(cid:89)\n\nn=1\n\nHere n indexes the neurons up to the last neuron N, and y<n is the (n\u20131)-vector [y1 . . . yn\u22121]. For\na Gaussian-distributed y, the conditionals P (yn|y<n) would be linear-Gaussian. Thus, we propose\nthe \u201ccascaded generalised linear model\u201d (CGLM) in which each such one-dimensional conditional\ndistribution is a generalised-linear model:\n\n\u02c6yn = f(cid:0)\u00b5n + ST\n\nn y<n\n\n(cid:1)\n\nP (yn|y<n) = ExpFam (\u02c6yn)\n\n(3)\n(4)\n\nand in which the linear weights Sn take on a structured form developed below.\nThe equations 3 and 4 subsume the Gaussian distribution with arbitrary covariance in the case that\nf is linear, and the ExpFam conditionals are Gaussian. In this case, for a joint covariance of \u03a3, it is\nstraightforward to derive the expression\n\n1\n\n(\u03a3\u2264n,\u2264n)\u22121\n\nn,n\n\nSn =\n\nn,<n .\n\n(\u03a3\u2264n,\u2264n)\u22121\n\n(5)\nwhere the subscripts < n and \u2264 n restrict the matrix to the \ufb01rst (n \u2212 1) and n rows and/or columns\nrespectively. Thus, we might construct suitably structured linear weights for the CGLM by applying\nthis result to the covariance matrix induced by the low-dimensional Gaussian model known as factor\nanalysis [12]. Factor analysis assumes that data are generated from a K-dimensional latent process\nx \u223c N (0, I), where I is the K\u00d7K identity matrix, and y has the conditional distribution P (y|x) =\nN (\u039bx, \u03a8) with \u03a8 a diagonal matrix and \u039b an N \u00d7 K loading matrix. This leads to a covariance\nof y given by \u03a3 = \u03a8 + \u039b\u039bT . If we repeat the derivation of equations 3, 4 and 5 for this covariance\nmatrix, we obtain an expression for Sn via the matrix inversion lemma:\n\nSn =\n\n=\n\n1\n\n(\u03a3\u2264n,\u2264n)\u22121\n\nn,n\n\n1\n\n(\u03a3\u2264n,\u2264n)\u22121\n\nn,n\n\n= \u2212\n\n1\n\n(\u03a3\u2264n,\u2264n)\u22121\n\nn,n\n\n(cid:0)\u03a8\u2264n,\u2264n + \u039b\u2264n,\u00b7\u039bT\u2264n,\u00b7(cid:1)\u22121\n(cid:16)\n(cid:16)(cid:0)\u03a8\u22121\u039b(cid:1)\n\u2264n,\u00b7 (\u00b7\u00b7\u00b7)(cid:0)\u039b\u03a8\u22121(cid:1)T\n\n\u03a8\u22121\u2264n,\u2264n \u2212 \u03a8\u22121\u2264n,\u2264n\u039b<n,\u00b7 (\u00b7\u00b7\u00b7) \u039bT\n<n,\u00b7\u03a8\u22121\n(cid:17)\n\n\u2264n,\u00b7\n\nn,<n\n\nn,<n\n\n(cid:17)\n\nn,<n\n\n(6)\n\n<n,<n\n\nif we arrange all Sn as the upper columns of an N \u00d7 N matrix S, we can write S = upper(cid:0)zwT(cid:1)\n\nwhere the omitted factor (\u00b7\u00b7\u00b7 ) is a K \u00d7 K matrix. The \ufb01rst term in equation 6 vanishes because it\ninvolves only the off-diagonal entries of \u03a8. The surviving factor shows that Sn is formed by taking\na linear combination of the columns of \u03a8\u22121\u039b and then truncating to the \ufb01rst n \u2212 1 elements. Thus,\nfor some low-dimensional matrices z = \u03a8\u22121\u039b and w, where the operation upper extracts the\nstrictly upper triangular part of a matrix. This is the natural structure imposed on the cascaded\nconditionals by factor analysis. Thus, we adopt the same constraint on S in the case of generalised-\nlinear observations. The resulting (CGLM) is shown below to provide better \ufb01ts to binarized neural\ndata than standard Ising models (see the Results section), even with as few as three latent dimensions.\nAnother useful property of the CGLM is that it allows stimulus-dependent inputs in equation 3. The\nCGLM can also be used in combination with the generalised-linear RLM, with the CGLM replacing\nthe otherwise independent observation model. This approach can be useful when large bins are used\nto discretize spike trains. In both cases the model can be estimated quickly with standard gradient\nascent techniques.\n\n5 Alternative models\n\n5.1 Alternative for temporal interactions: causally-coupled generalised linear model\n\nOne popular and simple model of simultaneously recorded neuronal populations [3] constructs tem-\nporal dependencies between units by directly coupling each neuron\u2019s probability of \ufb01ring to the past\n\n5\n\n\fspikes in the entire population:\n\nyt \u221d Poisson(f(\u00b5t +\n\nN(cid:88)\n\ni=1\n\nBi (hi (cid:63) yt)))\n\nHere, hi (cid:63) yt are convolutions of the spike trains with a set of basis functions hi, and Bi are pairwise\ninteraction weights. Each matrix Bi has N 2 parameters where N is the number of neurons, so the\nnumber of parameters grows quadratically with the population size. This type of scaling makes the\nmodel prohibitive to use with very large-scale array recordings. Even with aggresive regularization\ntechniques, the model\u2019s parameters are dif\ufb01cult to identify with limited amounts of data. Perhaps\nmore importantly, the model does not have a physical interpretation. Neurons recorded in cortex\nare rarely directly-connected and retinal ganglion cells almost never directly connect to each other.\nInstead, such directly-coupled GLMs are used to describe so-called \u2019functional\u2019 interactions between\nneurons [3]. We believe a much better interpretation for the correlations observed between pairs of\nneurons is that they are caused by common inputs to these neurons which seem often to be con\ufb01ned\nto a small number of dimensions. The models we propose here, the RLM and the CGLM, are aimed\nat discovering such inputs.\n\n5.2 Alternative for instantaneous interactions: the Ising model\n\nInstantaneous interactions between binary data (as would be obtained by counting spikes in short\nintervals) can be modelled in terms of their pairwise interactions [1] embodied in the Ising model:\n\nP (y) =\n\n1\nZ\n\neyT Jy.\n\n(7)\n\nwhere J is a pairwise interaction matrix and Z is the partition function, or the normalization constant\nof the model. The model\u2019s attractiveness is that for a given covariance structure it makes the weakest\npossible assumptions about the distribution of y, that is, like a Gaussian for continuous data, it\nhas the largest possible entropy under the covariance constraint. However, the Ising model and\nthe so-called functional interactions J have no physical interpretation when applied to neural data.\nFurthermore, Ising models are dif\ufb01cult to \ufb01t as they require estimates of the gradients of the partition\nfunction Z; they also suffer from the same quadratic scaling in number of paramters as does the\ndirectly-coupled GLM. Ising models are even harder to estimate when stimulus-dependent inputs\nare added in equation 7, but for data collected in the retina or other sensory areas [1], much of the\ncovariation in y may be expected to arise from common stimulus input. Another short-coming of\nthe Ising model is that it can only model binarized data and cannot be normalized for integer y-s [6],\nso either the time bins need to be reduced to ensure no neuron \ufb01res more than one spike in a single\nbin or the spike counts must be capped at 1.\n\n6 Results\n\n6.1 Simulated data\n\nWe began by evaluating RLM models \ufb01t to simulated data where the true generative parameters were\nknown. Two aspects of the estimated models were of particular interest: the phenomenology of the\ndynamics (captured by the eigenvalues of the dynamics matrix A) and the relationship between the\ndynamical subspace and measured neural activity (captured by the output matrix C). We evaluated\nthe agreement between the estimated and generative output matrices by measuring the principal\nangles between the corresponding subspaces. These report, in succession, the smallest angle achiev-\nable between a line in one subspace and a line in the second subspace, once all previous such vectors\nof maximal agreement have been projected out. Exactly aligned n-dimensional subspaces have all\nn principal angles equal to 0\u25e6. Unrelated low-dimensional subspaces embedded in high dimensions\nare close to orthogonal and so have principal angles near 90\u25e6.\nWe \ufb01rst veri\ufb01ed the robustness of maximisation of the generalised-linear RLM likelihood by \ufb01tting\nmodels to simulated data generated by a known RLM. Fig. 2(a) shows eigenvalues from several sim-\nulated RLMs and the eigenvalues recovered by \ufb01tting parameters to simulated data. The agreement\nis generally good. In particular, the qualitative aspects of the dynamics re\ufb02ected in the absolute val-\nues and imaginary parts of the eigenvalues are well characterised. Fig. 2(d) shows that the RLM \ufb01ts\n\n6\n\n\f(a)\n\n(d)\n\n(b)\n\n(e)\n\n(c)\n\n(f)\n\nFigure 2: Experiments on 100-dimensional simulated data generated from a 5-dimensional latent\nprocess. Generating models were Poisson RLM (ad), Poisson LDS with random parameters (cf) and\nPoisson LDS model with parameters \ufb01t to neural data (cf). The models \ufb01t were PCA, LDS with\nGaussian (LDS/GLDS) or Poisson (PLDS) output, and RLM with Poisson output (RLM). In the\nupper plots, eigenvalues from different runs are shown in different colors.\n\nalso recover the subspace de\ufb01ned by the loading matrix C, and do so substantially more accurately\nthan either principal components analysis (PCA) or GLDS models. It is important to note that the\nlikelihoods of LDS models with Poisson observations are dif\ufb01cult to optimise, and so may yield\npoor results even when \ufb01t to within-class data. In practice we did not observe local optima with the\nRLM or CGLM.\nWe also asked whether the RLM could recover the dynamical properties and latent subspace of data\ngenerated by a latent LDS model with Poisson observations. Fig. 2(b) shows that the dynamical\neigenvalues of the maximum-likelihood RLM are close to the eigenvalues of generative LDS dy-\nnamics, whilst Fig. 2(e) shows that the dynamical subspace is also correctly recovered. Parameters\nfor these simulations were chosen randomly. We then asked whether the quality of parameter identi-\n\ufb01cation extended to Poisson-output LDS models with realistic parameters, by generating data from\na Poisson-output LDS model that had been \ufb01t to a neural recording. As seen in \ufb01gs. 2(c) and 2(f),\nthe RLM \ufb01ts remain accurate in this regime, yielding better subspace estimates than either PCA or\na Gaussian LDS.\n\n6.2 Array recorded data\n\nWe next compared the performance of the novel models on neural data. The RLM was compared\nto the directed-coupled GLM (\ufb01t by gradient-based likelihood optimisation) as well as LDS models\nwith Gaussian or Poisson outputs (\ufb01t by EM, with a Laplace approximation E-step). The CGLM\nwas compared to the Ising model. We used a dataset of 92 neurons recorded with a Utah array\nimplanted in the premotor and motor cortices of a rhesus macaque monkey performing a delayed\ncenter-out reach task. For all comparisons below we use datasets of 108 trials in which the monkey\nmade movements to the same target.\nWe discretized spike trains into time bins of 10ms. The directed-coupled GLM needed substantial\nregularization in order to make good predictions on held-out test data. Figure 3(a) shows only\nthe best cross-validation result for the GLM, but results without regularization for models with\n\n7\n\n00.20.40.60.81\u22120.4\u22120.200.20.4RealImaginaryRLM recovers eigenvaluesof simulated dynamics  Ground truthIdentified0.20.40.60.81\u22120.4\u22120.200.20.4RealImaginaryRLM identifies the eigenvaluesof diverse PLDS models  Generative PLDSIdentified by PLDSIdentified by RLM0.850.90.951\u22120.1\u22120.0500.050.1RealImaginaryRLM identifies the eigenvaluesof a PLDS model fit to real data  Generative PLDSIdentified by RLMPCAGLDSRLM04590Principal anglesbetween ground truth (RLM)and identified subspacesDegreesPCALDSRLMPLDS04590Principal anglesbetween true and identified subspacesDegreesPCAGLDSRLM04590Principal anglesbetween PLDS fit to dataand identified subspacesDegrees\f(a)\n\n(b)\n\nFigure 3: a. Predictive performance of various models on test data (higher is better). GLM-type\nmodels are helped greatly by self-coupling \ufb01lters (which the other models do not have). The best\nmodel is an RLM with three latent dimensions and a low-rank model of the PSTH (see the supple-\nmentary material for more information about this model). Adding self-coupling \ufb01lters to this model\nfurther increases its predictive performance by 5 (not shown). b. The likelihood per spike of Ising\nmodels as well as CGLM models with small numbers of hidden dimensions. The CGLM saturates\nat three dimensions and performs better than Ising models.\nlow-dimensional parametrisation. Performance was measured by the causal mean-squared-error in\nprediction subtracted from the error of a low-rank smoothed PSTH model (based on a singular-\nvalue decomposition of the matrix of all smoothed PSTHs). The number of dimensions (5) and the\nstandard deviation of the Gaussian smoothing \ufb01lter (20 ms) were cross-validated to \ufb01nd the best\npossible PSTH performance. Thus, our evaluation is focuses on each model\u2019s ability to predict\ntrial-to-trial co-variation in \ufb01ring around the mean.\nA second measure of performance for the RLM was obtained by studying probabilistic samples\nobtained from the \ufb01tted model. Figure 4 in the supplemental material shows averaged noise cross-\ncorrelograms obtained from a large set of samples. Note that the PSTHs have been subtracted from\neach trial to reveal only the extra correlation structure that is not repeated amongst trials. Even with\nfew hidden dimensions, the model captures well the full temporal structure of the noise correlations.\nIn the case of the Ising model we binarized the data by replacing all spike counts larger than 1 with\n1. The log-likelihood of the Ising model could only be estimated for small numbers of neurons, so\nfor comparison we took only the 30 most active neurons. The measure of performance reported in\n\ufb01gure 3(b) is the extra log-likelihood per spike obtained above that of a model that makes constant\npredictions equal to the mean \ufb01ring rate of each neuron. The CGLM model with only three hidden\ndimensions achieves the best generalisation performance, surpassing the Ising model. Similar results\nfor the performance of the CGLM can be seen on the full dataset of 92 neurons with non-binarized\ndata, indicating that three latent dimensions suf\ufb01ce to describe the full space visited by the neuronal\npopulation on a trial-by-trial basis.\n\n7 Discussion\n\nThe generalised-linear RLM model, while sharing motivation with latent LDS model, can be \ufb01t more\nef\ufb01ciently and without approximation to non-Gaussian data. We have shown improved performance\non both simulated data and on population recordings from the motor cortex of behaving monkeys.\nThe model is easily extended to other output distributions (such as Bernoulli or negative binomial),\nto mixed continuous and discrete data, to nonlinear outputs, and to nonlinear dynamics. For the\nmotor data considered here, the generalised-linear model performed as well as models with further\nnon-linearites. However, preliminary results on data from sensory cortical areas suggests that non-\nlinear models may be of greater value in other settings.\n\n8 Acknowledgments\n\nWe thank Krishna Shenoy and members of his lab for generously providing access to data. Funding\nfrom the Gatsby Charitable Foundation and DARPA REPAIR N66001-10-C-2010.\n\n8\n\nGLM \u2212 SCGLMPLDS10 LDS10LDS20RLM10RLM20             RLM3+PSTH5678910111213MSEbaseline \u2212 MSEbaseline = PSTH (low rank)Filtering prediction on test dataIsingrank=1r=2r=3r=4r=500.010.020.030.040.050.060.07Likelihoodperspike\u2212baseline(bits)\fReferences\n[1] E Schneidman, MJ Berry, R Segev, and W Bialek. Weak pairwise correlations imply strongly correlated\n\nnetwork states in a neural population. Nature, 440:1007\u20131012, 2005.\n\n[2] Gyorgy Buzsaki. Large-scale recording of neuronal ensembles. NatNeurosci, 7(5):446\u201351, 2004.\n[3] J. W. Pillow, J. Shlens, L. Paninski, A. Sher, A. M. Litke, E. J. Chichilnisky, and E. P. Simoncelli. Spatio-\ntemporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207):995\u2013\n999, 2008.\n\n[4] Mark M. Churchland, Byron M. Yu, Maneesh Sahani, and Krishna V. Shenoy. Techniques for extract-\ning single-trial activity patterns from large-scale neural recordings. CurrOpinNeurobiol, 17(5):609\u2013618,\n2007.\n\n[5] BM Yu, A Afshar, G Santhanam, SI Ryu, KV Shenoy, and M Sahani. Extracting dynamical structure\nembedded in neural activity. Advances in Neural Information Processing Systems, 18:1545\u20131552, 2006.\n[6] JH Macke, L Bsing, JP Cunningham, BM Yu, KV Shenoy, and M Sahani. Empirical models of spiking in\n\nneural populations. Advances in Neural Information Processing Systems, 24:1350\u20131358, 2011.\n\n[7] R.E. Kalman. A new approach to linear \ufb01ltering and prediction problems. Journal of Basic Engineering,\n\n82(1):35\u201345, 1960.\n\n[8] JL Elman. Finding structure in time. Cognitive Science, 14:179\u2013211, 1990.\n[9] L Buesing, JH Macke, and M Sahani. Spectral learning of linear dynamics from generalised-linear obser-\nvations with application to neural population data. Advances in Neural Information Processing Systems,\n25, 2012.\n\n[10] DE Rumelhart, GE Hinton, and RJ Williams. Learning internal representations by error propagation. Mit\n\nPress Computational Models Of Cognition And Perception Series, pages 318\u2013462, 1986.\n\n[11] T Mikolov, A Deoras, S Kombrink, L Burget, and JH Cernocky. Empirical evaluation and combination\nof advanced language modeling techniques. Conference of the International Speech Communication\nAssociation, 2011.\n\n[12] Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006.\n\n9\n\n\f", "award": [], "sourceid": 1434, "authors": [{"given_name": "Marius", "family_name": "Pachitariu", "institution": "Gatsby Unit, UCL"}, {"given_name": "Biljana", "family_name": "Petreska", "institution": "UCL"}, {"given_name": "Maneesh", "family_name": "Sahani", "institution": "Gatsby Unit, UCL"}]}