{"title": "Extracting Dynamical Structure Embedded in Neural Activity", "book": "Advances in Neural Information Processing Systems", "page_first": 1545, "page_last": 1552, "abstract": "", "full_text": "Extracting Dynamical Structure Embedded in\n\nNeural Activity\n\nByron M. Yu1, Afsheen Afshar1,2, Gopal Santhanam1,\n\nStephen I. Ryu1,3, Krishna V. Shenoy1,4\n\n1Department of Electrical Engineering, 2School of Medicine, 3Department of\nNeurosurgery, 4Neurosciences Program, Stanford University, Stanford, CA 94305\n{byronyu,afsheen,gopals,seoulman,shenoy}@stanford.edu\n\nManeesh Sahani\n\nGatsby Computational Neuroscience Unit, UCL\n\nLondon, WC1N 3AR, UK\n\nmaneesh@gatsby.ucl.ac.uk\n\nAbstract\n\nSpiking activity from neurophysiological experiments often exhibits dy-\nnamics beyond that driven by external stimulation, presumably re\ufb02ect-\ning the extensive recurrence of neural circuitry. Characterizing these\ndynamics may reveal important features of neural computation, par-\nticularly during internally-driven cognitive operations. For example,\nthe activity of premotor cortex (PMd) neurons during an instructed de-\nlay period separating movement-target speci\ufb01cation and a movement-\ninitiation cue is believed to be involved in motor planning. We show\nthat the dynamics underlying this activity can be captured by a low-\ndimensional non-linear dynamical systems model, with underlying re-\ncurrent structure and stochastic point-process output. We present and\nvalidate latent variable methods that simultaneously estimate the system\nparameters and the trial-by-trial dynamical trajectories. These meth-\nods are applied to characterize the dynamics in PMd data recorded\nfrom a chronically-implanted 96-electrode array while monkeys perform\ndelayed-reach tasks.\n\n1 Introduction\n\nAt present, the best view of the activity of a neural circuit is provided by multiple-electrode\nextracellular recording technologies, which allow us to simultaneously measure spike trains\nfrom up to a few hundred neurons in one or more brain areas during each trial. While the\nresulting data provide an extensive picture of neural spiking, their use in characterizing the\n\ufb01ne timescale dynamics of a neural circuit is complicated by at least two factors. First,\nextracellularly captured action potentials provide only an occasional view of the process\nfrom which they are generated, forcing us to interpolate the evolution of the circuit between\nthe spikes. Second, the circuit activity may evolve quite differently on different trials that\nare otherwise experimentally identical.\n\n\fThe usual approach to handling both problems is to average responses from different trials,\nand study the evolution of the peri-stimulus time histogram (PSTH). There is little alter-\nnative to this approach when recordings are made one neuron at a time, even when the\ndynamics of the system are the subject of study. Unfortunately, such averaging can obscure\nimportant internal features of the response. In many experiments, stimulus events provide\nthe trigger for activity, but the resulting time-course of the response is internally regulated\nand may not be identical on each trial. This is especially important during cognitive pro-\ncessing such as decision making or motor planning. In this case, the PSTH may not re\ufb02ect\nthe true trial-by-trial dynamics. For example, a sharp change in \ufb01ring rate that occurs with\nvarying latency might appear as a slow smooth transition in the average response.\n\nAn alternative approach is to adopt latent variable methods and to identify a hidden dy-\nnamical system that can summarize and explain the simultaneously-recorded spike trains.\nThe central idea is that the responses of different neurons re\ufb02ect different views of a com-\nmon dynamical process in the network, whose effective dimensionality is much smaller\nthan the total number of neurons in the network. While the underlying state trajectory may\nbe slightly different on each trial, the commonalities among these trajectories can be cap-\ntured by the network\u2019s parameters, which are shared across trials. These parameters de\ufb01ne\nhow the network evolves over time, as well as how the observed spike trains relate to the\nnetwork\u2019s state at each time point.\n\nDimensionality reduction in a latent dynamical model is crucial and yields bene\ufb01ts beyond\nsimple noise elimination. Some of these bene\ufb01ts can be illustrated by a simple physical\nexample. Consider a set of noisy video sequences of a bouncing ball. The trajectory of\nthe ball may not be identical in each sequence, and so simply averaging the sequences to-\ngether would provide little information about the dynamics. Independently smoothing the\ndynamics of each pixel might identify a dynamical process; however, correctly rejecting\nnoise might be dif\ufb01cult, and in any case this would yield an inef\ufb01cient and opaque rep-\nresentation of the underlying physical process. By contrast, a hidden dynamical system\naccount could capture the video sequence data using a low-dimensional latent variable that\nrepresented only the ball\u2019s position and momentum over time, with dynamical rules that\ncaptured the physics of ballistics and elastic collision. This representation would exploit\nshared information from all pixels, vastly simplifying the problem of noise rejection, and\nwould provide a scienti\ufb01cally useful depiction of the process.\n\nThe example also serves to illustrate the two broad bene\ufb01ts of this type of model. The \ufb01rst is\nto obtain a low dimensional summary of the dynamical trajectory in any one trial. Besides\nthe obvious bene\ufb01ts of denoising, such a trajectory can provide an invaluable representa-\ntion for prediction of associated phenomena. In the video sequence example, predicting the\nloudness of the sound on impact might be easy given the estimate of the ball\u2019s trajectory\n(and thus its speed), but would be dif\ufb01cult from the raw pixel trajectories, even if denoised.\nIn the neural case, behavioral variables such as reaction time might similarly be most easily\npredicted from the reconstructed trajectory. The second broad goal is systems identi\ufb01ca-\ntion: learning the rules that govern the dynamics. In the video example this would involve\ndiscovery of various laws of physics, as well as parameters describing the ball such as its\ncoef\ufb01cient of elasticity. In the neural case this would involve identifying the structure of\ndynamics available to the circuit: the number and relationship of attractors, appearance of\noscillatory limit cycles and so on.\n\nThe use of latent variable models with hidden dynamics for neural data has, thus far, been\nlimited. In [1], [2], small groups of neurons in the frontal cortex were modeled using hidden\nMarkov models, in which the latent dynamical system is assumed to transition between a\nset of discrete states. In [3], a state space model with linear hidden dynamics and point-\nprocess outputs was applied to simulated data. However, these restricted latent models\ncannot capture the richness of dynamics that recurrent networks exhibit.\nIn particular,\nsystems that converge toward point or line attractors, exhibit limit cycle oscillations, or\n\n\feven transition into chaotic regimes have long been of interest in neural modeling. If such\nsystems are relevant to real neural data, we must seek to identify hidden models capable of\nre\ufb02ecting this range of behaviors.\n\nIn this work, we consider a latent variable model having (1) hidden underlying recurrent\nstructure with continuous-valued states, and (2) Poisson-distributed output spike counts\n(conditioned on the state), as described in Section 2. Inference and learning for this nonlin-\near model are detailed in Section 3. The methods developed are applied to a delayed-reach\ntask described in Section 4. Evidence of motor preparation in PMd is given in Section 5.\nIn Section 6, we characterize the neural dynamics of motor preparation on a trial-by-trial\nbasis.\n\n2 Hidden non-linear dynamical system\n\nA useful dynamical system model capable of expressing the rich behavior expected of\nneural systems is the recurrent neural network (RNN) with Gaussian perturbations\n\nxt | xt\u22121 \u223c N (\u03c8(xt\u22121), Q)\n\n\u03c8(x) = (1 \u2212 k)x + kW g(x),\n\n(1)\n(2)\nwhere xt \u2208 IRp\u00d71 is the vector of the node values in the recurrent network at time\nt \u2208 {1, . . . , T}, W \u2208 IRp\u00d7p is the connection weight matrix, g is a non-linear activa-\ntion function which acts element-by-element on its vector argument, k \u2208 IR is a parameter\nrelated to the time constant of the network, and Q \u2208 IRp\u00d7p is a covariance matrix. The\ninitial state is Gaussian-distributed\n\nx0 \u223c N (p0, V0) ,\n\n(3)\nwhere p0 \u2208 IRp\u00d71 and V0 \u2208 IRp\u00d7p are the mean vector and covariance matrix, respectively.\nModels of this class have long been used, albeit generally without stochastic pertubation,\nto describe the dynamics of neuronal responses (e.g., [4]). In this classical view, each node\nof the network represents a neuron or a column of neurons. Our use is more abstract. The\nRNN is chosen for the range of dynamics it can exhibit, including convergence to point\nor surface attractors, oscillatory limit cycles, or chaotic evolution; but each node is simply\nan abstract dimension of latent space which may couple to many or all of the observed\nneurons.\n\nThe output distribution is given by a generalized linear model that describes the relationship\nt \u2208 IR of neuron i \u2208 {1, . . . , q} in\nbetween all nodes in the state xt and the spike count yi\nthe tth time bin\nci\u00b7 xt + di\n\n(4)\nwhere ci \u2208 IRp\u00d71 and di \u2208 IR are constants, h is a link function mapping IR \u2192 IR+,\nand \u0394 \u2208 IR is the time bin width. We collect the spike counts from all q simultaneously-\nrecorded physical neurons into a vector yt \u2208 IRq\u00d71, whose ith element is yi\nt. The choice of\nthe link functions g and h is discussed in Section 3.\n\nt | xt \u223c Poisson\nyi\n\n(cid:2)\n\n\u0394\n\n,\n\n(cid:3)\n\n(cid:3)\n\n(cid:2)\n\nh\n\n3\n\nInference and Learning\n\nThe Expectation-Maximization (EM) algorithm [5] was used to iteratively (1) infer the\nunderlying hidden state trajectories (i.e., recover a distribution over the hidden sequence\n{x}T\n1 ), and (2) learn the model parameters (i.e.,\nestimate \u03b8 =\n\nW, Q, k, p0, V0,{ci},{di}(cid:5)\n1 corresponding to the observations {y}T\n\n), given only a set of observation sequences.\n\n(cid:4)\n\n\ffor each\nInference (the E-step) involves computing or approximating P\nsequence, where \u03b8k are the parameter estimates at the kth EM iteration. A variant of the\nExtended Kalman Smoother (EKS) was used to approximate these joint smoothed state\nposteriors. As in the EKS, the non-linear time-invariant state system (1)-(2) was trans-\nformed into a linear time-variant sytem using local linearization. The difference from EKS\narises in the measurement update step of the forward pass\n\n1 , \u03b8k\n\n(cid:2)\n\nxt | {y}t\n\n(cid:3) \u221d P (yt | xt) P\n\n(cid:2)\n\n(cid:3)\n\nxt | {y}t\u22121\n\n1\n\n(cid:2){x}T\n\n1 | {y}T\n\n(cid:3)\n\n1\n\nP\n\n(5)\nBecause P (yt | xt) is a product of Poissons rather than a Gaussian, the \ufb01ltered state pos-\nterior P (xt | {y}t\n1) cannot be easily computed. Instead, as in [3], we approximated this\nposterior with a Gaussian centered at the mode of log P (xt | {y}t\n1) and whose covariance\nis given by the negative inverse Hessian of the log posterior at that mode. Certain choices\nof h, including ez and log (1 + ez), lead to a log posterior that is strictly concave in xt. In\n(cid:8)(cid:9)\nthese cases, the unique mode can easily be found by Newton\u2019s method.\n\n(cid:7)\n\n(cid:6)\n\n.\n\nlog P\n\nLearning (the M-step) requires \ufb01nding the \u03b8 that maximizes E\n,\nwhere the expectation is taken over the posterior state distributions found in the E-step.\nNote that, for multiple sequences that are independent conditioned on \u03b8, we use the sum\nof expectations over all sequences. Because the posterior state distributions are approxi-\nmated as Gaussians in the E-step, the above expectation is a Gaussian integral that involves\nnon-linear functions g and h and cannot be computed analytically in general. Fortunately,\nthis high-dimensional integral can be reduced to many one-dimensional Gaussian integrals,\nwhich can be accurately and reasonably ef\ufb01ciently approximated using Gaussian quadra-\nture [6], [7].\n\n{x}T\n\n1 ,{y}T\n\n1 | \u03b8\n\nWe found that setting g to be the error function\n\n(cid:10) z\n\n0\n\ng(z) =\n\n2\u221a\n\u03c0\n\n\u2212t2\ne\n\ndt\n\n(6)\n\nmade many of the one-dimensional Gaussian integrals involving g analytically tractable.\nThose that were not analytically tractable were approximated using Gaussian quadrature.\nThe error function is one of a family of sigmoid activation functions that yield similar\nbehavior in a RNN.\n\nIf h were chosen to be a simple exponential, all the Gaussian integrals involving h could be\ncomputed exactly. Unfortunately, this exponential mapping would distort the relationship\nbetween perturbations in the latent state (whose size is set by the covariance matrix Q)\nand the resulting \ufb02uctuations in \ufb01ring rates. In particular, the size of \ufb01ring-rate \ufb02uctations\nwould grow exponentially with the mean, an effect that would then add to the usual linear\nincrease in spike-count variance that comes from the Poisson output distribution. Since\nneural \ufb01ring does not show such a severe scaling in variability, such a model would \ufb01t\npoorly. Therefore, to maintain more even \ufb01ring-rate \ufb02uctuations, we instead take\n\nh(z) = log (1 + ez) .\n\n(7)\n\nThe corresponding Gaussian integrals must then be approximated by quadrature methods.\nRegardless of the forms of g and h chosen, numerical Newton methods are needed for\nmaximization with respect to {ci} and {di}.\nThe main drawback of these various approximations is that the overall observation likeli-\nhood is no longer guaranteed to increase after each EM iteration. However, in our simula-\ntions, we found that sensible results were often produced. As long as the variances of the\nposterior state distribution did not diverge, the output distributions described by the learned\nmodel closely approximated those of the actual model that generated the simulated data.\n\n\f4 Task and recordings\n\nWe trained a rhesus macaque monkey to perform delayed center-out reaches to visual tar-\ngets presented on a fronto-parallel screen. On a given trial, the peripheral target was pre-\nsented at one of eight radial locations (30, 70, 110, 150, 190, 230, 310, 350\u00b0) 10 cm\naway, as shown in Figure 1. After a pseudo-randomly chosen delay period of 200, 750, or\n1000 ms, the target increased in size as the go cue and the monkey reached to the target. A\n96-channel silicon electrode array (Cyberkinetics, Inc.) was implanted straddling PMd and\nmotor cortex (M1). Spike sorting was performed of\ufb02ine to isolate 22 single-neuron and\n109 multi-neuron units.\n\ns\n \n/\n \ns\ne\nk\np\ns\n\ni\n\n100\n\n0\n\n 200 ms\n\nDelay \nDelay \nActivity\nActivity\n\nPeri-Movement \nPeri-Movement \nActivity\nActivity\n\nFigure 1: Delayed reach task and average action potential (spike) emission rate from one\nrepresentative unit. Activity is arranged by target location. Vertical dashed lines indicate\nperipheral reach target onset (left) and movement onset (right).\n\n5 Motor preparation in PMd\n\nMotor preparation is often studied using the \u201cinstructed delay\u201d behavioral paradigm, as\ndescribed in Section 4, where a variable-length \u201cplanning\u201d period temporally separates an\ninstruction stimulus from a go cue [8]\u2013[13]. Longer delay periods typically lead to shorter\nreaction times (RT, de\ufb01ned as time between go cue and movement onset), and this has been\ninterpreted as evidence for a motor preparation process that takes time [11], [12], [14], [15].\nIn this view, the delay period allows for motor preparation to complete prior to the go cue,\nthus shortening the RT.\n\nEvidence for motor preparation at the neural level is taken from PMd (and, to a lesser\ndegree, M1), where neurons show sustained activity during the delay period (Figure 1,\ndelay activity) [8]\u2013[10]. A number of \ufb01ndings support the hypothesis that such activity\nis related to motor preparation. First, delay period activity typically shows tuning for the\ninstruction (i.e., location of reach target; note that the PMd neuron in Figure 1 has greater\ndelay activity before leftward than before rightward reaches), consistent with the idea that\nsomething speci\ufb01c is being prepared [8], [9], [11], [13]. Second, in the absence of a delay\nperiod, a brief burst of similarly-tuned activity is observed during the RT interval, consistent\nwith the idea that motor preparation is taking place at that time [12].\n\nThird, we have recently reported that \ufb01ring rates across trials to the same reach target\nbecome more consistent as the delay period progresses [16]. The variance of \ufb01ring rate,\n\n\fmeasured across trials, divided by mean \ufb01ring rate (similar to the Fano factor) was com-\nputed for each unit and each time point. Averaged across 14 single- and 33 multi-neuron\n\u221210) from\nunits, we found that this Normalized Variance (NV) declined 24% (t-test, p <10\n200 ms before target onset to the median time of the go cue. This decline spanned \u223c119 ms\njust after target onset and appears to, at least roughly, track the time-course of motor prepa-\nration.\n\nThe NV may be interpreted as a signature of the approximate degree of motor prepara-\ntion yet to be accomplished. Shortly after target onset, \ufb01ring rates are frequently far from\ntheir mean. If the go cue arrives then, it will take time to correct these \u201cerrors\u201d and RTs\nwill therefore be longer. By the time the NV has completed its decline, \ufb01ring rates are\nconsistently near their mean (which we presume is near an \u201coptimal\u201d con\ufb01guration for the\nimpending reach), and RTs will be shorter if the go cue arrives then. This interpretation\nassumes that there is a limit on how quickly \ufb01ring rates can converge to their ideal values\n(a limit on how quickly the NV can drop) such that a decline during the delay period saves\ntime later. The NV was found to be lower at the time of the go cue for trials with shorter\nRTs than those with longer RTs [16].\n\nThe above data strongly suggest that the network underlying motor preparation exhibits\nrich dynamics. Activity is initially variable across trials, but appears to settle during the\ndelay period. Because the RNN (1)-(2) is capable of exhibiting such dynamics and may\nunderly motor preparation, we sought to identify such a dynamical system in delay activity.\n\n6 Results and discussion\n\nThe NV reveals an average process of settling by measuring the convergence of \ufb01ring\nacross different trials. However, it provides little insight into the course of motor planning\non a single trial. A gradual fall in trial-to-trial variance might re\ufb02ect a gradual convergence\non each trial, or might re\ufb02ect rapid transitions that occur at different times on different\ntrials. Similarly, all the NV tells us about the dynamic properties of the underlying network\nis the basic fact of convergence from uncontrolled initial conditions to a consistent pre-\nmovement preparatory state. The structure of any underlying attractors and corresponding\nbasins of attraction is unobserved. Furthermore, the NV is \ufb01rst computed per-unit and\naveraged across units, thus ignoring any structure that may be present in the correlated\n\ufb01ring of units on a given trial. The methods presented here are well-suited to extending the\ncharacterization of this settling process.\nWe \ufb01t the dynamical system model (1)\u2013(4) with three latent dimensions (p = 3) to training\ndata, consisting of delay activity preceding 70 reaches to the same target (30\u00b0). Spike\ncounts were taken in non-overlapping \u0394 = 20 ms bins at 20 ms time steps from 50 ms\nafter target onset to 50 ms after the go cue. Then, the \ufb01tted model parameters were used to\ninfer the latent space trajectories for 146 test trials, which are plotted in Figure 2. Despite\nthe trial-to-trial variability in the delay period neural responses, the state evolves along\na characteristic path on each trial.\nIt could have been that the neural variability across\ntrials would cause the state trajectory to evolve in markedly different ways on different\ntrials. Even with the characteristic structure, the state trajectories are not all identical,\nhowever. This presumably re\ufb02ects the fact that the motor planning process is internally-\nregulated, and its timecourse may differ from trial to trial, even when the presented stimulus\n(in this case, the reach target) is identical. How these timecourses differ from trial to trial\nwould have been obscured had we combined the neural data across trials, as with the NV\nin Section 5.\n\nIs this low-dimensional description of the system dynamics adequate to describe the \ufb01ring\nof all 131 recorded units? We transformed the inferred latent trajectories into trial-by-trial\n\n\fTarget onset + 50 ms\nGo cue + 50 ms\n\n50\n\n0\n\n3\n\n \n\ni\n\nn\no\ns\nn\ne\nm\nD\n\ni\n\n -50\n\n \nt\n\nn\ne\n\nt\n\na\nL\n\n-2\n\nL\nate\n\n-1\nnt Dim\n\n0\n\ne\n\nnsio\n\nn 1\n\n60\n\n40\n\nn\n\ne\n\nsio\n\nn 2\n\n1\n\n-20\n\n0\n\na t e\n\nL\n\n20\n\nn t D i m\n\nFigure 2: Inferred modal state trajectories in latent (x) space for 146 test trials. Dots\nindicate 50 ms after target onset (blue) and 50 ms after the go cue (green). The radius of\nthe green dots is logarithmically-related to delay period length (200, 750, or 1000ms).\n\ninhomogeneous \ufb01ring rates using the output relationship from (4)\n\n(cid:2)\n\n(cid:3)\n\n\u03bbi\nt = h\n\nci\u00b7 xt + di\n\n,\n\n(8)\n\nwhere \u03bbi\nt is the imputed \ufb01ring rate of the ith unit at the tth time bin. Figure 3 shows the\nimputed \ufb01ring rates for 15 representative units overlaid with empirical \ufb01ring rates obtained\nby directly averaging raw spike counts across the same test trials. If the imputed \ufb01ring rates\ntruly re\ufb02ect the rate functions underlying the observed spikes, then the mean behavior of\nthe imputed \ufb01ring rates should track the empirical \ufb01ring rates. On the other hand, if the\nlatent system were inadequate to describe the activity, we should expect to see dynamical\nfeatures in the empirical \ufb01ring that could not be captured by the imputed \ufb01ring rates. The\nstrong agreement observed in Figure 3 and across all 131 units suggests that this simple\ndynamical system is indeed capable of capturing signi\ufb01cant components of the dynamics\nof this neural circuit. We can view the dyamical system approach adopted in this work as a\nform of non-linear dynamical embedding of point-process data. This is in contrast to most\ncurrent embedding algorithms that rely on continuous data. Figure 2 effectively represents\na three-dimensional manifold in the space of \ufb01ring rates along which the dynamics unfold.\n\nBeyond the agreement of imputed means demonstrated by Figure 3, we would like to di-\nrectly test the \ufb01t of the model to the neural spike data. Unfortunately, current goodness-\nof-\ufb01t methods for spike trains, such as those based on time-rescaling [17], cannot be ap-\nplied directly to latent variable models. The dif\ufb01culty arises because the average trajectory\nobtained from marginalizing over the latent variables in the system (by which we might\nhope to rescale the inter-spike intervals) is not designed to provide an accurate estimate of\nthe trial-by-trial \ufb01ring rate functions. Instead, each trial must be described by a distinct\ntrajectory in latent space, which can only be inferred after observing the spike trains them-\nselves. This could lead to over\ufb01tting. We are currently exploring extensions to the standard\nmethods which infer latent trajectories using a subset of recorded neurons, and then test\nthe quality of \ufb01ring-rate predictions for the remaining neurons. In addition, we plan to\ncompare models of different latent dimensionalities; here, the latent space was arbitrarily\nchosen to be three-dimensional. To validate the learned latent space and inferred trajecto-\nries, we would also like to relate these results to trial-by-trial behavior. In particular, given\nthe evidence from Section 5, how \u201csettled\u201d the activity is at the time of the go cue should\nbe predictive of RT.\n\n\f80\n\n40\n\n0\n\n80\n\n40\n\n0\n\n80\n\n40\n\n0\n\n)\ns\n/\ns\ne\nk\np\ns\n(\n \n\ni\n\ne\n\nt\n\na\nr\n \n\ng\nn\ni\nr\ni\nF\n\n500\n\n0\nTime relative to target onset (ms)\n\n1000\n\n0\n\n500\n\n1000\n\n0\n\n500\n\n1000\n\n0\n\n500\n\n1000\n\n0\n\n500\n\n1000\n\nFigure 3: Imputed trial-by-trial \ufb01ring rates (blue) and empirical \ufb01ring rates (red). Gray\nvertical line indicates the time of the go cue. Each panel corresponds to one unit. For\nclarity, only test trials with delay periods of 1000 ms (44 trials) are plotted for each unit.\n\nAcknowledgments\n\nThis work was supported by NIH-NINDS-CRCNS-R01, NSF, NDSEGF, Gatsby, MSTP,\nCRPF, BWF, ONR, Sloan, and Whitaker. We would like to thank Dr. Mark Churchland for\nvaluable discussions and Missy Howard for expert surgical assistance and veterinary care.\n\nReferences\n\n[1] M. 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J Neurophysiol, 61:534\u2013549, 1989.\n[12] D. Crammond and J. Kalaska. J Neurophysiol, 84:986\u20131005, 2000.\n[13] J. Messier and J. Kalaska. J Neurophysiol, 84:152\u2013165, 2000.\n[14] D. Rosenbaum. J Exp Psychol Gen, 109:444\u2013474, 1980.\n[15] A. Riehle and J. Requin. J Behav Brain Res, 53:35\u201349, 1993.\n[16] M. Churchland, B. Yu, S. Ryu, G. Santhanam, and K. Shenoy. Soc. for Neurosci.\n\nAbstr., 2004.\n\n[17] E. Brown, R. Barbieri, V. Ventura, R. Kass, and L. Frank. Neural Comput, 14(2):325\u2013\n\n346, 2002.\n\n\f", "award": [], "sourceid": 2823, "authors": [{"given_name": "Byron", "family_name": "Yu", "institution": ""}, {"given_name": "Afsheen", "family_name": "Afshar", "institution": ""}, {"given_name": "Gopal", "family_name": "Santhanam", "institution": ""}, {"given_name": "Stephen", "family_name": "Ryu", "institution": ""}, {"given_name": "Krishna", "family_name": "Shenoy", "institution": ""}, {"given_name": "Maneesh", "family_name": "Sahani", "institution": ""}]}