{"title": "Probabilistic Inference of Hand Motion from Neural Activity in Motor Cortex", "book": "Advances in Neural Information Processing Systems", "page_first": 213, "page_last": 220, "abstract": null, "full_text": "Probabilistic Inference of Hand Motion from Neural\n\nActivity in Motor Cortex\n\nY. Gao M. J. Black\u0001\n\nE. Bienenstock\u0003\u0002\n\nS. Shoham\u0004\n\nJ. P. Donoghue\u0002\n\n Division of Applied Mathematics, Brown University, Providence, RI 02912\n\n\u0001 Dept. of Computer Science, Brown University, Box 1910, Providence, RI 02912\n\n\u0004 Princeton University, Dept. of Molecular Biology Princeton, NJ, 08544\n\n\u0002 Dept. of Neuroscience, Brown University, Providence, RI 02912\n\ngao@cfm.brown.edu, black@cs.brown.edu, elie@dam.brown.edu,\n\nsshoham@princeton.com, john donoghue@brown.edu\n\nAbstract\n\nStatistical learning and probabilistic inference techniques are used to in-\nfer the hand position of a subject from multi-electrode recordings of neu-\nral activity in motor cortex. First, an array of electrodes provides train-\ning data of neural \ufb01ring conditioned on hand kinematics. We learn a non-\nparametric representation of this \ufb01ring activity using a Bayesian model\nand rigorously compare it with previous models using cross-validation.\nSecond, we infer a posterior probability distribution over hand motion\nconditioned on a sequence of neural test data using Bayesian inference.\nThe learned \ufb01ring models of multiple cells are used to de\ufb01ne a non-\nGaussian likelihood term which is combined with a prior probability for\nthe kinematics. A particle \ufb01ltering method is used to represent, update,\nand propagate the posterior distribution over time. The approach is com-\npared with traditional linear \ufb01ltering methods; the results suggest that it\nmay be appropriate for neural prosthetic applications.\n\n1 Introduction\n\nThis paper explores the use of statistical learning methods and probabilistic inference tech-\nniques for modeling the relationship between the motion of a monkey\u2019s arm and neural\nactivity in motor cortex. Our goals are threefold: (i) to investigate the nature of encoding\nin motor cortex, (ii) to characterize the probabilistic relationship between arm kinematics\n(hand position or velocity) and activity of a simultaneously recorded neural population, and\n(iii) to optimally reconstruct (decode) hand trajectory from population activity to smoothly\ncontrol a prosthetic robot arm (cf [14]).\n\nA multi-electrode array (Figure 1) is used to simultaneously record the activity of 24 neu-\nrons in the arm area of primary motor cortex (MI) in awake, behaving, macaque monkeys.\nThis activity is recorded while the monkeys manually track a smoothly and randomly mov-\n\n\f\u0004\u0006\u0005\n\n\u0012\u0014\u0013\n\u0015\u0017\u0016\n\n\u001e\u0017\u001f\n\u0018\u001a\u0019\u001b\u0019\n\nC.\n\nAcrylic\n\nConnector\n\nSilicone\n\nBone\n\n!\u0006\"\n\nWhite Matter\n\n,.-0/\u001b1\n\n564\n\nFigure 1: Multi-electrode array. A. 10X10 matrix of electrodes. Separation 4007 m (size\n4X4 mm). B. Location of array in the MI arm area. C. Illustration of implanted array\n(courtesy N. Hatsopoulos).\n\ning visual target on a computer monitor [12]. Statistical learning methods are used to derive\nBayesian estimates of the conditional probability of \ufb01ring for each cell given the kine-\nmatic variables (we consider only hand velocity here). Speci\ufb01cally, we use non-parametric\nmodels of the conditional \ufb01ring, learned using regularization (smoothing) techniques with\ncross-validation. Our results suggest that the cells encode information about the position\nand velocity of the hand in space. Moreover, the non-parametric models provide a better\nexplanation of the data than previous parametric models [6, 10] and provide new insight\ninto neural coding in MI.\n\nDecoding involves the inference of the hand motion from the \ufb01ring rate of the cells. In par-\nticular, we represent the posterior probability of the entire hand trajectory conditioned on\nthe observed sequence of neural activity (spike trains). The nature of this activity results in\nambiguities and a non-Gaussian posterior probability distribution. Consequently, we repre-\nsent the posterior non-parametrically using a discrete set of samples [8]. This distribution\nis predicted and updated in non-overlapping 50 ms time intervals using a Bayesian estima-\ntion method called particle \ufb01ltering [8]. Experiments with real and synthetic data suggest\nthat this approach provides probabilistically sound estimates of kinematics and allows the\nprobabilistic combination of information from multiple neurons, the use of priors, and the\nrigorous evaluation of models and results.\n\n2 Methods: Neural Recording\n\nThe design of the experiment and data collection is described in detail in [12]. Summa-\nrizing, a ten-by-ten array of electrodes is implanted in the primary motor cortex (MI) of\na Macaque monkey (Figure 1) [7, 9, 12]. Neural activity in motor cortex has been shown\nto be related to the movement kinematics of the animal\u2019s arm and, in particular, to the\ndirection of hand motion [3, 6]. Previous behavioral tasks have involved reaching in one\nof a \ufb01xed number of directions [3, 6, 14]. To model the relationship between continuous,\nsmooth, hand motion and neural activity, we use a more complex scenario where the mon-\nkey performs a continuous tracking task in which the hand is moved on a 2D tablet while\nholding a low-friction manipulandum that controls the motion of a feedback dot viewed on\na computer monitor (Figure 2a) [12]. The monkey receives a reward upon completion of\na successful trial in which the manipulandum is moved to keep the feedback dot within a\npre-speci\ufb01ed distance of the target. The path of the target is chosen to be a smooth random\nwalk that effectively samples the space of hand positions and velocities: measured hand\npositions and velocities have a roughly Gaussian distribution (Figure 2b and c) [12]. Neu-\nral activity is ampli\ufb01ed, waveforms are thresholded, and spike sorting is performed off-line\nto isolate the activity of individual cells [9]. Recordings from 24 motor cortical cells are\nmeasured simultaneously with hand kinematics.\n\n\n\u0001\n\u0002\n\u0003\n\u0007\n\b\n\t\n\n\u000b\n\f\n\n\u000b\n\u000e\n\n\u000f\n\u0010\n\t\n\u000e\n\n\b\n\u0005\n\f\n\u0011\n\u0010\n\u001c\n\u001d\n \n!\n#\n$\n%\n&\n'\n(\n)\n)\n*\n+\n2\n3\n4\n4\n4\n4\n5\n\fMonitor\n\nTarget\n\nTablet\n\nTrajectory\n\nManipulandum\n\na\n\n25\n\n20\n\n15\n\n10\n\n5\n\n0\n\nb\n\nc\n\n16\n\n14\n\n12\n\n10\n\n8\n\n6\n\n4\n\n2\n\n0\n\nFigure 2: Smooth tracking task. (a) The target moves with a smooth random walk. Distri-\nbution of the position (b) and velocity (c) of the hand. Color coding indicates the frequency\nwith which different parts of the space are visited. (b) Position: horizontal and vertical\n\naxes represent and \u0001 position of the hand. (c) Velocity: the horizontal axis represents\ndirection, \u0002\u0004\u0003\u0006\u0005\b\u0007\n\t\u000b\u0003\n\n, and the vertical axis represents speed, \f .\n\n3\n\n\u0011\u0013\u0012\n\n\u000f\u000e\n\n\u001bcell 3\n\ncell 16\n\ncell 19\n\n2.5\n\n2\n\n1.5\n\n1\n\n0.5\n\n0\n\nFigure 3: Observed mean conditional \ufb01ring rates in 50 ms intervals for three cells given\nhand velocity. The horizontal axis represents the direction of movement,\n\n0 cm/s to 12 cm/s. Color ranges from dark blue (no measurement) to red (approximately 3\nspikes).\n\n\u0007 , in radians\nto\u0003 ). The vertical axis represents speed, \f , and ranges from\n\n(\u201cwrapping\u201d around from \u0002\u0004\u0003\n\n3 Modeling Neural Activity\n\nFigure 3 shows the measured mean \ufb01ring rate within 50 ms time intervals for three cells\nconditioned on the subject\u2019s hand velocity. We view the neural \ufb01ring activity in Figure 3\nas a stochastic and sparse realization of some underlying model that relates neural \ufb01ring\nto hand motion. Similar plots are obtained as a function of hand position. Each plot can\nbe thought of as a type of \u201ctuning function\u201d [12] that characterizes the response of the cell\nconditioned on hand velocity.\nIn previous work, authors have considered a variety of\nmodels of this data including a cosine tuning function [6] and a modi\ufb01ed cosine function\n[10]. Here we explore a non-parametric model of the underling activity and, adopting a\nBayesian formulation, seek a maximum a posterior (MAP) estimate of a cell\u2019s conditional\n\ufb01ring.\n\nAdopting a Markov Random Field (MRF) assumption [4], let the velocity space,\n\n\u001c\u001e\u001d\n, be discretized on a &(')'\n*+&(')' grid. Let g be the array of true (unobserved) condi-\n\f! \"\u0007!#%$\ntional neural \ufb01ring and , be the corresponding observed mean \ufb01ring. We seek the posterior\n\nprobability\n\n(1)\n\n-/. g 01,123\u001d5476\n\n.98:-/.9;\n\n-/.\n\n6<0>=?6@2A4CB\n\nDFE/G\n\n=?6<0H=?6?IJ2K2\n\n\u000e\n\u0010\n\u0014\n\u0015\n\u0016\n\u0017\n\u0018\n\u0019\n\u001a\n\u001f\n\f0.18\n\n0.16\n\n0.14\n\n0.12\n\n0.1\n\n0.08\n\n0.06\n\n0.04\n\n0.02\n\n0\n\n\u22122\n\n\u22124\n\n\u22126\n\n\u22128\n\n\u221210\n\n0\n\u22123\n\n\u22122\n\n\u22121\n\n0\n\na\n\n1\n\n2\n\n3\n\n\u221212\n\u22123\n\n\u22122\n\n\u22121\n\n0\n\n1\n\n2\n\n3\n\nb\n\n'\u0003\u0002\u0005\u0004\u0007\u0006 .\n\n:\n\n.9;\n\n0H=\n\n0>=\n\n;\u000f\u000e\n\n\u0010\u0012\u0011\u0012\u0013\u0015\u0014\n\nrespectively, =\n\n6 and =\n\n6 are the observed and true\n\nth neighboring\n\n= ). (a) Probability of \ufb01ring variation com-\n\nThe \ufb01rst term on the right hand side represents the likelihood of observing a particular \ufb01ring\n. Here we compare two generative models of the neural\n\n(b) Logarithm of the distributions shown to provide detail.\n\nis a normalizing constant independent of g, ;\n\nwhere 8\nmean \ufb01ring at velocity \u001c\nvelocity of \u001c\nrate ;\nspiking process within 50 ms; a Poisson model,-\n\nFigure 4: Prior probability of \ufb01ring variation (\nputed from training data (blue). Proposed robust prior model (red) plotted for\u0001\nI represents the \ufb01ring rate for the\b\n, and the neighbors are taken to be the four nearest velocities (\t\n\u001d\u000b\n ).\n\u0002\u0006=A2 \u001f\n\u0004\u0012\u0001\n\n6 given that the true rate is =\u000f6\n\n, and a Gaussian model,-\n\n,\nthe variation of neural activity in velocity space. The MRF prior states that the \ufb01ring,\n\ncorresponds to an assumption that the \ufb01ring rate varies smoothly. A robust prior assumes\n, (derivatives of the\n\n, at velocity \u001c depends only on the \ufb01ring at neighboring velocities. We consider two\n= : Gaussian and \u201crobust\u201d. A Gaussian prior\n\ufb01ring rate in the \f and \u0007 directions) and implies piecewise smooth data. The two spatial\n\nThe second term is a spatial prior probability that encodes our expectations about\npossible prior models for the distribution of\na heavy-tailed distribution of the spatial variation (see Figure 4),\n\u0004?\u0003\u0017\u0001\u0019\u0018\u001b\u001a\u001d\u001c\n\nsian+Gaussian, and Poisson+Robust) are \ufb01t to the training data as shown in Figure 5.G\n\nThe various models (cosine, a modi\ufb01ed cosine (Moran and Schwartz [10]), Gaus-\n\nIn the case of the Gaussian+Gaussian and Poisson+Robust models, the optimal value of\n\n\u0004?\u0003\u0017\u0001\u0019\u0018\u001b\u001a\u001d\u001c\n\n2(\u001f\n\u001f)!\n\n\u0004\u0012\u0001\n\n=?6\n\npriors are\n\n\u0004\u0007\u0001%$\n\u001f'&\n\n-#\"\n\nthe\u0001 parameter is computed for each cell using cross validation. During cross-validation,\n\neach time 10 trials out of 180 are left out for testing and the models are \ufb01t with the remain-\ning training data. We then compute the log likelihood of the test data given the model. This\nprovides a measure of how well the model captures the statistical variation in the training\nset and is used for quantitative comparison. The whole procedure is repeated 18 times for\ndifferent test data sets.\n\nThe solution to the Gaussian+Gaussian model can be computed in closed form but for\nthe Poisson+Robust model no closed form solution for g exists and an optimal Bayesian\nestimate could be achieved with simulated annealing [4]. Instead, we derive an approximate\n\n\u201cPoisson+Robust\u201d implies a Poisson likelihood and robust spatial prior.\n\n* By \u201cGaussian+Gaussian\u201d we mean both the likelihood and prior terms are Gaussian whereas\n\n\u001d\n6\n\f\n\n-\n\f\n.\n;\n2\n\u001d\n&\n=\n \n-\n\n2\n\u001d\n&\n\u0016\n\u001e\n\u0002\n.\n;\n\u001f\n!\n\u0002\n=\n=\n.\n\n=\n2\n\u001d\n\u0003\n.\n\u0001\n\n=\n\u001f\n2\n\u001f\n \n-\n\n.\n\n=\n2\n\u001d\n&\n\u0016\n\u001e\n\u0002\n.\n\n=\n\u0002\n\f1.2\n1\n0.8\n0.6\n0.4\n0.2\n1.2(cid:13)\n1(cid:13)\n0.8(cid:13)\n0.6(cid:13)\n0.4(cid:13)\n0.2(cid:13)\n0.8(cid:13)\n\n0.6(cid:13)\n\n0.4(cid:13)\n\n0.7\n\n0.5\n\n0.3\n\n1.5(cid:13)\n\n1(cid:13)\n\n0.5(cid:13)\n\n1.5(cid:13)\n\n1(cid:13)\n\n0.5(cid:13)\n\n0.9(cid:13)\n\n0.8(cid:13)\n\n0.7(cid:13)\n\n1\n\n0.8\n\n0.6\n\n1(cid:13)\n\n0.5(cid:13)\n\n0.8(cid:13)\n0.6(cid:13)\n0.4(cid:13)\n0.2(cid:13)\n\n0.75(cid:13)\n0.7(cid:13)\n0.65(cid:13)\n\n0.8\n0.7\n0.6\n0.5\n\nCosine\n\nMoran&Schwartz\n(M&S)\n\nGaussian+Gaussian\n\nPoisson+Robust\n\ncell 3\n\ncell 16\n\ncell 19\n\nFigure 5: Estimated \ufb01ring rate for cells in Figure 3 using different models.\n\nMethod:\nG+G over Cosine\nG+G over M&S\nP+R over Cosine\nP+R over M&S\n\nLog Likelihood Ratio\n\np-value\n\n24.9181\n15.8333\n50.0685\n32.2218\n\n7.6294e-06\n\n0.0047\n\n7.6294e-06\n7.6294e-06\n\nTable 1: Numerical comparison; log likelihood ratio of pairs of models and the signi\ufb01cance\nlevel given by Wilcoxon signed rank test (Splus, MathSoft Inc., WA).\n\nsolution for g in (1) by minimizing the negative logarithm of the distribution using standard\nregularization techniques [1, 13] with missing data, the learned prior model, and a Poisson\nlikelihood term [11]. Simple gradient descent [1] with deterministic annealing provides a\nreasonable solution. Note that the negative logarithm of the prior term can be approximated\nby the robust statistical error function\nextensively in machine vision and image processing for smoothing data with discontinuities\n[1, 5].\n\n=\u0002\u0001\n\n=A2\n\n=A2 \u001f12 which has been used\n\n\u0001\u0017\u001f\n\nFigure 5 shows the various estimates of the receptive \ufb01elds. Observe that the pattern of\n\n\ufb01ring is not Gaussian. Moreover, some cells appear to be tuned to motion direction, \u0007 , and\nnot to speed, \f , resulting in vertically elongated patterns of \ufb01ring. Other cells (e.g. cell 19)\n\nappear to be tuned to particular directions and speeds; this type of activity is not well \ufb01t by\nthe parametric models.\n\nTable 1 shows a quantitative comparison using cross-validation. The log likelihood ratio\n(LLR) is used to compare each pair of models: LLR(model 1, model 2) = log(\n(observed\n\n\ufb01ring 0 model 1)/Pr(observed \ufb01ring 0 model 2)). The positive values in Table 1 indicate\n\nthat the non-parametric models do a better job of explaining new data than the parametric\nmodels with the Poisson+Robust \ufb01t providing the best description of the data. This P+R\nmodel implies that the conditional \ufb01ring rate is well described by regions of smooth activity\nwith relatively sharp discontinuities between them. The non-parametric models reduce the\nstrong bias of the parametric models with a slight increase in variance and hence achieve a\nlower total error.\n\n\u0003\u0005\u0004\n\n\n.\n\n\u001d\n\n.\n&\n.\n\n\f4 Temporal Inference\n\nGiven neural measurements our goal is to infer the motion of the hand over time. Related\napproaches have exploited simple linear \ufb01ltering methods which do not provide a prob-\nabilistic interpretation of the data that can facilitate analysis and support the principled\ncombination of multiple sources of information. Related probabilistic approaches have\nexploited Kalman \ufb01ltering [2]. We note here however, that the learned models of neural\nactivity are not-Gaussian and the dynamics of the hand motion may be non-linear. Further-\nmore with a small number of cells, our interpretation of the neural data may be ambiguous\nand the posterior probability of the kinematic variables, given the neural activity, may be\nbest modeled by a non-Gaussian, multi-modal, distribution. To cope with these issues in\na sound probabilistic framework we exploit a non-parametric approach that uses factored\nsampling to discretely approximate the posterior distribution, and particle \ufb01ltering to prop-\nagate and update this distribution over time [8].\n\nD\u0005\u0004\n\nG\u0006\u0004\n\n7\u001d\n\n\f! \n\u0003\t\b\n\nbe the mean \ufb01ring rate of cell\n\nLet the state of the system be s\n\n\u0007?# at time \u0001 . Let\n# represent the \ufb01ring rate of all\n\nwindows. Also, let c\nSimilarly let\nC\n\n\b at time \u0001 where the mean \ufb01ring rate is estimated within non-overlapping 50 ms temporal\nrepresent the sequence of these \ufb01ring rates for cell\b up to time \u0001 and let\n\ncells up to time \u0001 .\nstate at time \u0001 depends only on the state at the previous time instant:\n. s\n\n# represent the \ufb01ring of all\n\nWe assume that the temporal dynamics of the states, s\n\n, form a Markov chain for which the\n\ncells at time \u0001 .\n\nD\t\u0004\n\u0002\f\u000b\n\n-/. s\n\n\u0002\u0007\u0002\n\nG\u0006\u0004\n\n\u0003\t\b\n\nwhere S\nobservation c\n\n\u001f s\n\nand the previous observations C\n\n(# denotes the state history. We also assume that given s\n\n, the current\n\nUsing Bayes rule and the above assumptions, the probability of observing the state at time\n\n0 S\n\n2\u0006\u001d\n\n\u0001 given the history of \ufb01ring can be written as\nwhere8\nlihood term-/. c\n\n. s\n\n. c\n\nH0 C\nD\t\u0004\n\nJ2\n0 s\n\nis a normalizing term that insures that the distribution integrates to one. The like-\n\n0 s\n\n\u001d\u000f\u000e\n\ncells where the likelihood for the \ufb01ring rate of an individual cell is taken to be a Poisson\ndistribution with the mean \ufb01ring rate for the speed and velocity given by s\ndetermined by\nthe conditional \ufb01ring models learned in the previous section. Plotting this likelihood term\nfor a range of states reveals that its structure is highly non-Gaussian with multiple peaks.\n\n2 assumes conditional independence of the individual\n\nDFE/G\n\n(2)\n\nThe temporal prior term,-\n-/. s\nH0 C\n\n. s\n\n0 C\n\n2 can be written as\n-/. s\n-/. s\n\nH0 s\n\n0 C\n\n2\u0012\u0011\n\n. s\n\nwhere-\nto be constant with Gaussian noise; that is, a diffusion process. Note, -\nH0 s\nposterior distribution over the state space at time \u0001\nThe posterior,-\n\n2 embodies the temporal dynamics of the hand velocity which are assumed\n. s\n\nrandom sam-\nples which are propagated in time using a standard particle \ufb01lter (see [8] for details). Unlike\nprevious applications of particle \ufb01ltering, the likelihood of \ufb01ring for an individual cell in\n\nJ2 , is represented with a discrete, weighted set, of\n\n\u0013\u000f')'\u000f'\n\n0 C\n\n0 C\n\nis the\n\n. s\n\n& .\n\ns\n\n(3)\n\n\u0002F s\n\n0 s\nG are independent.\n-/. s\n\n0 s\n\n\"2\n\nH0 C\n\n\u001f\n\u0002\n\u0003\n\n\n\u001d\n\u001f\n\u0002\n\u0003\n\n\u0002\n\u0002\n\u0004\n\n\n\u000b\n\u0003\n\n\n\u001d\n\u001f\n\u000b\n\u0003\n\n\u0002\n\u0002\n\u0004\n\n\n\n\n\n\u0013\nG\n-\n\n\n\u0013\nG\n2\n \n\n\u001d\n\n \n\u0002\n\u0002\n\n\n\n\u0013\n-\n\u001d\n8\n\u001f\n-\n\n\n\u0013\nG\n2\n\u001f\n\n\n2\n\b\n-\n.\n\u0002\n\u0003\n\n\n\n\n\n\u0013\nG\n\n\u0013\nG\n2\n\u001d\n\u0010\n\n\u0013\nG\n2\n\n\u0013\nG\n\n\u0013\nG\n\n\u0013\nG\n \n\n\u0013\nG\n\n\u0013\nG\n\n\u0013\nG\n2\n\u0002\n\n\ftrial No. 8, Vx in cm/s, blue:true, red:reconstruction\n\ntrial No. 8, Vx in cm/s, blue:true, red:reconstruction\n\n10\n\n5\n\n0\n\n-5\n\n-10\n\n125\n\n10\n\n5\n\n0\n\n(cid:21)-5\n\n(cid:21)-10\n\n125\n\n126\n\n127\n\n128\n\n129\n\n130\n\n131\n\n132\n\n133\n\n134\n\n135\n\ntime in second\n\nVy in cm/s\n\n126\n\n127\n\n128\n\n129\n\n130\n\n131\n\n132\n\n133\n\n134\n\n10\n\n5\n\n0\n\n-5\n\n-10\n\n125\n\n10\n\n5\n\n0\n\n(cid:21)-5\n\n(cid:21)-10\n\n125\n\n126\n\n127\n\n128\n\n129\n\n130\n\n131\n\n132\n\n133\n\n134\n\n135\n\ntime in second\n\nVy in cm/s\n\n126\n\n127\n\n128\n\n129\n\n130\n\n131\n\n132\n\n133\n\n134\n\n135\n\na\n\nb\n\n, (top)\nFigure 6: Tracking results using 1008 synthetic cells showing horizontal velocity,\nand vertical velocity,\n(a) Bayesian\nestimate using particle \ufb01ltering. Red curve shows expected value of the posterior. The\n. (b) Linear \ufb01ltering method shown in\n\n, (bottom). Blue indicates true velocity of hand.\n\n\u0004\u0003\nfor\n\n\u0002\u0001\n\nerror is \f\nred; \f\n\nand\f\n\n\u0006\u0001\nand \f\n\n'\u0003\u0002\n\n\u0013\n\t\n\nfor\n\n\u0006\u0001\n\n\u0013\u0007\n\n'\u0003\u0002\b\u0007\u0004\u0007\n\nfor\n\nfor\n.\n\n\u0006\u0003\n\n\u0006\u0003\n\n\u0005\f\u000b\n\n'\u0003\u0002\n\n'\u0003\u0002\n\n\u0006\u000f'\n\n50 ms provides very little information. For the posterior to be meaningful we must com-\nbine evidence from multiple cells. Our experiments indicate that the responses from our\n24 cells are insuf\ufb01cient for this task. To demonstrate the feasibility of the particle \ufb01ltering\nmethod, we synthesized approximately 1000 cells by taking the learned models of the 24\n\ncells and translating them along the \u0007 axis to generate a more complete covering of the\n\nvelocity space. Note that the assumption of such a set of cells in MI is quite reasonable\ngive the sampling of cells we have observed in multiple monkeys.\n\nFrom the set of synthetic cells we then generate a synthetic spike train by taking a known\nsequence of hand velocities and stochastically generating spikes using the learned condi-\ntional \ufb01ring models with a Poisson generative model. Particle \ufb01ltering is used to estimate\nthe posterior distribution over hand velocities given the synthetic neural data. The expected\nvalue of the horizontal and vertical velocity is displayed in Figure 6a. For comparison, a\nstandard linear \ufb01ltering method [6, 14] was trained on the synthetic data from 50 ms in-\ntervals. The resulting prediction is shown in Figure 6b. Linear \ufb01ltering works well over\nlonger time windows which introduce lag. The Bayesian analysis provides a probabilistic\nframework for sound causal estimates over short time intervals.\n\nWe are currently experimenting with modi\ufb01ed particle \ufb01ltering schemes in which linear\n\ufb01ltering methods provide a proposal distribution and importance sampling is used to con-\nstruct a valid posterior distribution. We are also comparing these results with those of\nvarious Kalman \ufb01lters.\n\n5 Conclusions\n\nWe have described a Bayesian model for neural activity in MI that relates this activity to\nactions in the world. Quantitative comparison with previous models of MI activity indicate\nthat the non-parametric models computed using regularization more accurately describe\nthe neural activity. In particular, the robust spatial prior term suggests that neural \ufb01ring in\nMI is not a smooth function of velocity but rather exhibits discontinuities between regions\n\n\u001f\n\u001d\n\u0006\n\u0005\n\u001f\n\u001d\n\u0005\n'\n\u0013\n\u0013\n\u001f\n\u001d\n\u001f\n\u001d\n\fof high and low activity.\n\nWe have also described the Bayesian decoding of hand motion from \ufb01ring activity using a\nparticle \ufb01lter. Initial results suggest that measurements from several hundred cells may be\nrequired for accurate estimates of hand velocity. The application of particle \ufb01ltering to this\nproblem has many advantages as it allows complex, non-Gaussian, likelihood models that\nmay incorporate non-linear temporal properties of neural \ufb01ring (e.g. refractory period).\nUnlike previous linear \ufb01ltering methods this Bayesian approach provides probabilistically\nsound, causal, estimates in short time windows of 50ms. Current work is exploring correla-\ntions between cells [7] and the relationship between the neural activity and other kinematic\nvariables [12].\n\nAcknowledgments. This work was supported by the Keck Foundation and by the National\nInstitutes of Health under grants #R01 NS25074 and #N01-NS-9-2322 and by the National\nScience Foundation ITR Program award #0113679. We are very grateful to M. Serruya,\nM. Fellows, L. Paninski, and N. Hatsopoulos who provided the neural data and valuable\ninsight.\n\nReferences\n[1] M. Black and A. Rangarajan. On the uni\ufb01cation of line processes, outlier rejection, and robust\n\nstatistics with applications in early vision. IJCV, 19(1):57\u201392, 1996.\n\n[2] E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson. A statistical paradigm for neural spike\ntrain decoding applied to position prediction from ensemble \ufb01ring patterns of rat hippocampal\nplace cells. J. Neuroscience, 18(18):7411\u20137425, 1998.\n\n[3] Q-G. Fu, D. Flament, J. Coltz, and T. Ebner. Temporal encoding of movement kinematics in the\ndischarge of primate primary motor and premotor neurons. J. of Neurophysiology, 73(2):836\u2013\n854, 1995.\n\n[4] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions and Bayesian restoration\n\nof images. PAMI, 6(6):721\u2013741, November 1984.\n\n[5] S. Geman and D. McClure. Statistical methods for tomographic image reconstruction. Bulletin\n\nof the Int. Stat. Inst., LII-4:5\u201321, 1987.\n\n[6] A. Georgopoulos, A. Schwartz, and R. Kettner. Neuronal population coding of movement\n\ndirection. Science, 233:1416\u20131419, 1986.\n\n[7] N. Hatsopoulos, C. Ojakangas, L. Paninski, and J. Donoghue.\n\nInformation about movement\ndirection obtained from synchronous activity of motor cortical neurons. Proc. Nat. Academy of\nSciences, 95:15706\u201315711, 1998.\n\n[8] M. Isard and A. Blake. Condensation \u2013 conditional density propagation for visual tracking.\n\nIJCV, 29(1): 5\u201328, 1998.\n\n[9] E. Maynard, N. Hatsopoulos, C. Ojakangas, B. Acuna, J. Sanes, R. Normann, and J. Donoghue.\nNeuronal interaction improve cortical population coding of movement direction. J. of Neuro-\nscience, 19(18):8083\u20138093, 1999.\n\n[10] D. Moran and A. Schwartz. Motor cortical representation of speed and direction during reach-\n\ning. J. Neurophysiol, 82:2676-2692, 1999.\n\n[11] R. Nowak and E. Kolaczyk. A statistical multiscale framework for Poisson inverse problems.\n\nIEEE Inf. Theory, 46(5):1811\u20131825, 2000.\n\n[12] L. Paninski, M. Fellows, N. Hatsopoulos, and J. Donoghue. Temporal tuning properties for\n\nhand position and velocity in motor cortical neurons. submitted, J. Neurophysiology, 2001.\n\n[13] D. Terzopoulos. Regularization of inverse visual problems involving discontinuities. PAMI,\n\n8(4):413\u2013424, 1986.\n\n[14] J. Wessberg, C. Stambaugh, J. Kralik, P. Beck, M. Laubach, J. Chapin, J. Kim, S. Biggs, M.\nSrinivasan, and M. Nicolelis. Real-time prediction of hand trajectory by ensembles of cortical\nneurons in primates. Nature, 408:361\u2013365, 2000.\n\n\f", "award": [], "sourceid": 1997, "authors": [{"given_name": "Yun", "family_name": "Gao", "institution": null}, {"given_name": "Michael", "family_name": "Black", "institution": null}, {"given_name": "Elie", "family_name": "Bienenstock", "institution": null}, {"given_name": "Shy", "family_name": "Shoham", "institution": null}, {"given_name": "John", "family_name": "Donoghue", "institution": null}]}