{"title": "Nonlinear physically-based models for decoding motor-cortical population activity", "book": "Advances in Neural Information Processing Systems", "page_first": 1257, "page_last": 1264, "abstract": null, "full_text": "Nonlinear physically-based models for decoding\n\nmotor-cortical population activity\n\nGregory Shakhnarovich\n\nSung-Phil Kim Michael J. Black\n\n{gregory,spkim,black}@cs.brown.edu\n\nProvidence, RI 02912\n\nDepartment of Computer Science\n\nBrown University\n\nAbstract\n\nNeural motor prostheses (NMPs) require the accurate decoding of motor cortical\npopulation activity for the control of an arti\ufb01cial motor system. Previous work\non cortical decoding for NMPs has focused on the recovery of hand kinematics.\nHuman NMPs however may require the control of computer cursors or robotic\ndevices with very different physical and dynamical properties. Here we show\nthat the \ufb01ring rates of cells in the primary motor cortex of non-human primates\ncan be used to control the parameters of an arti\ufb01cial physical system exhibiting\nrealistic dynamics. The model represents 2D hand motion in terms of a point mass\nconnected to a system of idealized springs. The nonlinear spring coef\ufb01cients are\nestimated from the \ufb01ring rates of neurons in the motor cortex. We evaluate linear\nand a nonlinear decoding algorithms using neural recordings from two monkeys\nperforming two different tasks. We found that the decoded spring coef\ufb01cients\nproduced accurate hand trajectories compared with state-of-the-art methods for\ndirect decoding of hand kinematics. Furthermore, using a physically-based system\nproduced decoded movements that were more \u201cnatural\u201d in that their frequency\nspectrum more closely matched that of natural hand movements.\n\n1 Introduction\n\nNeural motor prostheses (NMPs) aim to restore lost motor function to people with intact cerebral\nmotor areas who, through disease or injury, have lost the ability to control their limbs. Central\nto these devices is a method for decoding the \ufb01ring activity of motor cortical neurons to produce\na voluntary control signal. A number of groups have recently demonstrated the real-time neural\ncontrol of 2D or 3D computer cursors or simple robotic limbs in monkeys [1, 13, 18, 20, 22] and\nhumans [6]. Previous work on decoding motor cortical signals however has focused on modeling\nthe relationship between neural \ufb01ring rates and simple hand kinematics including hand direction,\nspeed, position, velocity, or acceleration [2, 4, 8, 10].\nWhile the relationship between neural \ufb01ring rates and hand kinematics is well established in able-\nbodied monkeys, the situation of a human NMP is quite different. For a paralyzed human, the NMP\nrepresents an arti\ufb01cial motor system with different physical properties than the intact human motor\nsystem. In particular, a human NMP may involve the control of devices as different as computer\ncursors or robotic wheelchairs. It remains an open question whether motor cortical neurons can\nsuccessfully control such varied systems with dynamics that are quite different from human limbs.\nHere we propose a model that makes a \ufb01rst step toward neural control of novel arti\ufb01cial motor\nsystems. We show that motor cortical \ufb01ring rates can be nonlinearly related to the parameters of\nan idealized physical system. This provides an important proof-of-concept for human NMPs. Our\nmodel decodes the dynamics of hand movement directly from the neural activity. Ultimately, such a\n\n\fmodel should re\ufb02ect the actuator being controlled. For a biological actuator this means the activation\nof individual muscles; for a robotic one, the forces and torques produced by the motors in the system.\nA model incorporating direct cortical control of dynamics has been proposed in [19]. There are two\nmajor distinctions between that work and ours. First, we consider the task of controlling an arti\ufb01cial\nsystem, rather than the subject\u2019s real limb. Second, applying the model in [19] in practice would re-\nquire constructing a very complex biomechanical model and controlling its many degrees of freedom\nwith a limited neural bandwidth. Here we propose a much simpler approach, that does not attempt\nto accurately model the musculoskeletal structure of the arm. Instead, it provides a computationally\neffective framework to model the dynamics of the limb moving in two dimensional plane. Our ap-\nproach is inspired by the recent work of Hinton and Nair [5], that suggested a generative model for\nimages of handwritten digits. In that work, observed images were assumed to have been generated\nby a pen connected to a set of springs, the trajectory of the pen controlled by varying the stiffness of\nthe springs according to a digit-speci\ufb01c \u201cmotor program\u201d. The goal was to infer the motor program\nfrom an observed image, in order to classify the digit. In the context of neural decoding, the image\nobservation is replaced with the recorded neural signal, from which we need to recover the \u201cmotor\nprogram\u201d, and thus the intended movement. This is where the parallels between our work and [5]\nend. One particularly important difference is that the neural decoder may be learned in a supervised\nprocedure, where the groud truth for the movement associated with a given neural signal is known.\nAn advantage of this spring-based model (SBM) over previous kinematics-based decoding methods\nis that the realistic dynamics of the model produce smoother recovered movement. We show that\nthe motions are more natural in that they better match the power spectrum of true hand movements.\nThis suggests that the control of a physical system (even an arti\ufb01cial one) may prove more natural\nfor a human NMP.\nThe experimental setup we consider in this paper involves an electrode array, implanted in the\narm/hand area of the MI cortex of a behaving monkey [17]. The animals are trained to control\nthe cursor by moving the endpoint of a two-link manipulandum constrained to a plane, much like\na human would use a computer mouse [11, 13]. Neural data and hand kinematics were recorded\nfrom two monkeys performing two different tasks. The data was separated into training and testing\nsegments and we quantitatively compared a variety of popular linear and nonlinear algorithms for\ndecoding hand kinematics and the spring coef\ufb01cients of our SBM. As expected, nonlinear meth-\nods tend to outperform linear ones. Moreover, movement reconstructed with the SBM has a power\nspectrum signi\ufb01cantly closer to that of natural movement. These results suggest that the control of\nidealized physical systems with real-time nonlinear decoding algorithms may form the basis for a\npractical human NMP.\n\nFigure 1: Sketch of the spring-based\nmodel.\nThe outer endpoints of the\nsprings are assumed to slide without\nfriction, so that A and B are always or-\nthogonal to C and D. The rest length\nis assumed to be zero for all springs.\nMovement is controlled by varying the\nstiffness coef\ufb01cients kA, kB, kC and\nkD.\n\nABCDx2x1y2y1ay=kCy1\u2212kDy2\u2212\u03b2vyax=kAx2\u2212kBx1\u2212\u03b2vxa=[,]ayax\u2212LL\f2 The spring-based model\n\nDecoding neural activity in N cells involves estimating the values of a hidden state X(t) at time\nt given an observed sequence of \ufb01ring rates Z(0) . . . Z(t) up to time t, with each Z(i) being a\n1 \u00d7 N vector. The state here is typically taken to be either hand position, velocity, etc. Methods\ndescribed in the literature can be roughly divided into two classes. Generative methods formulate the\nlikelihood of the observed \ufb01ring rates conditioned on the state and use Bayesian inference methods\nsuch as the Kalman \ufb01lter [21] or particle \ufb01lter [3] to estimate the system state from observations.\nIn contrast, direct (or discriminative) methods learn a function that maps \ufb01ring rates over some\npreceding temporal window into hand kinematics. Various methods have been explored including\nlinear regression [1, 13], support-vector regression [15] and neural network algorithms [12, 20].\nAll these previous methods have focused on direct decoding of kinematic properties of the hand\nmovement and have ignored the arm dynamics.\n\n2.1 Parametrization of dynamics\n\nOur approach to incorporating dynamics into the decoding process has been inspired by the follow-\ning model of [5], sketched out in Figure 1. Without loss of generality, let the work area (that fully\ncontains the movement range) be an axis-aligned square [\u2212L, L] \u00d7 [\u2212L, L]. The endpoint of the\nlimb (wrist) is assumed to be connected to one end of four imaginary springs, the other end of which\nis sliding with no friction along rails forming the boundaries of the \u201cwork area\u201d. Thus, at every time\ninstance each spring is parallel to one of the axes. The analysis of dynamics therefore can be easily\ndecomposed to x and y components. Below we focus on the x component.\nAll four springs are assumed to have rest length of zero. Suppose that the position of the wrist at\ntime t is [x(t), y(t)]. Then the springs A and B apply forces determined by Hooke\u2019s law, namely,\nkA(t) (L \u2212 x(t))) and \u2212kB(t) (x(t) + L), where kA(t) and kB(t) are the stiffness coef\ufb01cients of A\nand B at time t. To re\ufb02ect physical constraints on movement in the real world, the model presumes\na point mass m in the center of the wrist (i.e. at the cursor location). Furthermore, it is assumed that\nthe movement is damped by a viscous force proportional to the instantaneous velocity, \u2212\u03b2vx(t).\nThe viscosity is meant to represent both the medium resistance and the elasticity of the muscles. In\nsummary, according to Newton\u2019s second law the acceleration of the hand at time t is given by\n\nm \u00b7 ax(t) = kA(t) \u00b7 (L \u2212 x(t)) \u2212 kB(t) \u00b7 (L + x(t)) \u2212 \u03b2 \u00b7 vx(t),\n\n(1)\n\nwhere vx(t) is the instantaneous velocity of the wrist at time t along the x axis.\nControl of movement in this model is realized through varying the stiffness coef\ufb01cients of the\nsprings: given the current position of the wrist x, the desired acceleration a is achieved by set-\nting kA(t) and kB(t) so as to solve (1). This solution is not unique, in general. We note, however,\nthat the physiological meaning of the k\u2019s requires them to be non-negative, since the muscles can\nnot \u201cpush\u201d. This motivates us to introduce the total stiffness constraint\n\nkA + kB = \u03ba,\n\n(2)\n\nwhere \u03ba is a constant chosen so that no feasible acceleration would yield negative kA or kB.\nWe can now recover the underlying parameters K = [kA, kB, kC, kD] for the observed movement\nby applying (1) at each time step as follows. First we estimate the velocities \u02c6vx(t) = x(t + 1)\u2212 x(t)\nand accelerations \u02c6ax(t) = \u02c6vx(t + 1) \u2212 \u02c6vx(t). Then, we substitute (2) into (1), yielding\n\n\u02c6kA(t) = m \u00b7 \u02c6ax(t) + \u02c6vx(t) + \u03ba \u00b7 (L + x(t))\n\n2L\n\n.\n\n(3)\n\nThe value of kB(t) is then uniquely determined from (2). Repeating these calculations for the y-axis\nproduces the coef\ufb01cients for springs C and D.\n\n2.2 Decoding neural activity\n\nWe now turn to our main goal: inferring the desired movement from a recorded neural signal. We\ntreat this as a supervised learning task. In the training stage, we take a data set in which we have\nboth the recorded neural signal Z(t) and the observed trajectory of hand positions X(t) associated\n\n\fwith that signal. From this, we can learn a mapping g from the neural signal to the desired repre-\nsentation of movement. For direct kinematic decoding this means inference g : Z(t) \u2192 X(t). For\ndecoding with the SBM, this means inference of spring coef\ufb01cients in the SBM, g : Z(t) \u2192 K(t),\nfollowed by the calculation K(t) \u2192 X(t) as described above. The SBM formulation also requires\na preprocessing step for the training data: we need to convert the observed position trajectory X to\nthe trajectory through K, acording to (3).\nWe have focused on two ways of constructing g, described below.\n\nLinear \ufb01lter. The linear \ufb01lter (LF) approach [13] consists of modeling the mapping from \ufb01ring\nrate to movement by a linear transformation W that is applied on a concatenated \ufb01ring rate vector\nfor a \ufb01xed history depth l:\n\n(4)\n\nwhere x0 is a constant (bias) term and\n\nX(t) = x0 + W\u02dcZ(t),\n\n\u02dcZ(t) = (cid:2)ZT (t \u2212 l), . . . , ZT (t)(cid:3)T\n\n(5)\nThe dimension of \u02dcZ(t) for a recording from N channels, is 1 \u00d7 lN. The transformation W is \ufb01t to\nthe training data by solving the least squares problem, and then used at the decoding stage to predict\nvalues of X. Application of the LF to the SBM is straightforward: the target of the mapping is in\nthe space of coef\ufb01cients K, rather than position X.\n\n.\n\nSupport vector regression. Support Vector Machines (SVM) are a widely popular learning archi-\ntecture that relies on two key ideas: mapping the data into a (possibly in\ufb01nite-dimensional) feature\nspace using a kernel function, and optimizing the bound on generalization error. In the context of\nregression [16] this means using an \u0001-insensitive loss function, that does not penalize training errors\nup to \u0001, to \ufb01t a linear function in the feature space. SVMs also aim at reducing model complexity\nby penalizing the objective for the norm of the resulting function. The solution is \ufb01nally expressed\nin terms of kernel functions involving a subset of the training examples (the support vectors). The\nkey parameters that affect the performance of SVMs are the value of \u0001, the tradeoff c that governs\nthe penalty of training error, and parameters of the kernel function.\nSVMs have been widely successful in many applications of machine learning. However, their appli-\ncation to the task of neural decoding has been limited to the directional center-out task [15]. Here\nwe evaluate SV regression as a method for decoding more general 2D movement. Again, the SVM\nformulation is readily extended to the SBM (with the target functions being components of K).\n\nAlternative decoders. A variety of other decoding approaches has been proposed in the literature.\nWe conducted experiments with three additional algorithms: Kalman \ufb01lter [21], Multilayer Percep-\ntrons [20] and Echo-state Networks, a recurrent neural network architecture [7]. The Kalman \ufb01lter\nuses a linear model of the mapping of neural signals to movement, while the models underlying the\nother two methods are nonlinear. Our \ufb01ndings can be summarized as follows, for both kinematic\ndecoding and decoding with the spring-based model. First, nonlinear methods perform signi\ufb01cantly\nbetter than linear ones. Second, there was a trend for the Kalman \ufb01lter to perform better than the\nlinear \ufb01lter. Third, among nonlinear methods SVM tended to perform better than the two neural net-\nwork architectures. However, these latter differences could not be established with signi\ufb01cance. In\nthe following section, we focus on experiments with the linear \ufb01lter (the de-facto standard decoding\nmethod today) and SVM, which achieved the overall best results in our experiments.\n\n3 Experiments\n\nWe evaluated the performance of the proposed approach on data sets obtained from two behaving\nmonkeys (Macaca Mulatta). The neural signal was obtained with a Cyberkinetics microelectrode\narray [9] (96 electrodes) implanted in the arm/hand area of MI cortex. The experimental animals\nperformed the tasks described below.\n\nSequential reaching movement\n, described in [13] Reach targets and a hand position feedback\ncursor were presented on a video screen in front of the monkey. When a reach target was presented\n\n\fTable 1: Details of experiments. Units: number of\ndistinct units identi\ufb01ed after spike sorting. Train, test:\nlength of train and test sequences in seconds.\n\nSession\nCL-sequential\nLA-continuous\nCL-continuous\n\nUnits Train Test\n49\n140\n165\n96\n55\n140\n\n623\n244\n448\n\nthe animal\u2019s task was to move a manipulandum so that the feedback cursor moved into the target\nand remained in the target for 500ms, at which time that target was extinguished and a new reach\ntarget was presented in a different location. Target locations were drawn i.i.d. from the uniform\ndistribution over the screen surface. This was repeated for up to 10 targets per trial. Upon successful\ncompletion of a trial the animal received a juice reward. Hand kinematics and neural activity were\nsimultaneously recorded while the animal performed the task.\n\nContinuous tracking , described in [14] Monkey was viewing a computer screen on which a\nvisual target appeared in a random, but smooth, sequence of locations. The monkey was trained to\nfollow the target\u2019s position with a cursor, using a manipulandum, and received a reward for each\nsuccessful trial (i.e. when the cursor remained within the target for a duration drawn for each target\nrandomly between 3 and 10 seconds).\nThe recorded neural activity was converted to spike trains by computer-assisted spike-sorting soft-\nware, and the spike counts were calculated in non-overlapping 70ms windows. The hand kinematics\n(obtained by recording the 2D position of the manipulandum) were averaged within each window,\nto produce an aligned representation.\n\n3.1 Evaluation protocol\n\nIn each of the data sets, we selected a segment of the recording to train all the decoders, and a\nsubsequent segment to test the decoding accuracy. Tuning of parameters (the kernel parameters\nof the SVM or the mass and viscosity of the spring model) was done on a held-out portion of the\ntraining segment. We built the \ufb01ring rate history matrix by concatenating for each time step the \ufb01ring\nrates for 15 bins. For instance, for monkey CL, continuous tracking, the dimension of the neural\nsignal representation was 825 (55 channels \u00d7 15 history bins). This \ufb01ring rates were then normalized\nso that all values would be within [-1,1]. Basic statistics of the data used in the experiments are given\nin Table 1.\nWe considered three evaluation criteria:\n\nCorrelation coef\ufb01cients (CC): between the estimated and true value for each of the two spatial\n\ncoordinates over the entire trajectory:\n\nP\nqP\nt(xt \u2212 \u00afxt)2P\nt(xt \u2212 \u00afxt)(\u02c6xt \u2212 \u00af\u02c6xt)\n\nt(\u02c6xt \u2212 \u00af\u02c6xt)2\n\n.\n\nCC =\n\nPN\nt=1 kX(t) \u2212 \u02c6X(t)k.\n\n1\nN\n\nMean absolute error (MAE):\n\nin the estimated position versus the ground truth: MAE =\n\nPower spectrum reconstruction : One of the objectives of a practical decoding algorithm, espe-\ncially in the context of assistive technology, is to produce movement that appears \u201cnatu-\nral\u201d. As a criterion for evaluating the degree of \u201cnaturalness\u201d we use the similarity between\npower spectrum densities of the true movement and the reconstructed one. Speci\ufb01cally, we\ncalculated the L1 norm between the energy distributions over normalized angular frequen-\ncies, taken in the log domain (see Figure 2 for illustration).\n\n3.2 Results\nThe reported results for SVM were obtained with quadratic kernel, k(x, y) = (x\u00b7y +1)2; the trade-\noff term c was \ufb01xed to 100, and the insensitivity parameter \u0001 was set to 5 for the spring coef\ufb01cient\nand 2 for direct position decoding. The number of support vectors was between 20% and 65% of\nthe training set size.\n\n\fCL/sequential\n\nLA/continuous\n\nCL/continuous\n\nDecoder\n\nLinear-kinematics\nLinear-SBM\nSVM-kinematics\nSVM-SBM\n\n0.69\n0.64\n0.80\n0.76\n\n0.79\n0.74\n0.85\n0.81\n\nMAE CCx CCy MAE CCx CCy MAE CCx CCy\n0.83\n5.3\n0.81\n5.7\n0.86\n4.45\n4.91\n0.84\n\n0.5\n0.46\n0.60\n0.55\n\n0.75\n0.72\n0.82\n0.80\n\n5.03\n5.26\n4.44\n4.69\n\n6.66\n6.82\n3.82\n4.05\n\n0.80\n0.77\n0.86\n0.83\n\nTable 2: Summary of results on the three datasets. MAE is given in cm, over workspace of roughly\n30\u00d730 cm.\n\nTable 2 summarizes the MAE and CC measured on the test segment for each method. One obser-\nvation is that SVM tends to outperform the linear \ufb01lter, in line with previous observations [12, 15].\nWe believe that this is due to inherent nonlinearity in the underlying relationship, which is better\ncaptured by the SVM. Moreover, it is apparent that the decoding accuracy of the SBM is on par\nwith that of the conventional kinematic decoding (the observed differences were not signi\ufb01cant at\nthe 0.05 level, measured over the per-bin position errors).\n\nFigure 2: Example of power spec-\ntrum densities for true hand trajec-\ntory (dotted black), reconstruction with\nSVM-kinematics (dashed blue) and re-\nconstruction with SVM-SBM (solid\nred). Estimated using Burg\u2019s algorithm\n(pburg in Matlab, order 4). Data from\nx coordinate, LA-continuous.\n\nFigure 3: A 1.5 second path segment, true (circles) and reconstructed (squares). Left: SVM on\nkinematics, right: SVM with SBM. Markers show position averaged in each 70ms bin. Note the\nragged form of the SVM-kinematics trajectory.\n\nResults in Table 2, however, tell only a part of the story. Figure 3 shows, for a segment of 4.2 sec,\na typical example of the movement reconstructed with SVM on kinematics versus SVM on spring\ncoef\ufb01cients. The accuracy in terms of deviation from ground truth is similar, however the estimate\nproduced by the direct kinematic decoding is signi\ufb01cantly more \u201cragged\u201d. Such discrepancy is not\n\n00.20.40.60.81\u221260\u221240\u22122002040Normalized frequency, \u03c0 rad/samplePower/freq. unit, db/sample\fnecessarily re\ufb02ected in the standard measures of accuracy such as CC or MAE. Quantitavely, this\ncan be assessed by calculating the L1 norm between the power spectrum densities of the true and\nreconstructed hand trajectories. The estmated values of this quantity in our experiments are shown\nin Table 3. These results re\ufb02ect the relationship shown in Figure 2 (a typical case).\n\nTable 3: Estimated L1 norm between power spectrum density of true and reconstructed trajectories.\n\nDecoder\n\nLinear-kinematics\nLinear-SBM\nSVM-kinematics\nSVM-SBM\n\n4 Discussion\n\nCL-sequential\nx\n\ny\n\n147.41\n71.58\n143.78\n51.45\n\n154.80\n68.24\n151.35\n52.31\n\nLA-continuous\n\nCL-continuous\n\nx\n\n199.24\n72.99\n188.65\n53.05\n\ny\n\n206.61\n80.37\n196.14\n66.20\n\nx\n\n49.68\n35.68\n33.96\n20.83\n\ny\n\n44.37\n43.72\n28.44\n21.15\n\nThe spring-based model proposed in this paper represents a \ufb01rst attempt to directly incorporate\nrealistic physical constraints into a neural decoding model. Our experiments illustrate that the co-\nef\ufb01cients of an idealized physical system can be decoded from motor cortical \ufb01ring rates, without\nsttistically signi\ufb01cant loss of decoding accuracy compared to more standard direct decoding of kine-\nmatics. An advantage of such an approach is that the physical properties of the system damp high\nfrequency motions resulting in decoded movements that inherently have the properties of natural\nmovement, with no ad-hoc smoothing.\nFuture work should consider more sophisticated physical models such as a simulated robotic arm\nand a biophysically motivated musculoskeletal system. With the current state of the art in neural\nrecording and decoding, recovering the parameters of such models may be challenging. In contrast,\nthe approach presented here \u201csummarizes\u201d the effect of a more complicated system with just a few\nidealized muscle-like elements.\nAdditional experiments are also warranted. In particular using a robotic feedback device we can\nsimulate the physical system of springs presented here such that the monkeys control a device with\nthe properties of our model. We hypothesize that the accuracy of decoding spring coef\ufb01cients from\nmotor cortical activity in this condition will improve. This would suggest that matching the decoding\nmodel to the physical system being controlled will improve decoding accuracy.\nFinally, the real test of physically-based models will come in human NMP experiments. We plan\nto test human cursor control with kinematic and physically-based decoders. We hypothesize that\nthe dynamics of the physically-based model will make it easier to control accurately (and perhaps\nprovide a more satisfying experience for the user). This could be a \ufb01rst step toward the neural control\nof mechanical actuators in the physical world.\n\nAcknowledgments\n\nThis work is partially supported by NIH-NINDS R01 NS 50867-01 as part of the NSF/NIH Col-\nlaborative Research in Computational Neuroscience Program and by the Of\ufb01ce of Naval Research\n(award N0014-04-1-082). We also thank the European Neurobotics Program FP6-IST-001917. We\nthank Matthew Fellows and John Donoghue for providing data, and Reza Shadmehr for helpful\nconversations.\n\nReferences\n[1] J. M. Carmena, M. A. Lebedev, R. E. Crist, J. E. O\u2019Doherty, D. M. Santucci, D. F. Dimitrov,\nP. G. Patil, C. S. Henriquez, and M. A. L. Nicolelis. Learning to control a brain-machine\ninterface for reaching and grasping by primates. PLoS, Biology, 1(2):001\u2013016, 2003.\n\n[2] D. Flament and J. Hore. Relations of motor cortex neural discharge to kinematics of passive\nand active elbow movements in the monkey. 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