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Motor Cortical Decoding Using an Autoregressive Moving Average Model

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Motor Cortical Decoding Using an Autoregressive Moving Average Model Motor Cortical Decoding Using an Autoregressive Moving Average Model Jessica Fisher and Michael J. Black Department of Computer Science, Brown University Providence, Rhode Island 02912 Email: {jfisher, black}@cs.brown.edu Abstract— We develop an Autoregressive Moving Average An ARMA model was suggested by [1] in the context (ARMA) model for decoding hand motion from neural firing of decoding a center-out reaching task. They extended the data and provide a simple method for estimating the parameters method to model a non-linear relationship between firing of the model. Results show that this method produces more accurate reconstructions of hand position than the previous rates and hand motions using Support Vector Regression. Kalman filter and linear regression methods. The ARMA model Here we apply the simpler ARMA model to a more complex combines the best properties of both these methods, producing task involving arbitrary 2D hand motions. We show that a reconstructed hand trajectories that are smooth and accurate. very simple algorithm suffices to estimate the parameters of This simple technique is computationally efficient making it the ARMA process, and that the resulting decoding method appropriate for real-time prosthetic control tasks. results in reconstructions that are highly correlated to the true I. INTRODUCTION hand trajectory. We explore the choice of parameters and pro- One of the primary problems in the development of prac- vide a quantitative comparison between the ARMA process, tical neural motor prostheses is the formulation of accurate linear regression, and the Kalman filter. We found that the methods for decoding neural signals. Here we focus on the simple ARMA process provides smooth reconstructions that problem of reconstructing the trajectory of a primate hand are more accurate than those of previous methods. given the extracellularly recorded firing activity of a popu- lation of neurons in the arm area of primary motor cortex. II. METHODS Various machine learning techniques, mathematical models, A. Recording and decoding algorithms have been applied to this problem [1]–[7]. The simplest and most common of these is the linear Our experiments here use previously recorded and reported regression method which represents hand position at a given data [9] in which a Bionic Technologies LLC (BTL) 100- time instant as a linear combination of population firing rates electrode silicon array [10] was implanted in the arm area over some preceding time interval [2], [3], [6]. While simple of the primary motor cortex of a macaca mulatta monkey. and relatively accurate, the resulting reconstructions require The monkey was trained to move a two-joint planar manip- post hoc smoothing to be practical in a neural prosthesis [8]. ulandum to control a feedback cursor on a computer screen. Alternatively, Bayesian decoding methods have been used, The position of the manipulandum and the neural activity including the Kalman filter [4] and particle filter [5], [7]. In were recorded simultaneously, and the neural activity was contrast to the linear regression method, Bayesian methods summed into 70-ms bins. The task the monkey performed include an explicit temporal smoothness term that models was a “pinball” task [4] in which he moved the cursor to hit the prior probability of hand motion. The Kalman filter is a target on the screen, and when he succeeded, a new target particularly appropriate for prosthetic applications given its appeared. Neural signals were detected on 42 electrodes, and accuracy and efficiency (for both training and decoding) a simple thresholding operation was used to detect action [8]. Unlike the linear regression method which uses a large potentials. The spikes on each electrode were treated as one history of firing data to reconstruct hand motion at every (potentially) multi-unit channel of data. In [11] it was found time instant, conditional independence assumptions in the that multi-unit data provided decoding accuracy on a par with standard Kalman filter restrict it to using only the current the best single unit data obtained by human spike sorting. firing rates of the neurons. While the hand trajectories The dataset was divided into separate training and testing sets decoded with the Kalman filter do not require post hoc of approximately 6.2 minutes and one minute respectively. smoothing, they still lack the smoothness of natural hand B. Linear Regression motion [8]. Here we develop a simple Autoregressive Moving Average The linear regression method is the most common method (ARMA) process for the neural decoding task that combines used for motor cortical decoding and assumes that the the linear regression method with the smoothness term of current hand state (position, velocity, and acceleration) can the Kalman filter. By using more data at each time instant be represented as a linear combination of the current firing than the Kalman filter, accuracy is improved, and by adding rates of a population of neurons. Least-squares regression is a spatial smoothing term to the linear regression method, performed to determine the coefficients (the “filter”) for this smooth trajectories are obtained without post hoc smoothing. linear combination based on training data, and the filter is then used on testing data to decode the state at each time history of firing rates and the preceding hand states, given instant [2], [6]. The linear relationship is described by by t−1 t xt = Ax − + Fz − (2) X = ZF (1) t m t k where A ∈ <D×Dm (D is the dimensionality of the state where X is a state matrix containing the (xt,yt) hand vector x) and F ∈ <D×Nk, and k and m are parameters positions at time instants t = 1 ...T , Z is a matrix of firing determining how many previous time steps (of hand state rates of the N = 42 multi-units over the same time period, and neural firing respectively) to include in the calculation. and F is the linear filter matrix relating hand positions and t−1 Dm×1 xt−m ∈ < is a column vector containing the concate- firing activity. In particular, t nation of the states from times t − m to t − 1, and zt−k ∈ <Nk×1 is a column vector containing the concatenation of xT yT the firing rates for all neurons from times t − k to t. We xT −1 yT −1 call this an Autoregressive Moving Average (ARMA)(m, k) X = . , . process [12]. A x1 y1 The parameters to be estimated in this model are and F. We alternate learning these (beginning with F) using the 1 1 2 n same method as linear regression: zT ... zT −k zT ... zT −k 1 1 1 2 n T + zT −1 ... zT −k−1 zT −1 ... zT −k−1 1 F = (Z ) (X2 − AX1) (3) Z = . , . T + . A = (X1 ) (X2 − FZ) (4) 1 1 2 n zk ... z1 zk ... z1 1 D×T −1 where X2 ∈ < is a matrix of states from times 2 to T Nk×T −1 fx1 fy1 (T being the total number of time steps), Z ∈ < is a t fx2 fy2 firing rate matrix where each column is of the form zt−k as Dm×T −1 F . described above, and X1 ∈ < is a matrix in which = . t−1 each column is of the form xt−m as described above. fxkN fykN Initially, we set A to a matrix of zeros and learn F (note f1 f2 that this is simply the standard linear regression method). where xt represents the hand x position at time t (analogous To determine when convergence has occurred, we use the i for y), zt represents the firing rate of neuron i at time t, mean squared error of the training data [4]. Convergence has th fxp represents the p filter coefficient for x (same for y), occurred when the difference between the previous iteration’s and the fp’s represent constant offset terms. Note that k is error and the current error is less than some parameter . In a constant representing a time window of neural firing rates our experiments, we used = 0.001. to be considered. Also, the column of ones in X provides D. Kalman Filter a constant bias term. This model can easily be expanded to include velocity and acceleration. The Kalman filter has been proposed for decoding motor We solve for F by minimizing the squared error kX − cortical data [4]. Like the above methods it assumes a 2 linear relationship between neural firing and hand kinematics ZFk2 which gives the solution for the filter matrix F as (though formulated as a generative model) and like the − F = (ZT Z) 1ZT X = Z+X ARMA model assumes the hand motion evolves linearly, as follows: where Z+ is the pseudo-inverse of Z. The hand position xt = Axt−1 + w (5) xt = (xt,yt) at a particular time instant can be reconstructed as zt = Hxt + q (6) x = zt F t t−k where w ∼ N(0, W) and q ∼ N(0, Q). A recursive t 1×Nk x z where zt−k ∈ < is a vector representing a k time bin Bayesian method is used to predict t given t. While the history of neural firing preceding time instant t (a row of the reader is referred to [4] for details, it is worth noting that Z matrix above). the Kalman filter reconstructs hand kinematics as C. Autoregressive Moving Average (ARMA) Model xt = Axt−1 + Kt(zt − H(Axt−1)) The linear filter models how hand position is related to were Kt is the Kalman gain matrix. Note that the first term neural activity over some time interval, but it does not ex- is the same as in the ARMA model but is restricted to using plicitly model anything about how hands move. The motion only the previous time instant due to a first order Markov of the hand is constrained by physics and the properties of assumption used to derive the filter.
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