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1Center for Neural Science, New York University, New York, New York 10003; 2Department of Neuroscience, Brown University, Providence, Rhode Island 02912; and 3Committee on Computational Neuroscience, University of Chicago, Chicago, Illinois 60637
Submitted 22 July 2003; accepted in final form 8 September 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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; Kettner et al. 1988), speed (Ashe and Georgopoulos 1994; Moran and Schwartz 1999a), direction of motion (Ashe and Georgopoulos 1994; Fu et al. 1995; Georgopoulos et al. 1982), and force (Sergio and Kalaska 1998; Taira et al. 1996). Most MI neurons appear to combine information about multiple movement features (Ashe and Georgopoulos 1994; Moran and Schwartz 1999a) that may be specified separately in time (Fu et al. 1995). The temporal aspects of the encoding process are important both for understanding the neuronal processing of dynamic signals (Buracas et al. 1998; Mainen and Sejnowski 1995; Rieke et al. 1997) and for the problem of decoding information from populations of neurons (Humphrey et al. 1970; Warland et al. 1997), yet previous work has not differentiated temporal patterns imposed by task demands from the underlying temporal dynamics of encoding.
Investigation of the spatiotemporal encoding of motor variables presents several challenges. Tasks used to study movement have most often involved point-to-point movements to a limited number of well-rehearsed targets. Step-tracking tasks, as typically implemented, allow only limited control over kinematic variables because hand motion is a function of the subject's strategy rather than of the experimental design. For example, in a typical point-to-point movement task, any hand velocity can be used to reach a target as long as target acquisition falls within a maximum allotted time. In addition, typical step-tracking tasks limit the size of the parameter space sampled for each variable; studies of target location encoding are typically limited to a small subset of possible locations (8, in the widely used "center-out" task; Ashe and Georgopoulos 1994; Georgopoulos et al. 1982; Kalaska et al. 1989; Moran and Schwartz 1999a). Furthermore, because hand position and velocity are strongly interdependent in these tasks, it is difficult to determine their relative contributions to MI firing. For example, in the standard radial task, any given peripheral position is associated with just one single direction of motion, and with a highly stereotyped set of velocity profiles.
Another problem— especially significant for studies of temporal dynamics—in tasks typically used to study motor coding is that neural and behavioral variables (such as firing rate and hand speed) are statistically nonstationary. Distributions of these measures vary systematically as a function of trial time, so that, for example, peak firing occurs within a narrow interval after a cue to move. Nonstationarities in the underlying data distributions greatly complicate the analysis of temporal encoding processes because lag-dependent interactions (those related to coding delays) are confounded with trial-time-dependent modulations in activity.
In earlier studies in motor cortex, behavioral variables were treated as static, scalar quantities such as average hand direction or speed, and the concomitant time-varying neural activity was summarized as a single number—the mean firing rate. The data were averaged over many trials and/or fit to highly parametric tuning models (e.g., cosine functions), thereby collapsing what may be more information-rich tuning functions (Sanger 1996). These multiple averages eliminate most of the dynamic, trial-specific information needed to characterize spatiotemporal encoding properties. In contrast, more recent studies have explicitly examined the temporal aspects of kinematic coding in MI using center-out-type tasks (Ashe and Georgopoulos 1994; Fu et al. 1995; Moran and Schwartz 1999a; Sergio and Kalaska 1998) or curved drawing tasks (Moran and Schwartz 1999b; Schwartz and Moran 1999), and treating the kinematics and neural activity as time-varying data. These studies avoid the issues of collapsing data across time but still suffer from the inherent statistical constraints described above, that is, interactions between variables of interest, and the confounding of time-dependent with lag-dependent properties, where lag is the delay between spiking and its manifestation as behavior. Temporal dynamics, as they have been studied in the context of these tasks, could be an indication of the temporal evolution of task demands and are not necessarily an indication of the underlying dynamics of encoding.
Finally, the serial recording techniques employed in previous work preclude the direct comparison of spatiotemporal encoding properties between neurons because units are recorded under behavioral and state conditions that vary from trial to trial (and therefore from cell to cell). Furthermore, serial recordings of neural data necessitate assumptions of statistical independence between neurons (because the dependencies cannot be observed without simultaneous recording), and these assumptions have been shown to be inaccurate in general (Maynard et al. 1999; Oram et al. 2001).
The present study characterized spatiotemporal encoding of hand motion using a random, continuous pursuit-tracking task (PTT) designed to facilitate evaluation of the spatial and temporal characteristics of MI neurons, while minimizing dependencies and nonstationarities. Using continuous tracking of a randomly moving stimulus, position and velocity encoding is characterized within a systems analysis framework. In this context, hand trajectory is viewed as a random "stimulus" to the system and neural activity is the "response." Each stimulus is drawn from an experimenter-determined distribution that broadly and continuously covers velocity and position space, and is stationary with respect to trial time. This design effectively controls hand motion at all times and reduces statistical dependencies among variables across the experiment. These statistical properties of the PTT permit the rigorous application of information-theoretic and signal-processing methods to the analysis of position and velocity coding. The relationship between kinematics and firing rate can be characterized in a nonparametric (model-free) manner, without assumptions about the underlying tuning properties of the sampled neurons. The multielectrode recording approach taken here allows quantitative comparisons of encoding between cells, because multiple neurons are recorded under completely identical conditions. Finally, the systems analysis approach further permits a direct quantification of hand trajectory information using signal reconstruction methods that can demonstrate planned motions from population activity. In this paper we describe the spatiotemporal tuning functions of MI neurons for velocity and position during pursuit tracking and we compare the information coded within single cells and across the population. We also demonstrate that MI neurons contain sufficient position and velocity information to reconstruct novel hand trajectories based on information available from the firing of a small sample of MI neurons.
Part of this work appeared in abstract form (Society for Neuroscience Meeting 1999; abstract 665.9; Society for Neuroscience Meeting 2001; abstract 940.1).
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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Three monkeys (one Macaca fascicularis and 2 M. mulatta) were operantly conditioned to track a smoothly and randomly moving visual target. The monkey viewed a computer monitor and gripped a two-link, low-friction manipulandum that constrained hand movement to a horizontal plane. Manipulandum position was sampled on a 30 x 30-cm digitizing tablet (Wacom Technology, Vancouver, WA) at 167 Hz, with an accuracy of 0.25 mm, and recorded to disk. Hand position was continuously reported on the monitor by a black, 0.2° visual angle circle (0.5 cm radius on the tablet) (Fig. 2A).
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; Maynard et al. 1999 for details). In these experiments, radial and tracking trials were randomly interleaved. The monkeys used in this study had been trained on the center-out task before introduction to the continuous tracking task. All 3 animals were able to perform the tracking task within the first 2 days of training, with varying degrees of proficiency; performance (as measured by the length of time for which the monkey could consistently track a target of given mean speed; see Table 1) continued to improve throughout the training period. Data analyzed here were collected 8-11 mo after introduction to the tracking task.
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Details of the basic recording hardware and protocols are available elsewhere (Donoghue et al. 1998; Maynard et al. 1999). After task training, a Bionic Technologies LLC (BTL, Salt Lake City, UT) 100-electrode silicon array was implanted in the arm representation of MI. The array was placed on the precentral gyrus medial to a line extending from the genu of the arcuate sulcus posteriorly to the central sulcus, and parallel to the sagittal fissure, a region previously localized as the MI arm representation (Georgopoulos et al. 1982). The neurons showed modulations around movement commonly observed in radial task experiments, thus confirming, physiologically, placement of the array in the MI arm area. The BTL arrays consisted of 100 platinizedtip silicon probes (about 200-1,000 k
at 1 kHz; Nordhausen et al. 1996), arranged in a square grid (400 µm on center). The electrodes were 1 mm in length, corresponding in MI to recordings near the layer III/V boundary. In the 3 monkeys there were 74 (Monkey Ra), 47 (Er), and 24 (Co) possible active recording electrodes, a number limited by the connectors used. All procedures were in accordance with Brown University Institutional Animal Care and Use Committee-approved protocols and the Guide for the Care and Use of Laboratory Animals (National Institutes of Health publication no. 85-23, revised 1985).
Signals were amplified and sampled at 30 kHz/channel using a commercial recording system (Bionic Technologies, Salt Lake City, UT). All waveforms that crossed a manually set threshold were digitized and stored (from 0.33 ms before to 1.17 ms after threshold crossing); spike sorting to isolate single units was performed off-line. Single units with signal-to-noise (SNR) ratios >2.5 were stored as spike times referenced to the stimulus signal for further analysis. Analysis of spiking was confined to data recorded from 1 s after tracking began to 1 s before the end of trial, to eliminate nonstationarities associated with trial beginning and end.
Analysis
SPATIOTEMPORAL TUNING. We summarized the spatiotemporal tuning of the recorded cells as follows. We computed functions N(
, 
) and N(
, ) to describe the firing rate as a function of position () and velocity (), respectively, at a series of time leads and lags (). These functions are defined as the conditional mean firing rate of a cell at time t, given that a particular kinematic value ( or ) occurred at time t + . That is
 | (1) |
 | (2) |
defines the delay between the spike count bin and the kinematic bin [i.e., N(, -100 ms) gives the expected firing rate 100 ms after the particular hand position was observed].
To compute these tuning functions, data were taken at all times {ti} when the hand was moving with a particular velocity (or was located at a particular position) (
± d, 
± d) cm/s, for some (, ) in polar coordinates. The bin widths 2d and 2d were chosen to be just large enough to ensure adequately sampled data in all bins; we typically took >50 samples per bin. For example, we set bin widths in one experiment to 0.4 radians x 0.7 cm/s (velocity), and 0.4 radians x 0.5 cm (position). We then calculated the mean firing rates at {ti - } for the lags shown. We represent this lag variable by the symbol throughout, reserving t for the time since the beginning of the behavioral trial.
We used polar instead of rectangular coordinates for the discretization for 3 reasons. First, polar coordinates respect the radial symmetry of the (properly scaled) observed Gaussian joint distributions of hand position and velocity (Fig. 4): all bins at a given radius are roughly equiprobable, whereas the corresponding statement is false for any fixed value of horizontal or vertical position or velocity. Second, the size of the bins in polar coordinates (approximately d d) grows with , partially correcting for the falloff of the probability distribution of these behavioral variables at the extremes of their ranges. Finally, in polar coordinates firing rates are represented as a function of direction, a convention that facilitates comparisons with prior studies. The origin for these curves was taken to be the mean of the distributions of the behavioral variable; for the velocity tuning functions, the origin was at (0, 0) cm/s, whereas for position the origin was at the center of the tablet. We fit planes and other parametric families to the tuning curves by a standard least-mean-squares optimization procedure (Nelder-Mead simplex search). In addition, we used a Monte Carlo procedure to obtain conservative significance levels for the presence of good fits, under the null hypothesis that the spike trains were homogeneous Poisson processes (i.e., that the apparent fluctuations in firing rate observed in Fig. 2 were random, had a trivial probabilistic structure, and were independent of the behavior of the hand). We simulated spike trains (homogeneous Poisson processes with rates matched to the observed individual neural firing rates), estimated N(, ) and N(, ) using real kinematic data for each instantiation of these simulated spike trains, and computed the mean-square deviation for each resulting fit. The significant fit level was taken as the point at which the cumulative empirical probability distribution of the random goodness-of-fit value reached 0.99.
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)
 | (1) |
X is the integral over some space X. Information is difficult to compute in general because full knowledge of the joint distribution p(N; S) (where N and S are functions of time) is needed. This presents a possibly infinite-dimensional learning problem; in the present experiment one would be required to know the probability of a given spike train given any time-varying position signal. Consequently, we do not attempt to estimate the information rate between spike trains (denoted N, for neuron) and the behavioral signal (S); rather, we address the simpler problem of computing the information between the observed neuronal firing rate and the behavioral signal (hand velocity or position, here) at discrete (single) time lags ; that is
 | (2) |
) denotes the state of the behavioral signal (e.g., the position of the hand) at time lag after the present time; computing Eq. 2 requires only an integral in 2-D space (one dimension each for horizontal and vertical), instead of the high-dimensional integral required to compute the full information (Eq. 1) between spike trains and the time-varying position signal.
To simplify Eq. 2 even further, we modeled the conditional distributions of the behavioral signal given an observed spike count per bin, p[S() | N(0) = i], i 
0, 1, 2,..., as Gaussian, with mean µ,i and covariance matrix 
,i. This simplification makes the computation of Eq. 2 tractable, given the size of the available data set. Thus for Eq. 2, we calculate
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) is the (2-D) Gaussian density with mean µ,i and covariance ,i. The Gaussian model was motivated by empirical observations and gave a sufficient fit to the data for many observed cells and spike count bins, according to a 2-D Kolmogorov-Smirnov test (bivariate Kolmogorov-Smirnov-type test; Press et al. 1992; P < 0.05). In the cases in which the Gaussian fit was inadequate, we applied a nonparametric binning approach (computing the integral in Eq. 2 as a finite sum) instead; the Gaussian and binned-information estimates were highly correlated (correlation coefficient > 0.95) across all cells and all time bins, indicating that the Gaussian method provides an adequate information estimator for this set of data.
A Monte Carlo procedure identical to the one described in the previous section was used to estimate significance levels for the observed information values. This procedure produced information values <10-4 bits. A different procedure, in which we shuffled the neural data with respect to the behavioral data, so that neural data from one trial was associated, in a random manner, with the behavioral data from a different trial, led to similar results. The significance bound was therefore defined as I[N(0); S()] > 10-4 (see Fig. 12).
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): our estimate R (for reconstruction) of the position at the current time t is given by a linear combination
 | (4) |
). Two filters were generated, one each for the horizontal and the vertical positions.
The analytical solution to the optimal linear estimation problem in the time domain involves the inversion of a correlation matrix (NTN) that can be fairly large [matrix size = D2, where D = 1 + C(Tpre + Tpost)/dt]; we used standard singular value decomposition (Press et al. 1992) techniques to check the numerical stability of this matrix inversion. The data showed no evidence of overfitting such as a decrease in performance as D became large. None of the results shown was smoothed, nor were any relevant parameters subjectively selected (e.g., to select the "best" neurons for analysis). Cross-validation methods were used to estimate the expected error of our reconstructions: we fit the regression model to a "training" set consisting of all but 10 trials of the data set, then computed the mean-square error of the regression on this "test" set, the 10 held-out trials. This process was iterated multiple times as successive, disjoint blocks of 10 trials were used to test the regression; we report the regression coefficient computed by this procedure, where this coefficient is defined as usual as r2 = 1 - {E[(R - S)2]/E(S2)}, where R is the reconstructed hand position and S is the true hand position.
A frequency domain regression analysis (Haag and Borst 1998; Rieke et al. 1997) was used to estimate a lower bound on the frequency content of the information contained in the MI population (Fig. 15). Neural and position signals were Fourier transformed, and the neural Fourier coefficients at a given frequency 
, 
(), were regressed onto the coefficients of position, 
(), to obtain the reconstruction of S at , 
(). Goodness of reconstruction was plotted as the SNRs obtained at each frequency
 | (5) |
; Rieke et al. 1997).
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, which gives the expected r2 given that the true covariance matrix of S is ss and the cross-correlation between N and S is ns; N here is a vector-valued signal, with each element corresponding to the firing rate of a single cell, and E(·) denotes expectation. In practice, ss and ns must be estimated from data, and because of sampling error, the r2 computed by cross-validation tends to be of lower magnitude than the E(r2) calculated here; therefore we normalize the curves in Fig. 16 by the maximal observed E(r2).
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Cells exhibiting significant trends in rate over experimental time were further tested for significant changes in their spatiotemporal tuning functions over experimental time. Those cells with significant rate changes and significant tuning changes were discarded. Cells exhibiting significant intratrial rate changes were not excluded (see RESULTS). Of an original 120 cells, we excluded 7 because of nonstationarities, leaving the 113 we use in all subsequent analyses.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
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Pursuit-tracking task
The pursuit-tracking task (PTT) and typical point-to-point movement tasks vary considerably in the extent of parametric space explored, the dependencies among variables, and the stationarity of kinematic and neural signals. Figure 1 illustrates kinematic and neural activity data obtained from one monkey performing the center-out task, to provide explicit comparison with the PTT. The center-out task, by design, results in movements from a constant location to one of a fixed set (here, 8) of discrete locations. Although there is no specific trajectory requirement, the need to end at a specific location within task-time constraints generally results in roughly straight, stereotyped hand trajectories. Figure 1A shows hand paths for trials to each of the 8 directions. This task design results in strong dependencies between horizontal and vertical position (Fig. 1A) and horizontal position and velocity (Fig. 1B). Note, also, that many (x, y) pairs, even near the center of the workspace, are never sampled. Figure 1, C and D illustrate the nonstationarity of kinematic and neural variables in the center-out task: mean hand speed shows a sharp transient increase with movement onset, irrespective of target location (Fig. 1C), and mean firing rates show similar large t-dependent modulations (recall that t denotes time relative to the start of the trial).
By contrast, the PTT covers the kinematic space more fully and achieves considerably improved independence of kinematic variables and stationarity of kinematic and neural activity (Figs. 2, 3, 4). Figure 2A provides an example of PTT performance for a single trial. Tracking was smooth, with continuous modulation of hand speed and direction. Mean hand speed, which followed that of the visual target set in the experimental design, ranged from 2.5 to 4.7 cm/s across this set of experiments (Table 1). Tracking movements were largely determined by the visual stimulus, as demonstrated by the close temporal relationship of the hand and visual cue (Fig. 2A, inset). The peak of this cross-covariance was consistently located within 50 ms of zero with a peak correlation coefficient that exceeded 0.97 in each data set, consistent with the conclusion that the animals tracked the stimulus. The short visuomotor "reaction time" indicates that the animal is at times actively predicting the smoothly evolving stimulus trajectory. The relatively high tracking accuracy over time can also be appreciated in the individual plots of x and y position versus time across a trial (Fig. 2, B and C). The overall smoothness of hand movement during tracking is evident in the autocovariogram (Fig. 3A), and in the power spectrum of hand position (Fig. 3B); most of the power in the hand position signal was below 1 Hz (Fig. 3B; the autocovariogram and power spectra in Fig. 3 were computed from data from a single experiment, but these functions were qualitatively similar in each other data set). For comparison the power spectrum of the horizontal position of the stimulus signal is shown (Fig. 3C); again, most of the power is below 1 Hz.
Figure 4 presents the statistical properties of the PTT for comparison with those of the center-out task (cf. Fig. 1). The joint distributions of 2-D hand position and 2-D velocity in the PTT were well approximated by Gaussian distributions with zero covariance (modified Kolmogorov-Smirnov test; P < 0.05), as expected given the task design. No significant correlation was observed between any of the pairs of velocity and position variables (Pearson test; P < 0.05). Thus the PTT samples the kinematic space more densely than does the center-out task. In addition, kinematic variables such as hand speed and position are effectively stationary across the task. Mean hand speed does not vary as a function of trial time (P < 0.05; compare Figs. 4C and 1C) and average firing rate does not depend on the time relative to the start of tracking for the cells shown (test on correlation with linear trend over the first or last 2.5 s of the trial; P < 0.05; compare Figs. 1D and 4D). Figure 4D is shown for illustrative purposes because, for some cells in our database, the average firing rate was not constant over trial time (e.g., some cells displayed anticipatory "ramp-up" activity near the end of successful trials). Any intratrial rate nonstationarities during the PTT cannot be explained as a function of the variables of interest (i.e., the kinematics) because these variables are stationary. The comparison between Figs. 1D and 4D is meant to show that the center-out task induces rate nonstationarities, whereas the PTT does not.
Neural activity during tracking
Figure 2D shows a representative example of the spiking patterns of 21 cells recorded simultaneously during a single pursuit-tracking trial. Qualitatively, randomly selected MI neurons typically showed varying modulation patterns in the PTT; these same neurons showed marked mean rate modulations in step-tracking tasks (compare Figs. 1D and 4D). Mean firing rates during the PTT ranged over 1.5 log units (about 2-40 Hz; Fig. 5) and were not correlated with overall mean hand speed (Spearman rank-order correlation coefficient; P < 0.05). The relationship between the spike count mean and variance (per 50-ms bin) is largely linear with unity slope, except at the highest mean firing rates, where the Fano factor (the ratio of the variance to the mean) falls slightly below the unity level.
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Our results depend on the stationarity of the underlying data. By construction, the stimulus (i.e., the motion of the tracking target) is stationary; thus the animals' hand motions are approximately stationary. This does not, however, guarantee the stationarity of the neural activity associated with these motions. In averaging over the entire experimental time period to derive our tuning measures we are implicitly assuming that tuning is constant on this time scale. Because the subjects are well trained on the task before recording, and the task requirements are held constant across the experiment, there is good reason to think that this is true—no learning is likely to be occurring. However, changes in the animal's overall behavioral state (e.g., motivation) might cause average spike rates to drift up or down over a recording session. To test for this we looked for linear trends in the average spike rate for each cell across experimental time.
Cells with a linear trend whose slope was not significantly different from zero, or with less than a 20% change in rate, were deemed stationary on the experimental time scale and included in the other analyses. Cells with a significant nonzero slope and a change in rate of >20% over the experiment were further tested for trends in their spatiotemporal tuning functions (see following text). Of an original 120 cells we found 44 (37%) with significant (by bootstrap shuffling of time bins, P < 0.05) rate trends over the experiment. Of these, 7 (5%) were found to have tuning functions that differed significantly (see METHODS) over experimental time. These cells were excluded from further analysis, leaving the 113 reported here.
We also tested for stationarity of rate as measured across trial time. For each experiment we aligned trials on the beginning of the tracking phase and averaged the neural activity for each cell across trials to get a mean firing rate for each time bin. We tested for linear trends in the average rate over the course of trial time. We found 27 (23%) of 120 cells with significant (by bootstrap shuffling of time bins, P < 0.05) rate trends of >20% over trial time. No cells were excluded based on these intratrial rate trends. Because the kinematics are stationary over trial time these intratrial trends in rate are unlikely to be linked to the tuning that we report. The fact that intratrial trends, when they were present, were different for different, simultaneously recorded cells (e.g., some cells had a positive rate trend, whereas others showed a negative rate trend) also supports the idea that it is not the kinematics that are inducing these changes. It is likely that other, uncontrolled and unobserved variables (e.g., reward expectation) are inducing these rate trends. For these reasons, we argue that these effects may be interesting in their own regard, but do not detrimentally influence the results reported here.
Spatiotemporal tuning
The spatiotemporal tuning properties of MI neurons were defined from the time (lag)-varying tuning of the cell with respect to velocity or position signals (see METHODS). Conceptually, using each spike time as a reference point for sampling of the kinematic variable, one can determine the spatial information provided by firing about that variable at any time in the future or the past, relative to that spike time. Spatiotemporal tuning functions for 113 single MI neurons were generated for velocity and position [denoted N(, ) and N(, ), respectively]. These functions summarize a neuron's instantaneous firing rate dependency on hand velocity or position, , at different delays , where is the time difference between a particular hand motion variable sample and the observed firing rate sample. A lead ( > 0) is the amount of time the neuron was firing in advance of that kinematic measurement, whereas a negative represents a lag.
Figure 6 illustrates the spatial features of velocity [N(, )] and position [N(, )] tuning, at a single value of , for 2 different neurons. Tuning functions are plotted first in rectangular coordinates (Fig. 6, A1, B1) and then transformed into polar coordinates (Fig. 6, A2, B2; see METHODS). Polar coordinates are adopted for the remaining figures to simplify comparisons between position and velocity tuning. The origin for these tuning surfaces is taken as (0, 0) for velocity, and the center of the tablet workspace for position (in each case, the origin was the mean and mode of the observed kinematic distribution (see Fig. 4).
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) and direction (); = 0 corresponds to movement to the right. The cell shown in Fig. 6A is approximately sinusoidally (i.e., cosine-) tuned for direction [i.e., the function Nv(, , ) can be fit by a cosine for any speed ]. The phase of this cosine is constant as a function of , so that the direction tuning curve
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 | (6) |
PD is the cell's "preferred direction." Because Eq. 6 defines a plane in velocity space, we will refer to this model as the "planar model," with a1 termed the "planar slope" parameter and PD the "major axis." This model has been shown to apply to MI firing during reaching (center-out) movements as well (Moran and Schwartz 1999a). For our data, the planar model for velocity gave a significant fit for 99% of the neurons in our sample (see METHODS). The data for Fig. 6A were recorded during an experiment in which pursuit-tracking and center-out trials were interleaved; by plotting the center-out target location tuning curve (Fig. 6A3) next to the PTT velocity tuning function (Fig. 6, A1, A2), we see that, for this neuron—although not necessarily for all neurons—the 2 concepts of tuning effectively coincide.
Neurons in MI were also tuned for hand position (Fig. 6B) during the PTT. For the position tuning functions in polar coordinates, the firing rate is plotted against distance from the origin () and direction (), where = 0 corresponds to rightward locations. Sinusoidal tuning in , similar to that observed in Fig. 6A for velocity, is evident. The firing rate increases linearly with but maintains constant phase; that is, tuning functions for position are significantly fit by planes as well (98% of neurons). A planar model significantly fit MI tuning functions for both velocity and position for 90% of the cells in our database. In comparison, Kettner et al. (1988) found that 64% of neurons they recorded in the motor cortex arm area showed a linear relationship between firing rate and hand position, although, in their case, the hand was held static at each position. To examine whether tuning peaked at a particular value (e.g., akin to tuning of hippocampal place cells), we tested the fit of 2-D Gaussian functions for these tuning curves. The Gaussians provided a better fit to the position tuning functions for only 5 (4%) of the cells, and a better fit to velocity tuning for only 2 (2%) of the cells, despite the fact that the Gaussian function had 4 extra free parameters. Moreover, in each of these 7 cases, the width parameter in the Gaussian function was quite large, indicating the shallowness of the observed "peaks." Thus the simple planar model in Eq. 6 appears to be a reasonable first-order description of the 2-D tuning of MI cells for both position and velocity. The distribution of R2 values for fits to Eq. 6 are shown in Fig. 9, D and E. In the following, the fit parameters of the planar model are used to summarize the tuning properties of the observed MI population.
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), which fails to show the temporal dynamics of this tuning. Consequently, tuning was examined over multiple lags and leads . Figures 7 and 8 each show an example of spatiotemporal tuning functions for velocity N(, ) and position N(, ) for a single cell. These figures illustrate the heterogeneity of the temporal dynamics of MI tuning for these variables. Figure 7 depicts the most common MI tuning type. First, the cell is strongly velocity-tuned, especially at nonnegative delays ( 
0). Second, velocity tuning peaks at approximately = 100 ms, a lead consistent with the hypothesis that these cells signal upcoming observed hand velocity. Tuning begins to emerge several hundred milliseconds before this time and fades several hundred milliseconds afterward. Throughout this time the overall tuning structure remains essentially phase () invariant. The temporal structure of this velocity tuning function N(, ) is, for many cells, largely explained by a modification of Eq. 6, expressed as
 | (7) |
) is a smooth function of , with a maximum at 100 ms, such that a1() 
0 for > 1 s. Equation 7 is a useful heuristic for understanding how tuning evolves for most cells, in that it implies a fixed orientation (PD) over all . In no case do we see a smooth shift in PD over . That is, over , the gain (i.e., a1) may go from positive to zero to negative—thus effectively abruptly flipping the PD by 180°— but the PD term does not vary as a function of .
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, ) of the neuron in Fig. 7 can be explained in terms of the inherent dependencies between velocity and position (when considered as time-varying signals, not as static variables; cf. Fig. 4). To see why, assume that this cell's firing rate depends only on hand velocity. Nevertheless, hand velocity and position are necessarily correlated for most nonzero lags (although for PTT data this correlation is fairly weak for all lags, and zero for zero lag, as shown in Fig. 4). Whenever the hand is moving to the right at time t = 0, the mean position at time t = -
will be to the left of the mean position at time t = +, for all sufficiently small positive times . Thus if we have a neuron signaling rightward velocity of the hand at 100 ms, as does the cell shown in Fig. 7, we should expect this neuron to signal the leftward position of the hand at negative time lags ( = -1 s) and the rightward position at more positive lags ( = +1 s), as observed here. Thus in this case, the position "tuning" of this cell can be explained parsimoniously in terms of its velocity tuning.
In contrast, Fig. 8 shows an example of a neuron whose position tuning cannot be readily explained from velocity tuning, suggesting that it specifically encodes position separately from velocity. In this example, position tuning is more pronounced and more temporally invariant than velocity; peak position tuning remains stable at 
/4, whereas the velocity tuning peak changes from /4 to -2/3 between = -1 and = 0.88 s. Note that this change in phase is not a continuous shift, with peaks at intermediate angles, but a bimodal function in which, at intermediate values of , the tuning diminishes and then reappears. As described above, and consistent with Eq. 7, phase shifts of a more continuous (i.e., rotational) nature were not observed in this population. Having recorded this cell during an experiment in which pursuit-tracking and center-out trials were interleaved, we can observe that the center-out target location tuning (Fig. 8, inset) matches closely that predicted by integrating the spatiotemporal tuning function for position, but not velocity, over .
Figure 9 summarizes the spatial aspects of these velocity- and position-tuning functions. The distribution of the optimal planar angle (a1 in Eq. 7) and major axis (PD) is shown for both position (Fig. 9A) and velocity (Fig. 9B). The distributions of PD were indistinguishable from uniform on [0, 2] for both variables (Kolmogorov-Smirnov test); that is, even within the small patches of MI sampled by the electrode array, a broad representation of hand position and velocity is present. The position and velocity major axes are weakly statistically dependent: when the differences modulo between the major axes (Fig. 9C) are plotted, the position and velocity major axes for a neuron tend to be close [Kolmogorov-Smirnov deviation from uniformity (i.e., independent velocity and position PD), P < 0.0001], as shown by the peak at 0. Position and velocity appear, for about half our recorded population, to be encoded essentially independently (
PD > /8). For the other half (corresponding to the peak at zero in Fig. 9C) position and velocity tuning mirror each other, as in Fig. 7.
Temporal dynamics of encoding
An information-theoretic analysis was used to provide a direct measure of position and velocity information available from the recorded neurons and to describe more quantitatively the temporal evolution of this encoding. The results in Figs. 6, 7, 8 demonstrate that by observing the position or velocity of the hand it is possible to derive information about the activity of a given MI neuron. The converse, by Bayes's rule, is also true: information about position or velocity can be decoded from MI firing rates. Figure 10 shows the conditional probability distributions, with corresponding Gaussian fits, of the horizontal hand velocity at t + , = 100 ms, given that this cell fired zero (Fig. 10A), one (B), 2 (C), or 3 (D) spikes within a 50-ms window around time t. The marked overlap in the set of curves demonstrates that the firing rate of MI neurons typically conveys highly ambiguous information with the small numbers of spikes observed in a narrow time window.
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, I[N(0); S()]. Here N(0) represents the cell's activity in a given short time interval (here, 5 ms; the interval is taken to be short to avoid redundancy effects induced by the fact that the hand position and velocity change relatively slowly) and S() denotes the value of position or velocity some time before or after the current time, t = 0. This information statistic is an objective measure of how well these neurons are tuned for these behavioral variables; the more tuned a given cell is at a given value of , the more highly separated are the probability distributions corresponding to those shown in Fig. 10, and the higher the value of I[N(0); S()]. Because this quantity is calculated directly from the underlying probability distributions it does not depend on any underlying assumptions about the linearity of the relationship between the neural firing rate and the behavioral variable, as do standard correlational statistics. The resulting curves, as functions of , discard all spatial tuning properties (e.g., preferred direction) and therefore show only temporal (-dependent) tuning features.
Figure 11 shows examples of information curves for hand velocity (Fig. 11, A1-C1) and position (Fig. 11, A2-C2), for 3 experiments. Individual curves within a panel (A, B, or C) and between panels (e.g., A1 vs. A2, etc.) can be directly compared because the neurons shown were recorded simultaneously (and therefore the information curves were constructed using identical kinematic data). These temporal tuning curves were heterogeneous, especially in the position domain; some are unimodal, others multimodal, some peak at > 0 and others at < 0, all within the same set of simultaneously recorded data. The widths and shapes of the curves vary widely (note that the position curves change more slowly than do the velocity curves, partially because of the autocorrelation structure, as discussed above) and there does not appear to be any simple rule relating the curves for velocity and position. The width of the velocity information curves is uncorrelated with those of the corresponding position curve (Spearman's rank-order correlation coefficient, P < 0.05; test performed only on the 77 cells with significant velocity and position information content). This analysis also showed differences in the time at which peak information was available about position and velocity (Fig. 11, D and E). Temporal tuning peaks are always markedly more clustered for velocity than for position, with velocity curves consistently peaking near = 100 ms (i.e., firing leads behavior by 100 ms), and position peaks more temporally dispersed, suggesting that cells carry feedback as well as advance position information.
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on this log-log scale); the Poisson line fits the data well at rates up to about 10 Hz; variance is slightly sub-Poisson at higher rates.
