Motor variability arises from a slow random walk in neural state

Abstract

Even well practiced movements cannot be repeated without variability. This variability is thought to reflect "noise" in movement preparation or execution. However, we show that, for both professional baseball pitchers and macaque monkeys making reaching movements, motor variability can be decomposed into two statistical components, a slowly drifting mean and fast trial-by-trial fluctuations about the mean. The preparatory activity of dorsal premotor cortex/primary motor cortex neurons in monkey exhibits similar statistics. Although the neural and behavioral drifts appear to be correlated, neural activity does not account for trial-by-trial fluctuations in movement, which must arise elsewhere, likely downstream. The statistics of this drift are well modeled by a double-exponential autocorrelation function, with time constants similar across the neural and behavioral drifts in two monkeys, as well as the drifts observed in baseball pitching. These time constants can be explained by an error-corrective learning processes and agree with learning rates measured directly in previous experiments. Together, these results suggest that the central contributions to movement variability are not simply trial-by-trial fluctuations but are rather the result of longer-timescale processes that may arise from motor learning.

Figure 1.

Examples of movement variability. a, Spread of release points for all four-seam fastball pitches thrown by Los Angeles Dodgers pitcher Clayton Kershaw versus the New York Mets on May 8, 2011. The location is defined in the frontal plane of the pitcher but in coordinates fixed relative to the field. Baseball circumference shown for scale. b, Reach paths from all tuning block trials in a single experimental session (monkey E, color coded by target).

Figure 2.

Quantification of behavioral and neural drift. a, Mean-subtracted speed for reaches to a single target during one experimental session (monkey E); the black trace shows the best polynomial fit of drift (Eq. 4). b, Breakdown of macaque reach speed variability for each session target into drift and residual components, expressed as SDs for both monkeys. Solid points indicate session targets with significant (p < 0.05, permutation test) drift. c–h, Analogous to a and b for initial reach direction (c, d), neural firing rate (e, f; neural firing rates were not mean subtracted, and in f, no distinction is made for cells with significant vs insignificant drift) and principal components (PC) of pitcher release point for 2011 MLB data (g, h).

Figure 3.

Time course for all behavioral drifts. Each trace corresponds to the fourth-order polynomial fit (Eq. 4) of the drift for a given behavioral metric (speed or direction) for a given target in a single session. Vertical pairs correspond to the speed and direction drifts measured in the same session. Each of the eight panels contains data from the corresponding target direction, with color indicating the animal's identity.

Figure 4.

Relating trial-by-trial fluctuations in neural activity and behavior. a, b, Histogram (with bins logarithmically spaced) of R² between trial-to-trial fluctuations in behavior (reach speed, a; initial direction, b) and trial-to-trial responses of individual neurons, with drift removed from both. Control histogram for null hypothesis of no correlation (gray) was determined by correlating neural activity with behavioral data from different sessions with matched experimental conditions. Data are combined from both monkeys, but results hold for each. c, d, Analogous to a and b, but correlating behavior to fluctuations in PMd/M1 population response via optimal linear estimator (see Materials and Methods). e, f, Summary of population behavior R² analysis (c, d) but correlating behavior to neural firing rates observed in different trial epochs. Each data point represents distribution median, and error bars denote the smallest interval containing 68% of data points; colors are analogous to a–d. “pre-trial” is a 1 s window before the trial begins. “trial start” begins with the appearance of a central start target; “hold” begins when the monkey has moved to that target. “motor prep.” begins with the appearance of the visual reach target and ends with the go cue. We also repeated the analysis using the neural firing rates measured during each of the four quartiles of the preparatory period. (For more details on the trial timeline, see Materials and Methods.)

Figure 5.

Relating neural and behavioral drift magnitudes. a, Correlation across sessions and targets between the amounts of variability attributable to drift in reach speed and in preparatory neural activity. b, Analogous to a but comparing residual trial-to-trial variability instead of variability attributable to drift. c, d, Analogous to a and b but with reach direction as the behavioral variable. In all panels, the behavioral variability is expressed as SD; neural variability is expressed as coefficient of variation (SD/mean), averaged across all neurons. All values are mean subtracted for matched experimental conditions (see text). The gray trace shows the regression slope for both monkeys combined. For both behavioral metrics, the magnitude of behavioral drift correlates positively with the magnitude of neural drift for each monkey (a, c) and is significant with both monkeys combined (p < 0.05, t test).

Figure 6.

Autocorrelation functions and the error-corrective learning model. a–c, Mean (and SE) autocorrelation functions of reach speed (a), direction (b), and neural activity (c), averaged across target and session for behavior and across targets, sessions, and neurons for neural activity. The data were fit to a linear dynamical learning model characterized by either one decay time constant (first-order model, Eq. 5a; gray lines) or two decay time constants (second-order model, Eq. 5b; colored lines). d, Comparison of fit decay time constants using the second-order model; fit errors are from the bootstrapping technique. These time constants are also compared with analogous time constants found in the 2011 MLB pitching data. (To reduce our sensitivity to noise, we only included the 64, of 160, pitcher games in which measured drift is found to be significant at p < 0.003, permutation test.) We also show motor learning time constants from Smith et al. (2006), who directly measured the time course of learning in response to external perturbations. (There, learning is described with a two-state learning model; we used Equation 3 to express their empirical parameters as autocorrelation time constants. We assumed that their stated measurement errors were independent but verified that this does not significantly affect the size of the errors on the time constants.)

Figure 7.

Relating neural and behavioral autocorrelation time constants. Correlation across sessions and targets between the autocorrelation time constants, fit using the first-order model (Eq. 5a). a, Correlation between time constants for movement direction and speed. b, Correlation between time constants for neural firing rate and movement speed. c, Correlation between time constants for neural firing rate and movement direction. For all plots, the gray line is the regression fit for data from both monkeys combined. Correlation coefficients, R, are shown for data from each monkey separately, with p values denoted as follows: *p < 0.05; **p < 0.005; ***p < 0.001.