When reaching movements are made from various starting points to the same target, we know of no model that predicts a consistent relationship between the reach period commands, which depend on reach direction, and the subsequent hold period commands, which depend on target location. Yet, if control of holding depends on the movement period (Figure 1A, right), a single function must exist that transforms the commands that were generated during the movement period to the subsequent commands that are generated to hold the arm. To explore the plausibility of such a relationship, we used intramuscular electrodes to measure activity of 20 shoulder and elbow muscles during point-to-point reaching in the vertical plane (nine muscles in monkey A, 11 muscles in monkey B). On each trial, monkeys first moved their hand from a central location to one of eight targets, held their hand at that target for at least 0.5 s, then reached back to the central location and again held their hand for at least 0.5 s (Figure 1B). We normalized muscle activity by setting to zero its average activity at the central hold location, and setting to one its maximum activity in the task.

Consider the activity of the anterior deltoid (Figure 1B) as the arm reached from the central position to a target. For some directions, this muscle exhibited a burst of activity during the reach, and then sustained activity during holding (Figure 1B, targets 3 and 5). For other targets, the muscle exhibited a smaller burst of activity during movement, and little or no activity during holding (target 1). Let us imagine that the measured EMG, denoted as u(t), is actually the sum of two signals: a ‘hold’ command h(t) (brown traces in Figure 1B) and a ‘move’ command m(t) (red traces in Figure 1B). The hold command is computed by adding the real-time integral of the move command to the initial hold activity that precedes movement (see Materials and methods Section A).

If the hold commands are computed from the move commands in this way, the two commands should be related by a common function across different types of reaching movements: the change in muscle activity from before the movement to after the movement, should be related to the intervening move period muscle activity (see Materials and methods A for derivation). We measured muscle activity with respect to its pre-movement hold period, $u\left(h_1\right)$, and then integrated that change with respect to time until the end of the reach, $t=T$.

This equation predicted that change in the hold period activity of a given muscle from before reach onset $h_1$ to the target $h_2$ should be approximately proportional to the integral of its activity during the reach period.

To test the validity of Equation (1), we defined the hold period at the target $h_2$ starting at 300 ms after reach end (Figure 1D, hold 2), thus allowing time for muscle activation dynamics to settle. Indeed, for most muscles (14/20), change in the hold period activity was well predicted by Equation (1). For example, when the target location was constant (Figure 1D, reach back) across various movement directions, the changes in the hold period activities of many muscles were proportional to the integral of their respective activity during the preceding reach (red lines, Figure 1E). When the target position varied (Figure 1D, reach out), Equation (1) was again a good predictor of the change in hold period activity, despite the fact that both direction and endpoint of the reach changed (green lines, Figure 1E).

Figure 1F presents the data across all muscles, directions, and endpoints. Remarkably, we found that integration of the reach period activity was a reasonable predictor of the change in hold period activity across all conditions (R² = 0.58 for Monkey A and R² = 0.54 for Monkey B). Within each muscle, the integration gain k was no different for outward reaches and return reaches (Figure 1G, paired t-test on single muscle regression slopes, p=0.943). In addition, the same integration gain predicted hold activity for fast and slow movements (Figure 1G, fast vs. slow, two-sample t-test, p=0.30) which differed modestly in terms of movement duration (two-sample t-test, p<0.001, fast movement duration of 350.3 ± 11.1 ms and slow movement duration of 453.6 ± 4.9 ms, mean ± SEM). This indicated that a single function (Equation (1)) could account for various movements and speeds, despite the differing dynamics of these reaches.

Notably, despite these general trends, some muscles (6/20) did not exhibit the pattern described by Equation (1). These muscles shared a specific property: they had little to no activity during the hold periods (Figure 1—figure supplement 1). Thus, Equation (1) seemed to apply primarily to those muscles that modulated their activity during the hold period, contributing to maintenance of arm posture in this task. However, to not bias our results, we included all muscles in our regressions in Figures 1F, G and H.

We considered an alternative hypothesis: a muscle that is more active to lift the arm will be also be more active to hold the arm. Perhaps, the correlations are driven mostly by biomechanical constraints of the reaching movements. Thus, we separated movements based on directions that were not affected by a change in gravitational forces (horizontal reaches), vs. other directions (Figure 1B). If the relationship between move period and hold period was solely due to the gravitational field, we would expect that the integration function would differ for horizontal versus vertical movements. However, the gain of integration was similar for the two groups of movements (Figure 1G, ANCOVA, movement type by moving EMG integral interaction effect on holding activity, F = 0.37, p=0.54). To broaden the scope of this alternative hypothesis, we considered the possibility that the change in hold period activity in each muscle depended on the maximum or minimum activity of that muscle during the reach period, not the integral of the entire reach period activity. This alternative hypothesis also proved to be significantly less accurate than Equation (1) (within muscle comparison, paired t-test, p=0.005, Figure 1H). In 17/20 muscles, integration of the reach period activity was a better predictor of the hold period activity than either the maximum or minimum muscle activity.

In summary, for reaching across various directions and endpoints, the change in a muscle’s activity from the pre-movement hold period to the post-movement hold period was partially predicted by the integral of that muscle’s activity during the intervening reach period.

According to Equation (1), trial-to-trial variation in the move period commands should lead to consistent trial-by-trial changes in the subsequent hold period commands, even if the target location remains constant. That is, if the integral of a muscle’s activity is greater on some trials, then that muscle should also be more active during the hold period that immediately follows.

It is difficult to test this prediction for reaching because there are numerous configurations of the wrist, elbow, and shoulder joints that would maintain the hand at a target location. Therefore, to more precisely examine within-trial covariance between move and hold periods, we simplified the problem to a single degree of freedom: finger flexion (Soteropoulos et al., 2012).

Monkeys were trained to use their index finger to track a visual target that moved at 12 deg/sec over a 1 s period between a start (12°) and an end location (24°) against a spring load that resisted flexion (Figure 2A). At the start location, the load was 0.026 N⋅m. As the finger flexed, the load increased linearly, reaching 0.048 N⋅m at the target. We measured muscle activity using subcutaneous electrodes implanted over 17 muscles (eight muscles in monkey R, nine in monkey D). For each session, we normalized each muscle’s activity by setting the average activity at the start location to 0 (period $h_1$, 400 ms in duration, began 1 s before movement onset, Figure 2C), and setting its maximum activity in the task to 1. The hold period $h_2$ at the target was 200 ms in duration and began 700 ms after movement end, thus allowing time for muscle dynamics to settle.

Figure 2

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The EXCEL source data file contains holding EMG activity and the integral of moving EMG activity for all muscles shown in Figure 2.

The data provide trial-by-trial measures of each quantity. In addition, the position of the finger during hold 1 and hold 2 are provided for all trials in each session. The ‘readme’ tab describes all quantities reported in the ‘data’ tab.

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The finger accurately tracked the target on each trial, moving along very similar trajectories (Figure 2B). To both flex the finger and support the mechanical load, muscle activity during each hold period was strongly modulated by finger location (Figure 2C). For example, the flexor digitorum profundus (FDP) muscle, one of the prime movers in this task, was more active when the finger was at the target as compared to the start location (Figure 2C, right panel). However, from one trial to the next, hold period EMG exhibited marked variability (Figure 2C, note the vertical spread of the trial distribution at hold 2).

One possibility is that the variability in hold period EMG is due to position-related variability in the spring force applied to the finger (larger displacements lead to larger spring forces on the finger). To assess this possibility, for each muscle we regressed its activity onto its position during the target hold period. We found that variation in hold position accounted for less than 2% (R² = 0.0107 ± 0.0041, mean ± SEM across all muscles in both monkeys, individual regressions for each muscle) of the trial-to-trial variability in hold period EMG (we do not mean for this to give the impression that EMG was not strongly modulated by position, only that trial-by-trial differences in EMG were poorly explained by differences in position; see Materials and methods Section B2 for more information).

If variability in position could not explain the trial-to-trial changes in hold period EMG, what was the source of hold period variability? Inspection of muscle activity in Figure 2C suggested two possibilities: trial-by-trial changes in the hold period activity at the target (the hold 2 period) could be explained by (1) modulation in the initial holding activity prior to movement onset (the hold 1 period), or (2) modulation in move period activity. We first investigated the former possibility that when a muscle was more active during the initial hold period, it was also more active during the final hold period. This hypothesis stated that $u\left(h_2\right)\approx ku\left(h_1\right)+a$. To evaluate this hypothesis, we regressed the activity of each muscle during hold 2 onto its activity during hold 1. The left panel in Figure 2D shows the strength of this correlation in the FDP muscle for a single session. Each ellipse in the middle panel of Figure 2D represents the 95% confidence boundary for the trial-by-trial joint distribution between hold period activity at the target and hold period activity at the starting position, for each muscle, across all trials and sessions. Overall, hold activity at the start appeared to be a rather poor predictor of hold activity at the target, accounting for about 8% (monkey R) and 7% (monkey D) of the variance in the data (Figure 2D).

We next considered the hypothesis that variation in hold period activity may be due to variation in preceding move period activity. Using Equation (1), we integrated the move period activity in the muscle, $u\left(t\right)$ with respect to its pre-movement activity, $u\left(h_1\right)$, and asked if this integral could predict the change in hold period activity from the start location, $u\left(h_1\right)$, to the target location, $u\left(h_2\right)$. This comparison could be confounded by trial-to-trial differences in movement displacement. Under spring forces, a larger displacement might lead to greater move period as well as greater hold period activity. This did not appear consistent with the data: trial-to-trial displacement explained less than 1% of the variance in both the integral of move period activity (R² = 0.006 ± 0.0018, mean ± SEM), and the change in hold activity (R² = 0.005 ± 0.001, mean ± SEM). On the other hand, the integral of move period activity exhibited robust correlation with the change in hold period activity, as shown for an example recording session in the left panel of Figure 2E. For this session, about 56% of the trial-by-trial variance in FDP hold period activity $u\left(h_2\right)-u\left(h_1\right)$ was accounted for by Equation (2) (linear fit for this session, p<0.001).

To determine if all muscles exhibited similar within-trial correlations between moving and holding, we considered the data across all muscles and sessions (Figure 2E, middle and right panels). Each ellipse in Figure 2E represents the 95% confidence boundary for the within-trial joint distribution between holding activity and the integral of moving activity, for each muscle. Strikingly, the orientations of various muscle distributions (the angle of the major axis) were roughly parallel with each other. Thus, the gain of the integration function was similar across muscles, and a single function accounted for approximately 50% and 40% (monkeys R and D) of the trial-by-trial variability in holding activity.

Finally, we considered another potential source of correlation between move and hold periods: co-contraction. If finger stiffness varied from one trial to another, we would observe correlations between move and hold periods. To change finger stiffness, agonist and antagonist pairs of muscles would exhibit coordinated increases or decreases in their activities. In other words, we should be able to predict the change in hold period activity in one muscle based on the activity of other muscles. To test this idea, we regressed the change in hold period activity in each muscle onto the integral of move period EMG in other muscles. Roughly 10% of the variability in the change in hold period EMG could be explained by the integral of move period activity of other muscles (R² = 0.10 ± 0.02, mean ± SEM). Therefore, while some of the variance in hold period activity could be explained by the move period activity in other muscles, move period activity in a given muscle remained a much better predictor of the change in hold period activity in that same muscle (R² = 0.42 ± 0.031, mean ± SEM).

In summary, for a constant target location, on trials in which a muscle moved the finger with greater activity, it also produced greater activity during the subsequent hold period. About 45% of the trial-to-trial variation in the change in hold period activity could be associated with the integral of the preceding move period activity.

These EMG patterns illustrated a correlation between move and hold period commands, but did not test whether there was a causal link between the two. That is, trial-to-trial coupling between move and hold period commands arose from variability that was internally generated by the animal. To rigorously test if hold period commands directly followed from move period commands, we next imposed external changes on move period commands and measured if hold period commands changed in a manner consistent with integration. To do this, we instructed participants to reach to a target while holding the handle of a robotic arm (Figure 3A), and adapted their move period commands by imposing a velocity-dependent force field (Albert and Shadmehr, 2016).

Figure 3 with 1 supplement see all

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Figure 3—source data 1

The EXCEL source data file contains holding forces and the integral of moving forces as shown in Figure 3.

The data are divided into two tabs, ‘data-panels-B-D’ and ‘data-panels-E-G’. For the former, data are provided for all channel trials appearing in the baseline null field period for all healthy participants (n = 220). Here, we also indicate the movement duration and whether or not the arm was supported against gravity. For the latter, data are provided for force field adaptation trials for the participants shown in the corresponding figure panels (n = 32). The adaptation paradigm for these trials is shown in Figure 3E (bottom). The ‘readme’ tab describes all quantities reported in the two tabs containing participant data.

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With the force field our goal was to gradually bias the move period commands through the process of adaptation. If hold period commands partly depended on the preceding move period, then biases in move commands should lead to biases in hold commands, even though (1) the hold period commands were never perturbed, and (2) the hold location remained constant. To ensure that hold period commands were never perturbed, on all force field trials the hand was placed in a ‘well’ that held the hand still at the end of the movement for at least 1.5 s (Figure 3A, target-hold well). Next, we used a channel to prevent the hand from suddenly moving off the target while participants transitioned from the target-hold well to the next reaching movement (not shown in Figure 3A; see ‘partial channel’ in Materials and methods).

Unlike the monkey experiments, we did not record EMG during these experiments. Instead, we measured changes in the forces that participants exerted against the handle. These forces served as a low-dimensional proxy for the motor commands sent to the arm muscles. Critically, we drove the adaptation process with forces that acted perpendicular to the trajectory of the hand. Because the learning axis was orthogonal to the primary movement, we could cleanly isolate the component of the motor commands that varied in response to the field, from the motor commands responsible for the primary movement. To measure the forces perpendicular to the primary movement, on some trials the robot produced a stiff channel that connected the start position to the target via a straight line (Figure 3A, channel trial). To test the integration hypothesis, we recorded forces perpendicular to the direction of the target during the reach and hold periods, and asked if they were related through the following integration function (i.e. the force analogue of Equation 1):

At the start of each experiment, participants (n = 220 in total) reached to the target in a neutral (i.e. null field) condition in which the robot did not produce any forces on the hand (Figure 3A, null field trial). Even in the null field period, we observed significant trial-by-trial variability in the perpendicular reach period forces, as shown for a representative subject in Figure 3B. During the reach period, the hand pushed slightly to the right on some trials, left on others, or exhibited a bimodal profile. However, because the movements were guided within a channel, the hand followed a straight line that ended at the center of the target (Figure 3A, bottom left). Notably, following conclusion of the reach, we observed that the arm generated forces during the hold period (period $h_2$, Figure 3B) that were often different than baseline (period $h_1$). Indeed, the change in the hold period forces $F\left(h_2\right)-F\left(h_1\right)$ was well predicted by Equation (2), as illustrated by data for the representative subject in Figure 3B (regression at right), and the entire population in Figure 3C. On a within-trial basis, the integral of move period force accounted for 39 ± 2% (mean ± SEM) of the variance in hold period force. Thus, just as EMG patterns exhibited a within-trial relationship between the reach and hold periods, so did the force patterns.

Like the EMG patterns, the force patterns did not appear to be trivially related to biomechanical constraints imposed on the arm due to gravity: the gain of the integration function was the same whether or not the weight of the arm was supported by a frictionless air-sled (Figure 3B, two-sample t-test, p=0.90). Remarkably, the relationship between move and hold period forces remained unchanged when we divided the null period reaches of each subject into fast and slow movements (Figure 3D, ANCOVA, movement type by move force integral interaction effect on hold force, F = 0.007, p=0.935). Thus, in the null field, forces during the reach and subsequent hold periods showed natural variability. However, on a within-trial basis, the integral of the move period forces appeared to influence the subsequent hold period forces.

At the conclusion of the null field period, we gradually imposed a velocity-dependent force field during the reach (Figure 3E). The velocity-dependent forces perturbed the hand perpendicular to the reach trajectory, thus leading to adaptation of reach period forces (Figure 3F). The gradual onset of the perturbation produced gradual changes in reach period forces (while also minimizing positional errors throughout the trajectory). Remarkably, as the reach period force changed from one trial to the next, so did the hold period force (Figure 3F and G). Notably, the hold period forces were not transient, but sustained, persisting up to 6 s during the entire hold period interval (Figure 3—figure supplement 1). The relationship between the change in hold period force and the (now externally-driven) reach period force was again consistent with integration: over the course of adaptation, Equation (2) accounted for 48 ± 3% (mean ± SEM across 32 participants) of the variance in the change in hold period forces.

As an alternative hypothesis, we considered the possibility that hold period forces may be a trivial continuation of the forces produced near the end of the preceding reach, not an integration of the entire history of the reach period. To test this idea, we conducted a pair of experiments. In the first experiment (Figure 4A), participants (n = 11) reached in a force field that was active only during the second half of the reach. In Phase 1 of the experiment, trial after trial we gradually increased the magnitude of the force field (Figure 4A, Phase 1). As expected, participants produced hold period forces that increased with the integral of the reach period forces. In Phase 2, we maintained the force field at a constant magnitude for hundreds of additional trials. The change in hold period forces and the integral of reach period forces remained correlated during all trials (Figure 4A, Phase 2).

Figure 4

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The EXCEL source data file contains holding forces and the integral of moving forces as shown in Figure 4.

The data are divided into two tabs, ‘data-panel-A’ and ‘data-panel-B’. The former contains the data for Figure 4A and the latter contains the data for Figure 4B. Data are provided for all channel trials during Phase 1 and Phase 2 of adaptation as shown in Figure 4. The ‘readme’ tab describes all quantities reported in the two tabs containing participant data.

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In the next experiment (Figure 4B), we again exposed participants to a force field that was active only during the second half of the movement. But then, in Phase 2, we gradually added a second force field that was active during the first half of the movement, but in the opposite direction (Figure 4B, Phase 2). In this way, the reach period forces should integrate to approximately zero. If holding forces were a simple continuation of the reach period forces, then the addition of the force field during the first half of the movement should not alter the hold forces at the end of the movement. However, this is not what we observed: as the integral of reach period forces approached zero, hold period forces gradually vanished (Figure 4B, Phase 2). That is, even though reach period forces were matched just before the end of movement (Figure 4C), the ensuing hold period force depended on the entire history of the reach. In addition, we observed no difference in the integration function between each phase of the experiment (paired t-test on slope, p=0.22, paired t-test on intercept, p=0.09).

Together, these observations demonstrated that on a within-trial basis, as the reach period forces changed, so did the ensuing hold period forces. The change in hold period force was partially described via a function that integrated in time the temporal history of the preceding reach period forces.

Equations (1) and (2) use initial hold activity ($u\left(h_1\right)$ or $F\left(h_1\right)$) both to calculate the change in hold activity, and to estimate move period activity (Materials and methods Section A). We were concerned that some or all of the observed correlations may be due to this common factor that appears on both sides of the equation, and not the integral of the move period activity.

To consider this problem, we noted that for outwards reaching movements in Figure 1, this was not a concern because the trial-averaged EMGs were nearly identical before movement onset at the center location, but differed greatly during the movement to various directions. To address this concern for our data in Figures 2 and 3, we re-analyzed the correlation between move and hold activity, but only on trials in which hold 1 activity fell within one standard deviation of the mean. This criterion reduced the variance in hold 1 activity by 75% for our finger movement dataset, and 88.6% for our reach force dataset. Despite this dramatic reduction in hold 1 variability, we found little effect on the measured correlations: the correlation coefficient between integral of move activity and change in hold activity remained 58.4% for muscles in the finger (compared to 64.2% if all trials are included) and 48.5% for reach forces on channel trials in the null field (compared to 52.2% if all trials are included).

In summary, the observed trial-by-trial correlations between move activity and change in hold activity were almost entirely driven by the integration of the move commands, with little dependence on variability in the initial hold activity.

Thus far, we have described the state of the limb during the hold period in terms of muscle activity or force generation. However, in order to hold the limb still, the postural controller does not simply produce a force, rather it generates a converging field of position-dependent forces (Mussa-Ivaldi et al., 1985; Shadmehr et al., 1993). We next asked if this postural field also relied on the commands generated during the preceding reach.

We designed a new experiment in which we measured the postural field following completion of a reach. Participants (n = 27) reached to a target as before, but now, during the hold period they were engaged in a short-term memory task (2-back, Figure 5A). As they performed the memory task, the robot slowly displaced their arm in a random direction. In response to the displacement, the postural controller produced restoring forces against the handle, thus allowing us to measure the postural field (Figure 5B).

Figure 5

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The EXCEL source data file contains parameters for our postural field model, as shown in Figure 5D.

We fit a linear spring model (Equations 10 and 11) to the postural fields measured before (Figure 5B left) and after adaptation (Figure 5B right). Here we provide parameters pertaining the null point of the two-dimensional spring and the stiffness of the postural field for individual participants. The ‘readme’ tab describes all quantities reported in the ‘data-panel-D’ tab.

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As expected, the postural field’s null position was near the target (Figure 5B, null point of postural field). However, after participants were exposed to a force field, the postural field changed: the null position was no longer aligned with the target (Figure 5B, right). Rather, it shifted by approximately 1 cm (Figure 5D; paired t-test, p<10⁻⁴) in the direction of the force produced during the reach. In contrast, the orientation (Figure 5D, paired t-test, p=0.84) and stiffness (Figure 5D, paired t-test, p=0.62) of the postural field remained unchanged.

After the reach had ended and the cursor was at the target, we slowly displaced the hand toward the postural null position. We observed that the hold period forces gradually approached zero, and then switched direction and grew larger as the hand was displaced beyond the null position (Figure 5E). The holding force at the hand scaled linearly with the distance between hand position and the postural null position (Figure 5F). This implied that the hold period forces we had measured in previous experiments (Figures 3–4) were a proxy for the location of the null position of the postural controller: the larger the hold period force at the target, the farther the null position of the postural field.

Thus, as the reach period forces changed, so did postural control: the null position of the postural field shifted in the same direction as the change in the preceding move period forces.

These results create a puzzling scenario. In the presence of a velocity-dependent force field, the reach controller readily adapts and changes the move period forces. However, changes to the move period forces are integrated and cause the hold system to program an entirely different null position. This implies that postural stability will be compromised in the face of an adapting reach controller. To solve this problem, the integrator must also be adaptive: the integration function must change when there is an error between its current output and the desired movement endpoint (Figure 6A). Presumably, this error-based adaptation would be driven by unexpected deviations from the hand’s desired trajectory as the reach period ends and the hand arrives at the target location.

Figure 6

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The EXCEL source data file contains data for panels D and E of Figure 6.

In the ‘data-panel-D-force-field’ tab, we report the endpoint error (labeled ‘endpoint correction’ in Figure 6B) measured on all trials in the force field period for each participant. In the ‘data-panel-D-channel’ tab, we report the holding forces and the integral of moving forces measured on channel trials at the end of the adaptation period for each participant. In the ‘data-panel-D-gain-and-error’, we provide single measures of endpoint error and integration gain for each participant, obtained from post-processing of the data provided in the prior two tabs. In ‘data-panel-E’, we provide the integration gain measured for each subject during the null period, early during force field adaptation, and late in force field adaptation. The ‘readme’ tab describes all quantities reported in the other four tabs containing participant data.

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To test this idea, we examined reach trajectories of individual subjects as they gradually adapted to velocity-dependent force fields. These trajectories exhibited two primary types of errors. The first error happened midway through the movement and was caused by incomplete compensation for the velocity-dependent perturbation (Figure 6B, the large negative mid-movement error). The second error happened near the end of the movement and was oriented in the opposite direction (Figure 6B, ‘endpoint correction’). This near endpoint error possessed two properties that were well-suited for integrator adaptation: (1) they occurred late in the movement as the participant attempted to stop their hand within the target, and (2) they were oriented in the direction opposite the shift in postural null point reported in Figure 5. To quantify the size of these endpoint errors, we measured the largest ‘positive’ deviation (or ‘negative’ if the lateral deviation was in the opposite direction) from the terminal hand position, after the hand exceeded 80% of its reach displacement.

To determine if endpoint errors caused integrator adaptation, we compared the size of these errors during the adaptation process, to the gain of integration observed at the end of adaptation. The size of endpoint error was heterogeneous across our subjects; some participants exhibited large endpoint errors (Figure 6B, S11) while others exhibited small endpoint errors (Figure 6B, S5). Critically, we found that participants with larger endpoint errors ultimately produced smaller holding forces (Figure 6C). In other words, these errors appeared to reduce the gain of integration. In fact, about 40% of the variability in integration gain could be explained by the magnitude of the late endpoint errors (Figure 6D).

This adaptation of integration gain progressed over time (Figure 6E, repeated measures ANOVA, F = 12.60, p<0.001). In the early part of training, the integration gain was no different than that of the pre-adaptation null trials (Figure 6E, post-hoc comparison, p=0.88). However, late in training, the integration gain had decreased substantially (Figure 6E, post-hoc tests, p<0.001 for comparison of late adaptation with both null field trials as well as early adaptation trials).

These data suggest that Equation (2) alone cannot predict the change in holding forces. In certain conditions, for example velocity-dependent force fields, the reach period commands change, but if one is to hold the hand at the target, then the hold period commands cannot simply integrate the preceding move period commands. Rather, as the move period commands adapt, so too must the integration function. Like the adaptation of movement commands, the adaptation of hold commands does not occur instantaneously, but appears to emerge gradually as errors near the end of the movement are experienced repeatedly. This process may also explain why holding forces gradually diminish during adaptation to an abrupt force field, where endpoint errors are large (Sing et al., 2009).

The CST conveys the cortically generated reach commands to downstream motor structures. Does this same pathway also convey postural signals, or does a separate, downstream structure receive and then integrate the reach commands? If both reaching and holding commands are conveyed in the CST, then damage to the CST above the level of the brainstem should disrupt both the generation of forces during reaching, and its integration during the hold period. However, if the integrator is downstream to this level, then damage to the CST might result in deficient reach commands, but spare the process of integration, resulting in hold commands that reflect the within-trial integration of the now deficient reach commands.

To examine these possibilities, we recruited stroke patients (n = 14) who had suffered lesions affecting the CST pathway from the cortex through the internal capsule (Supplementary file 1). The patients exhibited profound impairments, as demonstrated by difficulty with extension of their arm during unsupported reaching (Zackowski et al., 2004; Roh et al., 2015; Figure 7A, patient S015). To improve their reach capacity, we supported the weight of their arm in the horizontal plane (frictionless air sled), which allowed them to better extend their arm at the elbow, enabling them to make planar, point-to-point reaching movements while holding the handle of the robot arm (as in Figure 3A).

Figure 7

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Figure 7—source data 1

The EXCEL source data file contains holding forces and the integral of moving forces as shown in Figure 7.

The data are divided into two tabs, ‘data-null-field-trials’ and ‘data-force-field-trials’. For the former, data are provided for all channel trials appearing in the baseline null field period for the paretic and non-paretic arms of stroke patients, as well as their age-matched controls. For the latter, the same data are provided but for the force field adaptation period. The ‘readme’ tab describes all quantities reported in the two tabs containing participant data.

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As has been noted before (Scheidt and Stoeckmann, 2007; Coderre et al., 2010), movements of the paretic arm exhibited erratic trajectories (Figure 7B), increased movement duration (paretic vs. control) of approximately 41% (paretic vs. non-paretic, paired t-test, p<0.01; paretic vs. control, two-sample t-test, p<10⁻⁴), and reach endpoints that terminated nearly 89% (paretic vs. control) further away from the target location (paretic vs. non-paretic, paired t-test, p<0.01; paretic vs. control, two-sample t-test, p<0.001). The reaching impairment coincided with a marked increase in the trial-to-trial variability of move period forces (traces in Figure 7C solid lines; Figure 7D paretic vs. control, Wilcoxon rank-sum, p<0.001; Figure 7D paretic vs. non-paretic, Wilcoxon signed-rank, p=0.058). However, like healthy subjects, these move period forces, no matter how variable, terminated with stable holding forces (traces in Figure 7C dashed lines). The trial-by-trial variability of the hold period forces, like the move period forces that preceded them, was significantly greater in the paretic arm of the patients (Figure 7E; paretic vs. control, two-sample t-test, p=0.01; paretic vs. non-paretic, paired t-test, p=0.06).

If move period commands were integrated into hold period commands, the increased variability in holding forces (Figure 7E) could arise indirectly from the normal integration of the highly variable moving commands. If this were true, the variability in the moving and holding commands would be similarly structured in both healthy subjects and stroke patients. To test this idea, we quantified the within-trial correlation between change in hold period forces and the integral of the preceding move period forces (Figure 7F, left column, representative subjects). Remarkably, in the null field trials, the coupling between move and hold periods was intact in stroke patients (Figure 7H, paretic vs. control, two-sample t-test, p=0.14; paretic vs. non-paretic, paired t-test, p=0.63).

We next used adaptation to systematically manipulate move period forces. Because force field adaptation is largely a cerebellar-dependent process (Smith and Shadmehr, 2005), despite damage to the CST the patients learned to alter their move period forces. As the move period force changed in the paretic arm, so did the change in hold period force (Figure 7F, right column, example subjects). Once again Equation (2) provided a reasonable account of the within-trial relationship between the move period and the change in hold period forces for the paretic arm, non-paretic arm, and the dominant arm of age-matched control subjects (paretic vs. control, two-sample t-test, p=0.08; Figure 7I: paretic vs. non-paretic, paired t-test, p=0.24). Notably, the integration gain was not significantly different across the paretic and non-paretic limbs of the patients, nor across the patients and age-matched controls (Figure 7J; paretic vs. control, two-sample t-test, p=0.86; paretic vs. non-paretic, paired t-test, p=0.91). In other words, the integration function was similar in healthy participants and stroke patients.

In summary, damage to the CST severely affected the reach period commands, resulting in high trial-to-trial variability. However, the change in hold period commands remained coupled to the integral of the preceding reach commands in both null field and force field trials. The gain of this integration in the stroke patients was not different than that of healthy controls, suggesting that CST damage impaired the reach commands, but not the process of integration that may have generated the hold commands.

The model presented in Figure 7G represents our conjecture regarding architecture of the reach and hold controllers. In this conjecture, movements are encoded by the cortex, resulting in the move signal, termed $m\left(t\right)$, which is integrated in real-time by an unknown subcortical area. The integral of move commands is then added to initial hold activity, yielding a feedforward estimate of the commands required for holding still, termed $h\left(t\right)$. During a movement, the commands that arrive at the motoneurons are a sum of commands for moving and holding, $u\left(t\right)$.

There are a number of predictions that arise from this model. Motor commands required to move the arm to a target position are not fixed, but vary because of interactions with external objects, interaction torques that arise when the body is in motion, and over time as our bodies change. We know that the move system continuously adapts to these novel dynamics. A similar process of adaptation would also be required of the proposed reach integrator.

Indeed, in the oculomotor system, move period and hold period commands can both undergo adaptation, and this adaptation depends on the cerebellum (Shadmehr, 2017). However, different regions of the cerebellum are required for adaptation of the move and hold periods (Shadmehr, 2017). Optican and Miles (1985) demonstrated that the oculomotor integrator could be adapted by translating the target position as the eye transitioned from a saccade to gaze holding. In our reaching experiments, we found evidence for a similar adaptation mechanism. Errors near the end of the reach that were consistently encountered as the arm attempted to stop within the target location reduced the gain of the integration function (Figure 6). Through adaptation of the hold system, the arm would be able to cope with novel dynamics during the reach while also maintaining the ability to hold the arm at the target.

While this adaptation mechanism would achieve endpoint stability over the sequence of many movements, immediate corrections in the integrator output would be handled through the parallel operation of sensory feedback (Figure 7G, sensory feedback). This scheme would resemble a neural integrator for control of the head in the interstitial nucleus of Cajal (Crawford et al., 1991; Klier et al., 2002; Shaikh et al., 2013), which relies on proprioceptive and visual feedback (Shaikh et al., 2013). The importance of these feedback pathways is further illustrated by conditions in which the arm is passively moved to a new position.

We think that this new model of reaching might shed light on a number of interesting puzzles. For example, transient inhibition of the motor cortex during a reach results in ‘freezing of the arm’ at its current posture, and not loss of muscle tone (Guo et al., 2015). That is, despite near complete removal of output from the motor cortex during a reach, commands of unknown origin continue to sustain arm position against the force of gravity. Our model suggests that a distinct structure, possibly located in a subcortical area, integrates the cortically-generated reach commands up until the moment of cortical inhibition, and thus maintains posture despite removal of reach commands. This model is also consistent with the observation that cortical stimulation not only moves the limb, but also produces specific postures in primates (Graziano et al., 2002) and rodents (Harrison et al., 2012). Moreover, the total displacement of the limb appears to scale with the duration of stimulation, consistent with the idea that displacement is produced due to integration of motor commands over time.

The idea that integration occurs outside of the motor cortex potentially explains why successful decoders of cortical activity are designed to control cursor velocity, as opposed to cursor position (Sachs et al., 2016; Kim et al., 2011; Gilja et al., 2012). To hold the cursor still, these decoders assume that the cortex encodes a zero-velocity command rather than a position-based command. This observation is consistent with the possibility that the cortex is primarily concerned with dynamic quantities, that is changes in the state of the limb, rather than the maintenance of a specific limb state over time. This idea would account for the observation that neurons in the motor cortex modulate their activity when there is a change in force production, but less so when the constant force is maintained over time (Shalit et al., 2012; Georgopoulos et al., 1992). To maintain a constant force over time, our conjecture states that a subcortical area integrates phasic activity from the cortex, and maintains its output over time. This idea is consistent with evidence that transient stimulation of the brainstem in decerebrate cats produces sustained (timescale of minutes) changes in extensor muscle force (Mori et al., 1982).

Equations (1) and (2) describe how the integration of move activity may relate to changes in hold commands, but does not specify the hold command at the target. This reflects the reality that move period commands alone will not determine the terminal position of the arm: the initial arm position must also be taken into account. In other words, to hold the limb at the target, the integrator must not only integrate move commands, it must add this integral to the hold period activity that preceded movement (see Equation (6) in Materials and methods; illustrated in Figure 7G). We do not know if the integrator internally performs this addition, or if a downstream structure handles this adjustment for initial limb position.

Without a biomechanical model of the arm, it is not obvious why the linear functions in Equations (1) and (2) robustly predicted the relationship between change in hold activity and the integral of move activity. It may be that as we test movements that are more complex than simple point-to-point reaches, the observed linearity will break down. For example, how would the reach integrator maintain endpoint accuracy when the arm grasps an object, thus altering its mass? Such a scenario would require an increase in muscle activity to move the larger mass, and then hold it still against the forces of gravity, so it may be the case that integration would naturally account for these positive correlations. Furthermore, over successive movements, inaccuracies in the output of the integrator could be reduced through adaptation of the integration gain as in Figure 6. Finally, as is the case for the move controller (Sheahan et al., 2016; Heald et al., 2018; Howard et al., 2013), the reach integrator may store object-specific or environment-specific memories and implicitly alter its integration properties when an often-encountered dynamical context is re-experienced. These mechanisms, along with the parallel output of visual and proprioceptive controllers would allow the arm to be stabilized under different inertial conditions.

These possibilities could be tested in the future, by altering the mass of the arm and recording EMG from various arm muscles. With regard to EMG recording, it should be re-emphasized that the reach period forces measured in our human subject experiments were perpendicular to the direction of movement, and thus represented only part of the complete motor command. Because perturbations were applied perpendicular to the reach trajectory, our measure captured the primary dimension of adaptation. In future studies, a more complete picture of the integration process would be provided by recording EMG during adaptation.

It is unlikely that the motor cortex has no role in postural control. At the very least, the monosynaptic projections from corticomotoneurons (Griffin et al., 2015) to alpha-motoneurons in the spinal cord are likely to be active during periods of holding still. We do not know if these hold period signals in the cortex arise from recurrent connections with a subcortical integration system, or from a separate position controller.

Understanding the differential contributions of the cortical systems and the putative subcortical integrator is essential to improve our understanding of neurological disorders such as stroke and dystonia (Sadnicka et al., 2018). Identifying the locus of the reach integration may help explain why these patients exhibit abnormal postures at rest. These abnormalities could arise from a lesion to the integrator circuit, or perhaps more provocatively, from the normal integration of chronically abnormal moving commands.

Muscles are engaged during the period of moving as well as the subsequent period of holding still. We hypothesized that for arm and finger muscles, activity during the movement $u\left(t\right)$ could be decomposed into contributions from a move controller $m\left(t\right)$ and a hold controller $h\left(t\right)$:

If the move and hold controllers are connected in series, like the control system present for the eye and the head, the hold controller produces its output by integrating in real-time the output of the move controller. An example of this is shown in Figure 1B, in which EMG of ant. deltoid is decomposed into move and hold commands. Here, we explain this decomposition and derive some of its predictions.

For a movement from position $h_1$ to position $h_2$ the hold controller first produces commands for holding at $h_1$ and then transitions to holding commands at $h_2$ through integration of $m\left(t\right)$. Here is an integration function that could accomplish this task:

In the above equation, $u\left(h_1\right)$ represents hold activity at position 1. Combination of Equations 3 and 4 yields:

Given the measurement $u\left(t\right)$, for example EMG from a given muscle, we can decompose the measurement into its constituents $m\left(t\right)$ and $h\left(t\right)$ by solving the above equation iteratively for $m\left(t\right)$. This is what we did to plot the traces in Figure 1B (we assumed k = 1).

To evaluate the predictions of this equation, we measured movements of duration T, between two positions $h_1$ and $h_2$. At the end of the movement, the move commands go to zero yielding:

Unfortunately, we do not have an a priori estimate of the integration gain k. Thus, during movement we cannot uniquely estimate $m\left(t\right)$ and $h\left(t\right)$. To proceed, we made a simplifying assumption: the move commands could be approximated by taking the overall EMG signal $u\left(t\right)$ and subtracting off the hold commands measured at the start of movement:

Here, we added a bias term, a, to account for systematic error introduced by our approximation. Rearranging the terms in Equation (7) yields the prediction of the hypothesis:

This states that the change in hold period activity should be roughly a linear function of the integral of muscle activity during the preceding movement with a gain k. Thus, from this approximation we arrive at Equations (1) and (2) in the main text.

We performed four sets of experiments: (1) reaching movements in non-human primates, (2) finger movements in non-human primates, (3) reaching movements in healthy humans, and (4) reaching movements in stroke patients.

All human subject experiments were approved by the Institutional Review Board at the Johns Hopkins School of Medicine. Our healthy human cohort consisted of n = 223 individuals. Healthy participants ranged from 18 to 61 years of age (mean ± SD, 25.2 ± 7.9) and included 128 males and 95 females.

Our stroke patient cohort consisted of n = 14 adults that had suffered damage to the corticospinal tract (CST). The stroke patients ranged from 30 to 80 years of age (mean ± SD, 56.4 ± 15.2) and included six males and eight females. For comparison, we recruited a cohort of healthy age-matched controls who ranged from 28 to 81 years of age (mean ± SD, 60.6 ± 16.3) and included five males and five females. There was no significant difference in age between the patient and older healthy control populations (two-sample t-test, p=0.53).

The stroke patients we recruited had survived a stroke affecting cortical or subcortical white matter associated with the CST. Patients were selected based on MRI or CT scans, and/or available radiologic reports. Scans and/or reports were corroborated to determine the level at which the white matter of the corticospinal tract (CST) was lesioned. Supplementary file 1 provides the level of the brain at which the white matter was damaged.

We measured the degree of motor impairment in the patients using the Fugl-Meyer Assessment (FMA) and the Action Research Arm Task (ARAT). Two separate raters scored these assessments in each patient. In each limb, we measured the strength of elbow flexion and extension and shoulder horizontal adduction and abduction using a dynamometer (microFet 2). During measurements, participants rested their arm on a side table so the arm rested slightly below shoulder level. Strength measurements were repeated twice, the maximal force was recorded on each effort, and forces were averaged over repetitions. FMA scores, ARAT scores, strength, and other patient characteristics are reported in Supplementary file 1. Missing entries in table indicate that the patient was unable to perform the desired action.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

This work was supported by grants from the National Institutes of Health (R01NS078311, 1DP2NS083037, R01NS100066, 1U19NS104649, F31NS095706, and F32NS092350), the National Science Foundation (1723967), the Simons Foundation (SCGB#542957), the UK Medical Research Council (MR/P023967 and MR/K023012/1), and the Sheikh Khalifa Stroke Institute. We thank Kahori Kita for help with performing and scoring the assessments for the stroke patients, and Jennifer Keller for help performing some of these assessments and allowing us to borrow dynamometry equipment. We also thank all study participants, especially stroke survivors, for their time and contribution to this work.

Human subjects: Informed consent was obtained from all participants. All human subjects work was approved by the Johns Hopkins School of Medicine Institutional Review Board, protocol number NA_00037510.

Animal experimentation: All animal procedures in the U.S. were conducted in accord with the US National Institutes of Health guidelines and were approved by the Columbia University Institutional Animal Care and Use Committee (AC-AAAQ7409). These data were originally published in Lara, Cunningham, & Churchland (2018) as well as Lara, Elsayed, Zimnik, Cunningham, & Churchland (2018). All procedures in the U.K. were carried out under appropriate UK Home Office licenses in accordance with the Animals (Scientific Procedures) Act 1986, and were approved by the Local Research Ethics Committee of Newcastle University. These data were originally published in Soteropoulos, Williams, & Baker (2012).

© 2020, Albert et al.

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