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. 2014 May 7;137(7):1931–1944. doi: 10.1093/brain/awu115

Predicting and correcting ataxia using a model of cerebellar function

Nasir H Bhanpuri 1,2,3,4, Allison M Okamura 5,6, Amy J Bastian 2,3,✉
PMCID: PMC4065021  PMID: 24812203

Is cerebellar dysmetria due to a malfunctioning internal dynamic model? By applying forces to the arms of patients via a robotic exoskeleton, Bhanpuri et al. identify patient-specific deficits in reaching performance, and produce improvements in movement accuracy. Mathematical models reveal that bias in ataxic movements can be explained by an internal misestimate of arm inertia.

Keywords: cerebellum, ataxia, dysmetria, internal model, computational model

Abstract

Cerebellar damage results in uncoordinated, variable and dysmetric movements known as ataxia. Here we show that we can reliably model single-joint reaching trajectories of patients (n = 10), reproduce patient-like deficits in the behaviour of controls (n = 11), and apply patient-specific compensations that improve reaching accuracy (P < 0.02). Our approach was motivated by the theory that the cerebellum is essential for updating and/or storing an internal dynamic model that relates motor commands to changes in body state (e.g. arm position and velocity). We hypothesized that cerebellar damage causes a mismatch between the brain’s modelled dynamics and the actual body dynamics, resulting in ataxia. We used both behavioural and computational approaches to demonstrate that specific cerebellar patient deficits result from biased internal models. Our results strongly support the idea that an intact cerebellum is critical for maintaining accurate internal models of dynamics. Importantly, we demonstrate how subject-specific compensation can improve movement in cerebellar patients, who are notoriously unresponsive to treatment.

Introduction

People normally make smooth and accurate reaching movements with minimal effort. This ability is thought to rely on the existence of adaptable internal models of limbs and objects in the brain. Such models relate motor commands to changes in limb/object states (i.e. position and velocity) to manipulate them in an accurate, predictable manner (Wolpert et al., 1998; Kawato, 1999; Wolpert and Flanagan, 2001; Miall et al., 2007; Chen-Harris et al., 2008). For example, an internal model of the forearm might use estimates of forearm inertia, damping and other dynamic properties to specify how forces applied at the elbow joint would move the forearm.

It is well known that damage to the cerebellum disrupts the ability to make accurate movements (Holmes, 1917; Hore and Flament, 1988; Manto et al., 1994, 1998; Moschner et al., 1994; Bastian et al., 1996, 2000; Trouillas et al., 1997; Martin et al., 2000) and adapt to different dynamics (Manto et al., 1994; Maschke et al., 2004; Smith and Shadmehr, 2005; Donchin et al., 2012). These lesion studies, in combination with imaging (Diedrichsen et al., 2005), electrophysiology (Shidara et al., 1993; Pasalar et al., 2006; Hewitt et al., 2011), and magnetic stimulation (Miall et al., 2007; Galea et al., 2011) studies suggest that the cerebellum is important for updating and/or storing internal models used in motor control.

Is cerebellar dysmetria (overshooting or undershooting targets) due to a malfunctioning internal dynamic model (Manto, 2009)? It is not understood how a damaged internal model could lead to patient-specific movement characteristics. This is, in part, because individuals with cerebellar lesions (humans and animals) are idiosyncratic in their behaviours—some tend to overshoot, whereas others undershoot, when reaching to targets (Flament and Hore, 1986; Manto et al., 1994, 1998; Topka et al., 1998; Martin et al., 2000; Manto, 2009). Further, adding loads to the arm can produce counterintuitive effects: patients systematically overshoot more with added inertia, rather than reducing their movement amplitude (Manto et al., 1994).

Here we investigated whether specific internal dynamic model biases could account for differences in patient behaviour by: (i) inducing patient-like behaviour in control subjects with unexpected dynamic perturbations to the arm; (ii) creating a computational model of reaching control that could simulate patient movements and control subject movements; and (iii) providing dynamic changes (compensation) to reduce patient dysmetria. Taken together, our findings indicate that the cerebellum is critical to upholding unbiased internal dynamic models of the arm needed for accurate movement control.

Materials and methods

Subjects

Ten patients with cerebellar deficits (age: 51.8 ± 14.9 years, eight males, eight right hand dominant) and 11 control subjects (mean age 51.5 ± 12.8 years) without any known neurological impairments participated in Experiment 1 (Table 1). The dominant hand was defined as the hand used to write and throw a ball. The 11 control subjects also participated in Experiment 2, and 9 of 10 patients participated in Experiment 3. Subjects gave informed consent according to the Declaration of Helsinki and the experimental protocols were approved by the Institutional Review Board at the Johns Hopkins University School of Medicine (originally, 13 patients were recruited, but three were not included in our analysis as they were unable to meet the task requirements, see ‘Experiment 1: Null Reaching’ section). All subjects used their dominant arm throughout the study.

Table 1.

Subject demographics

ICARS
Age Gender Dominant hand Total /100 Kinetic /52 Diagnosis
CRB01 38 F R 5 2 SCA8
CRB02 65 F R 35 18 Sporadic
CRB03 25 M R 36 19 Sporadic
CRB04 38 M L 43 19 SCA8
CRB05 47 M R 48 25 Sporadic
CRB06 58 M R 50 28 ADCA
CRB07 75 M R 55 19 SCA6
CRB08 53 M R 58 26 Sporadic
CRB09 57 M L 66 25 ADCA
CRB10 62 M R 66 30 SCA14
CRB Group 51.8 ± 14.9 F = 2 L = 2 46.2 ± 18.1 21.1 ± 8.0
CNT Group 51.5 ± 12.8 F = 3 L = 2
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ICARS = International Cooperative Ataxia Rating Scale; CRB = cerebellar subject; CNT = control subject; F = female; M = male; R = Right; L = Left; ADCA = autosomal dominant cerebellar ataxia; MSA = multiple system atrophy; SCA = spinocerebellar ataxia; sporadic = sporadic adult-onset cerebellar ataxia; Group data: mean ± SD.

Four cerebellar participants were diagnosed as having a genetically defined spinocerebellar ataxia (SCA8, n = 2; SCA6, n = 1; SCA14, n = 1). Two patients presented with symptoms of autosomal dominant cerebellar ataxia (ADCA), a pure form of inherited ataxia, and four patients were diagnosed with sporadic ataxia, a pure form of ataxia with no clear hereditary causes (details in Table 1). Diagnosis of ataxia was based on patient signs and clinical MRI scans. Detailed anatomical information about the extent of cerebellar atrophy was unavailable. The severity of ataxia was quantified with the International Cooperative Ataxia Rating Scale (ICARS) (Trouillas et al., 1997). The ICARS assigns higher scores to patients with greater impairments. The mean total ICARS of the patient group was 46.2 [standard deviation (SD) 18.1, maximum possible score = 100]. There was no evidence of white matter damage or spontaneous nystagmus among these patients. Nine of the 10 patients displayed saccadic dysmetria; only the least impaired subject, as measured by total ICARS score, did not.

Experimental setup

All of the experiments were performed using the KINARM exoskeleton robot system (BKIN Technologies Ltd.; Scott 1999). Kinematic information and robot-generated torque were sampled at 1 kHz. A schematic of the setup and the visual targets are shown in Fig. 1A and B. Experiments were subdivided into blocks (groups of trials) and subjects were instructed to maximize the number of points scored in each block. For each trial, a single point was awarded when the ‘target entry time’, the time between leaving the start circle and entering the end circle, was less than 800 ms. A second point was awarded for on-time movements that were also accurate—i.e. when subjects were able to keep the cursor inside the end target for 1500 ms. The instructions prioritized timing over accuracy to discourage patients from making overtly slow, presumably feedback-guided movements that could mask dysmetria (Topka et al., 1998).

Figure 1.

Figure 1

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Task overview and explanation of metrics. (A) Overhead view of a right-handed subject arm (measurements from left-handed individuals were reflected across the midline). Throughout all tasks, the shoulder angle (θs) was fixed at 30° flexion by mechanically clamping the robot arm. Elbow angle (θe) was measured as the angle between the upper arm and forearm in the direction of flexion. (B) Subjects saw a white dot (0.25-cm radius) over their index finger during the entire experiment but were unable to see their arm due to a metal screen. At the start of each trial, subjects held the start target (yellow, 0.4-cm radius) at an elbow angle of 75° and were instructed to make fast, accurate movements to the end target (purple, 0.4-cm radius) located 30° away (θe = 105°). Note visual targets are enlarged for illustrative purposes. (C) Position profile for an example hypermetric movement. Dysmetria was calculated as the difference between the final position and the position at the time of first correction (time of first correction: the time where the velocity or acceleration crossed thresholds of 2°/s or −2°/s2). Target entry time was computed by subtracting the time at which the fingertip first exited the start target from the time at which the fingertip first entered the end target. (D) Velocity profile for the same movement depicted in (C). Early velocity was the velocity 150 ms after movement onset (movement onset: the first time when the velocity exceeded 10°/s for five consecutive milliseconds). See also Supplementary Table 1.

A subset of control subjects (‘duration-matched controls’) were required to have a target entry time >500 ms but >800 ms to receive a point, so as to match the slower patients. Subjects received visual feedback based on the speed and accuracy of their reaching movements (Supplementary Table 1). Between trials, the robot provided torques to smoothly move the subject’s arm back to the start position. Subjects were instructed to move at any time after appearance of the end target.

Experiment 1: Null reaching

This experiment was done to examine the kinematics of single-joint movements and quantify dysmetria of patients relative to controls. Both control and patient subjects performed two blocks (40 trials each) of the elbow flexion trials described above. The robot motors did not provide any force on the arm during the null trials, but did provide force in between trials to assist subjects back to the start target. The importance of correct speed at the expense of accuracy was emphasized for subjects to avoid excessively slow movements. As mentioned above, 3 of 13 patients who were initially recruited were unable to meet the timing requirement (>50% of trials were too slow) and were excluded from further analysis. All control subjects were able to satisfy the timing requirements.

Experiment 2: Dynamic perturbations of controls

This experiment explored if changes to arm dynamic properties could induce controls to over/undershoot in a fashion similar to patients (in the null condition). Each block consisted of 40 trials, 36 of which were null trials. During the four remaining trials (termed ‘perturbation trials’), the robot provided forces to mimic changes in dynamic properties of the arm: an increase in inertia (+i of 0.039 kgm2), decrease in inertia (−i of 0.032 kgm2), increase in damping (+b of 0.30 Nms/rad), or decrease in damping (−b of 0.31 Nms/rad). Recall that inertia describes how the acceleration of an object is related to the applied force (e.g. the mechanics of swinging a baseball bat are dominated by inertia), while damping (i.e. viscosity) describes how the velocity of an object is related to the applied force (e.g. the mechanics of moving a spoon through honey are dominated by viscosity). Thus, changes in damping and inertia were rendered by producing torques based on the measured velocity and acceleration of the robot arm. We used a −i value of 0.032 kgm2 because it was the largest value that was safe. During experiment preparation, we found that larger magnitudes of −i led to instability and rapid oscillations of the robot arm, due to noise in the acceleration sensors and delays in the robot control loop. We chose the listed +i, +b, and –b-values based on preliminary experiments where these perturbation magnitudes resulted in approximately the same absolute value of changes in early velocity (∼10°/s). System identification was used to verify that the controlled damping and inertia of the robot arm were accurate (Hollerbach et al., 2008). The trials were pseudorandomly ordered such that at least five null trials occurred between two perturbation trials to mitigate effects of adapting (i.e. learning) from a previous perturbation. Subjects performed 10 blocks and thus, did each perturbation type 10 times. Only controls participated in Experiment 2, which was conducted immediately after Experiment 1.

Experiment 3: Dynamic compensation of patients

Based on control subject reaches with perturbations (Experiment 2) and modelling of patient reaches in the null condition (Experiment 1), we predicted that, for a given patient, specific dynamic changes would reduce dysmetria while other changes would exacerbate it. Subjects were given blocks of trials with altered dynamics (+i or −i) that were predicted to improve their dysmetria and early velocity. As a control, blocks of trials with dynamics that should not improve dysmetria and early velocity were also given (+b, −b). The order of blocks was randomized across subjects. Only patients participated in Experiment 3, which was conducted either immediately after Experiment 1 or during a return visit. It is also important to note that cerebellar patients typically do not adapt to abrupt changes in dynamics (Smith and Shadmehr, 2005; Criscimagna-Hemminger et al., 2010). Therefore, their average behaviour should stay constant over a block with the same bias in dynamics. As expected, there was no adaptation during blocks of 20 trials with biased dynamics.

Computational models of reaching control

Computational models of reaching were developed to describe and predict movement patterns under different conditions. All modelling was performed using MATLAB and Simulink (Mathworks, Inc.). The first computational model incorporated an internal model of inverse dynamics, body and robot dynamics, and sensory-driven feedback corrections with physiologically plausible delays. The model can explain specific characteristics of control subject and patient kinematics in the null field (no imposed virtual dynamics) and across multiple dynamic perturbations. The second computational model, which incorporated an internal model of forward dynamics and feed-forward gain (instead of an internal model of inverse dynamics), to generate motor commands (Miall et al., 1993), was developed and compared with the results of the first model (Supplementary Fig. 1). In theory, a forward model in combination with high controller gain approximates an inverse model, and thus, we anticipated both models would produce similar results as long as the correct gains were implemented. If the gain is too high, external perturbations and inherent noise can lead to instability and unwanted oscillations. More details on computational modelling are provided in the Supplementary material.

Kinematics analysis

For a given subject and trial type, movements were aligned by maximum velocity before averaging. The first 10 trials of Experiment 1 were considered the familiarization phase and were not used for analysis. Aberrant trials were discarded (typically <5%) for the following reasons: the target entry time (described below) was longer than 1500 ms, the subject returned to the start without pausing at the end target for 1500 ms, or if a trial was an outlier (>2 SD from mean velocity value).

Before averaging the kinematics, three important metrics were extracted from each movement trajectory: dysmetria, early velocity and target entry time (Fig. 1C and D). First, dysmetria, the magnitude of overshoot (hypermetria) or undershoot (hypometria), was computed as the distance in degrees between the position at the ‘time of the first correction’ and the final position achieved. The time of first correction was defined as time when the velocity or acceleration crossed predefined thresholds of 2°/s or −2°/s2 (Fig. 1C). These criteria result in control subject behaviour having dysmetria near 0° because velocity and acceleration for these subjects approach zero when they arrive at the target. Second, early velocity was defined as the velocity 150 ms after the threshold of 10°/s was surpassed for five consecutive milliseconds (Fig. 1D). Early velocity reflects the preplanned behaviour, before corrections based on sensory feedback. Third, target entry time was calculated as the time between leaving the start target and entering the end target, indicating movement duration (Fig. 1C).

Several additional metrics were extracted from the smoothed average acceleration profiles. Acceleration profiles were first averaged across trials and smoothed by computing a forward-backward five-point moving average. The moving average was computed to reduce artefacts introduced when computing the numerical second derivative of position data as measured by the sensors embedded in the KINARM. The peak acceleration, peak deceleration, and ratio between the two values (termed ‘acceleration ratio’) were computed. The time of peak acceleration and peak deceleration were also recorded.

Statistical analysis

Statistical analyses were performed using the MATLAB statistics toolbox and Stastica. To determine if dynamic perturbations caused significant changes to control subject behaviour, we performed two-sided t-tests on the movement metrics (i.e. dysmetria and early velocity). For the entire patient group, model fits were compared with t-tests for dependent samples. For the patient subgroups (hypermetric and hypometric), effects of different dynamic compensations on metrics were subjected to t-tests for dependent samples. Correlation analyses were performed by computing Pearson’s correlation coefficient and corresponding significance levels.

Results

Dysmetria and early movement velocity are correlated among patients

In the null field, healthy control subjects made smooth, accurate movements to the target with stereotypical sigmoidal-shaped position profiles (Hogan, 1984; Richardson and Flash, 2002; Fig. 2A). Nine of 10 patients, however, either displayed a tendency to overshoot (hypermetric exemplary subject, Fig. 2B) or undershoot the target (hypometric exemplary subject, Fig. 2C). Previous studies have described this behaviour after cerebellar lesions with regard to both single joint (forearm, wrist and finger) movements (Hore and Flament, 1988; Manto et al., 1994; Martin et al., 2000; Manto, 2009) and eye movements (Zee et al., 1976; Moschner et al., 1994). Interestingly, among the patients we found a significant correlation (R = −0.785, P = 0.007) between dysmetria and early velocity, which has not been previously reported (Fig. 2D). Somewhat surprisingly, overshooting patients moved slower early in the movement relative to the undershooting patients. This contrasts the simple explanation that over/undershooting may result from over/underproduction of force throughout the entire movement. Control subjects, however, did not show this relationship and exhibited little dysmetria for a wide range of early velocities. Another notable feature of patient behaviour is larger variability than controls (c.f. spread of trials in Fig. 2A with Fig. 2B and C).

Figure 2.

Figure 2

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Comparison of controls and patients during null field reaches. (A–C) Position traces of three subjects making 30° elbow flexion movements. (A) Movements of a typical control subject were smooth, accurate and exhibited low variability across trials. (B) Some patients showed a tendency to overshoot the target (hypermetric). (C) Other patients showed a tendency to undershoot the target (hypometric). Individual trials are aligned by maximum velocity and color-coded by the extent of dysmetria according to the colour bar. The average of the kinematics (black, solid) and the target (grey, dashed) are also shown. (D) Dysmetria versus early velocity for controls (normal: light blue open vertical rectangles; duration-matched: light blue filled vertical rectangles) and patients (brown symbols; horizontal rectangles, circles, diamonds, triangles). These metrics were negatively correlated among the patient group (R = −0.785, P = 0.007) but not the control group. The shaded region indicates the average dysmetria ± 1 SE (standard error) for the duration-matched controls.

Inertial perturbations of control subjects resemble patient null movements

We hypothesized that a mismatch between the brain’s internal model estimates of arm dynamics and the actual arm dynamics could cause dysmetria and perturb early velocity in a systematic fashion. To examine this further, we had control subjects perform targeted forearm movements where 90% of trials were null field trials and 10% were one of four perturbations: increased moment of inertia (henceforth referred to as inertia, +i), decreased inertia (−i), increased rotational damping (referred to as damping, +b), or decreased damping (−b). The magnitudes of the different perturbations are listed in the ‘Experiment 2: Dynamic perturbations of controls’ section. The different dynamic perturbations induced specific changes to dysmetria and early velocity. Figure 3A shows the average position traces of null reaches and perturbed reaches for a typical control subject. Effects of the different perturbations across the entire control group are shown in Fig. 3C (circles). Dysmetria and early velocity values during perturbation trials were computed relative to null averages. We found that controls with +i perturbations behaved similarly to hypermetric patients, while controls with −i perturbations resembled hypometric patients (c.f. brown symbols in Fig. 2D and blue and green circles in Fig. 3C). Specifically, the +i perturbation decreased early velocity (P < 0.001) and caused overshoot (P < 0.0005), whereas the −i perturbation increased early velocity (P < 0.00005) and caused undershoot (P < 0.005). The damping perturbations did not resemble patient behaviour: the −b perturbation increased early velocity (P < 0.00005) and caused overshoot (P < 0.005), and the +b perturbation decreased early velocity (P < 0.00005), but did not have a large effect on dysmetria (P > 0.27).

Figure 3.

Figure 3

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Irregular trajectories of control subjects are predicted by the computational model. (A) Position traces of average reaches in null and perturbation conditions for a typical control subject. Shaded regions indicate standard deviation. (B) Model output of different conditions for same subject shown in A. Thin arrows indicate time of peak acceleration and thick arrows indicate time of peak deceleration. (C) Control group (n = 11) averages of dysmetria difference and early velocity difference for reaches (round symbols) in the various conditions. Dysmetria difference and early velocity difference were computed for each subject as the average dysmetria and early velocity values of each condition minus the average dysmetria and early velocity values of the null condition. All conditions resulted in significant differences from null in both metrics (all P < 0.005) except for dysmetria difference in the increased damping (+b) condition (P > 0.27). Model output was computed for each subject and then averaged for each condition (diamond symbols). Error bars indicate standard deviation. Note the inertial perturbations (green long dash and blue short dash) have a similar pattern to the patient reaches in the null condition (Fig. 2D, brown symbols; horizontal rectangles).

Control subject movements can be predicted by a computational model across perturbations

We developed a computational model of the motor control system that can account for the major characteristics of perturbed movements (see Supplementary material for details). Briefly, the model combined a feedforward predictive motor command and delayed error-driven feedback to explain how subjects differ in behaviour for the various perturbations (Fig. 4). The model had four major components: (i) desired trajectory (goal); (ii) internal inverse model (command generator); (iii) estimated arm dynamics (plant model); and (iv) delayed position and velocity feedback (feedback model). With this model, we could reproduce movement trajectories across the various perturbations by changing elements of the estimated arm (i.e. plant) dynamics.

Figure 4.

Figure 4

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Block diagram of the computational model of reaching control. The blocks represent computations that could be done by the brain as well as physical dynamics. Lines are the signals (torques and kinematics) sent from one brain region to another or signals going to/returning from the periphery. The model consists of several computations. First, a desired trajectory (Goal) is computed with the following requirements: it must have a bell-shaped velocity profile, cause a positional change of 30°, and last a specified duration. Second, the trajectory information is sent to an internal inverse dynamic model (Command Generator) that contains the internal model estimates of inertia and damping. In the Supplementary material, there is information about how a similar computational model could be constructed with a forward dynamic model and feedforward gain as the command generator (Supplementary Fig. 1). For a given desired trajectory and internal model, the appropriate torque at each time point can be computed to match the desired trajectory. Third, the dynamic model of the arm and robot (Plant) translates the torque to changes in the kinematics (i.e. position, velocity and acceleration) of the plant. When the internal model of the command generator perfectly matches the dynamics of the plant, then the desired trajectory is perfectly achieved and there is no error. However, if the internal model values are biased and do not match the values of the plant, the plant will deviate from the desired trajectory. Fourth, sensory feedback (Sensory System) relays the actual changes in the plant, which are compared to the desired trajectory, to compute errors. These errors are multiplied by the feedback error gains (KFB) to provide on-line corrections. u = motor command; t = time step; FF = feedforward; FB = feedback; pos. = position; vel. = velocity; Δ = delay. This model framework was used for simulations shown in Figs 3 and 5, and Supplementary Figs 4 and 5. See the Supplementary material for more details.

For each subject, we used the average null trajectory to estimate the desired trajectory duration. In other words, we used the average behaviour in the null condition to estimate the self-selected movement duration. Next, the desired trajectory was supplied to the model to simulate the various perturbations. First, the model output for the −b perturbation (where plant damping was less than the command generator damping; also the condition that produced movements most different from null) was compared to the actual −b perturbation average trajectory to find the values of three feedback parameters (delay, feedback position-error gain, and feedback velocity-error gain) that best recreated the observed movements. Importantly, we were able to both accurately describe the −b condition (single subject: Fig. 3B, red trace; group: Fig. 3C, red diamond) and accurately predict the behaviour of the other three conditions, +i, −i, and +b, using the same duration and feedback parameters with different types of dynamic perturbations applied to the plant (single subject: Fig. 3B, green, blue and purple traces; group: Fig. 3C, green, blue and purple diamonds). Additionally, this modelling framework was used to simulate patient movements observed in this study and previous studies conducted by other investigators (see below).

Patient reaches can be reproduced with internal model inertial biases

As mentioned above, the patient behaviour in the null condition most closely resembled the inertial perturbations of control subjects. Specifically, the hypermetric patients (overshooters) were similar to the +i condition whereas the hypometric patients (undershooters) were similar to the −i condition, with regard to dysmetria and early velocity. In other words, one class of patients (hypermetric) appears to have an internal model bias causing an underestimate of arm inertia, which causes them to move slowly early in the movement, but eventually overshoot the target. The other class of patients (hypometric) seems to have internal model bias causing an overestimate of arm inertia, which causes them to move faster early in the movement but eventually undershoot the target. However, these only describe two specific points during the movements; it is possible that other points along the trajectories differed from the control conditions. To test whether an internal model with a biased inertia value could explain the entire movement profile of a given patient, we used the same computational model to reproduce patient behaviour in the null condition. In this case, the internal model inertia value, the duration of the desired trajectory and the three feedback parameters were all set as free parameters to find the best fit between the model output and the average null trajectory.

This model was able to fit the kinematics of both hypermetric patients and hypometric patients throughout the movement (Fig. 5A and B). In agreement with control subject perturbations, the model suggested that overshooters were using an internal model value of inertia less than the actual plant inertia, whereas undershooters had the opposite problem (Fig. 5C). Furthermore, the predicted internal model biases were highly correlated with extent of dysmetria, indicating that larger inaccuracies corresponded to greater internal model biases (R = 0.887, P = 0.0006). To compare the predictive ability of different model fits (i-fit, b-fit, i&b-fit, see Supplementary material), the root mean square error (RMSE) was computed for the time window of 900–1300 ms—a period of time that was not used for fitting. Across the group, the i-fit and i&b-fit models performed similarly (P > 0.26), whereas the b-fit was significantly worse (both P < 0.01; Fig. 5D). This suggests that an inertia mismatch alone can account for much of the observed patient behaviour.

Figure 5.

Figure 5

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Patient dysmetria is modelled as internal model inertial bias. (A and B) Comparison of average patient null movements (orange, solid) to best model fit (green, dotted) for the same (A) hypermetric patient and (B) hypometric patient shown in Fig. 2B and C. Thin arrows indicate time of peak acceleration and thick arrows indicate time of peak deceleration. (C) The internal model inertia bias determined by the computational model was highly correlated with dysmetria (R = 0.887, P = 0.0006). (D) The i&b-fit was best during the fit period, but the i-fit was comparable to the i&b-fit during the predicted period, suggesting that i bias alone can account for much of the observed behaviour. i = inertia compensation; b = damping compensation; i&b = simultaneous inertia and damping compensation. *P < 0.05, **P < 0.01. See also Supplementary Fig. 2. IM = internal model; PL = plant.

In addition, according to our model fits, patients had lower positional feedback gain (P < 0.0001) than duration-matched control subjects (i.e. patients made smaller corrections for the same amount of error), whereas other parameters were similar (Supplementary Fig. 2A–D). This is perhaps surprising because one might predict that patients would rely more heavily on the presumably intact feedback system (and thus have higher feedback gains) given poor feedforward control. A possible explanation for these findings related to stability and tremor reduction is considered in the ‘Discussion’ section and Supplementary material.

Compensation for purported patient biases reduces dysmetria

The behavioural and modelling results suggest that internal misestimates of inertia cause the idiosyncratic patient deficits. Here we asked whether this information could be used to correct patient-specific inaccuracies. As we could not alter the internal model representation in patients, we opted to change the plant dynamics in a way that matched the internal biases identified by the model. We had patients conduct multiple reaches with biased plant dynamics to the same 30° target. Note that it is well known that patients with cerebellar damage do not adapt to new dynamics (Smith and Shadmehr, 2005; Criscimagna-Hemminger et al., 2010), which was also the case here. Dysmetria and early velocity changed in a manner consistent with our hypothesis concerning internal model biases. Importantly, dysmetria was decreased by: (i) reducing plant inertia of hypermetric patients (P < 0.003; Fig. 6A); and (ii) increasing plant inertia of hypometric patients (P < 0.02; Fig. 6C). Additionally, early velocity was also modified in a predictable way (−i, P < 0.04; Fig. 6B; +i, P < 0.07; Fig. 6D) such that the patient group averages were more similar to the average value of duration-matched controls. These results further support the hypothesis that internal model biases underlie dysmetria. Our findings are inconsistent with the notion that other deficits, such as alterations in desired trajectory, fully explain patients’ movements.

Figure 6.

Figure 6

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Patient dysmetria is reduced by modifying arm inertia. (A and B) Results from hypermetric patients, individual and group data. Decreasing arm inertia (A) reduced overshoot (P < 0.003) and (B) increased early velocity (P < 0.04) of hypermetric patients. (C and D) Results from hypometric patients. Increasing arm inertia (C) reduced undershoot (P < 0.02) and (D) decreased early velocity (P < 0.07) of hypometric patients. The variability (black error bars indicate standard deviation) was similar with or without inertial compensation. Brown markers, left of the tick marks, indicate baseline performance (individual subjects and means), and blue and green markers, right of the tick marks, show performance with increased and decreased inertia perturbation (individual subjects and group means). Shaded regions indicate the duration-matched control average ± 1 standard error. SE = standard error; i = inertia; PL = plant; IM = internal model. *P < 0.05, **P < 0.01. See also Supplementary Fig. 3.

We also examined whether increasing plant damping for hypermetric patients and decreasing plant damping for hypometric patients would lead to improvements in dysmetria and early velocity. If movement abnormalities were due to over- or underdamping, then with appropriate compensation, dysmetria and early velocity should be systematically modified in a manner such that patients behave more similarly to controls. However, we did not observe this to be the case (Supplementary Fig. 3). Our simulations suggested that the damping that would improve the dysmetria would also produce an undesirable change in the early velocity. For example, increased damping (+b) for hypermetric patients would reduce overshoot but further slow the movement. By comparing results shown in Fig. 6 and Supplementary Fig. 3, it is clear that changing inertia systematically improves patient movements, whereas changing damping does not.

Simulating other characteristics of cerebellar ataxia

Although our behavioural results are primarily focused on the accuracy of rapid single-joint movements with altered limb dynamics, our theory that patients suffer from internal model biases accurately predicts several other aspects of ataxic behaviour. First, our model predicts that patients who make hypermetric movements (because of an internal model that underestimates arm inertia and cannot adapt) would overshoot a greater distance when additional mass is added to the limb (Supplementary Fig. 4A). This agrees with a previous study where hypermetria of patient wrist movements was exacerbated with increased inertial load (Manto et al., 1994). Second, the model predicts that dysmetria would be larger for faster movements than slower movements (Supplementary Fig. 4B), which agrees with previous studies of arm movements (Topka et al., 1998; Manto, 2009).

Researchers who studied natural multijoint pointing movements in cerebellar patients found that patients exhibited greater dysmetria for faster movements (i.e. movements for the same target distance but shorter duration) (Topka et al., 1998). We simulated a similar set of conditions by taking the model parameters for the exemplary hypermetric patient (Fig. 5A) and increasing the duration by 50% to simulate a slow movement, or decreasing the duration by 50% to simulate a fast movement. Our model predictions matched the observation that hypermetria increases with movement speed (Supplementary Fig. 4B).

Patients are more variable than control subjects

Endpoint variability was significantly higher for patients in both the null condition and the compensation condition as compared to controls in the null condition (both P < 0.01; Fig. 7A). Thus, although inertia compensation improved the average dysmetria, the variability of dysmetria was not reduced. This suggests that variability is not simply secondary to inaccuracy, but perhaps a more fundamental problem. We simulated how different sources of noise may lead to the variability of patient movement observed (Supplementary Fig. 5). Our simulations suggest that variability of the internal model (as opposed to variability in desired movement duration or feedback parameters) may underlie the increased variance of patient movements. In sum, our theory that patient internal models are both biased and variable can explain a wide range of behaviours in different dynamic environments.

Figure 7.

Figure 7

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Patients are more variable than controls and have persistent biases. (A) The group average of variance of dysmetria was higher for patients (P < 0.01) than controls in the null condition (Null). In addition, variance of dysmetria in the inertia compensation condition (Comp) was higher than controls (P < 0.01). See also Supplementary Fig. 5 for simulations of variability. (B) Patient dysmetria is similar within a 6–24 month period. Three patients were retested 6–24 months after their initial visit and average dysmetria did not change significantly with the passage of time. (C) Schematic of presumed internal model distributions of inertia for the three classes of subjects: controls (teal, solid), hypermetric patients (dark red, long dash) and hypometric patients (purple, short dash). Internal model estimates of patients are thought to be biased and more variable compared to controls. **P < 0.01. IM = internal model; PL = plant.

Patient biases persist over time

We were able to examine null reaches of two patients 6–12 months after their initial visit. A third patient returned to perform null reaches three times over a period of 6–24 months after the initial visit. For all three patients, the average dysmetria was very similar across the different visits (Fig. 7B). Thus, it appears that biases may not fluctuate much over a span of 1–2 years. For the patients, the interval between experiments ranged from ∼3% to 13% of the time since disease onset.

Acceleration ratios differ between dysmetria types

Examples of velocity and acceleration profiles for a typical subject are shown in Supplementary Fig. 6. The average acceleration ratio was <1 for hypermetric patients, which agrees with previously reported results related to hypermetric wrist movements of cerebellar patients (Manto et al., 1994). This ratio increased in the reduced inertia (−i) condition. For hypometric patients, the average acceleration ratio was >1, but was lower in the increased inertia (+i) condition. These results are consistent with the hypothesis that appropriate inertia compensation can help patient movements look more similar to healthy subjects. Among control subjects in the null condition, the peak acceleration was generally larger than the peak deceleration leading to acceleration ratios >1 (Supplementary Table 2). Thus, the trajectories were not exactly symmetric as assumed in the computational model. This difference may account for some the error between the actual and modelled kinematics shown in Fig. 3.

Discussion

We have shown that a fundamental deficit following cerebellar damage is the misrepresentation of arm dynamics, and that this information can be used to improve movement in a patient-specific manner. Patients who tend to overshoot are akin to control subjects who encounter a sudden increase in arm inertia: movements are initially slower and then overcorrected based on time-delayed feedback leading to overshoot. Patients who tend to undershoot are similar to controls that face a sudden decrease in arm inertia. These deficits can be systematically changed: for hypermetric patients, decreasing arm inertia reduces the extent of overshoot, and for hypometric patients, the extent of undershoot is reduced by increasing arm inertia. These findings suggest that cerebellar degeneration causes a bias in the nervous system’s representation of arm dynamics.

There is a wide body of experimental and computational research that suggests the nervous system uses internal dynamic models for skilled motor control and that the cerebellum is critical to updating and/or storing such models (Shadmehr and Mussa-Ivaldi, 1994; Wolpert et al., 1998; Kawato, 1999; Wolpert and Flanagan, 2001; Smith and Shadmehr, 2005; Bastian, 2006; Miall et al., 2007; Xu-Wilson et al., 2009; Shmuelof and Krakauer, 2011). Also, it is well known that cerebellar dysfunction causes over- or undershoot during targeted movement (Zee et al., 1976; Hore and Flament, 1988; Manto et al., 1994; Martin et al., 2000; Manto, 2009). Our work links these ideas in a specific way—we now understand what type of internal model bias (i.e. an increase or decrease in inertia) can account for different patterns of cerebellar dysmetria.

It has been suggested that the cerebellum may contain two types of internal models: inverse models that can compute the appropriate motor command given a desired change in state (e.g. position and velocity), and forward models which can predict the next state given the current state and the motor command (Wolpert et al., 1998; Kawato, 1999). For conciseness, the computational model described in the main body of this manuscript contains only an inverse model; dysmetric movements were modelled with misestimates of arm inverse dynamics. However, we believe that the misinterpretation of arm dynamics could be a problem with a forward model or inverse model or both. Thus, a second computational model containing a forward model was also examined (Supplementary Fig. 1). We cannot distinguish between these computational models based on the behaviours that we have studied here since similar movement patterns could be modelled using a biased inverse model (Fig. 3B, C and Fig. 5A–C) or a biased forward model (Supplementary Fig. 7). The purpose of the computational modelling conducted in this study was not to differentiate an inverse from a forward model, but to see if (i) an internal model (inverse or forward) mismatch could accurately reproduce the movements of different subjects; (ii) what specific parameter values best matched behaviour; and (iii) how parameter values may differ between patient and control groups. We focused on the simpler model (i.e. the model with fewest free parameters) in this study, but a forward model-based controller described in the Supplementary material is also a plausible representation of cerebellar function. Indeed, considering several contemporary studies in motor control, it is possible that the cerebellum acts as an adaptive forward model (Pasalar et al., 2006; Miall et al., 2007; Xu-Wilson et al., 2009) that informs other brain areas, such as primary or premotor cortical areas, which generate motor commands (Shadmehr and Krakauer, 2008).

Importantly, this study offers a parsimonious explanation for previous results regarding inertial loading as a method to reduce dysmetria and intention tremor (limb oscillations observed during targeted movements but not at rest). First, a pair of studies (Hewer et al., 1972; Morgan et al., 1975) showed that arm weights improved movement for only a subset of ataxic patients, and for each patient that benefited, there was an optimal amount of ‘helpful’ mass above which the movements deteriorated. This is supported by our results that hypometric cerebellar patients each have specific internal model overestimates of limb inertia, and that adding mass to the limb could bring the modified limb inertia closer to the biased internal model (thus improving movement accuracy), but adding too much mass could cause a bias in the opposite direction. Second, Manto and colleagues (1994) clearly demonstrated that, among cerebellar patients, hypermetria at the wrist increases with larger inertial loads during rapid targeted movements. This counterintuitive phenomenon is explained by our study, and we have shown that reducing limb inertia can decrease hypermetria. Furthermore, internal model biases would lead to muscle activity that is both inappropriate with regard to timing and magnitude, which is a common feature of patient reaching movements (Flament and Hore, 1986; Manto, 2009).

Why do patients have biases of varying magnitude in different directions (hypermetria versus hypometria)? One possible explanation is that these patients have non-uniform damage to the cerebellum. In cats, it has been shown that selective inactivation of the anterior interpositus deep cerebellar nucleus led to consistent hypometria, whereas inactivation of the posterior interpositus led to consistent hypermetria (Martin et al., 2000). The authors propose that the interpositus output signals are expressed through connections with rostral motor cortex based on previous findings (Martin and Ghez, 1993). Perhaps the anterior and posterior regions of the interpositus work together to modulate the command generator in motor cortex; the relative activity of the different regions could represent information regarding estimates of internal model parameters. In addition, in humans it has been shown that muscle activity and kinematics during hypometric movements can differ based on the topography of cerebellar lesions (Manto et al., 1998). These findings are in line with the notion that different lesion locations may cause different internal model biases. Another possibility is that following cerebellar damage, internal model values are fixed at a specific value and can no longer adapt to changes in physical properties of the body. Future studies involving detailed brain imaging and monitoring patient characteristics (i.e. height and weight) over time may help answer this question.

In addition to decreased accuracy, patients exhibited less precision (more variability) than controls. Increased uncertainty of internal model estimates could contribute to this variability. Patients did not show a decrease in variability following helpful dynamic changes, suggesting that variability does not arise from compensatory strategy employed by patients to deal with dysmetria, but rather, it is a fundamental symptom due to cerebellar disease. Thus, our interpretation is that, in patients, internal model estimates are both biased and more variable as compared to controls (Fig. 7C). As mentioned above, although we implemented an inverse dynamic model in the computational work presented here, it is possible that cerebellar patients have corrupt forward models instead of, or in addition to, inverse model impairments. If predictive forward dynamic models are indeed more variable following cerebellar damage, then patients may be less precise in perceptual tasks requiring action, which are thought to involve prediction of self-generated movement (Angel, 1980; Synofzik et al., 2008; Bhanpuri et al., 2012, 2013), but perform normally in perceptual tasks that are passive in nature (i.e. not involving self-generated movement) (Holmes, 1939; Dow and Moruzzi, 1958; Maschke et al., 2004; Bhanpuri et al., 2012, 2013). The reasoning behind this idea is that, during self-generated movement, the nervous system uses cerebellum-dependent forward dynamic models to make predictions of how motor commands will change the state of the arm, which are integrated with signals from peripheral sensors to enhance the final perception of arm position. During passive movement, the nervous system cannot make such predictions and thus has less information to guide perception.

Though our results indicate that patient dysmetria is well described by an internal model bias of only inertia, it is possible that damping values (and other dynamic parameters not included in our model) will also be biased. The reason inertial biases account for much of the irregular behaviour of patients is probably because inertia is the most dominant dynamic parameter of the forearm, relative to damping. In other words, for rapid point-to-point movements, inertia contributes to a much higher percentage of the overall elbow torque as compared to other dynamic terms. When both inertia and damping biases are free parameters, then model fits of patient null behaviour do not significantly improve. When only a damping bias is used for fitting, and we assume no inertia bias, then fits are worse. Thus, with regard to cerebellar dysmetria of arm movements, contributions from internal model damping biases are likely to be relatively small compared to the impact of inertial biases.

Symptoms of dysmetria are known to increase in multi-joint movements (Bastian et al., 1996). Internal biases of limb inertias would be compounded in multi-joint movements as it is possible that multiple limb inertias may be misestimated. For example, during whole arm reaching, cerebellar patients may have internal model misestimates of upper arm, forearm, and hand, which would lead to greater errors. Perhaps more importantly, multi-joint movements require proper compensation of interaction torques, which are composed of Coriolis, centripetal and inertial torques. If inertias were misestimated, then interaction torque compensation would also be erroneous and cause greater errors in multi-joint versus single-joint movements (Bastian et al., 1996). Expanding the computational model presented here to multi-joint movements may provide further insights.

Our model fit suggested that patients have lower feedback gains than controls. One possibility is that the cerebellum directly sets feedback gains and, following cerebellar damage, these gains decrease in value. Another possibility is that feedback gains are computed elsewhere in the nervous system, and the decrease in feedback gains observed in cerebellar patients is a compensatory strategy used by the nervous system when feedforward control is corrupted. Low feedback gains are consistent with control approaches for stabilizing systems with perturbed feedback or perturbed command generator computations (Ioannou and Sun, 1995). Thus, for patients who apparently have poor feedforward control, feedback gains might be reduced relative to healthy individuals to improve stability and decrease the magnitude of corrective oscillations. We examined this idea by once again starting with the model parameters that provided the best fit for the average movement of the exemplary hypermetric patient. The feedback gains were increased to the average control subject values (Kp = 6.8, Kv = 1.6) while all other parameters were unchanged. Indeed, our model predicts that if patients had feedback gains similar to controls, they would exhibit even greater intention tremor than what was observed (Supplementary Fig. 4C).

Whether the reduction in feedback gains is a direct result of cerebellar damage or a compensatory strategy to improve stability, this result is consistent with several observations that have been documented following cerebellar damage. Feedback gain is thought to be one of several components that contribute to joint impedance, others being muscle tone, reflexes, and co-contraction of agonist–antagonist pairs (Hogan, 1985). It is well documented that in acute cerebellar lesions, muscle tone can be decreased (Holmes, 1939; Blumenfeld, 2010). It has also been shown that, in certain scenarios, cerebellar patients have diminished long-latency responses to perturbations (Kurtzer et al., 2013). Recently, Gibo et al. (2013), found a deficit among patients to modify arm impedance in a perturbation-specific manner. Thus, although not the main goal of this study, the modelling results that suggest patients have abnormal feedback gains relative to controls agree with multiple studies suggesting deficits in impedance control among cerebellar patients.

In summary, a unifying theme across the behavioural and modelling studies described here and other lines of research involving cerebellar patients, is that movement and perceptual abnormalities following cerebellar damage can be attributed to malfunctioning internal models. We have demonstrated how internal model biases can cause dysmetria. In addition, we have discussed how internal model variability appears to underlie the lack of precision displayed during reaching. We also mentioned several studies of how internal model variability may disrupt different forms of active perception but not passive perception. Furthermore, we discussed how reductions in feedback gains might be a compensatory strategy by the sensorimotor system to improve stability with perturbed internal models and how modifications to feedback gain might be linked to a more general impairment in impedance control.

Perhaps most importantly, our findings have noteworthy implications for therapeutic intervention aimed at patients suffering from cerebellar dysmetria. Altering arm dynamics, perhaps with a portable compensatory device, may improve the accuracy of patients’ movements in their everyday lives. Going forward, it will be important to test whether modifying limb inertia provides consistent benefits for a wide range of natural movements. In this study, we have made significant strides to understanding the mechanism underlying cerebellar dysmetria, which is the foundation for creating appropriate devices and therapies for effective rehabilitation.

Supplementary Material

Supplementary Data
supp_137_7_1931__index.html (894B, html)

Acknowledgements

We thank the patients for volunteering their time. We thank Ed Connor and Daniel Wolpert for reviewing earlier versions of the manuscript. We thank D. Grow, T. Gibo, and S. Charles for suggestions related to robot control algorithms and computational modelling.

Funding

This work was supported by the Johns Hopkins University, the Kennedy Krieger Institute, National Institutes of Health (NIH) grants R21 NS061189 to A.M.O., R01 HD040289 to A.J.B, and NIH Fellowship F31 NS070512 to N.H.B.

Supplementary material

Supplementary material is available at Brain online.

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