Conceived and designed the experiments: MS RD. Performed the experiments: MS. Analyzed the data: MS FJVC RD. Contributed reagents/materials/analysis tools: MS. Wrote the paper: MS.
The authors have declared that no competing interests exist.
We analyzed age-related changes in motor response in a visuomotor compensatory tracking task. Subjects used a manipulandum to attempt to keep a displayed cursor at the center of a screen despite random perturbations to its location. Cross-correlation analysis of the perturbation and the subject response showed no age-related increase in latency until the onset of response to the perturbation, but substantial slowing of the response itself. Results are consistent with age-related deterioration in the ratio of signal to noise in visuomotor response. The task is such that it is tractable to use Bayesian and quadratic optimality assumptions to construct a model for behavior. This model assumes that behavior resembles an optimal controller subject to noise, and parametrizes response in terms of latency, willingness to expend effort, noise intensity, and noise bandwidth. The model is consistent with the data for all young (n = 12, age 20–30) and most elderly (n = 12, age 65–92) subjects. The model reproduces the latency result from the cross-correlation method. When presented with increased noise, the computational model reproduces the experimentally observed age-related slowing and the observed lack of increased latency. The model provides a precise way to quantitatively formulate the long-standing hypothesis that age-related slowing is an adaptation to increased noise.
In a hand-eye coordination task that requires continuous movement to correct for a disturbance, it turns out that signs of response to the disturbance appear no later in the elderly than in the young. The elderly motion is noisy and less efficient, however, and once movements in response to a disturbance begin, they are at a lower speed. One can model subject response by assuming that it results from combining noise and a response that is mathematically optimal given this noise, delay, and a least-squares sort of control objective. This modeling approach is appropriate for young and most elderly subjects. The model holds that increased noise should lead to no change in delay until response gets underway, but should make the response itself proceed at a slower speed. This is consistent with the data and with a causal link from the observed noise and disorder in elderly motor function to the observed age-related slowing.
The existence of a general phenomenon of age-related impairment of the sensorimotor system is widely accepted
The hypothesis that age-related slowing is an adaptive response to increased cortical disorder rather than a primary cause of impairment was put forward in a survey of work on aging and reaction times
Age-related sensorimotor impairment and slowing have been heavily investigated
We distinguish between two different kinds of “slowing” that are well characterized in the engineering field: (i) response latency, that is, delay until the onset of motor response, vs. (ii) slowed timescales of post-onset response. These are shown in
This figure gives an example of the distinction between slowing of the post-onset response vs. response latency. We use this distinction to analyze a more complex visuomotor response. The figure shows the response of two different systems to an impulse at time
It is important to note that our definition of endogenous noise is a more general concept than physiological motor noise. For the purposes of this paper we define endogenous noise to be any deviation from optimal behavior (possibly of sensory, motor, conduction, or neural processing origin). We address the roles of specific sources such as muscular “motor noise” or cortical “functional dysregulation”
The experimental procedure was identical to that described in
Representative time series data: All elderly subjects, trial 7. Note identical
The cursor's motion results from simple but marginally unstable dynamics:
Equation 1 states that the horizontal velocity of the cursor
The data for eleven of the young subjects appeared in
We used a computational model to improve the precision of the response latency estimates and to investigate quantitatively the slowing of the post-onset response. The structure of this continuous linear feedback control task is suited to modeling using concepts from Linear Quadratic Gaussian (LQG) optimal control
This block diagram shows the closed loop system model for the optimal control approach. Only the shaded parts are physically outside of the subject: the controlled dynamics
The standard LQG model introduced above is based on an additive noise model. It is well known that sensorimotor noise is signal-dependent, that is, its variance is a function of the amplitude of the corrupted signal
The optimal control model structure is shown in
response latency
intensity of the endogenous noise
endogenous noise bandwidth
a parameter
Once we fit parameters in the assumed modeling framework, the model is fully defined and we can predict properties of the control output
The first two results make no use of the optimal control model. The third result is a set of tests of the relevance of the optimal control model. The remaining results involve properties of the fitted optimal control models.
To determine the response latency in this task, we examined the cross correlation of the random signal driving the perturbation, and the subjects' response. The cross correlation
Application of our model-free cross-correlation method yields mean inferred response latencies
The mean cross-correlations for young and elderly show that there is no aggregate increase in response latency. This analysis is free of assumptions and makes it clear that there is a change in the post-onset response, but no change in the response latency. We removed spurious effects by subtracting
We observed increased disorder in the response of elderly subjects. Good performance in these tests corresponds to low root mean squared (RMS) cursor error
The left panel presents normalized RMS tracking error
The elderly also have a lower average RMS control input rate
Despite the use of a fitting process, there exist opportunities for the optimal control model to be tested. To be precise, the model parametrizes an infinitesimal fraction of all possible functional control strategies, and thus despite the fitting process it has opportunities to be inconsistent with either the data or with the results of accepted analysis methods. These are enumerated below.
First, a fitted model implies a certain level of RMS cursor error
The observed RMS tracking errors are consistent with those implied by the fitted model. Agreement is excellent if results are averaged across trials for each subject, as shown on the left. The amount of experimental scatter can be seen on the right. One subject-trial is off the right hand panel. Young subjects are denoted with circles unless they were poorly fit by the optimal control model (1,10), then they are shown with downward pointing triangles. Elderly subjects are denoted by squares except the poorly fit 19 and 24 denoted by upright triangles.
Second, the model would be inconsistent with the data if the fitted models did not obtain significantly different parameters for different subjects (consistent with their supposed meaning), in cases where direct inspection of the data shows that such differences exist. This is not a difficult test, as the modeling method is of course designed to capture this difference, but it is still useful to inspect the results. Summary statistics of the model parameters in
parameter | mean young | mean elderly | SD | MISSD | MITSD |
control cost/mult. noise |
−0.7257 | −0.3010 | 0.4552 | 0.1817 | 0.4446 |
latency | −0.5879 | −0.6252 | 0.1109 | 0.0268 | 0.1099 |
noise int. | −2.7556 | −2.6408 | 0.5756 | 0.4205 | 0.5729 |
noise bandwidth (BW) | 0.9495 | 0.9039 | 0.2713 | 0.2428 | 0.2668 |
BW*int./(Mean Sq. |
−1.8566 | −1.7332 | 0.3745 | 0.2607 | 0.3448 |
Statistics of inferred parameters and a derived measure of multiplicative noise intensity. Base-10 logarithms are used for normality. MISSD is median intra-subject SD, and MITSD is median intra-trial SD across trials 3–10. BW is bandwidth in radians per second of the endogenous noise filter
Third, the agreement between observed and fitted Fourier transformed auto- and cross- correlations
This shows representative spectral fits for young (left) and elderly (right) subjects. Discrete data points are circles, and lines are fits. They were selected by choosing the trials with median fit cost from each group. The top panel is the magnitude of the closed loop transfer function
Fourth, the optimal control model would be in doubt if its inferred latency values were inconsistent with the result from the cross-correlation method; this is not the case, as described in detail in the next subsection. Along similar lines, the increase in disorder visible in
The left panel shows that among a uniformly healthy young population whose multiplicative noise characteristics can be expected to be uniform, a relationship between the control cost/multiplicative noise parameter
Fifth, the optimal control model would be in doubt if more general linear system identification techniques were able to better fit the data. Even with a restriction to the set of all stabilizing linear controllers with similar state dimension, the optimal controllers represent an infinitesimal fraction of that set, due to restrictions on the LQG cost function implied by our parametrization
Sixth, the optimal control model predicts that there is a volitional degree of freedom in response, parametrized by the control cost component of the parameter
Last, we have a catch-all that if some phenomenon exists in the data that is not qualitatively consistent with the model, the model is flawed in that sense. We observe intermittent periods of stillness
Besides consistency with the data, a good model should be parsimonious, and we demonstrate this for young subjects in
Based on the above results, we conclude that the optimal control model is appropriate as a model of the behavior of all young subjects, and for the elderly except subjects 19, 24, and marginally 22. The model was consistent with the behavior of all but two young subjects during a first session, and with the behavior of all young subjects upon repeated testing
The inferred mean latencies using the model-based method for young and elderly groups are 260 ms and 247 ms, therefore we cannot attribute elderly impairment to increased response latency. The median intra-subject standard deviations are 14 ms for the young and 20 ms for the elderly, which is significantly less that the variability of the cross-correlation method applied to the same data. The difference in means is not significant: Averaging inferred latencies within each subject, grouping them into 12 young and 12 elderly, and applying the two-tailed t-test yields a p-value of 0.175. A histogram of the model-based inferred response latencies is presented in
Inferred latencies using the optimal control model, confirming the absence of increased response latency in the elderly. The young are thin black lines, the elderly are wide gray bars. This is a histogram, meaning that we compared inferred latencies to a set of evenly spaced reference values, and incremented a count of subject trials corresponding to the nearest reference value.
This is a computational result obtained by taking the models fitted to the young and increasing the noise bandwidth parameter, the noise intensity parameter, or the multiplicative noise/control cost parameter
Within our data, the behavioral changes seen in nine of the twelve elderly can be replicated by retuning of the optimal models found in the young to increase noise. This follows directly from the fact that nine of twelve elderly subjects' behavior is well fit by our model without significant changes from latency values from the young group.
We examined the normalized Hankel Singular Values (HSVs) of the optimal control models that we fit to the behavior of our subjects. We found that models fit to the elderly subjects have smaller third and fourth normalized HSVs as shown in histograms in
These histograms of normalized third and fourth Hankel Singular Values (HSVs) of inferred controllers suggest that the complexity of the behavior of the elderly is reduced (see the
The principal experimental result of this work is that age-related impairment in a dynamic compensatory tracking task takes the form of reduced efficiency of corrective motion without an increase in response latency. This is a robust result in that it can be obtained with a phenomenological cross-correlation method free of assumptions, and reproduced with greater precision by a more sophisticated model-based approach. The optimal control model presented in this paper has testable implications for behavior in this task as surveyed in the results. These indicate that it is relevant as a model of behavior for all young subjects and most elderly subjects. Therefore the model is relevant to analysis of at least the initial phases of age-related changes in sensorimotor response. We then have the computational result that if one adjusts the parameters used in the optimal control models fitted to the young to account for increased noise levels, the control strategy changes in a way consistent with slowing. Thus analysis of the computational model is consistent with the hypothesis that slowing of post-onset response is an optimal adaptation to increased endogenous noise. This hypothesis has been suggested before without formulating an explicit model
Both our empirical and model-based results are inconsistent with the generalized slowing hypothesis of aging as reviewed in
Elderly impairment is shown in
We emphasize that optimal modeling does not assume that the subjects behave perfectly. It is better understood as a modeling approach that treats behavior as the sum of the effects of an optimal controller and a random noise source, acting in a feedback loop. The series of successful predictions of the optimal model were detailed in the
Adaptation of motor control strategies with age has been reported elsewhere
We investigated the possibility that reduced complexity of elderly control strategies in this task caused reductions in response latency that offset expected aging effects, as suggested by well-documented age-complexity-latency interactions
Concepts from our modeling approach resemble those in contemporary work involving Parkinson's disease. It was found in
In conclusion, the major contribution of the optimal control model is that it quantitatively specifies the extent to which subjects should use slower control strategies to reduce the effects of noise. This is subtly different from less mathematically precise ways of arriving at this reasonable strategy. Specifically our model holds that the slowing should take the form of longer timescales during the response dynamics, and makes no prediction of longer latency until the onset of these response dynamics. This is consistent with the surprising experimental result that there is no increase in latency until onset of response in this relatively complex task.
All subjects completed consent forms approved by Cornell University's Committee on Human Subjects and brief health questionnaires.
The young subjects numbered 1 through 12 were healthy volunteers between 20 and 32 years of age. The elderly subjects numbered 13 through 24 were between 65 and 92 years of age. The manipulandum was a calibrated optical mouse modified to only record lateral hand motion, and mounted on ball bearings to reduce friction. Subjects were free to use the hand with which they felt most comfortable. Two self-reported left-handed subjects chose to use their right hands because they are accustomed to using computer mice with the right hand. This did not seem to affect the data, as the outliers were all using their right-dominant hand. There were ten 60-second trials. The first 9 seconds of data from each trial were discarded. Trials started every two minutes to prevent fatigue. Each trial consisted of a different random perturbation time history, and all subjects had the same perturbation time histories in the same order. The results of the first two trials are neglected to allow for learning effects. All young and all but one elderly subjects' behavior converged in this time, as measured by RMS cursor error and input velocity. The software sampled the control input
To assign a specific time value for response latency based on cross correlations, we looked at the peak of the second derivative of the cross correlation with respect to lag index
Consistent with today's literature, we assume quadratic optimality and Gaussian noises. We present a linear plant to be controlled. The dynamics of our task and the basic dynamics assumed to exist in the human are jointly described by the state space matrix equation:
The other sub-systems (endogenous noise prefiltering
The measurement noise
The matrix
We assume that the subject behaves so as to minimize the following quadratic cost:
Our model of the response,
Forms of LQG control are widely used in both control engineering and the modeling of sensorimotor response, as described in
Our model fitting procedure is done in the frequency domain. It is technically equivalent to using a weighted function of the discrepancy between the experimental and predicted values of the cross- correlation of
The sensitivity of the predicted closed loop transfer function
Further details can be found in
Our measure of control strategy complexity is size of the Hankel Singular Values (HSVs)
The controllability Gramian of the LQG controller in Eqns. 17, 18 is
Without altering input-output relationships from
While, to our knowledge, we are the first to use HSVs in the neurophysiological field, they have a long history of use in system and control theory, where they are the standard method to make decisions on what order of model is required to approximate the behavior of dynamical systems
Based on the preceding development, the complexity of implementing the inferred controllers may be quantified by examining the extent to which high order models are required to describe the behavior of the system - that is, whether the higher order HSVs are large. We normalized the magnitude of all HSVs by the subject's mean first HSV in order to eliminate the gross effect of altered gain, which we do not consider a form of complexity (omitting this refinement only amplifies the age differences). The dynamical significance of larger third and fourth HSVs in models fit to young subjects is that simpler low order models are less able to describe their response.
The HSVs are sensitive to all of our model parameters, but respond most strongly to multiplicative noise/control cost
Joern Diedrichsen, Gerald Loeb, Madhu Venkadesan, Lore Thaler, Maurice Smith, Evangelos Theodorou, Aldo Faisal, Emo Todorov, Rob Clewley, and anonymous reviewers made helpful comments. This article is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of Arthritis and Musculoskeletal and Skin Diseases (NIAMS), the National Institute of Neurological Diseases and Stoke (NINDS), the NIH or the NSF.