Noise, information transmission, and force variability

Journal of Experimental Psychology: Human Perception and Performance 1999, Vol. 25, No. 3, 837-851 Copyright 1999 by the American Psychological Association, Inc. 0096-1523/99/$3.00 Noise, Information Transmission, and Force Variability Andrew B. Slifkin and Karl M. Newell Pennsylvania State University This study was designed to test the hypothesis derived from information theory that increases in the variability of motor responses result from increases in perceptual-motor noise. Young adults maintained isometric force for extended periods at different levels of their maximum voluntary contraction. Force variability (SD) increased exponentially as a function of force level. However, the signal-to-noise ratio (M/SD), an index of information transmission, as well as measures of noise in both the time (approximate entropy) and frequency (power spectrum) domains, changed according to an inverted U-shaped function over the range of force levels. These findings indicate that force variability is not directly related to noise but that force output noisiness is positively correlated with the amount of information transmitted. In laboratory studies of motor behavior, individuals rapidly adjust the criterion dimension of their motor responses to levels required for goal achievement. Nevertheless, even after long periods of practice, performance continues to deviate within a certain range around the task requirement. Traditionally, such variability of movement and its outcome has been viewed as noise imposed on a deterministic signal (Fitts, 1951, 1954; Schmidt, Zelaznik, Hawkins, Frank, & Quinn, 1979). In this article we directly test the notion that enhanced force variability, following increments of force output, results from enhanced system noise. Information theory (Shannon, 1948) has had a strong influence on the modeling of perceptual-motor processes (Attneave, 1959; Broadbent, 1958; Fitts, 1954; Miller, 1956). In this view, performance variability results from a signal (the motor command) being transmitted through a noisy channel, with lower signal-to-noise ratios yielding higher levels of variability and, therefore, reduced channel capacity defined in terms of the number of bits of information processed per unit of time. In Fitts' (1954) landmark study, his index of human perceptual-motor channel capacity was explicitly derived from information theory (Shannon, 1948). The influence of the information-theoretic framework was sustained over the next 25 years, as exemplified by Schmidt et al.'s (1979) impulse variability theory, which proposed that increases in the variability of targeted motor responses resulted from the enhancement of peripheral neuromuscular noise. This view is also a central tenet of more recent theories of motor control in which reductions in movement outcome variability are achieved by a mechanism that reduces (Meyer, Abrams, Kornblum, Wright, & Smith, 1988) or damps out (van Galen & de Jong, 1995) the system's intrinsic neuromotor noise. The notion that noise and variability are equivalent has been perpetuated by studies that have had the explicit aim of identifying conditions under which actors perform optimally. Experimental conditions have often required participants to match performance levels to externally defined performance targets (e.g., Bilodeau, 1966). Thus, variation of the criterion dependent measure around a target has been viewed as a failure to comply with task requirements, as incorrect responses, or even as random errors. In this narrow context, variability may be identified with noise (task irrelevant and nonfunctional), but when viewed on a more lengthy time scale, such as is necessary in the study of exploratory behavior, variability may be found to have greater adaptive value (e.g., Staddon, 1983, pp. 73-74). For example, behavioral variability may be an expression of an exploratory process that allows the organism to pick up information about itself, the environment, and its capacity to operate within the environment (e.g., Brener, 1986; Gibson, 1988; Riccio, 1993). Another factor that has masked the distinction between variability and noise in the study of movement control is the limits of variables used to characterize response output. The predominant use of summary descriptive statistics, such as the within-subject standard deviation in force output and aiming studies, provides only a global, static image of behavior and neglects a characterization of the trial-to-trial or moment-to-moment relations of events in performance time series (cf. Newell & Corcos, 1993). For example, the standard deviation, which provides an index of the degree of deviation from a point in a distribution of scores, captures only the magnitude of fluctuations in system output. However, variations in system output can also be measured along another dimension, independently of the magnitude, that is, in the way system output changes over time: the structure of Andrew B. Slifkin and Karl M. Newell, Departments of Biobehavioral Health and Kinesiology, Pennsylvania State University. This research was supported in part by Grants F32-HD07885 and RO1-HD21212 from the National Institutes of Health. D. L. Gilden, G. P. van Galen, and N. Yamada provided feedback on an earlier version of this article. We are grateful for their thoughtful comments. Preliminary communications of this research were presented at the 26th annual meeting of the Society for Neuroscience in November 1996. Correspondence concerning this article should be addressed to Andrew B. Slifkin, 267P Recreation Building, Pennsylvania State University, University Park, Pennsylvania 16802-3903. Electronic mail may be sent to abs7@psu.edu. 837 838 SLIFKIN AND NEWELL system dynamics. We propose that measures of the structure of variability are also needed for the proper evaluation of the nature of noise in motor output variability. Such measures can be used to characterize the relations among system output values within a time series. If variability in motor output was solely the product of the intrinsic stochasticity (noisiness) of perceptual-motor processes, then one would anticipate that each event in a time series would be generated independently of the other events. In this case, the structure of the time series can be identified as being white Gaussian noise. Alternatively, analyses of the dynamics of behavior may also uncover orderly relations of events in a time series. A finding of order, and not white noise, in the time series would argue against the hypothesis derived from information theory that motor response variability arises from increased stochasticity of the processes underlying the generation of motor responses. Indeed, recent analyses indicate that fluctuations in response timing (Blackwell & Newell, 1996; Gilden, Thornton, & Mallon, 1995; Yamada, 1995a, 1995b), postural stability (Collins & DeLuca, 1993; Newell, van Emmerik, Lee, & Sprague, 1993), and errors in sequential distance estimation (e.g., Gilden et al., 1995) are not random or "errors," but rather exhibit a degree of orderliness that can be attributed to the operation of adaptive complex control systems. Our purpose in the current study was to examine how changes in the magnitude of force output variability are related to changes in the structure of force output variability and to test hypotheses that make predictions about the relationship between the variability and structure of forcetime series. This approach required an examination of the relationship between descriptive measures of variability that have usually been used alone to infer system noise and measures of system dynamics that provide actual measures of system output structure. In the experiment reported here, we required participants to maintain isometric force production over extended time periods over a broad range of force requirements. The force levels were based on percentages of maximum voluntary contraction (MVC; 5%, 15%, 25%, 35%, 45%, 55%, 65%, 75%, 85%, and 95% MVC). In discrete force-production protocols, where the matching of peak force to force targets is required, reliable increases in the within-subject standard deviation of force output as a function of increases in the force requirement have been found (Carlton & Newell, 1993; Newell, Carlton, & Hancock, 1984; Slifkin, Mitchell, & Brener, 1995). However, unlike in the majority of studies on the control of isometric responses, we required our participants to maintain isometric force output at target levels over a continuous, extended period (15 s). A primary consideration in the selection of a continuous force-production task was the fact that analyses aimed at assessing time-series structure generally require lengthy sequences of performance samples (e.g., Bassingthwaighte, Liebovitch, & West, 1994). This requirement is easier to meet in continuous than in discrete motor output tasks. Furthermore, in contrast to the literature on discrete force production, relatively few studies are available on changes in continuous force output variability as a function of increases in the force requirement (Barlow & Abbs, 1984; Caligiuri & Lohr, 1990; Loscher & Gallasch, 1993; Stephens & Taylor, 1974; Sutton & Sykes, 1967a, 1967b; Vrtunski, Alphs, & Meltzer, 1991), and this research has not examined continuous force production over a full range of levels. Instead, performance has been assessed mainly at levels of force less than 50% MVC, and in these studies, force variability has been shown to increase linearly over the range of forces examined. A goal of ours in the current study was to provide a more complete account than is available in the extant literature on force variability in continuous force-production tasks. Accordingly, we examined force variability over the full range of potential force production (5%-95% MVC). The primary measure of force variability has been the within-subject standard deviation, which indexes the average deviation of scores in the distribution from the distribution mean. However, the use of the standard deviation as the sole index of system variability seems insufficient because information theory clearly specifies that the quality of system performance, or the amount of information transmitted, is directly related to the signal-to-noise proportionality. In studies of movement control, the quantity of the information transmitted in targeted responses has been calculated on the basis of required movement-amplitude-to-target-width ratios (Fitts, 1954; Meyer et al., 1988; Schmidt et al., 1979; van Galen & de Jong, 1995), which can be considered equivalent to a signal-to-noise ratio: ID = log2(2A/W). (1) Fitts (1954) used this index of difficulty (ID) to identify the bits of information required for signal transmission, and it was based on the movement amplitude (A: the signal) requirement divided by the target width (W: the noise, or the permissible range of endpoint scatter) of targeted movements. Regardless of the absolute values in the amplitude-totarget-width ratio, under conditions in which their proportionality remains constant, the required information transmission (ID) remains the same. Therefore, according to an information-theoretic analysis of the quality of system output, a measure that takes account of the amplitude-to-target-width ratio would seem to hold the greatest promise for indexing the information content of targeted motor responses. One candidate measure is the coefficient of variation (SD/M), which has been used to index the scatter of response outcomes (effective target width) relative to mean response amplitude. This measure has commonly been used in, for instance, discrete force-production tasks (Newell et al., 1984), although, to our knowledge, the coefficient of variation has never been directly identified as a measure of information (a noise-to-signal ratio). To stay within the parlance of information theory, we use here as the primary index of information transmission the mean force divided by the standard deviation. This amplitude-to-target-width ratio can be considered analogous to a signal-to-noise ratio in targeted motor control tasks (e.g., Fitts, 1954).1 1 There have been various formulations of informational measures, and many of them are based on the logarithm of a signal-to-noise ratio. Pitts' (1954, p. 388) index of difficulty, log2(2A/W), appears to have been based on Goldman's (1953) 839 NOISE, INFORMATION, AND VARIABILITY M = 1.00, SD = 0.25, ApEn = 0.09 4 _ M = 1.00, SD = 0.25, ApEn = 1.65 2 0 -2 - M = 1.00, SD = 0.50, ApEn = 1.65 M = 1.00, SD = 0.50, ApEn = 0.09 , JM = 1.00, SD = 1.00, ApEn = 1.65 M = 1.00, SD = 1.00, ApEn = 0.09 4 ? 2- H 200 400 600 1000 0 200 Time (.01 s) 400 600 800 1000 Figure 1. The independence of measures of time-series structure and measures of performance outcome. In all time series, the mean simulated force output was 1N. However, from top to bottom in both the left (white Gaussian noise) and right (sine waves) panels, there is a fourfold increase in the standard deviation and a fourfold decrease in the signal-to-noise ratio (M/SD). Nevertheless, the approximate entropy (ApEn) measure of signal structure did not change among the time series in the left or right panels. Comparisons of white-noise and sine-wave time series with equivalent means and standard deviations reveal large differences in structure, as evidenced by visual inspection and by the approximate entropy scores. These simulated force-time series were generated with Matlab 4.2 (1994; see Footnote 2 for details). In this study, we also examine measures of the structure of force variability through analysis of the power spectrum and the approximate entropy of the force-time series. Spectral analysis decomposes a signal into component frequencies so that the power assigned to each frequency in the spectral profile provides an index of the portion of total amplitude variability that can be attributed to each frequency. Thus, the profile of the power spectrum can provide clues about the frequency domain structure of a time series. For example, Equation 29, which identified the amount of information transmitted as log2(P/N). In Goldman's Equation 29, as well as in other informational measures (Shannon, 1948), P is the signal power and N represents white thermal noise. (In Pitts' [1954] adaptation, he multiplied A by 2 in order to minimize the possibility of the ED taking on a zero or negative value [when AAV £ 1]). In turn, Goldman (1953) based his equation on Shannon's (1948) Theorem 17, log2(P + N/N). When the ID is based on Pitts' adaptation of Goldman's Equation 29 or on an index closer to that originally proposed by Shannon, log2(A + WAV), both are highly predictive of the time taken to execute movements (movement time; MacKenzie, 1989). In fact, the index based on Shannon's Theorem 17 provides a slightly better description of the increases in movement time that accompany increases in the ID (see Table 1 in MacKenzie, 1989). white noise has equal power at all frequencies of the power spectrum (e.g., Schroeder, 1990), whereas the highly periodic structure of a pure sine wave yields a spectral profile with all power at a single frequency. Unlike the power spectrum, approximate entropy returns a single value based on the time-domain analysis (event-to-event relations) of the structure of the time series (Pincus, 1991; Pincus & Goldberger, 1994) and has been used to capture the complexity of signals produced by biological systems (e.g., Lipsitz, 1995; Lipsitz & Goldberger, 1992). Approximate entropy provides an index of the predictability of the value of future events in a time series based on past time-series events. The more random the signal output, the more information required to specify given future values, and this results in greater approximate entropy values (see Figure 1 and the Appendix for more details). By examining force output structure (approximate entropy and the power spectrum) in relation to the magnitude of variability (the within-subject standard deviation) and information transmission (signal-to-noise ratio), we tested two contrasting hypotheses. The first hypothesis is that the magnitude of variation in force output is positively correlated with increases in the noisiness of the structure of force output (e.g., Fitts, 1954; Schmidt et al., 1979). In this case, 840 SLIFKIN AND NEWELL increases in the within-subject standard deviation, induced by increases in the force requirement, should be positively correlated with approximate entropy and a broadening of the power spectrum. This follows from the expectation that increased signal complexity or noisiness reflects a greater influence of stochastic processes on signal output, which, in turn, is related to an increase in the magnitude of system output fluctuations (e.g., Schmidt et al., 1979). In this view, increments of signal noise are reflected in increments of both the magnitude and noisiness of the structure of force variability. The second hypothesis is that an inverse relation exists between the magnitude and noisiness of force variability. Recent studies have shown that decreases in system degrees of freedom through aging (e.g., Lipsitz, 1995; Lipsitz & Goldberger, 1992) and neuropathy (e.g., Newell, Gao, & Sprague, 1995) produce more periodic behavior, as indexed by lowered complexity in the output of the system under consideration. Thus, this hypothesis would predict that increases in force variability should be accompanied by decreases in approximate entropy and more peaked spectral profiles. A related prediction is that as information transmission in matching force output to the force requirements increases (the signal-to-noise ratio increases), approximate entropy should increase and profiles of the power spectrum should broaden. Thus, this hypothesis predicts that increases in the information content of force production (signal-tonoise) should be related to increases in motor output noisiness. In summary, the main purpose of this study was to provide a direct test of the hypothesis that increases in the magnitude offeree variability are related to increases in the noisiness of force output structure. Although the exact form of the force variability function has yet to be established in continuous force-production tasks, we anticipate that the magnitude of force variability (viz., the within-subject standard deviation) will increase with increments in the force requirement. Furthermore, examinations of the relations between measures of the magnitude and structure of the variability in force output will provide a stronger test of prevailing notions of the projected relationships between noise, information transmission, and variability of force output.2 Method Participants Nine healthy young adults from a university population served as participants. The group's mean age was 19.56 years (SD = 2.07). Four participants were women, and all participants identified themselves as right-hand dominant. They responded to advertisements for volunteers or requests for volunteers made in university classrooms. All participants gave informed consent to the experimental procedures, which were approved by the local institutional review board. Apparatus Participants were seated with their right arms resting on a flat desktop (75 cm high) that provided support for the forearm and hand. The position of the hand and forearm was adjusted so that the pad of the distal interphalangeal segment of the index finger covered the load cell (Entran ELF-500 [Entran Devices, Inc., Fairfield, NJ]; diameter = 1.27 cm) that transduced force production. The Entran ELF-500 load cell can transduce oscillations of frequencies over a range from 0 to 1000 Hz. Although no apparatus was used for physical restraint, during experimental trials participants were instructed to sit with their backs straight against the back of a chair and to keep their elbows, forearms, wrists, palms, and fingers flat against the surface of the desktop on which the load cell was mounted. To limit force production to the musculature controlling index finger flexion, we monitored compliance with these instructions during the familiarization trials (see below), and participants were reminded of these instructions during the experiment. Force applied to the load cell resulted in changes of the electrical resistance of strain gauges housed in the load cell. These resistance changes, directly related to force applied to the load cell, were amplified by a Coulbourn strain gauge amplifier (Model S72-25; Coulbourn Instruments, Allentown, PA). Amplifier output was sampled continuously at 100 Hz by a 16-bit analog-to-digital converter controlled by a 486/66 MHz microcomputer. Force was measured in units of 0.0018 N (0.18 g). A 14-in. (35.56-cm) computer monitor was placed on a tabletop so that its center was 100 cm high at a distance 85 cm from the center of participants' eyes. During experimental trials, the force target was presented as a horizontal red line spanning the width of the screen and set against a black background. The height of the line on the video monitor was proportional to the force requirement. During each trial, the force-time trajectory produced by participants was fed to a video monitor to provide online performance information. Force-time trajectories appeared as yellow lines on the screen unfolding over the duration of the trial. For each newton change in force, there was an approximately 0.65-cm excursion of the force-time trajectory on the screen. The force-time series of each trial was saved to a computer file for subsequent filtering and data analyses. 2 The testing of the hypotheses posed here depends on analysis of the relationship between measures of signal structure (noise) and measures of performance outcome extracted from the frequency distribution of force output scores: the standard deviation (variability) and signal-to-noise ratio (information). It should be noted that any relationship between measurements of signal structure (e.g., approximate entropy and the power spectrum exponents) and their associated distributional properties does not arise from mathematical dependence. This is illustrated in the computer-simulated time series of Figure 1. In the left panels are three separate generations of white Gaussian noise produced by the RANDN fimction of Matlab 4.2 (1994). In the right panels are three sine-wave generations. The means in all time series are identical, but from top to bottom in both the left and right panels there are a fourfold increase and a fourfold decrease, respectively, in the standard deviation and signal-to-noise ratio (M/SD). However, even given these large changes in measures of performance outcome, the structure of signal output, as measured by approximate entropy, remains the same. Comparisons of the left (white Gaussian noise) and right (sine wave) panels illustrate that when the standard deviation and signal-to-noise ratio are constant, signal output structure may greatly differ. In this case, structural differences are apparent from visual inspection and are also reflected by differences in approximate entropy. Thus, these simulations show that measures of signal structure vary independently of distributional measures of performance outcome (also see Lipsitz & Goldberger, 1992; Pincus & Goldberger, 1994, p. H1648). 841 NOISE, INFORMATION, AND VARIABILITY Procedure Assessment of MVC. In the first phase of each experimental session, the MVC of the dominant (right) index finger was assessed. Participants were asked to press as hard as they could on the load cell for a duration of 6 s on each of three trials, with force output sampled at 50 Hz during the MVC trials. Instructions to participants emphasized that they should allow sufficient time between trials to allow for dissipation of fatigue. During the MVC trials, as in all subsequent trials of the experiment, room lights were extinguished in order to reduce any possible influence of visual stimuli other than that provided by the target display. We programmed the computer to store the 10 highest force values achieved from each of the three MVC trials. The computer then returned an average of the 30 stored values when the third trial was complete. This average became the MVC value for each participant, and during all subsequent experimental trials it was used to calculate force target levels based on percentages of the MVC. General experimental instructions. Four levels of instruction were provided regarding compliance with the task goal of minimizing deviations of force from the target. First, following the MVC trials, participants were provided with instructions that the goal of the task was to match the force they produced to the force target displayed on the computer monitor. Second, after each trial, a knowledge-of-results summary reflecting the participants' accuracy in matching the force they produced to the force requirement was displayed on the computer monitor. The knowledge of results was the root mean square error (RMSE), and it indexed the average deviation of the force produced from the prevailing force target. The RMSE differs from the standard deviation only in that the latter is calculated by taking the difference between each sample (100 samples per second of the trial) in the distribution and the distribution's mean. The RMSE was based on the final 10 s of the 15-s trial: We omitted the first 5 s from the RMSE calculation to ensure that participants had adequate time to adjust their force output to the force requirement. Participants were asked to keep their RMSE scores as low as possible, with the best score, although virtually impossible to achieve, being 0. Third, to further engage participants in monitoring their performance, we required them to record the RMSE scores following each consecutive trial on a score sheet designed for recording the RMSE feedback. Fourth, on the basis of the recorded RMSE scores, when each block of five trials was complete, the experimenter provided a brief, verbal evaluation of performance during that block. Familiarization and experimental trials. During the familiarization trials, force was produced during a single trial at each of the 10 force levels used during the subsequent testing of force production. The familiarization trials were presented in random order for each participant. During the familiarization procedure, the experimenter observed that participants maintained the same posture during and between trials and understood the goal of the task. During the experiment, participants completed a block of five consecutive trials at each of the 10 force requirements. The order of presentation of the force requirements was randomized for each participant. A 5-10-min break was provided between the fifth and sixth blocks of trials. Data analyses. For purposes of economy in the presentation of results, and because our current interests are not related to assessing changes in performance as a function of practice, we have omitted the analyses of results related to changes in performance across trials. Overall, there were small changes in the absolute level of the dependent measures over trials, but the form of the functions describing changes in the dependent variables with increases in the force requirement were preserved at each of the five trials. Therefore, although the analysis of variance results reported here were based on a two-way Force Requirement (10) X Trials (5) design, only results related to the force requirement effect are reported. When results are identified as significant, it means there was less than a 5% chance of a Type I error (p < .05). Depending on which provided the better characterization, either first-, second-, or third-order polynomial regression was used to characterize changes in the dependent variable as a function of the force requirement (e.g., Hamilton, 1992, pp. 151-153). The polynomial regression equation for each dependent variable is reported in Table 1. We used Pearson product-moment correlation coefficients to identify the degree of association between the group mean data of selected dependent variables and a two-tailed t test to assess the reliability of these relationships (Minium & Clarke, 1982, pp. 297-299). The first 5 s of each force-time series was omitted from all analyses. The subsequent portion of the trial was conditioned with a ninth-order Butterworth filter having a 30-Hz lowpass cutoff. The descriptive statistics submitted to analyses were the mean force, the standard deviation, and the signal-to-noise ratio (mean force/ standard deviation of force). The structure of force output was analyzed in two ways: Approximate entropy was used as an index of force output noisiness in the time domain (see Pincus, 1991; Pincus & Goldberger, 1994; see the Appendix for more details), Table 1 Regression Equations Describing Changes in Dependent Variables as a Function of Force Level Polynomial regression equations Dependent variables Mean force (% MVC) SD (in newtons) Signal-to-noise ratio Approximate entropy Power function exponents Peak Power (in newtons2) Proportion of peak power 3.216652 0.107556 11.300761 0.490708 -1.738766 16.721134 0.268518 0.847112 -0.003519 2.290296 0.004951 0.005684 -1.661905 -0.001454 0.000254 -0.049213 -0.000054 -0.000084 0.030501 0.000018 — 0.000278 — — — — .991** .988** .912** .859** .854** .828** .721* Note. Polynomial regression equations describe changes in the group means of dependent variables as a function of force requirement. Dashes indicate that the second- or third-order term of the2 equation was not included in the polynomial regression. The significance levels associated with the r values were based on two-tailed t tests applied to the corresponding correlation coefficients (r). MVC = maximum voluntary contraction. *p<.01. **p<.001. 842 SLIFKIN AND NEWELL and spectral analysis was used to evaluate the profile of the frequency domain3 (e.g., Lipsitz, 1995). The power spectrum of each trial was divided into 30 equal bins, ranging from 0 to 11.72 Hz.4 The power in each frequency bin represented the portion of total power in the overall amplitude of force output oscillation that could be attributed to the frequencies specified by that bin. Changes in the magnitude of spectral power as a function of force requirement were estimated by identifying the frequency bin in which power reached a maximum value. This allowed for assessments of changes in power at the peak (peak power) as a function of force level. Peak power and the standard deviation are related: Increases in the amplitude of signal oscillation will result in concurrent increases in the standard deviation and an upward rescaling of the spectral profile. In the simulations presented in Figure 1, increases in the amplitude of either the sine-wave (right panels) or white-noise (left panels) time series resulted in increases in the standard deviation. The height, and therefore peak power, of the spectral profiles would increase too. On the other hand, measures of signal structure extracted from the spectral profiles—the proportion of peak power and the power spectrum exponents—vary independently of changes in the amplitude (magnitude) of signal oscillations. First, for each power spectrum, the peak power was divided by the total power in the spectrum. This provided a measure of the proportion of power at the modal frequency (proportion of peak power). Decreases in the proportion of peak power were taken as reflections of broadening spectral power and therefore of increases in signal noisiness. For the white-noise signals in the left panels of Figure 1, even given the large differences in signal amplitude among the panels (because in each case power would be evenly distributed across all frequencies of the spectrum), the proportion of peak power would remain constant. Second, a power function was used as a more global descriptor of changes in power as a function of frequency within each spectrum: (2) In this equation / is frequency, P is the predicted power, a is the y-intercept of the equation, and |3 identifies the slope or rate of change of spectral power as a function of frequency within a spectral profile (e.g., Gilden et al., 1995; Lipsitz, 1995). As the power spectrum becomes more broadband, and therefore closer to white noise, the absolute value of the power function exponents declines toward zero. In spite of the large differences in signal amplitude among the white-noise signals depicted in the left panels of Figure 1, the even distribution of power across frequency would result in all three power function exponents taking on a value of zero. In summary, like the standard deviation and signal-to-noise ratio, peak power can also be considered a measure of performance outcome. In contrast, approximate entropy, the proportion of peak power, and the power spectrum exponents are measures of signal structure that may vary independently of changes in the amplitude (magnitude) of system output oscillations. Results Force-Time Series Figure 2 provides an illustration of force-time series taken from an individual participant at the 5%, 35%, 65%, and 95% MVC requirements. In each panel of Figure 2 it is possible to identify a slow, relatively periodic oscillation of higher amplitude and superimposed oscillations of higher frequencies. The periodicity of the higher amplitude component of the rhythm is most evident at 65% MVC but can also be seen at 5% and 95% MVC. Such periodicity appears to be minimized at 35% MVC. The structure of these force-time series clearly differs from that of white Gaussian noise (cf. Figure 1, left panels). Distributional Properties of Force Production Figure 3 illustrates the mean force produced, the withinsubject standard deviation, and the signal-to-noise ratio (mean force / standard deviation) as a function of increases in the force requirement. Each data point represents an average across all participants and trials at each force requirement. The group mean MVC was 31.07 N (5£> = 9.34 N). As can be seen in the top panel of Figure 3, the mean force output increased significantly over the force range, .F(9,72) = 65.21, p < .001, and closely tracks increases in the force requirement. Furthermore, with each increment in the force requirement there was nearly an equivalent unit increase in force produced, with the y-intercept slightly exceeding zero (see Table 1). The reduction of the slope from unity reflected a slight but positive constant error (deviations of mean force from the force requirement) at lower force requirements, with a projection of zero constant error at 21% MVC (~ 6.58 N). With further increases in the force requirement, there were systematic, albeit small, increases in the degree to which mean force underestimated the force requirement. Nevertheless, overall, participants complied well with the instruction to scale force to the force requirement. 3 The power spectrum was computed with standard algorithms, activated by the SPECTRUM command in Matlab 4.2 (1994). The program initially fits a first-order polynomial regression equation to the time series and returns the residuals from the regression line. The resultant time series has a mean and a slope of zero. This detrending procedure removes any global nonstationarity but preserves the event-to-event structure of the time series. Then, using the Welch method of power spectrum estimation, the program divides each detrended force-time series into sections of 256 data points. The successive sections are Hanning windowed, submitted to fast Fourier transform, and accumulated. Results other than those taken from the spectral analyses were based on nondetrended time series: A comparison of functions describing changes of these dependent variables as a function of force requirement revealed that their forms were essentially the same before and after detrending. 4 When the frequencies included in the power spectra were extended to 30 Hz (77 bins), the location of the lowpass filter, the total power in the 5%, 35%, 65%, and 95% MVC power spectra (averaged across all participants and trials at each force requirement) increased by only 1.85%, 3.42%, 1.42%, and 0.97%, respectively, compared with spectra in which only the first 30 bins (up to 11.72 Hz) were examined. Therefore, the reported spectral analyses were based on the frequency range in which the majority of power was located (Bins 1-30). 843 NOISE, INFORMATION, AND VARIABILITY minimize deviations of force output from the required force levels. As anticipated on the basis of previous studies of discrete and continuous force variability (Newell et al., 1984; Slifkin et al., 1995; Stephens & Taylor, 1974; Sutton & Sykes, 1967b), the magnitude of variability in force output increased over the range of forces. However, unlike the results from the extant research on discrete force-production tasks, the trend for force variability in the continuous task increases exponentially over the range of force levels, not in a negatively accelerating fashion (e.g., Carlton & Newell, 1993). As illustrated in the bottom panel of Figure 3, the signal-to-noise ratio (mean force / standard deviation) increased sharply until the 25% MVC requirement (7.77 N), where it took on a value of 42.70, and then decreased systematically with increases in force level beyond 35% 100 £ 2 80- S 60 40 20 —i— 20 40 60 80 100 60 80 100 40 400 600 Time (.01 s) 1000 Figure 2. Examples of force-time series for continuous force production at force requirements based on 5%, 35%, 65%, and 95% of the maximum voluntary contraction (MVC). These examples were taken from the last force-production trial under the relevant force requirement. All four trials are taken from a single participant. Force output was sampled at 100 Hz or once every .01 s. The last 10 s of each of the 15-s trials are shown here. Force was measured in newtons. 0 40 45 o '•a 40 a: <B <n 'o From the middle panel of Figure 3 it may be observed that the within-subject standard deviation increased according to an exponential growth function. These increases were significant, F(9,72) = 19.39, p < .001, and were well described by a second-order polynomial regression equation (see Table 1). The y-intercept from the equation was close to zero. Thus, the predicted standard deviation of force at a force requirement of zero was near zero. Relative to the mean, there were only small increases in the standard deviation over the range of forces. At 95% MVC (29.52 N), where the standard deviation was at a maximum, it took on an absolute value of only 2.1 N, and this, relative to the mean force produced at that level, represented a range of variation of only 8.49%. Thus, overall, participants complied well with the instruction to 20 CO 35 30 25 - c en 20 35 15 0 20 40 60 80 100 Force Requirement (%MVC) Figure 3. Changes in mean force (top), standard deviation (middle), and signal-to-noise ratio (bottom) as a function of force requirement. Force requirement and mean force produced are expressed as percentages of the maximum voluntary contraction (% MVC). The standard deviation was measured in newtons, and the signal-to-noise ratio (mean force [N] / the standard deviation [N]) is a unitless index. Each data point represents a group mean based on an average across five trials at each force requirement. 844 SLIFKIN AND NEWELL MVC. Thus, over the force range, the signal-to-noise ratio changed according to an inverted U-shaped function that was well described by a third-order polynomial regression equation (see Table 1). These changes were highly reliable, F(9,72) = 11.15, p < .001. It is notable that the value of this informational index, which is the quotient of mean force and the standard deviation, reaches a maximum not at the lowest force requirement or the level requiring the "least effort," but rather at the 25% and 35% MVC requirements. From these points, information transmission is reduced as the force requirement approaches either the minimum or maximum force requirements. The Structure of Force Output As shown in Figure 4, approximate entropy increased systematically from the initial force requirement until reaching a projected maximum at about 40% MVC, indicating a point of maximum noisiness in the time-domain structure of force output. With further increases in force level there were systematic declines in approximate entropy. These changes in the time-domain structure of force output were reliable, F(9, 72) = 2.19, p < .05, and like the signal-to-noise ratio, could be described as an inverted U-shaped function (see Table 1). Thus, it can be concluded that changes in the time-domain structure of force output noisiness were not directly related to increases in force variability (the standard deviation; cf. Figure 3, middle panel, and Figure 4) but changed in parallel with the information transmitted (the signal-to-noise ratio; cf. Figure 3, bottom panel, and Figure 4) to achieve the task goal. The top panel of Figure 6 illustrates changes in peak power as a function of force requirement (also see Figure 5, left panel). The peak power increased exponentially over the 0.64 -, 0.62- 0.60 I 0.58 - I 0.56 "(0 £ 0.54 - | 0.52 0.50 0.48 0.46 - 20 40 60 80 100 Force Requirement (% MVC) Figure 4. Changes in approximate entropy as a function of force requirement (% MVC = percentage of maximum voluntary contraction). Approximate entropy is a time-domain measure of signal output structure. Increasing approximate entropy values reflect increases in signal complexity or noisiness. Each data point represents a group mean based on an average across the five trials at each force requirement. Details on the calculation of approximate entropy are provided in the Appendix. range offeree requirements, and these reliable changes, F(9, 72) = 5.83, p < .001, were well described by a second-order polynomial regression formula (see Table 1). Despite the large changes in peak power (see Figure 6, top panel), the frequency at which the peak power was located remained low (Af= 1.24 Hz) and changed little (SD = 0.35 Hz) across the force range. Increases in both the standard deviation (see Figure 3, middle panel) and peak power (see Figure 6, top panel) followed similar trends over the range of forces, and a correlation based on their group means was high, r(8) = .91, p < .001. Although these indices are calculated with the use of different methods, they both capture the magnitude of variability in force output. To provide a measure of spread of power in the power spectrum, we divided the peak power by the total power in the power spectrum. This provided a measure of the proportional contribution of the dominant frequency component to power in the overall oscillations in force output (the proportion of peak power). As illustrated in the bottom panel of Figure 6, the proportion of peak power decreased from the initial force level to the 35% MVC requirement (with a. projected minimum at about 40% MVC) and increased thereafter. These significant, F(9, 72) = 4.30, p < .001, changes were U-shaped and could best be described by a second-order polynomial regression formula (see Table 1). Because the frequency at which the peak power occurred remained essentially the same across the force range and was located at low frequencies, shifts in the proportion of peak power indicate that power was redistributed to higher frequencies up until about 40% MVC, at which point it returned to lower frequencies with further increases in force. The decreased concentration of power at the dominant frequency at the force midrange implies increased noisiness in the frequency structure of force output. This point is illustrated in the middle panel of Figure 5, where relative to the other depicted power spectra (5%, 65%, and 95% MVC), the proportion of peak power is the least, whereas the proportion of power at higher frequencies (>3 Hz) has the greatest elevation at the 35% MVC requirement. To provide a more global index of changes in the distribution of power, we fit power functions to the power spectra (Equation 2). Figure 7 illustrates changes in the exponents of power functions that were fit to the power spectra. The closer the exponents were to a value of zero, the greater the spread of power across the spectrum and, in particular, to the higher frequencies. The value of this variable became systematically less negative from the initial requirement (5% MVC) until the 35% MVC requirement, where it was closest to zero. Thereafter, the exponents became increasingly negative. The right panel of Figure 5 provides an illustration of power spectra averaged both across participants and across trials at the 5%, 35%, 65%, and 95% MVC force levels. This figure clearly shows that power is most broadly distributed at the 35% MVC requirement. The increased spread of power at this requirement is related to the observation (see Figure 5, right panel) that the smaller 6-8-Hz peak, apparent in each spectra, is most NOISE, INFORMATION, AND VARIABILITY Frequency (Hz) Frequency (Hz) 845 Hz) Figure 5. Three forms of the power spectrum: changes in absolute power (left panel) and proportional power (middle panel) as a function of spectral frequency and logic power as a function of logjo frequency (right panel). In each panel, power spectra from equally spaced force requirements (5%, 35%, 65%, and 95% maximum voluntary contraction [MVC]) are displayed. In the left panel, where power is measured in newtons2, there are large increases in power over the range of force requirements such that the spectra at 35% and, especially, 5% of the MVC are overshadowed. In the middle panel, by dividing the power in each frequency bin by the total power in the respective power spectrum, we rescaled (normalized) the spectra from their absolute values, which are depicted in the left panel. In the right panel, the data from the left panel are transformed by taking logic of frequency on the x-axis and logm of power on the y-axis. The slope of the first-order polynomial regression fit to the function describing changes in the Iog10 power as a function of the logic frequency, for each force requirement, was taken as the exponent of the power function. Identifying the exponent of the power function in this way is equivalent to obtaining it by using Equation 2 in the text. The exponent of the power function provided a global measure of the distribution of energy in each power spectrum (see text for details). There were 30 frequency bins, each with a bandwidth of 0.39 Hz. Power was measured in newtons2. The power in each frequency bin represents a group mean based on an average across the five trials and 10 participants at each force requirement. pronounced at the 35% MVC requirement. Across the range of forces examined, changes in the power function exponents (see Figure 7) over the range of forces were significant, F(9, 72) = 2.48, p < .05. The group mean data for approximate entropy and the power spectrum exponents were highly correlated, r(8) = .83, p < .01. This points to parallel changes in the frequency- and time-domain structures of force output over the range of permissible force levels. The results indicate that over the range of required force levels, the frequency- and time-domain structures of force output do not change in the same direction as do changes in the magnitude of variability. Namely, the standard deviation increased exponentially (see Figure 3, middle panel) while approximate entropy and the power function exponents changed according to inverted U-shaped functions over the range of forces. Thus, support is not provided for the first hypothesis (see the Introduction) that the magnitude and structure offeree output variability are positively correlated. This hypothesis was derived from studies on force variability which state that increases in the standard deviation are related to increased noisiness in the processing of taskrelated perceptual-motor information (Schmidt et al., 1979). Indeed, the correlations between the 10 group means for the standard deviation and the power function exponents, r(8) = -.87, p < .01, and for the standard deviation and approximate entropy, r(8) = -.55, p > .05, were negative, not positive. This provides support for the second hypothesis that the quality of performance outcome and the noisiness of system output increase together (e.g., Lipsitz, 1995). This hypothesis receives even greater support from examinations of the correlations between the structure of force output in the time domain (approximate entropy) and information (signal-to-noise ratio), r(8) = .88, p < .001, and between 846 SLIFKIN AND NEWELL 180 160 140 120 100 80 60 40 I 20 0 0 55 °- 20 40 60 80 100 force requirements. A better characterization is that changes in information transmission (signal-to-noise ratio) parallel changes in the noisiness of force output. Information transmission was directly related to the degree of noisiness in force production, with increases in the noisiness of force output structure being strongly related to increases, not decreases, in information transmission. The findings of the current study can be addressed in relation to several important theoretical issues on noise, information transmission, and force variability. 30 The Force Variability Function £ 0.28 - S 0.26 1 0.24 -| o | 0.22 -I 9 a- 0.20 0 20 40 60 80 100 Force Requirement (%MVC) Figure 6. Changes in power spectrum descriptors as a function of force requirement (% MVC = percentage of maximum voluntary contraction). Peak power (top panel, in newtons2) was the power in the frequency band with the maximum power, and the proportion of peak power (bottom panel) was calculated by dividing the peak power (N2) by the total spectral power (N2). The latter was a unitless index. Each data point represents a group mean based on an average across the five trials at each force requirement. structure in the frequency domain (power spectrum exponents) and information, r(8) = .83, p < .01. A relatively large literature has focused on examinations of the relationship between force output level and force variability. In most of these experiments, participants were required to match the peaks of brief impulses of force to particular force target levels (Newell et al., 1984; Carlton & Newell, 1993; Slifkin et al., 1995). The general finding has been that peak force variability increases as a negatively accelerating function of required force (e.g., Newell et al., 1984): Exponents from power functions describing changes in the within-subject standard deviation as a function of force fall below 1 (Carlton & Newell, 1993). A similar negatively accelerating relationship between required force level and the standard deviation of force was projected for continuous force production by van Galen and de Jong (1995) in a computer simulation model of changes in motor-unit recruitment with force level. However, at that time the prior empirical research on continuous force production had not examined the full range of potential force output to reveal the veridical force variability function (Barlow & Abbs, 1984; Loscher & Gallasch, 1993; Stephens & Taylor, 1974; Sutton & Sykes, 1967a, 1967b; Vrtunski et Discussion The main purpose of the current study was to examine the relationship between the magnitude and structure of force variability over the potential range of force production within a given effector action. In particular, we were interested in testing the long-held hypothesis, based on information-theoretic accounts of human performance (Fitts, 1951,1954; Shannon, 1948), that increases in force variability are related to increases in the noisiness of the perceptualmotor processes involved in force production. As anticipated on the basis of previous studies (Carlton & Newell, 1993; Slifkin et al., 1995), increasing the required level of force output was effective in inducing increments in the magnitude of variability (the standard deviation) over the range of forces. There were also reliable changes in the structure of the force-time signal in both the frequency and time domains. Namely, the noisiness in the structure of force output could be characterized as changing according to an inverted U-shaped function. These data on continuous isometric force production show that the concepts of variability and noise cannot be treated synonymously in force control because changes in the magnitude and structure of force variability did not change according to the same functions over the range of -1.5 -1.8 - -1.9 - -2.0 20 40 60 80 100 Force Requirement (%MVC) Figure 7. Changes in the distribution of power in the power spectrum as a function offeree requirement (% MVC = percentage of maximum voluntary contraction). These changes were indexed by taking exponents from power functions describing changes in power as a function of frequency within the power spectrum. This procedure provided a global measure of the distribution of power in the power spectrum (see text for details). Each data point represents a group mean based on an average of the exponents over five trials at each force requirement. NOISE, INFORMATION, AND VARIABILITY al., 1991). Also, in the few cases where performance had been examined over a reasonably broad range of forces, there were only a small number of participants in each study and the individual participant analysis presented in each study indicated linear relationships between the force requirement and standard deviation (Stephens & Taylor, 1974; Sutton & Sykes, 1967b). In the current experiment, in which force production was examined over essentially the full range of force output, the standard deviation of force increased exponentially. Why are there different variability functions for the discrete and continuous force-production protocols? There are two main distinctions between the discrete and continuous force-production protocols that may be responsible for the engagement of different force-production strategies, which in turn may yield different force variability functions. First, in the discrete force-production tasks, instructions often constrain both the peak force and the time to peak force of the impulses so that the time to peak force remains essentially constant and of short duration over the range of peak forces. For example, over essentially the full range of force requirements, force variability functions were examined with concurrent time-to-peak-force requirements of either 100, 200, or 400 ms (Carlton & Newell, 1985). The brevity of these responses was necessary because these studies were designed to test assumptions of the prevailing impulse variability theories (Schmidt et al., 1979). The responses were essentially ballistic impulses and, therefore, largely controlled by open-loop control processes. In contrast, and by definition, the durations of contractions in the current study and in the other continuous force-production studies were many times longer, which increased the likelihood of the engagement of closed-loop control in the maintenance of a given target force level. Second, in the discrete force-production studies, the protocol for obtaining the MVC required participants to press against a force-transducing manipulandum with as much force as possible and with the same time-to-peak-force requirement imposed during the subsequent experiment. Because it takes from 2 to 4 s for the absolute maximum force to develop (Kamen, 1983), the MVC estimates in the discrete force-production paradigm were necessarily less than the absolute MVC that can be produced by the effector action. Instead, MVC estimates were scaled to the time provided for force development, so briefer time-to-peakforce requirements yielded lower MVCs (Newell & Carlton, 1985). In the current continuous force-production protocol, estimates of the maximum force were generated during 6-s trials, which would allow for an estimation of the absolute MVC. Therefore, the range of force levels examined under the discrete paradigm was necessarily somewhat less than that examined under the continuous force-production paradigm, where force levels were based on percentages of the absolute MVC. It is also likely that differences in these task protocols engaged different neural mechanisms in force control, which in turn were responsible for differences between the force variability functions in the discrete and continuous force-production paradigms. 847 In the discrete force-production paradigm, the time provided for force to rise remained constant over the different peak force requirements. Thus, increases in the rate of rise of force (dF/dt) accompanied increases in required force output. To augment the rate of rise of force requires an increase in the number of active motor units (an alpha motor neuron and the muscle fibers it innervates) per unit of time. This follows because the force produced by simultaneously recruited muscle fibers sums, which results in an increase in the rate of rise of total force output. The recruitment of motor units follows the size principle (Henneman, 1957, 1979) in which fatigue-resistant motor units generating small contractions are recruited prior to the quickly fatiguing motor units that generate larger contractions. In fact, in some muscles, the most powerful motor units may be 100 times stronger than the weakest ones (Kernall, 1983). Thus, as the required force increases and more motor units are activated, the grain of precision in achieving a specific level of force becomes more coarse: Recruiting a given number of motor units beyond that necessary to match force output to the force requirement will result in a more substantial deviation from the target at high force requirements than will a situation in which the same number of additional motor units is recruited at lower force levels. Thus, the increase in variability over the range of forces in discrete forceproduction tasks can be accounted for by rules (viz., the size principle) that specify changes in motor-unit recruitment with increases in force level. In the discrete force-production task, the negatively accelerating force variability function appears to emerge from a corresponding negatively accelerating increase in the number of active units accompanying increases in force: Grillner and Udo (1971) found that 90% of the motor units in the available pool are recruited at 50% MVC. Thus, with only 10% of the motor units available beyond this force level, there are fewer ways in which the system can produce force and, therefore, deviate from the force requirement. This would explain the reliable observation in discrete force-production tasks that there are systematic increases in force variability up until 65% MVC, with sharp decreases in the rate at which variability increases with further increases in force output (Newell et al., 1984). In other words, as force output increases toward the maximum force level specified by the summated force of the total motor-unit pool, the increasing proximity offeree output to this ceiling results in a decrease in the range over which force can vary. In contrast, in the current study there was a positively accelerating, exponential increase in the magnitude of force variability over the force range. In continuous force production, at levels up to 3Q%-40% of the absolute MVC, force is increased by adding motor units, and beyond this level, further increases in force are achieved by increases in the discharge rate of the active motor units (Kamen, Sison, Du, & Patten, 1995). The force variability function from the current experiment appears to reflect the operation of these two neurophysiologically mediated rules for scaling force to increasing force requirements. Indeed, consistent with the operation of these rules, it is possible to describe the form of the current force variability function according to two 848 SLIFKIN AND NEWELL separate linear regression equations. For example, from the initial force requirement to about 45% MVC (see Figure 3, middle panel), the slope describing increases in force variability with force requirement is relatively shallow, and beyond 45% MVC, force variability increases more sharply. This finding suggests that the initial increases in force variability (to about 45% MVC) result from corresponding increases in the number, and consequently the size, of recruited motor units (the size principle). This provides a coarser grain for approximating the force requirement as force output increases (Kamen et al., 1995). Thereafter, when all motor units in the pool have been activated, the sharper increases in force may be seen as a product of increases in the discharge frequency of the motor units (Kamen et al., 1995). In addition, because the size principle specifies a fixed order of recruitment and derecruitment, maintaining force at higher target levels requires the sustained activity of the faster fatiguing, larger motor units. The faster fatigability of the higher tension motor units means that at higher forces, there are more punctuated fluctuations in force output as a consequence of their more intermittent, phasic activity. Thus, these factors, in addition to the engagement of discharge modulation beyond 30%-40% MVC, may be responsible for the sharper increases in force variability observed in the experiment reported here than in the discrete protocols. The Structure of Force Variability The structure of force variability changed systematically as a function of force level in the current experiment. This effect was revealed through analyses of both the frequency (spectral analysis) and time (approximate entropy) domains of the continuous force output. The analyses of the structure of force output variability showed that noisiness changes according to an inverted U-shaped function over the force range. The analyses of the frequency profile of force output revealed that most power was located at low frequencies with a mode of about 1 Hz. The shape of the spectral profile was generally retained but was upwardly rescaled with increases in the force requirement, and within spectra, power sharply declined with increases in frequency. These spectral profile features are highly consistent with those identified in prior continuous force-production studies (Allum, Dietz, & Freund, 1978, Figure 1; Stephens & Taylor, 1974, Figures 3 and 4; Sutton, 1957, Figure 6; Sutton & Sykes, 1967a, Figures 4-7). Thus, given the general retention of the narrow, peaked form of the power spectrum over levels of force, it can be concluded that there was a high degree of periodicity in force output. Nevertheless, there were increases in the contribution of higher frequency components from the initial force requirements up until about the 40% MVC requirement, and thereafter there was a systematic trend for the relative power to shift back from high to low frequencies (e.g., see Figure 5, middle and right panels). This trend was quantified in the present experiment through an examination of changes in the proportional contribution of power at the modal fre- quency to the overall power (proportion of peak power) and by the more global descriptor of shifts in spectral power, the exponent relating changes in spectral power to increase in frequency. First, the proportion of peak power declined across the initial force requirements, reaching a minimum at the 35% MVC requirement and increasing thereafter to give rise to an U-shaped relationship. Second, an inverted U-shaped function described changes in power function exponents across the range of forces, with the maximum broadbandedness occurring at a level just below the force midrange. The analyses of both the proportion of peak power and the power spectrum exponents demonstrated increased noisiness of system output from the initial force requirement until just below the midrange, with noisiness decreasing with further increases in the force requirement. The results of the time-domain analyses of the structure of the force-time profile also revealed a high degree of regularity in force output because approximate entropy (Pincus, 1991) values were generally low. Thus, relatively low amounts of information were required to specify the value of future samples in the force-time series on the basis . of the value of prior samples (see the Appendix). However, there were small but systematic changes in the regularity of force output, as reflected by changes in the value of approximate entropy over the force range. Namely, the noisiness of force output increased from low force levels until just below the midrange of the force requirements, and then it decreased again over the upper half of the force range. These findings show that noisiness in the time domain of force output is an inverted U-shaped function over the force range and is paralleled by observations made from analyses in the frequency domain. A high correlation between group mean data from the power function exponent and approximate entropy trends was identified. Why should the complexity of force output change as an inverted U-shaped function over the range of force levels? The evidence converges on the fact that there is a region of force production where information transfer related to targeted force production is optimized, and this in turn is related to maximized noisiness in the structure of force output. The minimum and maximum target forces can be viewed as boundaries that constrain the dynamics of force output as the task demands approach these extreme force requirements. The boundaries act to constrain the available "degrees of freedom" for the flexible assembly of solutions to the task requirements. In the current situation, the units or degrees of freedom whose assembly is constrained are the motor units. Up to 3Q%-40% MVC, force is increased by increasing the number of active motor units, but once all units are recruited, the generation of higher force levels depends on modulation of discharge frequency (Kamen et al., 1995). Thus, at about 30%-40% MVC, the constraints imposed by the extreme boundaries are relaxed, and force may be adjusted by using either or both neural control strategies, whereas above or below this region, only a single strategy may be engaged. When the force requirement coincides with this region, performance should be optimized. Indeed, this is the region in the current experiment where information transmission was maximized and where NOISE, INFORMATION, AND VARIABILITY the maximum noisiness in force production was found. Increases in the noisiness of the structure of system output may be related to an increase in the number of components or processes contributing to system output. The finding that the point of optimal information transmission was related to the point of maximum noisiness in the structure of force output is consonant with the research in a number of different domains that has shown that system adaptiveness increases as its output becomes more noisy in structure. First, noisiness (complexity) in the structure of sequences of interbeat intervals (the time between heartbeats) declines with age (Lipsitz, 1995; Lipsitz & Goldberger, 1992) and other forms of cardiac pathology (Bassingthwaighte, Leibovitch, & West, 1994, chap. 13; Pincus & Goldberger, 1994). The increased complexity in the interbeat interval time series of young, healthy individuals may emerge from the modulation of cardiac activity by multiple subsystems acting over different time scales, from seconds (sinoatrial, baroreflex, and chemoreflex activity) to minutes and hours (e.g., endocrine processes) and even days (e.g., circadian rhythms) and beyond (e.g., seasonal rhythms). These processes and their possible interactions converge and determine the time-series dynamics of the beating heart, which in the young has been characterized by broadband interbeat-interval power spectra and high approximate entropy (e.g., Lipsitz, 1995). Second, research on individuals with neuromuscular disease (e.g., Parkinson's disease, essential tremor) provides complementary evidence that system degeneration of known neural inputs to the final common pathway results in reduced complexity in postural control (e.g., Freund & Hefter, 1993). Third, in studies by van Galen and his colleagues (van Galen & Schomaker, 1992; van Galen, van Doom, & Schomaker, 1990), kinematic profiles of brief movements generated under conditions of easy compared with difficult task requirements revealed a narrowing of spectral profiles. Spectral profiles from continuous force output (bilabial lip closure) were also found to be more narrow for those with a speech disorder (stuttering) than for individuals without such a disorder (Grosjean, van Galen, de Jong, van Lieshout, & Hulstijn, 1997). In turn, the more narrow power spectra of the stutterers was associated with attenuated precision in the matching of force to the force targets. Thus, by the definition adopted in the present article, the data of van Galen and colleagues show that performance improves as system output becomes more noisy. In the current experiment we have demonstrated that information transmission in goal-directed continuous force production is at a maximum when force output is most noisy and that this occurs at a point just below the midrange of maximum force production. Thus, greater noisiness in force output is associated with an improvement in information transmitted in task performance. This finding contrasts with the traditional viewpoint in which variability has been interpreted as the reflection of noise in the system. Furthermore, the presence of noise has been viewed as detrimental to the realization of task requirements, so that adaptive motor control requires the minimization or elimination of noise (Grossman & Goodeve, 1963/1983; Fitts, 1954; Schmidt et al., 1979; van Galen & de Jong, 1995). 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Step 1 Two input parameters, m and r, need to be specified prior to the calculation of approximate entropy. The value of r is multiplied by the standard deviation (SD) of the time series and, as is discussed in more detail, provides tolerance limits for assessing the nearness of adjacent data points in the time series; m represents a vector length comprising m consecutive data points. In the calculations presented in the present article and in prior examples (e.g., Bassingthwaighte et al., 1994, pp. 309-312), r = 0.2 and m = 2. Step 2 The calculation of approximate entropy begins with the identification of the first vector in the time series, [w(l), «(2)], which comprises the first and second data points. This vector is termed the template vector. The calculation of approximate entropy entails a determination of the similarity of other vectors in the time series to the template vector. Beginning with the first data point, «(1), limits are established around it that extend across the time series. The limits are determined by multiplying r by the SD. If, for example, r = 0.2 and the SD = 0.10, a value of ±0.02 around the first data point is established. These limits are extended over the entire length of the time series. Then, according to the same procedure, limits are established for the second element, «(2). Step 3 The next step is to identify all other vectors of adjacent m length points, x(i) = [u(i), u(i + 1)], in the time series that are "componentwise close" to the template vector, [«(!), «(2)]. The first element, w(i), in all other vectors, x(0, is close to w(l) if the element falls within the limits around «(1). The second element, u(i + 1), of each vector is close if it falls within the limits around u(2). Thus, all vectors that are componentwise close to [«(!), «(2)] will have their first element within the boundaries of «(1) and their second element within the boundaries of u(2). These are called the conditioning vectors. The number of conditioning vectors becomes the denominator of a ratio that is the central calculation for approximate entropy. Another way of thinking about this procedure is that it identifies other vectors that are similar to the template vector, and the value of r determines the threshold or criterion for stating that other vectors are similar. For instance, decreasing the value of r decreases the range of the limits around template vector elements and would result in fewer vectors qualifying as conditioning vectors. Step 4 The next element of the template vector is identified, which in this example is u(3). According to the procedure in Step 2, limits are constructed around it as previously described for u(\) and u(2). Then the conditioning vectors with u(i + 2) values that are close to H(3)—within the limits around «(3)—are identified. The initial conditioning vectors, \(i), that were close to the template vector, [u(l), «(2)], and that still have u(i + 2) values close to u(3) of the template vector become the numerator of the ratio. StepS The ratio forms a conditional probability of the likelihood that runs of two adjacent data points in the conditioning vectors that were close to the first two values of the template vector will remain close to the template vector on the next incremental comparison—to «(3). In other words, given the number of runs of adjacent values that are a similar distance away (A), what is the likelihood that those runs will still be similar when the number of values have been incremented to three (B)? The natural logarithm of the conditional probability, A/B, is then calculated. Thus, compared with a highly periodic process (e.g., a sine wave), for white Gaussian noise there will be a low probability of finding conditioning vectors in the time series that are componentwise close to the first two elements in the template vector and that will still be close at the next incremental comparison. Step 6 If, as in the case of the current study, there are 1,000 data points in a time series, then there are N — m + 1, or 999, successive vectors of length m in the time series. Each of these vectors serves as a template vector for the process described in Steps 2 through 5. Therefore, the process is iterated 999 times. Then, the natural logarithms of the individual conditional probabilities are averaged, and the negative of this value is taken (to ensure a positive approximate entropy value). For example, the individual conditional probabilities potentially range from positive values near 0 to 1. In the case of white noise, most of the conditional probabilities should approach 0, whereas for a sine wave they will be near 1. The natural logarithm of. 1 is — 2.30, and the natural logarithm of 1 is 0. Therefore, when the negative of such values is taken, it results in increasing, positive approximate entropy values as the complexity or noisiness in the time series increases. (In recent simulations we found that the range over which approximate entropy values vary depends on the number of data points in the time series. For example, in Figure 1 the white noise simulations of 1,000 data points had approximate entropy values of 1.65. However, a white noise simulation extended to 10,000 data points yields an approximate entropy value of 2.20.) Received April 25, 1997 Revision received November 13, 1997 Accepted May 21, 1998