Dynamical signatures of isometric force control as a function of age, expertise, and task constraints

Stochastic and deterministic dynamical components contribute to force production. Dynamical signatures differ between force maintenance and cyclic force modulation tasks but hardly between age and expertise groups. Differences in both stochastic and deterministic components are associated with group differences in behavioral variability, and observed behavioral variability is more strongly task dependent than person dependent.

Abstract

From the conceptual and methodological framework of the dynamical systems approach, force control results from complex interactions of various subsystems yielding observable behavioral fluctuations, which comprise both deterministic (predictable) and stochastic (noise-like) dynamical components. Here, we investigated these components contributing to the observed variability in force control in groups of participants differing in age and expertise level. To this aim, young (18–25 yr) as well as late middle-aged (55−65 yr) novices and experts (precision mechanics) performed a force maintenance and a force modulation task. Results showed that whereas the amplitude of force variability did not differ across groups in the maintenance tasks, in the modulation task it was higher for late middle-aged novices than for experts and higher for both these groups than for young participants. Within both tasks and for all groups, stochastic fluctuations were lowest where the deterministic influence was smallest. However, although all groups showed similar dynamics underlying force control in the maintenance task, a group effect was found for deterministic and stochastic fluctuations in the modulation task. The latter findings imply that both components were involved in the observed group differences in the variability of force fluctuations in the modulation task. These findings suggest that between groups the general characteristics of the dynamics do not differ in either task and that force control is more affected by age than by expertise. However, expertise seems to counteract some of the age effects.

NEW & NOTEWORTHY Stochastic and deterministic dynamical components contribute to force production. Dynamical signatures differ between force maintenance and cyclic force modulation tasks but hardly between age and expertise groups. Differences in both stochastic and deterministic components are associated with group differences in behavioral variability, and observed behavioral variability is more strongly task dependent than person dependent.

force control is fundamental for many daily activities, for example, self-care skills and job routines. It is the product of the temporary coalition of multiple subsystems (neural, cognitive, muscular, etc.) as well as of the integration of different sensory feedback loops acting on different time scales (Hong, Lee, and Newell 2007; Vaillancourt and Newell 2003). These complex interactions are expressed in the produced force that contains fluctuations with a certain magnitude and time structure (Vieluf, et al. 2015b). These fluctuations are greatly affected by task constraints, such as target profile and force level, as the interactions of subsystems and feedback loops can be differentially structured depending on the tasks (Slifkin and Newell, 1999; Temprado et al. 2015; Vieluf et al., 2015b). Furthermore, the fluctuations can change due to organismal constraints (e.g., age and expertise) that may affect the individual subsystems as well as the subsystems’ interactions and, therefore, contribute to force control (see Morrison and Newell 2012 for an overview on aging; and Vieluf et al. 2012 for an experimental study on expertise). Presently, the bulk of research on force control has been devoted to study its statistical properties, including the structure of the force fluctuations (Slifkin and Newell 1998; Slifkin and Newell 1999; Vieluf et al. 2015b). In contrast, to our best knowledge, only one study (Frank, et al. 2006) has aimed to identify the dynamics, describing the underlying components contributing to variability in the produced force over time. These dynamics provide a phenomenological description of the system at hand and thus a better understanding of the system at that level of description. Therefore, here we investigate the dynamics underlying force control with respect to task constraints as well as organismal or internal constraints. Specifically, we assessed the dynamics associated with two force production tasks, force maintenance and cyclic force modulation, and how they change as a function of age and expertise.

The present study was conducted according to the conceptual and methodological framework of the dynamical systems approach. Central to this approach is the search for generic phenomenological laws, typically cast in terms of low-dimensional differential equations (see Kelso 1995 and references therein). The corresponding dynamics, describing the attractor structures, are investigated in terms of system stability and loss thereof. In that regard, an attractor is a dynamical structure to which the system invariantly evolves from and (thus) returns to when driven away from it by perturbations or noise. Well-known stable attractors are fixed points and limit cycles. A fixed (or equilibrium) point denotes a position in the system’s state space where the rate of change is zero. Stable fixed points thus describe systems that, when not at the fixed point, evolve toward it and remain at the corresponding value of the relevant variable (unless perturbed). In terms of a force control task, this would be expressed as the force level that is approached from all other force levels and that is maintained for a period of time. Limit cycles are orbits in the phase space, and thus require at least two state variables spanning the state space, and describe oscillatory behavior. Oscillatory behavior, however, can also be described by systems that contain one state variable only (Strogatz 1994), and such reduced systems are sometimes used (as we do here) to capture limit cycle behavior (Huys et al. 2010). In terms of a force control task, this would mean that for a force modulation task with a sinusoidal pattern, the dynamics can be captured in relation to the target force profile for examples by the Hilbert phase and would result in a stable line, as the rate of change is constant. Of particular interest for our current purposes, applied to force control tasks, is that this approach posits that the complex neuro-musculo-skeletal system can be formally conceived as a stochastic dynamical system (Wilmer et al. 2007). Such systems contain deterministic and stochastic dynamical components that interact with each other (van Mourik et al. 2006) to give rise to a highly organized and adaptable behavior. The deterministic component determines the behavioral solutions; this is the dynamical component that contains the attractor structures alluded to above, such as, for instance, whether the system will converge to a fixed point, and provides information about the attractor strength (Frank et al. 2006) or whether the system exhibits sustained oscillations. The stochastic component represents random fluctuations or dynamical noise in the system, which causes the system to fluctuate around its attractor structure. Typically, in the studies on force control, the structure and contribution of these two sources of behavioral variability have widely been neglected. However, their estimation provides valuable information about the underlying dynamics (Stepp and Frank 2009) and thus knowledge about the studied system; in the present context, this is the system underlying force control.

To our knowledge, only very few studies have explicitly investigated the dynamics associated with force control (Danion and Jirsa 2010; Frank et al. 2006); only Frank et al. (2006) studied both the deterministic and stochastic components. Specifically, these authors investigated the contribution of the deterministic and stochastic components in an isometric force control task where the participants were asked to maintain a relative force level [10, 20, 40, 60, and 70% of their individual maximum voluntary contraction force (MVC)] for 15 s. Specifically, in this force maintenance task, which was shown to be governed by stable fixed point dynamics, behavioral fluctuations increased with increasing force level. The authors assumed that this was due to a combination of a weaker deterministic component and higher noise. Regardless, both the deterministic and the stochastic components determined the observed behavioral variability. Frank et al. (2006) pointed out that the dynamical components were scaled as a function of constraints related to the task context (magnitude of the produced force required) as well as the participants (i.e., MVC). Consequently, decomposing the behavior into its deterministic and stochastic components allows us to 1) uncover whether different task constraints entail different dynamic behaviors and 2) test whether organismal differences are expressed in one and/or the other component.

Classically, two task paradigms are considered relevant to study force control, namely the isometric force maintenance task (maintaining a given constant force level over time) and the cyclic force modulation task (producing a periodic time-varying force level). Each of these tasks imposes specific constraints on the participants. However, it remains an open question whether they effectively result from similar or distinct generating mechanisms. Because the force maintenance task implies the production of a stationary force with fluctuations around that mean, it is assumed that it is generated by a stable, linear, fixed-point dynamic (Frank et al. 2006). In contrast, the cyclic task requires a time-varying force generation and thus exhibits fluctuations around a periodically varying required force, typically a sine wave. Such behavior can in theory be generated in various ways, including: 1) a harmonic oscillator, 2) (nonlinear) limit cycle dynamics, 3) two stable fixed points (corresponding to the minimal and maximal force) separated by an unstable fixed point, and 4) a stable fixed point driven by an external sinusoidal driving force. In any case, a single (linearly) stable fixed point cannot account for sinusoidal force production. Therefore, we expected different dynamical signatures to be revealed when studying the deterministic and stochastic components in the two tasks. Hence, we contend that the findings of Frank et al. (2006) on force maintenance cannot simply be extrapolated to describe cyclic force modulation or even generalized to different populations.

In that latter regard, numerous studies have indicated that various features of force production change with age as well as with training and expertise (Diermayr et al. 2011; Keogh et al. 2010; Keogh et al. 2007; Morrison and Newell 2012; Vieluf et al. 2012). As a main characteristic of age-related differences, variability of the produced force has been shown to increase for both force maintenance and time-varying force modulation (Vaillancourt and Newell 2002), although age effects were reported to be more prominent in the latter than in the former (Hu and Newell 2010; Keogh et al. 2006). Aging appears to render the fluctuations in force output more regularly in the maintenance task but less regularly in the cyclic task (Vaillancourt and Newell 2002). Furthermore, older adults have been shown to be less able to adapt to task constraints (Vaillancourt and Newell 2002). One factor that may slow down age-related deterioration, at least in specific domains, is the long-lasting engagement in domain-specific activities (Ericsson and Smith 1991; Horton et al. 2008). For instance, long-term practice in precision mechanics labor was shown to improve the performance in a force maintenance task at low force levels and to reduce the amplitude of force fluctuations (Vieluf et al. 2012). Furthermore, age-related differences were also shown to be less pronounced in these experts than in novices (Vieluf et al. 2012). However, it is unknown how expertise-related organismal changes are expressed in the dynamics underlying force control. In fact, the observed increased variability in older populations is commonly interpreted as the signature of increased neural noisiness (Liwt al. 2004). The approach followed by Frank et al. (2006), however, exemplifies that increased variability may be grounded in changes in the deterministic as well as stochastic dynamical component. Indeed, next to identifying the dynamics associated with the different force production tasks, we also aim to explore the dynamical source of the increased variability observed in older populations.

Furthermore, among the age- and expertise-related organismal characteristics, tactile sensitivity (Cole 1991) and MVC (Sosnoff and Newell, 2006) may also contribute to differences of force control. Hand afferent signals, for instance, are used to adapt and maintain forces (Johansson and Westling 1987; Westling and Johansson 1984; Westling and Johansson 1987). Reduced hand sensitivity alters grip forces; mostly forces higher than necessary are applied to allow for a larger safety margin during grasping. This was shown for older compared with young adults (Cole 1991), as well as for people with diseased or damaged skin (Brand 1973; Brink and Mackel 1987) and anesthetized people (Johansson and Westling, 1984). Sosnoff and Newell (2006) showed that the effect of the individuals’ MVC on force variability was more robust than the age effect. Whether and how the two components relate to the dynamics underlying force control are yet unknown and were tested in this study.

Overall, in the present study, we aimed to identify the deterministic and the stochastic components in force maintenance and in the cyclic force modulation task. Furthermore, we aimed to investigate their contribution, if any, to observed changes under different organismal factors (age, expertise, MVC, and tactile sensitivity) by comparing young and late middle-aged novices as well as late middle-aged experts. In terms of the dynamics, we expected to observe for the force maintenance and the cyclical task signatures of fixed-point and oscillatory mechanisms, respectively. For group comparisons, we assumed that alterations in the dynamical signatures underlying force control would be already visible in the late middle-aged (Lindberget al. 2009; Vieluf et al. 2012; Vieluf et al. 2013) through more stochasticity and a weaker deterministic component. In contrast, we expected that middle-aged experts, despite their relatively advanced age, would remain closer to young novices compared with the older novices as a result of their continuous, deliberate use of the hands in daily working routines.

MATERIALS AND METHODS

Participants

Thirty-six healthy adults took part voluntarily in the experiment. All participants were right-hand dominant as determined by the Edinburgh Handedness Inventory (Oldfield 1971), and all reported having normal or corrected-to-normal vision. Participants were recruited by flyers, telephone calls, and newspaper announcements. They were compensated €8/h. The protocol was approved by the ethics committee of the German Psychological Society and was in agreement with the Declaration of Helsinki. Informed, written consent was obtained from all participants. The data were collected as a part of the Bremen-Hand-Study@Jacobs (Voelcker-Rehage et al. 2013). None of the participants had hobbies involving a high degree of manual dexterity (i.e., needlework, playing a musical instrument, or fine mechanical tasks). Furthermore, none of them reported any neurological disorders. All participants were given a demographic and health questionnaire to obtain information about characteristics of the sample. Selected relevant characteristics of the groups are reported in Table 1.

Table 1.

Group characteristics

	Means ± SD			Statistics
	YN	LMN	LME	Df	F	P	ηp2	YN-LMN	YN-LME	LMN-LME
Education, yr	13 ± 1.38	15 ± 3.46	16 ± 3.05	2,33	2.87	0.07	0.156		†
Handedness (%tasks performed with right hand)	97.92 ± 3.75	99.33 ± 2.42	93.75 ± 10.75	2,33	2.23	0.12	0.996
Subjective hand usage at job	16.25 ± 6.15	14.33 ± 6.17	30.75 ± 4.86	2,33	29.13	<0.01	0.932		*	*
MVC right	53.56 ± 16.33	59.90 ± 24.95	57.90 ± 20.06	2,33	0.29	0.75	0.017
Tactile threshold	75.86 ± 41.84	226.13 ± 251.10	144.88 ± 66.72	2,33	34.60	0.07	0.151	†
Physical activity	7.70 ± 1.23	7.78 ± 1.79	7.59 ± 1.79	2,33	0.04	0.75	0.017

YN, young novices; LMN, late middle-aged novices; LME, late middle-aged experts; MVC, maximum voluntary contraction force.

^*Significant post hoc and

^†marginally significant post hoc, result tested via Bonferroni corrected pairwise comparison following up an ANOVA.

Based on their age and their occupational field, participants were assigned to three subgroups: young novices (YN: 12, 20–26 yr, mean age 23.33 ± 1.92 yr, 8 females), late middle-aged novices (LMN: 12, 57–67 yr, mean age 60.91 ± 3.02 yr, 7 females), and late middle-aged experts (LME: 12, 57–67 yr, mean age 60.50 ± 3.00 yr, 8 females). The groups of young and old novices were formed by service employees, i.e., consultants, office clerks, insurance agents, and vocational trainees in these occupations. The group of experts included precision mechanics who manipulate small objects in a highly dexterous way as part of their daily work routines, i.e., opticians, goldsmiths, watchmakers, hearing care professionals (Reuteret al. 2012; Trautmannet al. 2011; Vieluf et al. 2012). Based on the definition of Ericsson and Smith (1991), experts were included only when they had ≥10 yr of work experience in the specific field. To verify the expertise, we used a questionnaire that assessed the frequency of hand use at work (see Table 1), which showed that our experts had a significantly higher frequency of dexterous hand use than the novices (P < 0.001).

Experimental Setup

A force transducer (Mini-40 Model; ATI Industrial Automation, Garner, NC) was affixed to the experimental table so that the participant could comfortably grasp it while being seated with the arms placed on arm rests. Participants were instructed to apply forces on the force transducer using a precision grip with their index finger and thumb only while the other fingers build a fist. The right thumb was placed on the force transducer. The arm position was neutral so that the index finger and thumb could grasp the force transducer that was affixed orthogonal to the table. The grip force was recorded with an amplitude resolution of 0.06 N and a sampling rate of 120 Hz. An online low-pass filter with a cutoff frequency of 200 Hz was applied. A customized LabView (National Instruments, Austin, TX) program was used to collect force data and provide on-screen visual feedback to the participants. The target force level and the actual grip force produced by the subjects were displayed in light green and yellow, respectively, with line thickness of 1 mm (see Fig. 1A for a black and white illustration), over a black background on a 19-inch monitor with a 60-Hz frame rate. The screen was placed at ~80 cm in front of the participants, resulting in a visual angle of ∼45° for the whole screen and 38° for the relevant area, showing the force curves.

Fig. 1.

Exemplary performance and binning. A–D: representative force-time curves from 1 representative participant (target line, gray lines; applied force, black lines) over the whole trial and zoomed into the part analyzed with exemplary binning for force maintenance (A and B) and cyclic force modulation, with highlighted inflection point (C and D) tasks.

Task and Procedure

Touch detection threshold was measured in a separate session before the data acquisition of the force task. The threshold was defined by the use of 18 von-Frey filaments (custom-made, calibrated filaments) representing a force range on a logarithmic scale from 0.177 to 63.743 mN. The tactile threshold was determined by use of the two-down, one-up procedure with six points of return (Leek 2001; see Reuter et al. 2012 for a detailed description of the procedure).

We first measured the MVC of the right hand with the index finger opposing the thumb. The MVC was determined in three maximum precision grip trials of 5 s each. Participants were given ≥2 min of rest between each maximal effort. The applied force was averaged for the last 3 s of each trial, and the highest value among the three trials was considered the MVC. No differences between YN (mean = 53.59 ± 16.33 N), LMN (mean = 59.90 ± 24.95 N), and LME (mean = 57.90 ± 20.05 N) MVCs were observed [F(1, 33) = 0.642, P = 0.75, η_p² = 0.017].

In the experiment, the participants’ task was to match their produced force with the target line as precisely as possible. Target force level and the produced grip force in time moved from the left to the right on the screen. The target line was displayed as 0.5 s in advance and up to 4.5 s after trial completion. The y-axis ranged from 0 to 14 N in both conditions. The target curve was either a straight line at 2 N or a sine wave ranging from 2 to 12 N with a frequency of 1 Hz. Note that for the force maintenance task, we chose the lowest force that was requested in the cyclic task because, following previous findings (Galganski et al. 1993; Lindberg et al. 2009; Slifkin and Newell, 2000, Vieluf et al. 2015a), we expected age effects to be higher for this force level than for the mean force level (7 N). It also allowed us to avoid fatigue. To fulfill these two tasks, participants were required to perform either constant force maintenance or cyclic force modulation. Each task included 40 trials of 5 s each. In between the trials, a fixation cross was presented for 5 s. An auditory stimulus together with the disappearance of the fixation cross signaled the start of the trial. Participants were instructed to reach the target line as quickly as possible. Because all of them were already familiar with the setup from previous experiments (Vieluf et al. 2012; Vieluf et al. 2013), only the first two trials of each condition were considered as task adaptation and were accordingly excluded from further analysis.

Data Analysis

Raw data processing.

Data were analyzed using Matlab R2012b (MathWorks, Natick, MA). Data were filtered offline with a fourth-order Butterworth filter at 30 Hz. The first 2 s of each trial were discarded to exclude the ramp phase. The analyses were consequently conducted on the last 3 s of each trial. All variables were determined per trial and then averaged per condition. Outliers were detected on a trial basis. For the force maintenance task, trials exceeding the mean variability, calculated per participant per condition by ± 2.5 times the standard deviation (SD), were excluded from further analysis (see Frank et al. 2006 for a similar procedure). For the sinusoidal force task, outliers were identified as either trials in which the force dropped below 0.05 N so as to exclude trials where force was released or trials in which the amplitude within a cycle was <5 N.

General characteristics of performance.

The mean of the produced force, based on real force values, and the SD of the deviations between the applied and the target force were calculated to globally capture accuracy and the amount of variability of force production.

For the cyclic force modulation task, we computed the Hilbert phase of the applied force, φ_AF, and the target, φ_T, for each trial to get the phase angle as a function of time. The relative Hilbert phase was next calculated as φ_rel = φ_AF − φ_T. Positive values thus indicate that the applied force lags the target. We next calculated the mean and uniformity, a measure of dispersion, of φ_rel using circular statistics (Mardia 1975). These measures provide information about the accuracy and the variability of the relation between the target curve and the applied forces. Descriptive statistics for all the general characteristics of performance are reported in Fig. 2.

Fig. 2.

General characteristics of force maintenance. Means and SD of the general properties of force production per group are provided. YN, young novices; LMN, late middle-aged novices; LME, late middle-aged experts.

Dynamics characterization.

We used the Kramers-Moyal expansion to investigate the dynamics associated with both force tasks (cf. Daffertshofer 2010; Frank et al. 2006; Friedrich and Peinke, 1997). Force production and human movement in general are inherently stochastic; the dynamics of force production comprises a deterministic and a stochastic component. By implication, the future (force) state is conditional upon the probability for the state to be at a certain instant at a specific point in the state space, which is described by probability distributions that can be calculated from experimental data. The Kramers-Moyal expansion allows for the identification of the deterministic and stochastic dynamical components via the conditional probability matrix. The conditional probability matrix thus describes transition probabilities. For the force modulation task, the analysis was done on the Hilbert phase transformed data. For each trial, for both tasks separately, the two-dimensional conditional probability matrix P(AF′,t + Δt|AF,t), which denotes the probability to find the system at state AF′ at time t + Δt given its state AF at an earlier time step t, was computed using a bin size of [5 × SD(AF)]/N; with n = 7, the range of the AF space sampled was from −2.5 × SD to 2.5 × SD, n = 11, and the range of the AF space sampled was from −pi to pi, for the maintenance and dynamic task, respectively. Next, for each participant the average conditional probability matrix across all trials was computed. Each participant and hands’ deterministic and stochastic dynamics (also referred to as drift and diffusion coefficients) were then calculated based on P(AF′,t + Δt|AF,t):

$${\text{D}}^{\text{n}}=\underset{\Delta t⃗0}{\lim }\frac{1}{\Delta t}{\displaystyle \int \frac{{\left(\mathrm{A}\mathrm{F}\prime -\text{AF}\right)}^n}{n!}}P\left(\mathrm{A}\mathrm{F}\prime ,t+\Delta t|\text{AF},t\right)\mathrm{d}\text{AF}\prime .$$ (1)

The deterministic (drift) and stochastic (diffusion) dynamics were obtained for n = 1 and 2, respectively. To evaluate the fixed point’s stability in the force maintenance task, the slope of the deterministic dynamical component (the drift coefficient) across the three middle bins (i.e., where the coefficient changed sign) was determined by a linear regression.

After a first exploration and based on the observed results for the deterministic dynamics in the cyclic task, we followed up by testing for the possible existence of two fixed points that might have been falsely found to be absent in the deterministic dynamics. Fixed points are identified by a change in sign in the sequence of drift coefficients that capture the deterministic dynamics. If the dynamics are not fully stationary, and an actually existing fixed point’s location (slightly) changes from cycle to cycle, the drift coefficients may locally approach zero but never change sign. If this is the case, that is, if fixed points are present (but not detected by the Kramers-Moyal expansion), then it can be expected that the Hilbert phase, which continually increases for a (nonlinear) oscillator, locally reveals phase reversals at the location of the fixed point(s). Stochastic fluctuations will cause the system to overshoot the fixed point, which, due to its stability, will attract the system toward it. To test for the possibility that the attractor is shifted in time between different trials, we identified inflection points in the Hilbert phase evolution (see Fig. 1D) per phase of the sine wave’s cycle. Occurrences are given in percentage relative to the total number of cycles (38 trials × 3 cycles = 114). Note that more than one inflection point can occur per cycle. However, there was never more than one inflection point per phase detected.

Statistical analyses.

Statistical analyses were conducted in Statistica (StatSoft, Tulsa, OK). Analyses of variance (ANOVA) with the between-factor group (3; YN, LMN, and LME) were calculated for the variables describing the general performance of the task (mean force level of the applied forces, SD, mean relative phase, and variability of relative phase). To characterize the dynamics, group (3; YN, LMN, and LME) × bins (7; equally spaced from −2.5 × SD to 2.5 × SD) repeated-measures ANOVAs were conducted on the drift and diffusion coefficients (i.e., those representing the deterministic and stochastic component of the dynamics) of the force maintenance task as well as an analysis of variance by group for the slope around the fixed point (indicating the fixed point). Group (3; YN, LMN, LME) × bins (11; equally spaced from −pi to pi) repeated-measures ANOVAs were conducted on the drift and diffusion coefficients. For the statistical analysis of the number of inflection points of the cyclic task, we calculated a group (3; YN, LMN, and LME) × phase (4; minimum, ascending phase, maximum, and descending phase) repeated-measures ANOVA. Effect sizes are given as partial Eta squares (η_p²). Whenever sphericity was violated, Greenhouse-Geisser correction was applied. The level of significance was set to P < 0.05. Significant effects were followed by Newman Keuls’ post hoc test. To gain insight into a potential relation between tactile sensitivity and muscular strength with the level of noise, we correlated the tactile threshold and the MVC with the mean stochastic impact (represented by the diffusion coefficients) of force maintenance and force modulation. Additionally, the slope around the fixed point was correlated with the tactile threshold. Note that correlations are reported for all participants irrespective of their group, as preliminary results showed no difference between groups. Merging the groups still allowed us to get a global picture of possible relations between strength and tactile sensitivity with the characteristics of force control.

RESULTS

Force Maintenance Task

General characteristics of performance.

The mean applied force differed between groups [F(2, 33) = 3.56, P = 0.04, η_p² = 0.177; Fig. 2A]. LMN overshot the target force level more than YN (P = 0.03). However, a followup one-sample t-test showed that all groups overshot significantly (YN: P = 0.02; LMN: P < 0.01; LME: P = 0.03). SD did not differ between groups [F(2, 33) = 0.48, P = 0.62, η_p² = 0.028; Fig. 2A].

Dynamics: deterministic and stochastic components.

For the deterministic (i.e., drift) component, the curve across bins shows a nearly straight line that crosses the horizontal axis at 0, indicating the presence of a fixed-point attractor at that force level (Fig. 3A). Statistical analysis showed that the coefficients differed between bins [F(2, 33) = 373.73, P < 0.01, η_p² = 0.205 (all post hoc comparisons P < 0.01)]. The slope around the fixed point, which quantifies the strength of the attractor, did not differ between groups [F(2, 33) = 0.73, P = 0.49, η_p² = 0.042]. The bin-by-group interaction was not significant [F(12, 198) = 0.40, P = 0.96, η_p² = 0.024]. With regard to the stochastic (i.e., diffusion) component of the dynamics, analysis revealed significant main effect of bins [F(6, 198) = 28.00, P < .01, η_p² = 0.607]. Values were the lowest at the fixed point and increased away from it on both sides toward the outer bins (all P < 0.01, except for the comparison of bins 2 and 6 as well as bins 3 and 5; Fig. 3B). Thus, in the force maintenance task for the examined force level, the behavior of all groups was generated by a fixed-point dynamics with equivalent attractor strength. Additionally, all participants, independent of their group [F(2, 33) = 1.11, P = 0.34, η_p² = 0.063], presented a behavior with a comparable level of stochasticity (i.e., noise). Again, the interaction was not significant [F(12, 198) = 0.20, P = 0.97, η_p² = 0.012].

Fig. 3.

Dynamics of force maintenance and cyclic force modulation. Group means of the drift (A and C) and diffusion coefficients (B and D) for force maintenance (A and B) and cyclic force modulation (C and D) tasks are presented for young (YN) and late middle-aged novices (LMN) as well as late middle-aged experts (LME).

Correlations of dynamical signatures with tactile threshold and MVC.

The correlation between the slope through the fixed point and the tactile threshold was significant (r = 0.370, P = 0.026; Fig. 4A). In contrast, the slope through the fixed point correlated only marginally with MVC (r = 0.294, P = 0.081; Fig. 4B).

Fig. 4.

Correlations. The slope around the fixed point (left) and the mean diffusion coefficients (middle) of force maintenance task as well as the mean diffusion coefficients of the cyclic force modulation (right) were plotted against the tactile threshold (A) and the maximum voluntary contraction force (MVC; B). Asterisks indicate that the correlation was significant (P < 0.05).

Cyclic Force Modulation Task

General characteristics of performance.

The mean force level differed between groups [F(2, 33) = 4.25, P = 0.02, η_p² = 0.205]. LMN applied lower mean forces than YN (P = 0.02) and marginally lower than LME (P = 0.06). LMN (P < 0.01) and LME (P = 0.03) were more variable than YN. Mean forces were significantly lower than prescribed for LMN (P < 0.01) and LME (P = 0.03) but not for YN (P = 0.07) (Fig. 2B). Variability differed between groups [F(2, 33) = 7.12, P < 0.01, η_p² = 0.302]. Furthermore, the mean relative phase between target and applied force did not differ between groups [F(2,33) = 2.24, P = 0.12, η_p² = 0.120]. However, its variability (uniformity) differed between groups [F(2, 33) = 5.01, P = 0.01, η_p² = 0.233], as it was lower for YN than LMN (P = 0.01) and LME (P = 0.03) (Fig. 2C).

Dynamics: deterministic and stochastic components.

The deterministic dynamical component, the corresponding drift coefficients plotted as a function of bins based on the Hilbert phase, revealed a bimodal structure with no zero crossing of the horizontal axis (Fig. 3C). Thus, the observed profiles offered no evidence for a fixed-point dynamic and are suggestive of a limit cycle dynamics (see below). Post hoc analysis, following up on the significant effect of bins [F(10, 330) = 127.73, P < 0.01, η_p² = 0.795], revealed that all P values were <0.01, except for the comparison of bins 1 and 10, 3 and 9, and 4 and 7, as well as 5 and 6. Furthermore, group differences were revealed [F(2, 33) = 7.20, P < 0.01, η_p² = 0.304], which indicated that YN showed lower drift coefficients (representing the deterministic dynamics) than LMN (P < 0.01) and LME (P = 0.01). The group-by-bin interaction was not significant [F(20, 330) = 1.19, P = 0.31, η_p² = 0.067].

The diffusion coefficient, which captures the stochastic component of the dynamics, showed a similar M pattern as the drift component (representing the deterministic dynamics) in all groups (Fig. 3D). Again, the main effect of bins [F(10, 330) = 110.50, P < 0.01, η_p² = 0.770] reached significance with all P values <0.01, except for the comparison of bins 1 and 10, 2 and 3, 3 and 9, and 4 and 7, as well as 5 and 6. The main effect of group reached significance [F(2, 33) = 7.17, P < 0.01, η_p² = 0.303]. YN showed lower coefficients than the LMN (P < 0.01) and LME (P < 0.01), suggesting overall less noisy dynamics in the young group. The interaction of group and bin was not significant [F(20, 330) = 1.08, P < 0.38, η_p² = 0.061].

Analysis of inflection points revealed a significant interaction of phase and group [F(6, 99) = 2.46, P = 0.03, η_p² = 0.130; Fig. 5]. Overall and within each group the number of inflection points was higher for the minima than for the maxima as well as for the descending and the ascending phases (all P < 0.01). The number of inflection points observed for the minima was the highest for YN, followed by LMN, and the lowest for LME (all P < 0.01). The frequency of occurrence for the minima was between 30 and 40% of the cycles, and for all the other phases, inflection points occur during ~3 to 10% of the trials. For the ascending phase, LMN showed the most inflection points and LME more than YN (all P < 0.01). In the descending phase and around the maxima, most inflection points were found for the LME and more by the LMN than by the YN (all P < 0.01). This last analysis clearly distinguishes the minima phase, that is, around the reversal point, and shows the presence of group specificities with regard to how the dynamics are expressed.

Fig. 5.

Number of inflection points. Showing no. of inflection points relative to the no. of performed cycles for each of the phases of the sine wave per group.

Correlations of dynamical signatures with tactile threshold and MVC.

Mean stochastic (i.e., diffusion) coefficients correlated positively with the tactile threshold (r = 0.374; P = 0.025) but not with MVC (r = 0.227, P = 0.183), suggesting that subjects with higher discrimination capacities had lesser stochastic dynamics (see Fig. 4).

DISCUSSION

We studied the dynamical signatures of isometric force control in terms of extracted deterministic and stochastic dynamics in a constant force maintenance and a cyclic force modulation task performed by young and late middle-aged novices (YN and LMN) as well as late middle-aged experts (LME). To allow for comparison with the bulk of the existing literature, we also performed more conventional analysis characterizing the overarching force control properties. In the sections below, we first discuss the latter results, followed by those pertaining to the influence of force task constraints as well as those to age- and expertise-related differences in separate sections.

Effects of Age and Expertise on the Force Accuracy and Variability

Consistent with previous findings, we found that some age-related differences in force control occur already during the late middle-aged life span (Lindberg et al. 2009; Vieluf et al. 2013). This age effect was more pronounced in the arguably more demanding cyclic force modulation task than in the force maintenance task (see Diermayr et al. 2011 for an overview of consistent findings). In the force maintenance task, LMN applied higher forces than the other groups but revealed no differences in variability. Note, however, that we examined force maintenance at very low levels, which the LMN did not fully comply with; they showed higher overshooting forces than the other two groups (see Lindberg et al. 2009 for similar results for a force level of 3 N). Converging findings in this regard have been also found during lifting (Cole et al. 1999), where it supposedly expresses the safety margin to prevent the object from slipping (Cole and Beck 1994). Furthermore, the higher forces might be a way to compensate for tactile sensitivity loss with aging (Cole 1991), as observed in this group of participants (Reuter et al. 2012). Indeed, it has been shown that force control was more variable at these low levels than at higher levels (Galganski et al. 1993; Lindberg et al. 2009; Slifkin and Newell, 2000; Vieluf et al. 2015a). These previous results motivated the choice of a low force level. However, it remains of interest to test the dynamics at a comparable mean force level for both tasks. Going back to our results, the higher applied forces in the LMA might have been a strategic way of compensating for age-related deficits and, therefore, reduced some of the age effects especially for variability measures.

In the cyclic task, where, in contrast to the force maintenance task, the LMN applied lower mean forces than the other two groups, they showed higher variability (e.g., Voelcker-Rehage and Alberts 2005) as well as higher variability of the relative phase. Finally, the influence of fine motor expertise on the dynamics of force control was investigated to gain further insights into how motor functioning can be stabilized. According to the use-it-or-lose-it hypothesis (Salthouse 1985; Salthouse 2006) and the deliberate practice approach (Ericsson and Smith 1991), a frequent and continuous use contributes to maintaining skills. Continuous use of the hands to manipulate small objects in a dexterous way has been shown to lead to higher performance (Cannonieri et al. 2007; Jäncke et al. 1997; Krampe 2002). As expected, our findings confirmed that age effects were less pronounced for experts. Their performance was in between young and late middle-aged novices. Consequently, LME showed weaker age effects. However, based on the limitations that emerge in group comparisons (as a young expert group does not exist due to the 10 yr of experience criterion), we can only conclude that continued specific activities seem to postpone or counteract to some degree some age-related changes. In fact, LME did not show differences in accuracy, i.e., higher mean force levels in the maintenance task and lower mean force levels in the cyclic task, but were more variable in terms of force production and relative phasing than the young novices during the cyclic task. This might indicate a weaker coupling between the applied force and the stimulus with increasing age, as reported previously for a bimanual force modulation task (Vieluf et al. 2015a). However, overall we did not observe strong age and expertise effects. We assume that this could be at least partly due to the fact that participants were already familiar with the task, as parts of the age- and expertise-related differences may result from different strategies or different amounts of attention allocation to complete force control tasks. Furthermore, it may well be that age- and expertise-related differences would surface under different force levels and, for the cyclic task, frequency of the requested force modulation. Furthermore, our findings cannot be systematically extended to more senior elderly subjects. This remains to be explored in future work.

Dynamical Signatures of Force Maintenance and Cyclic Force Modulation Tasks

To investigate the tasks’ dynamics, we computed the deterministic and stochastic dynamical components associated with both force control tasks. Additionally, we identified inflection points in the Hilbert phase for the cyclic task as a mean to explore the possible existence of (moderately) nonstationary fixed points. In line with our hypothesis, we observed different dynamics for both tasks. The force maintenance task revealed a clear fixed-point dynamic, with smaller magnitude of stochastic fluctuations (noise) around the fixed point than away from it (see Frank et al. 2006 for consistent results). In other words, the magnitude of the random fluctuations was proportional to the strength of the flow (and thus was smallest near the fixed point). That is, large force deviations relative to the requirement are counteracted with more vigor than small ones, which intuitively seems to be a smart solution to the task at hand.

For the cyclic force modulation task, clearly different dynamics were observed. Specifically, no fixed points could be firmly established as attested by the deterministic dynamics; the corresponding drift coefficients were always positive. This result suggests that the sinusoidal force modulation is governed by an oscillatory, likely limit cycle, generating mechanism. Note that in the case of a perfect harmonic oscillator, the drift coefficients are of equal (non-zero) value across the entire space. This was clearly not the case; the coefficients revealed two local maxima and two minima. These local peaks reflect the fast ascending and descending phases of force production and the slower evolution in the cyclic force production around the force maxima and minima, respectively.

Consequently, if the underlying dynamics are generated by an oscillator, it is non-harmonic (Stepp and Frank, 2009). These local minima in the deterministic dynamics may indicate a location in phase space that just fails to be (in this case) a stable fixed point, in more technical terms, a ghost attractor (cf. Strogatz, 1994; Collins et al. 1998; Huys et al. 2010). In its presence, the system locally slows down considerably. Under that premise, one could predict that when modulating the frequency of the sinusoidal target (most likely by decreasing it), at some critical value (a bifurcation point) the drift coefficients would end up crossing the horizontal zero line, as fixed points are created. A similar scenario was found in movement tasks when the movement frequency was slowed down for a cyclic movement task (see Huys et al. 2010 in the context of circle drawing). One way to investigate the dynamics in further detail would be to systematically vary target frequency and the force levels required with the aim to identify the potential bifurcation(s).

The absence of identified fixed points may also be explained otherwise. For one, it may be that the force production is governed by two stable fixed points (corresponding to the extrema of the target force), but these fixed points slightly drift over time. In that case, the minima at the target extrema will be less pronounced, and zero-crossing may vanish as a result (Huys et al. 2010). However, inflection points, indicating no rate of change in the force profile, were observed mostly around the minimum and were relatively rare at the maximum. This finding is in line with findings by Masumoto and Inui (2010), who reported higher variability at the minima than for the maxima, but argues against the existence of a fixed point at the maximum and thus against a (symmetric) bistable system. Alternatively, the dynamics may adhere to a fixed point that is driven by the sinusoidal target so that the phase flow changes at the time scale of the force production. Under this scenario, the dynamics could be expected to be symmetrical and show an approximately homogenous distribution of inflection points. The pronounced difference in the occurrence of inflection points around the force minima and maxima thus argues against this hypothesis. Furthermore, in line with the ideas derived from the study by Danion and Jirsa (2010), the bimodal structure might be related to the combination of feed-forward and feed-backward control, and these control modes are differently strong, are involved during different phases of the sine wave, and lead to different relations whether the applied force leads or lags the target force.

Regardless, the dynamics governing the sinusoidal target force task could not be unambiguously identified and await future investigation. However, at this point we can state that they are clearly different from those observed in the static force production task and are asymmetric in terms of the up/down vs. minima/maxima phases as well as in terms of the “depth” of the two minima (see Fig. 3C). Note that although we cannot make definitive statements about the attractor structures involved, the current results favor a nonlinear oscillator, although the drift coefficients still clearly indicate the deterministic dynamics of the participants’ behavior. This is expressed by a strong slowing down, almost plateauing, at the force minimum, and to a lesser extent at the force maximum, and a faster rate of force change during the ascending and descending phases.

Noise was lower around the local minima in the drift coefficient than away from them. This finding indicates that for the sinusoidal force modulation task, the two components of the fast dynamics show higher noise than the slow dynamics. In other words, the magnitude of the stochastic fluctuations was proportional to the deterministic force. Thus, at least qualitatively, the relation between the deterministic and stochastic component of the dynamics appeared to be similar in both tasks, although their dynamics were qualitatively distinct.

Interestingly, we found a significant correlation between the tactile threshold and the degree of stability of the fixed point, i.e., the lower the tactile threshold the more stable the fixed point (in the static force production task), indicating that sensory capacities and stochasticity are somehow related. This suggests that tactile sensitivity might contribute to the stabilization of force control. In other words, it links perceptual ability to the deterministic dynamics of force control. In contrast, no evidence that tactile sensitivity is related to the stochastic component was found. Additionally, the marginally significant correlation between the stability of the fixed point and the MVC provided the first indication that stronger subjects may be more prone to generating more stable force dynamics. Taken together, individual organismal characteristics seem to influence force control in a way that the stronger and the more sensitive the participant, the more stable his/her expressed dynamics would be. However, further research is needed to gain deeper insights into a potential causal relationship between these components of force control and the underlying dynamics.

Effects of Age and Expertise on the Expressed Dynamics

In general, some group-related dynamical differences were expressed in a task-dependent manner, but the nature of the dynamics underlying force control did not differ between groups. Accordingly, for the examined populations, we conclude that the expressed behavior stemmed from the same generating mechanisms, which were determined by the task.

In the force maintenance task, no differences in dynamical signatures were observed between groups. In contrast, for the cyclic task, both the deterministic and stochastic components (i.e., drift and diffusion coefficients) were higher for older novices than for the two other groups. The higher drift coefficient for the older novices suggests that the rate of the change of their force production was higher than that of the other two groups. This result is somewhat puzzling given that all groups tracked a target force oscillating with 1 Hz and that the elderly are generally known for their slower rate of force production (Ng and Kent-Braun 1999; Stelmach et al. 1989). One potential explanation for this finding is that the older novices were slower in adapting to the stimulus frequency and, therefore, were still catching up with the target in the 3 s of data analyzed. Alternatively, it might be indicative of a different deficit, that is, the incapability of continuous slow force tracking. In line with this idea, older adults would be less capable of smoothly ramping up or down their produced forces according to the displayed sine wave. Such an assumption could be grounded in age-related alterations in force production smoothness, which has been found to be more apparent at low force levels, as are the ones used in the present study (Brown 1996; Galganski et al. 1993; Kinoshita and Francis, 1996). Notice that the group differences in the mean drift components, however, were very small (means ± SD of 7.58 ± 6.91 vs. 7.40 ± 6.51 for the young and novice elderly, respectively). The limited length of our data (recall, the last 3 s of 5 recorded seconds were analyzed) did not allow us to meaningfully verify this result using further analyses, e.g., power spectrum, but it certainly deserves to be explored in future studies, including longer trials and different movement frequencies as well as different force levels. Regardless, our finding suggests that concomitant changes in the deterministic as well as stochastic dynamical components could be causing motor behavior to become more variable in numerous tasks with increasing age (Christou and Tracy 2006; Vaillancourt and Newell 2003). It should be noted that in the force maintenance task no dynamical differences were found between groups. Unless the older novice participants were successfully compensating by applying higher forces, this could imply that the latter task lacks the sensitivity to bring this to the fore, at least for the force level tested here. Another explanation could be that aging does not increase stochasticity per se and thus that stochasticity is not a personal or age-related task-independent property but rather a property described over the performer and task in combination.

Finally, we found that in both tasks the experts had similar dynamics (both deterministic and stochastic) as the young novices. We conclude that continued specific activities seem to postpone or counteract to some degree some age-related changes in the components of force control dynamics.

Taken together, our findings suggest that although the nature of the dynamical processes underlying force modulation can be preserved with age, at least up to a certain age, the behavior can be more or less prone to stochastic influences as well as a different parameterization of the attractor states. It should be noted that it remains unclear how the different dynamical components relate to the functional organization of the sensorimotor function. If we tentatively speculate that the deterministic and stochastic components reflect processes and interactions between them that are (more) directly task-relevant and those that are not, respectively, then the framework used here may potentially be linked to the idea of “dedifferentiation”, that is, “a process by which structures, mechanisms of behavior that were specialized for a given function, lose their specialization and become simplified, less distinct, or common to different functions” (Sleimen-Malkoun et al. 2014). Under this hypothesis, the consequences of structural age-related changes in the nervous system will be expressed differently in different task contexts. Moreover, there are suggestions that the behavioral repertoire decreases with age (see Sleimen-Malkoun et al. 2014). That is, age but also expertise may change the number and/or nature of behaviors of the repertoire. Analysis of the dynamics along the lines described here provides for the means to investigate these issues in future work.

Conclusion

The underlying dynamics of force control vary qualitatively in response to task constraints. Within this study, we confirmed that fixed-point dynamics underlie force maintenance, as shown previously by Frank et al. (2006), whereas the dynamics underlying the cyclic force modulation task were found to resemble those of a nonharmonic oscillator. The latter finding indicates the combination of one set of slow and one set of fast dynamics while tracking a regularly time-varying force target. Age effects were found to be more pronounced in the cyclic, arguably more complex task. Compared with young novices, the older experts mainly appeared more variable. Overall, the dynamics underlying force control appeared similarly organized between groups; the observed differences were limited to small differences in the deterministic and stochastic dynamics in the cyclic task only, suggesting that, among others, the adaptation to task constraints varies. This result suggests that behavioral fluctuations cannot be uniquely traced back to system noise. It may indicate that increased variability in aged populations is not solely a matter of a task-independent increase of noisiness but suggests that the magnitude and structure of noise is specific to “the actor in action”; that is, it pertains to the actor in a task-dependent fashion. However, it seems plausible that qualitative differences between different populations (age, expertise) may appear when task parameters are pushed toward their limits.

GRANTS

The research was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG, VO 1432/7-1) as a part of the DFG priority program “Age-differentiated Work Systems” (SPP 1184) as well as by the Investissements d’Avenir French Government program managed by the French National Research Agency (ANR)as a part of the CoordAge-A*MIDEX project (no. ANR-11-IDEX-0001-02).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

S.V., C.V.-R., E.-M.R., B.G., and R.H. conceived and designed research; S.V. and E.-M.R. performed experiments; S.V., R.S.-M., and R.H. analyzed data; S.V., R.S.-M., J.-J.T., and R.H. interpreted results of experiments; S.V. prepared figures; S.V., R.S.-M., J.-J.T., and R.H. drafted manuscript; S.V., R.S.-M., C.V.-R., V.J., E.-M.R., B.G., J.-J.T., and R.H. edited and revised manuscript; S.V., R.S.-M., C.V.-R., V.J., E.-M.R., B.G., J.-J.T., and R.H. approved final version of manuscript.