Dynamic Coordination Phase and Joint Profiles: Cluster and Fixed-point Shift Techniques

Background

Mutuality between the relative oscillators arises from the insight of perception-action coupling perspective with work on the coordination problem (Bernstein, 1967), and has become a major tenet in the physical sciences (Kugler et al., 1980; Shaw & Turvey, 1981). Evidently, when adopting this view, phase synchrony is a traditional paradigm in behavioral dynamics (Warren, 2006). In particular, a bi-manual 1:1 frequency locking in-phase and anti-phase pendulum can represent a wide range of control theories (Turvey, 1990; Fultot, 2020).

In this equation, the phase difference, θ2-θ1 ≈ 0, denotes a condition of nearly synchronized in-phase and θ2-θ1 ≈ π indicates this in an anti-phase. The observed relative phase or phase relation (ϕ) between two oscillators at ϕ ≈ 0 deg (in-phase) and ϕ ≈ 180 deg (anti-phase) have been modeled as point attractors in our limb system, as they are purely stable patterns (Pikovsky et al., 2003).

Several studies on the 1:1 frequency locking of the left-and-right-hand phase, defined as ϕ = (θ_L-θ_R): the difference between the left (L) and right (R) phase angles (ϕ), have led to the identification of important invariant human system features (Kelso, 1984).

In this equation, ϕ is the phase angle of the individual oscillator. In addition, α and b are coefficients that denote the strength of the coupling between the two oscillators. A relative 1:1 frequency-locked coordination phase [V(ϕ)] is determined by the differences between the two component oscillators’ continuous phase angle [-α cos(ϕ)- b cos(2ϕ)]; the stability of the point attractor can be varied by varying the pendulum’s dimensions (Kugler & Turvey, 1988). This function indicates that the minima of the potential are located at ϕ = 0, and ϕ=±π (Haken et al., 1985).

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Figure 1.

Synchronous diagrams of the possible point attractors

Gap statement

Motivation for the research question was derived from the above assumptions and various work given the following model: function can be estimated in how the potential will change in shape as the other measurements. Although the magnitude (or stability) of synchrony can differ across different conditions (Schaffert et al., 2019), the most used quantifications are the spatiotemporal and neuronal performance that occurs between the movements of the limbs (Mechsner et al., 2001). Indeed, there is a lack of verified quantification of the magnitude and stability of systems that can occur between different categories (Purpura et al., 2017), even though such assumptions play an important functional role in the assembly of different oscillations (Kelso et al., 1991). Cluster methods and the fixed-point shift process can be integrated with methodological techniques that allow for modeling of data (Chen & Huang, 2017), providing the ability to look at the exact level of patterns as well as different constraints (Soechting & Lacquaniti, 1981). Particularly, a homogeneous characteristic (or data) can be detected from heterogeneous observations by clustering data that is unlabelled and without a defined variable pool (Frank & Richardson, 2010; Lin et al., 2019). Further, a locally stable or ultimate compact state (attractor) can be defined by the fixed point shift process (Ren et al., 2013; Park, 2019). These applications with constraints may be imposed on the redundant degrees of freedom problem (Bernstein, 1967) embracing purely stable patterns.

Purpose

In this study, therefore, we address these questions by adapting and testing cluster and fixed-point shift techniques (Frank & Richardson, 2010). We propose that the nature of this method is called radial calculation (Richardson et al., 2012), primarily suggesting which phase is more stable as a particularly well-suited physical model. In addition, we further expand such a functional linkage into a topological constraint that can be used to examine any other phase of synchronization of different bodies of topological systems (Shih et al., 2019).

Theoretical Value

The above work designed on a systemic investigation may elicit degree of freedom issues to deal with task-specific characteristics (Park, 2020). This will allow us to estimate a useful reference for system stability coordination tasks in which this functional pattern can be applied to all human movements, muscles, and even neural networks.

Participants

For this study, bimanual coordination embedding synchrony context data were collected from students (n = 24) at the University of Connecticut (n = 16) and Seoul National University (n = 8). They participated in this study to fulfill their course requirements. None of the participants were compensated for volunteering. All participants were not aware of the study’s purpose and had not previously participated in an experiment on rhythmic or synchrony movement coordination. They participated in different experiments in the three groups and were confirmed to have no physical or psychological problems. All of them provided written informed consent for the study, approved by the local ethics committee (SNUIRB No.1509/002-002). The study conformed to the ethical standards of the 1964 Declaration of Helsinki.

Experimental design

The present experiment was designed to verify whether a more stable bimanual synchrony variable exists between phases and anatomies. Although the in-phase on the wrist joint is properly used compared to other possible bimanual pendulums (Kelso, 1984; Turvey, 1990), whether this value is representative of the overall characteristics of a system must be assessed. Moreover, repeating the assessment for only one position under several different conditions with many trials is likely to be associated with learning (or fatigue) effects. The collected data were analyzed to compare different characteristics between phases (experiment 1: in-phase and anti-phase) as well as different limb joint performances (experiment 2: wrist, elbow, and shoulder).

Apparatus and procedure

Bimanual coordination without detuning was performed while each subject was seated in a chair holding a pendulum vertically without occluding their vision. The pendulums used here were two standard wooden rods (85 g, 1 m in length, 1.2 cm in diameter) with DC potentiometers attached. A 200 g weight was positioned 30 cm from the bottom of the rods. Each participant was asked to grasp the pendulum firmly 60 cm from the bottom so that the pendulum would not slip out of their hands; they were also asked not to rotate their finger joints. Their forearms were fixed voluntarily so that the pendulum motion was restricted to the sagittal parallel plane and the joint vertical axes (i.e., each oscillation pertained to only one joint, with the other joints being held immobile) (see Appendix 1). The appenidx (1) shows experimental sessions that were tapped into the ongoing bimanual coordination, focusing on its design structure (each experiment section includes more detail about the apparatus and procedures).

Analysis

Expectation values

These are (real) numbers that are computed from a continuous mathematical model. It can be an infinitely large dataset (i.e., pendulum = x), probability distribution or probability density describing certain aspects of a distribution. For example, Mean is the center of a distribution of values:

(4)

$$P\left(X\in \left[a,b\right]\right)=\frac{1}{N}{\sum }_{k=1,...,N;X^{\left(k\right)}\in \left[a,b\right]}1$$

and Variance is the variability in the values of the distribution:

(5)

$$\begin{array}{c}{\sigma }_e\left[f\left(x|\mu ,{\sigma }^2\right)\right.=\hfill \\ \frac{1}{\sqrt{2\pi {\sigma }^2}}{exp }^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}}\left|x=individual,\right.\hfill \\ \mu \in R=mean\left(location\right),\hfill \\ {\sigma }^2>0=variance\left(squared scale\right)\hfill \end{array}$$

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Figure 2.

A normal distribution. The parameters are µ = mean and σ² = variance (width given by σ). Prob. = probability, a.u. = arbitrary unit

This is intuitive if we think about our empirical variable. We assume, here, that the variable we are interested in is X. The moments can then be the expected values of X: E(X), E(X²), E(X³), and so on.

Mean of the first moment:

(6)

$$m=\frac{1}{N}{\sum }_{k=1,\cdots ,N}{X}^{\left(k\right)}, N\to \infty$$

and that of the nth moment:

(7)

$$m_n=\frac{1}{N}\sum _{k=1}^N{\left(X^{\left(k\right)}\right)}^n, N\to \infty$$

For example, when we practice with moment values, the first moment of the mean is defined as

(8)

$$m=\underset{n\to \infty }{\lim } m_e\left(N\right), m_e\left(N\right)=\frac{1}{N}{\sum }_{k=1,\cdots ,N}{X}^{\left(k\right)}$$

where X^(k) are the realizations (i.e., statistically independent scores) from a statistical ensemble. The average value can be interpreted as the sample (i.e., relative phase) average. Figure 3 below illustrates the limiting process. It shows that sample means m_e taken from a normal distribution with parameter μ = 7 and σ = 1 for samples of increasing size. The sample sizes used in the figure are shown in below.

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Figure 3.

Statistical representation of the expectation function. The figure suggests that in the limiting case when N goes to infinity, the sample mean converges to the value 7 (i.e., we have m = μ = 7).

Given the function of the variables X: ɡ(X), we can assume which function value we would expect to observe on average (expectation value) with a notation E[ɡ(X)]. For example, first moment (mean value): ɡ(X) = X → m=E[X]=<X>, given a function ɡ of the process X: ɡ(X), we calculate E[ɡ(X)] as follows:

(9)

$$\begin{array}{c}E\left[g\right]=\frac{1}{N}\left\{g\left(X^{\left(1\right)}\right)+g\left(X^{\left(2\right)}\right)+\dots +g\left(X^{\left(N\right)}\right)\right\},\hfill \\ N\to \infty \hfill \end{array}$$

Thus, the notation of the expectation value of function ɡ;

(10)

$$\begin{array}{c}E\left[g\left(X\right)\right]=〈g\left(X\right)〉= \hfill \\ \underset{N\to \infty }{\lim } \frac{1}{N}{\sum }_{k=1,\cdots ,N}g\left(X^{\left(k\right)}\right)X^{\left(k\right)}\hfill \end{array}$$

where the equations E[g(X)] represent measures of central tendency and variability for continuous variables defined on a real line pendulum movement dynamics.

Results

We determined the cluster amplitudes r and phases θ for the data sets G1, phase data A and G1 phase data B. For each data set, we computed the amplitude and cluster phase and compared the results obtained in Table 2.

The cluster amplitude of the datasets seems to be variable, and the cluster phase appears to be anti and in-phase. Regarding the data, the G1_A data looked anti-phase as theta was 179 (left side of Figure 4), whereas the G1_B data had a theta of –.57 (in-phase; right side of Figure 4). The results showed that motor performance varied given the shift in and the of these behavioral variables. The main observation was that the in-phase was significantly associated with stability, while the other condition (anti-phase) was lower. Moreover, there was a significance, as the cluster amplitude from an anti-phase was significantly wider: F(2, 47) = 4656.999, (p < 0.001).

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Figure 4.

Synchronous diagrams of the cluster methods

Discussion

Formation and retention refer to propriospecific information about the states of the muscular-articular links, and these patterns are constrained by the dynamic criteria of stability patterns (Kelso, 1997; Park et al., 2015). Specifically, let us consider a qualitative physical system such as stiffness, damping, and position over time in a dynamic mass spring system, as given.

Here, m is the mass, b is the friction, and k denotes the stiffness. The variable t is time, χ denotes the position, ${\chi }^{\text{'}}$ is the velocity, and ${\chi }^{″}$ represents the acceleration. In general, because damping is produced by a process that dissipates the energy stored in the oscillations, the interplay between input and damping approaches a stationary fixed point in the long-time limit.

Such systems possess a static equilibrium point, which is called a point attractor (Kugler et al., 1980). The property of this dynamic has been applied not only to a physical system but also to descriptions of the human neuromuscular level (Kay et al., 1987). This function involves an investigation of the intact movement of a limb oscillator in terms of muscle–joint kinematic variations (kinematic position, velocity, and acceleration) over time (Park, 2018). As it is, when we are asked to swing two limbs comfortably, it can be characterized by the pendulum’s dimension (Kugler & Turvey, 1988; Turvey, 1990), simplifying the point attractor while restricting it to certain domains of phase space.

In the observed relative rhythmic segment patterns, some studies have reported that the in-phase ϕ = 0 condition is more stable than the anti-phase ϕ = π condition (Schmidt et al., 1998; Li et al., 2019), and other studies have highlighted that this is not always the case in certain conditions and analyses (Shih et al., 2019). In particular, the stability of coordination has been leaning toward in-phase movements (Carson & Kelso, 2004), although such coordination is influenced by a constraint. This investigation specifically considers these perspectives to determine a more reliable characteristic of different phases, based on which one would be better as the experimental dependent variable, to illustrate the different phase effects between the two conditions. We confirmed that while the anti-phase pattern entailed larger instability with the cluster methods and structure (Cattaert et al., 1999), in-phase was more associated with a preference for homologous activations.

Analysis

Calculating Relative Phase Coordination

Corresponding to the previously introduced model (see Background), the mechanism of oscillation on two different (left pendulum and right pendulum), but nearly identical, process phases was defined by the following dynamic:

(17)

$$\mathrm{\varphi }={\mathrm{\theta }}_L-{\mathrm{\theta }}_R$$

Here, ϕ is the phase of the strength between the left (θ_L) and right hand (θ_R). The degree of relative phases (0° ~ 180°) depends on the difference between the two oscillators. If each θ is defined as a sine function as follows:

(18.1)

$${\mathrm{\theta }}_1={\mathrm{\omega }}_1-\frac{\mathrm{\alpha }}{2}\mathrm{s}\mathrm{i}\mathrm{n}\left({\mathrm{\theta }}_1-{\mathrm{\theta }}_2\right)$$

(18.2)

$${\mathrm{\theta }}_2={\mathrm{\omega }}_2-\frac{\mathrm{\alpha }}{2}\mathrm{s}\mathrm{i}\mathrm{n}\left({\mathrm{\theta }}_1-{\mathrm{\theta }}_2\right)$$

the logic can simply be rewritten to ϕ as the same dynamic function.

(19)

$${\mathrm{\theta }}_1-{\mathrm{\theta }}_2=\mathrm{\varphi }$$

Then, each θ can be specified as the following equation:

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(20)

$$\begin{array}{c}\mathrm{\varphi }={\mathrm{\theta }}_2-{\mathrm{\theta }}_1={\mathrm{\omega }}_2-{\mathrm{\omega }}_1=∆\mathrm{\omega }\left(t\right)+\mathrm{\varphi }\left(0\right)\hfill \\ =\mathrm{\varphi }\left(t\right)\hfill \end{array}$$

Results

Participants were divided into different experimental groups for the following two reasons: (a) although the wrist point is commonly considered compared to other possible bimanual pendulum areas, such as the elbow and the shoulder, representing this value as the overall characteristic of the system had to be confirmed. Moreover, (b) there was uncertainty that repeating only one position under several different conditions through a number of trials can be associated with learning (or fatigue) effects. This investigation specifically considered these differences to determine the fundamental characteristics of different anatomical joints.

Figure 5 illustrates the average stability of each joint for the wrist, elbow, and shoulder. Topological asymmetry, couplings, and statistics are reflected, showing that they are significantly wider for the distal (wrist) than the proximal (shoulder) with different joint parameters (Figure 6). Specifically, the results showed that coordination performance varied with different anatomical parts given the fixed-point shift [ϕ_{ave -} ϕ₀] and the standard deviation of the relative phase [SDϕ] of the three behavioral variables. This indicates that the topology Pearson correlation R = .5 (p = 0.00041)]. There was hierarchical significance, as the [ϕ_{ave -} ϕ₀] and the [SDϕ] were significantly wider for distal anatomy: F (2, 47) = 4656.999, (p < 0.001).

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Figure 5.

Circular representation of the different joint coupled oscillations

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Figure 6.

Coupling strength of the different joint coupled oscillations

Next, the data were separated into phases for different conditions according to the trial. The data were then analyzed using two values regarding which one would be better as the experimental dependent variable to illustrate the different trial effects between the two conditions of the one joint (wrist) or the position of different joints (wrist, elbow, and shoulder).

The results showed that motor performance varied with different anatomical parts given the fixed-point shift [ϕ_{ave -} ϕ₀] and the standard deviation of the relative phase [SDϕ(rad)] of these three behavioral variables. The important fact from these observations is that repetition with one joint may be significantly associated with increased stability, akin to trial effects (see the left plot of Figure 7) [Pearson Correlation R = .284 (p < 0.025)], while the other condition (different topology) did not have a significant effect according to the trial (see the middle plot of Figure 7) [Pearson Correlation R = .110 (p < 0.236)]. However, there was hierarchical significance, as the mean relative phase from the [ϕ_{ave -} ϕ₀] and the [SDϕ(rad)] was significantly wider for distal anatomy (see the right plot of Figure 7): F (2, 47) = 4656.999, η² = 8.077 (p < 0.001).

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Figure 7.

In-phases for the different conditions according to the trial

Discussion

Inquiry into the possibility of relating perception-action to dynamics began in the 1970s with the problem of coordination: could a principled dynamical account of fundamental rhythmic capabilities be given involving multiple joints, scores of muscles, and millions of cells? Efforts to address this question invoked the concepts and tools of nonlinear dynamics (Kugler & Turvey, 1987). We suggested a method of data collection (including trial effects; recommended to create as a relevant dependent variable) that can be used to measure the internal source of stability (Frank & Richardson, 2010; Shih et al., 2019; Gruevski et al., 2017). In particular, collecting different values for three joints but using only wrist data for biological characterization was determined for the following reasons: (a) although there was a significant learning effect when the participants undertook a task repeatedly with only the wrist point, it led to significant typicality compared to the other two (elbow and shoulder) data sets. As it is, to overcome the learning (or fatigue) effects, data were collected for three joints randomly but used only the wrist data. (b) Even though investigations could use combined data that included all three different motor positions, representing the combined data as the characteristic of a biological system appears to remove the important value of representativeness, as this combination likely has too many variables to manage. Moreover, (c) there was an expectation that collecting different motor scales but using concretely represented data (i.e., wrist position) may meet both requirements of typicality as a well-defined system characteristic eliminating the learning effect stemming from numerous trials. This implies a demonstration of the question of whether data from only one condition in many trials can be represented as a well-defined characteristic of the internal source of a system (Thomaz et al., 2017).

Practical Implications

The results revealed that the cluster phase and amplitude measures of the relative phases can be used to identify synchrony patterning. The methods applied in the experiments demonstrated that wrist in-phase motion was suggested as a reliable order parameter with a value of amplitude and the trail effect compared with other stable relative-phase patterns (three joints). At present, the observation across constraints is diverse, but consistencies occur in the suppression of the wrist joint and the attraction to stable phasing patterns (Stergiou, 2020). The dynamics of the performance are reasonable, even though conditional differences can be presented. Many reasons may evolve for such a condition namely, constraint, level, ability, and experience in specific tasks (Kelso et al., 2018). This study, at least, was designed to address such details, mainly for investigating the reliability in the order coordination parameter dynamics (Warren & Whang, 1987). In addition, although the cluster and fixed-point shift method presented here cannot account for coupling delays or leading/following/detuning, it is plausible to assume that the method can be generalized to take such effects into account.

Regarding the practical implications, the present study has several insights that can be applied in future studies to strengthen the applicability of the present findings. First, the present study is based on data collected from a time dependent dynamic (expectation value based cluster statistics), which boosts the generalizability of the study findings to different synchrony (intra and interpersonal synchrony; Loras et al., 2019). For instance, the moderating effect of cluster amplitude may be more marked at transportation stochastic effects, where the impact of the time series on the performance relationship is much stronger for participants, as their functional flexibility is essential to maintain biological stability (Kaur et al., 2018). Thus, the findings of the present study can be replicated across different conditional effects to ensure their generalizability.

Further, instead of behavioral performance-based measurements from participants, objective outcome variables related to stability, such as neural/chemical functions and microstate severity, can be used as important variables in an examination of the variant and invariant effects (Klinshov et al., 2017). If similar findings are obtained with these objective neuro-chemical outcome variables, the practical importance of systematically managing biological adaptations can be further understood (Forestier & Nougier, 1998).

Finally, although the experimental group’s company was considered as an identical population with regard to studying at a university, other potential individual effects that were not seriously considered in this study may exist. The possibility must be left open that there may be important boundary conditions for the moderation of individual effects of biological stability on the relationships between external levels and internal behaviors (Vieluf et al., 2015). For instance, working hours and habit (i.e., how long participants worked and rested before participating in the experiment) could possibly interfere with biological performance by themselves or in conjunction with the experimental setting. The uniqueness of the moderating effect of individual tendency variability can be confirmed by examining the effects of potentially confounding variables.

Researchers may have some concerns regarding whether the tools are constrained to situations in which the points are measured during rocking. The analyses presented here all involve periodic coordination, and it is important to note that the techniques and results could be extended to situations where the movements involve natural coordination. The benefits of investigating these levels of synchrony, i.e., different anatomical position, are that they allow us to empirically test the extent with which the synchrony is a process that emerges because of relatively stable prototypes. Accordingly, this suggestion could be used in future research to determine whether and how the magnitude and stability of synchrony influences the coordination dynamics of movements (Richardson et al., 2010).