Comparing dynamical systems concepts and techniques for biomechanical analysis

Abstract

Traditional biomechanical analyses of human movement are generally derived from linear mathematics. While these methods can be useful in many situations, they do not describe behaviors in human systems that are predominately nonlinear. For this reason, nonlinear analysis methods based on a dynamical systems approach have become more prevalent in recent literature. These analysis techniques have provided new insights into how systems (1) maintain pattern stability, (2) transition into new states, and (3) are governed by short- and long-term (fractal) correlational processes at different spatio-temporal scales. These different aspects of system dynamics are typically investigated using concepts related to variability, stability, complexity, and adaptability. The purpose of this paper is to compare and contrast these different concepts and demonstrate that, although related, these terms represent fundamentally different aspects of system dynamics. In particular, we argue that variability should not uniformly be equated with stability or complexity of movement. In addition, current dynamic stability measures based on nonlinear analysis methods (such as the finite maximal Lyapunov exponent) can reveal local instabilities in movement dynamics, but the degree to which these local instabilities relate to global postural and gait stability and the ability to resist external perturbations remains to be explored. Finally, systematic studies are needed to relate observed reductions in complexity with aging and disease to the adaptive capabilities of the movement system and how complexity changes as a function of different task constraints.

Keywords: Adaptability; Complexity; Dynamical systems; Nonlinear dynamics; Stability; Variability.

Fig. 1

End-point and joint space variability in novice (A) and expert marksman (B). The spectra of the autocorrelation functions as an individual prepares to shoot show less variation in end-point (gun) but more variation in joint space for the expert. The solid black lines represent the gun, the gray lines are the wrist joint, and the dotted lines are the shoulder.

Fig. 2

Coordination variability of a thigh rotation/leg rotation during an unanticipated cutting performance comparing males vs. females. * Indicates differences between females and males across the entire stance phase as well as in the 35%–45% range of stance.

Fig. 3

Determination of dynamic stability using time-to-contact (TtC). This analysis evaluates the instantaneous position, velocity, and acceleration of the center of mass (COM) or center of pressure toward the physical base of support. Here shown for COM toward anterior and posterior boundaries of base of support in response to perturbation inducing forward body sway. Dashed arrow in lower panel indicates minimum TtC.

Fig. 4

Evaluation of local dynamic stability using the finite-time Lyapunov exponent. A time series signal (A) is transformed into an Nth dimension state space (B; shown here as 3-dimensional) by adding a time delay (T) to the original signal. The state space reveals neighboring trajectories diverging as the behavior evolves. This divergence is a result of internal fluctuations or perturbations. On a logarithmic scale (C), the rate of short-term divergence over the course of a single stride (λ_ST) and long-term divergence from 4–10 strides (λ_LT) can be quantified by the slope of the solid lines of best fit. Higher rates of divergence are indicated by steeper slopes of the lines and indicate greater local instability.

Fig. 5

The phase space analysis using a Poincaré section evaluates a signal's (S_k) evolution through a cycle to the subsequent cycle, S_k+1, in respect to the mean cycle (S*). If the signal is farther from the mean cycle than in the previous cycle (i.e., (S_k+1 – S*) > (S_k – S*)), the trajectory is diverging and the behavior is considered unstable.

Fig. 6

Variability vs. complexity. Variation in a signal, such as the high amplitude deviation around the mean for the sine wave, cannot be equated with complexity. A signal can be variable but not complex (left panel), complex but not variable (middle panel), or variable and complex (right panel).

Fig. 7

Example of multiscale sample entropy (MSE) output. The two curves represent the complexity of the center of pressure path length of an individual drumming a 1:1 anti-phase rhythm while standing on one and two feet. Time scale factors range from 12.5 ms (factor 1) to 150 ms (factor 12). It is important to note that the 1st scale factor is the sample entropy of the original time series, typically used in previous entropy analyses such as approximate entropy (ApEn). From the this graph it is clear to see that simply using a single time scale would underrepresent the processes operating at slower frequencies. The area under the curve is the overall complexity of the system (for more details, see reference⁶⁷).