The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).