Variability is a critical aspect of a dynamical systems analysis. Because there are a number of numerical techniques that can be used in such an analysis, the calculation of variability has several issues that must be addressed. The purpose of this paper is to present a variety of quantitative methods for investigating variability from a dynamical systems perspective. The paper is divided into two major sections covering discrete and continuous methods. Each of these sections is subdivided into two sections. Within discrete methods, we discuss, first, the calculation of the discrete relative phase from a time-series history of two parameters and, second, the use of return maps. Using continuous methods, we present procedures for using angle-angle plots in the evaluation of relative phase. We then discuss the use of phase plots in the calculation of the continuous relative phase. Each of these methods presents unique problems for the researcher and the method to be used is determined by the nature of the question asked.

BackgroundOver the last two decades, various measures of entropy have been used to examine the complexity of human postural control. In general, entropy measures provide information regarding the health, stability and adaptability of the postural system that is not captured when using more traditional analytical techniques. The purpose of this study was to examine how noise, sampling frequency and time series length influence various measures of entropy when applied to human center of pressure (CoP) data, as well as in synthetic signals with known properties. Such a comparison is necessary to interpret data between and within studies that use different entropy measures, equipment, sampling frequencies or data collection durations.Methods and FindingsThe complexity of synthetic signals with known properties and standing CoP data was calculated using Approximate Entropy (ApEn), Sample Entropy (SampEn) and Recurrence Quantification Analysis Entropy (RQAEn). All signals were examined at varying sampling frequencies and with varying amounts of added noise. Additionally, an increment time series of the original CoP data was examined to remove long-range correlations. Of the three measures examined, ApEn was the least robust to sampling frequency and noise manipulations. Additionally, increased noise led to an increase in SampEn, but a decrease in RQAEn. Thus, noise can yield inconsistent results between the various entropy measures. Finally, the differences between the entropy measures were minimized in the increment CoP data, suggesting that long-range correlations should be removed from CoP data prior to calculating entropy.ConclusionsThe various algorithms typically used to quantify the complexity (entropy) of CoP may yield very different results, particularly when sampling frequency and noise are different. The results of this study are discussed within the context of the neural noise and loss of complexity hypotheses.

Routine activities performed while standing and walking require the ability to appropriately and continuously modulate postural movements as a function of a concurrent task. Changes in task-dependent postural control contribute to the emergence, maturation, and decline of complex motor skills and stability throughout the lifespan.

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Three-year-olds were given a search task with conflicting cues about the target's location. A ball rolled behind a transparent screen and stopped behind one of four opaque doors mounted into the screen. A wall that protruded above one door provided a visible cue of blockage in the ball's path, while the transparent screen allowed visual tracking of the ball's progress to its last disappearance. On some trials these cues agreed and on others they conflicted. One group saw the ball appear to pass through the wall, violating its solidity, and another group saw the ball stop early, behind a door before the visual wall. Children's eye movements were recorded with an Applied Science Laboratories eye tracker during these real object events. On congruent trials, children tended to track the ball to the visible barrier and select that door. During conflict trials, children's eye movements and reaching errors reflected the type of conflict they experienced. Our data support Scholl and Leslie's (1999) hypotheses that spatio-temporal and contact mechanical knowledge are based on two separate, distinct mechanisms.