Table 3.

Measurements of auditory-motor coupling and synchronization during walking: definitions, formulas, and interpretations.

Measurements of auditory-motor coupling and synchronization	Definitions, explanations and interpretations
Tempo (Beats or steps per minute)	Tempo is a term that refers to the basic tempo of audio or movement and is typically expressed in number of steps or beats per min (SPM/BPM). SPM is calculated as the total number of steps divided by duration expressed in minutes: $$SPM=\frac{Sum(steps)}{Sum(minutes)}\text{Tempo matching occurs if}:SPM=BPM$$
Relative phase angle(measured in degrees)	This is a measure of the timing of the footfall relative to the closest beat. The relative phase angle can be expressed as either a positive (footfall after the beat) or a negative (footfall before the beat) angle in degrees. With the formula below, the relative phase angle for 1 step is calcuated. St represents the time point where the step investigated takes place, and Bn is the beat at the time prior to the St. $$\varphi =360*(\frac{S_t-B_n}{B_{(n+1)}-B_n})$$ To calculate the average relative phase angle, circular statistics (Berens, 2009) is then applied.
Resultant vector length (expressed as a value from 0 to 1)	This measure expresses the coherence or stability of the relative phase angles over time. If the distribution of the relative phase angles over time is steep, it results in a high resultant vector length (max value 1). If the distribution of the relative phase angle over time is not steep but broad or multimodal, it results in a low resultant vector length (min value 0). Consider S as a step and n as the nth step in the following formula: $$|R|=\left|\frac{1}{N}\sum _{n=1}^N{e}^{i\varphi S_n}\right|$$
Asynchrony(measured in ms)	This parameter is a measure of the timing expressed in milliseconds (ms) between the footfall and beat instants, i.e., the asynchrony between the beat and the footfall. While the phase angles express the relative differences between the steps and beats, the intervals between the steps and beats are absolute differences. In the below formula, St represents the time point where the step investigated takes place, and Bn is the beat at the time closest to the St. asynchrony = St−Bn
Tempo matching accuracy(measured in ms)	This parameter indicates the extent to which the overall tempo of the footfalls matches the overall tempo of the beats. Inter-beat deviation (IBD) was defined as a parameter that measures the tempo-matching accuracy, as expressed by the formula below, where n represents the nth step or beat. $$IBD=\frac{1}{N}\sum _{n=2}^N(\frac{(B_n-B_{(n-1)})-(S_n-S_{(n-1)})}{B_n-B_{(n-1)}})$$ The standard deviation of the IBD can also be calculated as a unit of variability of the tempo matching.
Detrended Fluctuation analysis (DFA)(measured by the scaling exponent alpha)	The DFA is a common mathematical method to analyse the dynamics of non-stationary time series. More specifically, it characterizes the fluctuation dynamics of the time series through looking into its scaling component alpha (Chen et al., 2002). It has been shown that in other physiological time series the current value possesses the memory of preceding values. This phenomenon is known as long-range correlations, long-term memory, long-range correlations and fractal process of 1/f noise. A healthy gait time series pattern consists of a fractal statistical persistent structure equivalent to a pure 1/f noise (Goldberger et al., 2002). Authors suggest that the analysis of this gives an insight into the neuro-physiological organization of neuro-muscular control and the entire locomotion system (Hausdorff, 2007). The 1/f noise is correlated with a scaling exponent alpha value between 0.5 and 1.0 (indicative of a walking pattern found in healthy gait time series). If alpha is ≤0.5, it signifies an anti-correlation, and is associated with unhealthy walking pattern (randomness). For details of calculating the scaling exponent alpha, the reader is referred to Chen et al. (2002) and Terrier et al. (2005). The underlying rationale of using this analysis method in gait is addressed in the discussion section of our review. The integrated time series of N is divided into boxes of equal length. Each box has a length “n” and in each box of length n, a least square line is fit on the data.
	The y- coordinate of the straight line segments is denoted by yn(k). The integrated time series y(k) is detrended by subtracting the local trend yn(k) in each box. The root mean square fluctuation in this integrated and detrended time analysis is calculated by: $$F(n)=\sqrt{\frac{1}{N}\sum _{k=1}^N{\left[y(k)-y_n(k)\right]}^2}$$ Thus, the fluctuations can be categorized by the scaling exponent (ALPHA), which is the slope of the line relating LogF(n) to log(n):
	F(n)∽nα