**Apendix**

#### **RQA formulation**

RQA is a nonlinear data analysis tool, that quantifies the recurrent data in the phase space tra‐ jectory of a time series data set, here EMG signals [15, 18]. For RQA analysis each time series is defined as the EMG activity recorded for each muscle during each gait cycle depicted by "*d* → ":

$$\vec{d} = \begin{bmatrix} d\mathbf{1}, d\mathbf{2}, \dots, d\mathbf{N} \end{bmatrix}^T \tag{9}$$

Where N represents the total number of collected data points in each bin (i.e. EMG data). In our study, N decreased as the gait speed increased, owing to the shorter gait cycles. In the next step the phase space vector is calculated based on the embedding dimension (m=6) found by False Nearest Neighbor method and time delay (τ= 0.005 second equal to 5 EMG data samples) found by Mutual Information technique [33], [34]. The equation of the phase space can be constructed as [15]:

SYNERGOS: A Multiple Muscle Activation Index http://dx.doi.org/10.5772/56168 147

$$\hat{\vec{s}}\_{\vec{j}} = \left[d\_{\vec{l}}, d\_{\vec{l}+\tau}, d\_{\vec{l}+2\tau}, \dots, d\_{\vec{l}+\begin{pmatrix} m-1\\m-1\end{pmatrix}\tau}\right]^T \tag{10}$$

resulting in a vector with Ns=N−(m−1)τ elements. Based on the above phase space vector a Distance Matrix (DM) is defined. The elements of the DM are the Euclidian norm of the distance of each of two generated elements of the phase space vector [13].

$$\mathbf{DM}\_{\begin{subarray}{c}\vec{i},\vec{j}\end{subarray}}^{\mathfrak{m}} = \left\| \vec{s}\_{\vec{l}} \cdot \vec{s}\_{\vec{j}} \right\| \qquad \qquad \vec{s}\_{\vec{j}} \in \mathsf{R}^{\mathfrak{M}} \qquad i,j = 1,\ldots,N\_{\vec{s}}\tag{11}$$

Here ℝ indicates the real numbers. In the next step, RQA assesses the proximity of each element in the DM with other elements. This proximity is tested based on a predefined threshold radius (εi ). In this study, the threshold radius is found by an algorithm to keep the recurrence rate less than 2 percent [18] resulting in 2<*ε*i<10 units of the normalized DM by the maximum element in the original DM. Next, the outcome of this assessment is converted into a binary matrix as representing the approximately close elements while 0 indicates the "not-close" elements:

, ( ) , , *<sup>m</sup> <sup>i</sup> <sup>m</sup> <sup>H</sup> ij i ij* e **R** = e**DM** (12)

In which H represents the Heaviside function defined as

EMG signal generated by neuromuscular activation, which might result in limited exposure of actual muscular activities to the RQA, hence dismissing 'true' subtle changes in the EMG signal [16, 32] Thus no further filtering was applied on the EMG signals used in this study. Additionally the effect of noise on the % DET as the inputs of SYNERGOS was also minimized by the careful selection of initial parameters (i.e. embedding dimension, delay, and radius) to

In conclusion, we have proposed a nonlinear multiple-muscle coactivation quantification tool, "SYNERGOS", that is sensitive to changes in both the magnitude and the timing of muscle activity caused by environmental or task related changes. In the future, this method may have application as a diagnostic tool for the evaluation of the therapeutic interventions in individ‐ uals with neuromuscular disorders, or those in rehabilitation settings. Further development, validation, and application of the SYNERGOS measure in clinical populations are currently being explored. Additionally, assessment of SYNERGOS's intra- and inter-day reliability is

We would like to extend our sincere thanks to all the University of Houston students who participated in the study. Additionally, we are thankful to Mr. Chris Arellano, Mr. Marius

RQA is a nonlinear data analysis tool, that quantifies the recurrent data in the phase space tra‐ jectory of a time series data set, here EMG signals [15, 18]. For RQA analysis each time series is defined as the EMG activity recorded for each muscle during each gait cycle depicted by "*d*

Where N represents the total number of collected data points in each bin (i.e. EMG data). In our study, N decreased as the gait speed increased, owing to the shorter gait cycles. In the next step the phase space vector is calculated based on the embedding dimension (m=6) found by False Nearest Neighbor method and time delay (τ= 0.005 second equal to 5 EMG data samples) found by Mutual Information technique [33], [34]. The equation of the phase space can be

<sup>r</sup> (9)

, ,, 1 2 *<sup>T</sup> <sup>d</sup>* = ¼ é ù ë û *dd dN*

→ ":

Dettmer, and Ms. Azadeh Khorram for their help with the data collection.

ensure the integrity of the algorithm (see "RQA consideration").

also underway.

**Apendix**

**RQA formulation**

constructed as [15]:

**Acknowledgements**

146 Electrodiagnosis in New Frontiers of Clinical Research

$$H(\mathbf{x}) = \begin{cases} 0 & \mathbf{x} < \mathbf{0} \\ 1 & \mathbf{x} \ge \mathbf{0} \end{cases} \tag{13}$$

Equation 12 can be summarized into equation 14 as

$$\mathbf{R}\_{\mathbf{i},\mathbf{j}} = \begin{cases} 0 & dm\mathbf{i}\otimes dm\mathbf{j} \\ 1 & dm\mathbf{i}\equiv dm\mathbf{j} \end{cases} \text{ i.e.} \begin{aligned} \mathbf{i}, \mathbf{j} = 1, \dots, N\_{\mathbf{s}} \end{aligned} \tag{14}$$

in which *dmi* and *dmj* are two elements on the DM matrix [15]. All elements compared with itself (i.e. i = j) results in the recurrence matrix element of 1. Recurrence plots which visualize the recurrence matrix can be generated based on the frequency distribution of the recurrent points (non-zero elements in recurrence matrix). To calculate the % DET the noncumulative frequency distribution of the constructed diagonal lines (recurrent points) in the Recurrence matrix is defined as

$$P^{\mathcal{E}}\left(l\right) = \left\{l\_1; i = 1, 2, \dots, Nq\right\} \tag{15}$$

[36, 37, 39, 40], therefore the latter algorithm has the advantage of demonstrating the nature

To test the null hypotheses, discriminating statistics were applied to investigate the changes in the dynamical pattern of surrogate data. These statistics should be sensitive to higher order nonlinearity of the signal [36, 37]. In this study, the null hypotheses were generated based on the changes in the approximate entropy (ApEn) and % DET of the surrogate data. ApEn is a single quantity by which the regularity of a signal can be measured [58, 59]. ApEn has been shown to classify underlying complexity of the signals while no significant changes in frequency and amplitude parameters were detected [40-42]. ApEn rages from 0 to 2 for completely predictable signals (i.e. sine wave) to white Gaussian noise respectively. A value of ApEn=2 indicates complete uncertainty in prediction of future behavior of a dynamical system. Therefore, in surrogate testing the expected outcome is a significant increase in ApEn value while a significant drop in % DET as a result of random shuffling procedure [35, 38]. ApEn value for both original EMG signals and surrogate data were calculated by applying the parameter settings of embedding-dimension of m=2 and r= 0.2 × standard deviation of the signal. In addition, after conducting the RQA analysis on the original EMG signals, a Matlab script was used to perform the RQA on the surrogate data using the same parameter settings ( m=6 and τ= 0.005 second). The script increased the radius to obtain similar level of % REC (the percentage of recurrent points in the recurrence plot graphs) for each set of muscles per

Finally, the ApEn and % DET of surrogate data were statistically compared to the ApEn and

= - = (17)

indicates ApEn or % DET statistic from original

SYNERGOS: A Multiple Muscle Activation Index

http://dx.doi.org/10.5772/56168

149

and *SDsurrogate* denote the average and standard

( ) ( ) *Qoriginal surrogate* / *SD* 1 *i to n <sup>i</sup> <sup>i</sup> surrogate <sup>i</sup>*

*i*

deviation of computed ApEn and % DET of the surrogate series. *φ<sup>i</sup>* indicates the amount of change in the ApEn and % DET of the original data in the scale of standard deviations. To reject the null hypothesis a minimum *φ<sup>i</sup>* >2 is required to obtain a 5% significance level

Nonlinear data analysis techniques are capable of revealing subtle changes in dynamical systems that may be ignored during linear data analysis. However they require more sophis‐ ticated analysis procedures and are generally more time consuming and costly, therefore the application of such techniques should be justified especially during clinical measurements. In the current study, three algorithms were used to test the state of nonlinearity in the EMG signals. The first two algorithms (time shuffled and FT) confirmed the fact that the recorded EMG signals contain higher order nonlinear dynamics. The use of the third surrogate testing method (i.e. IAAFT algorithm) expands our understanding from the nature of the dynamical

gait cycle and the % DET was calculated base on the modified radius.

 m

Where n is the number of muscles and *Qoriginal*

EMG for each muscle per gait cycle. *<sup>μ</sup>*¯(*surrogate*)

% DET of original EMG signal by defining the following statistics [37, 40]:

of nonlinearity in the signal.

j

(normality assumed) [39].

Where *Nl* represents the number of diagonal lines with the length of *l <sup>i</sup>* [17]. Due to the increase in the deterministic pattern of EMG signal resulted from increasing motor unit firing rate during more intense activities (i.e. running) *Nl* increased for longer diagonal lines and decreased for shorter lines. Finally, % DET was calculated based on equation (16).

$$\%DET = \frac{\sum\_{l=l\_{\rm min}}^{N} lP^{\mathcal{E}} \left(l\right)}{\sum\_{i,j}^{N} \mathbb{R}\_{i,j}^{m,\mathcal{E}\_i}} \tag{16}$$

in which *l min* is the minimum number of recurrent point in a diagonal line required to define a line [15]. *l min* =1 represents the condition generated by the tangential motion of phase space trajectory [15] that is not indicating the systematic determinism of the recorded EMG signal. In this study, *l min* =3 was chosen to demonstrate the deterministic pattern of the space trajectory based on the recommendation of previous studies [16, 18]. As the denominator of the equation (16) indicates the total recurrent points, % DET measures the proportion of recurrent points that define recurrent diagonal lines longer than lmin representing the determinism or pre‐ dictability of the dynamical system, i.e. EMG signal.

#### **Shuffled Surrogate Tests of EMG**

To obtain a one-tailed significance level of α = 0.05, nineteen (M=19) surrogate data series out of 20 should reject the null hypothesis (*<sup>M</sup>* <sup>=</sup> *<sup>K</sup> <sup>α</sup>* −1 in which K=1). The results of the surrogate testing were evaluated using a one-tail significance level because we had a directional hypothesis that predicted a reduction in the determinism of the surrogate signals after shuffling the original EMG [35, 36, 40, 42]. In the first series, temporally independent surrogate data were generated by random shuffling of the time ordering of EMG data which destroyed any time synchronization and correlation in the original data while saving statistical properties such as the mean and standard deviation [35, 38]. In this step, rejecting the null hypothesis that the EMG signals are originated from white noise is evidence of the existence of a dynamical system in the EMG signal [37]. Next, a phase randomized surrogate algorithm [37] was used to shuffle the original data by these three steps: 1) determining the Fourier transformation of the EMG signals 2) randomization of the phase of Fourier transform 3) applying the inverse Fourier transform to obtain a surrogate time series. The goal of this test is to reject the null hypothesis that the EMG signal has a linearly correlated Gaussian noise pattern. Iterated amplitude adjusted Fourier transform (IAAFT) was the third algorithm to generate surrogate data series [36, 37, 39]. IAAFT algorithm generates surrogate data, which resemble the rank ordering and power spectrum of the original EMG signals. Rejecting the null hypothesis by using IAAFT algorithm can be an indicator of deterministic chaos in the original time series [36, 37, 39, 40], therefore the latter algorithm has the advantage of demonstrating the nature of nonlinearity in the signal.

*P l li N* ( ) { ; 1,2, , *i l*}

in the deterministic pattern of EMG signal resulted from increasing motor unit firing rate

, ,

trajectory [15] that is not indicating the systematic determinism of the recorded EMG signal.

based on the recommendation of previous studies [16, 18]. As the denominator of the equation (16) indicates the total recurrent points, % DET measures the proportion of recurrent points that define recurrent diagonal lines longer than lmin representing the determinism or pre‐

To obtain a one-tailed significance level of α = 0.05, nineteen (M=19) surrogate data series out

testing were evaluated using a one-tail significance level because we had a directional hypothesis that predicted a reduction in the determinism of the surrogate signals after shuffling the original EMG [35, 36, 40, 42]. In the first series, temporally independent surrogate data were generated by random shuffling of the time ordering of EMG data which destroyed any time synchronization and correlation in the original data while saving statistical properties such as the mean and standard deviation [35, 38]. In this step, rejecting the null hypothesis that the EMG signals are originated from white noise is evidence of the existence of a dynamical system in the EMG signal [37]. Next, a phase randomized surrogate algorithm [37] was used to shuffle the original data by these three steps: 1) determining the Fourier transformation of the EMG signals 2) randomization of the phase of Fourier transform 3) applying the inverse Fourier transform to obtain a surrogate time series. The goal of this test is to reject the null hypothesis that the EMG signal has a linearly correlated Gaussian noise pattern. Iterated amplitude adjusted Fourier transform (IAAFT) was the third algorithm to generate surrogate data series [36, 37, 39]. IAAFT algorithm generates surrogate data, which resemble the rank ordering and power spectrum of the original EMG signals. Rejecting the null hypothesis by using IAAFT algorithm can be an indicator of deterministic chaos in the original time series

å **R**

*N lP l*

*N m i ij ij*

( )

e

e

*min* is the minimum number of recurrent point in a diagonal line required to define

*min* =1 represents the condition generated by the tangential motion of phase space

*min* =3 was chosen to demonstrate the deterministic pattern of the space trajectory

represents the number of diagonal lines with the length of *l*

decreased for shorter lines. Finally, % DET was calculated based on equation (16).

% ,

*DET l lmin*

<sup>å</sup> <sup>=</sup> <sup>=</sup>

= =¼ (15)

*<sup>i</sup>* [17]. Due to the increase

(16)

increased for longer diagonal lines and

*<sup>α</sup>* −1 in which K=1). The results of the surrogate

e

during more intense activities (i.e. running) *Nl*

148 Electrodiagnosis in New Frontiers of Clinical Research

dictability of the dynamical system, i.e. EMG signal.

of 20 should reject the null hypothesis (*<sup>M</sup>* <sup>=</sup> *<sup>K</sup>*

**Shuffled Surrogate Tests of EMG**

Where *Nl*

in which *l*

a line [15]. *l*

In this study, *l*

To test the null hypotheses, discriminating statistics were applied to investigate the changes in the dynamical pattern of surrogate data. These statistics should be sensitive to higher order nonlinearity of the signal [36, 37]. In this study, the null hypotheses were generated based on the changes in the approximate entropy (ApEn) and % DET of the surrogate data. ApEn is a single quantity by which the regularity of a signal can be measured [58, 59]. ApEn has been shown to classify underlying complexity of the signals while no significant changes in frequency and amplitude parameters were detected [40-42]. ApEn rages from 0 to 2 for completely predictable signals (i.e. sine wave) to white Gaussian noise respectively. A value of ApEn=2 indicates complete uncertainty in prediction of future behavior of a dynamical system. Therefore, in surrogate testing the expected outcome is a significant increase in ApEn value while a significant drop in % DET as a result of random shuffling procedure [35, 38].

ApEn value for both original EMG signals and surrogate data were calculated by applying the parameter settings of embedding-dimension of m=2 and r= 0.2 × standard deviation of the signal. In addition, after conducting the RQA analysis on the original EMG signals, a Matlab script was used to perform the RQA on the surrogate data using the same parameter settings ( m=6 and τ= 0.005 second). The script increased the radius to obtain similar level of % REC (the percentage of recurrent points in the recurrence plot graphs) for each set of muscles per gait cycle and the % DET was calculated base on the modified radius.

Finally, the ApEn and % DET of surrogate data were statistically compared to the ApEn and % DET of original EMG signal by defining the following statistics [37, 40]:

$$\left| \varphi\_{\dot{\imath}} \right| = \left| \mathsf{Q}\_{original} - \overline{\mu} \left( {surrogate} \right)\_{\dot{\imath}} \right| / \left| \mathrm{SD}\_{\left(surrogate\right)} \right|\_{\dot{\imath}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (17)$$

Where n is the number of muscles and *Qoriginal* indicates ApEn or % DET statistic from original EMG for each muscle per gait cycle. *<sup>μ</sup>*¯(*surrogate*) *i* and *SDsurrogate* denote the average and standard deviation of computed ApEn and % DET of the surrogate series. *φ<sup>i</sup>* indicates the amount of change in the ApEn and % DET of the original data in the scale of standard deviations. To reject the null hypothesis a minimum *φ<sup>i</sup>* >2 is required to obtain a 5% significance level (normality assumed) [39].

Nonlinear data analysis techniques are capable of revealing subtle changes in dynamical systems that may be ignored during linear data analysis. However they require more sophis‐ ticated analysis procedures and are generally more time consuming and costly, therefore the application of such techniques should be justified especially during clinical measurements. In the current study, three algorithms were used to test the state of nonlinearity in the EMG signals. The first two algorithms (time shuffled and FT) confirmed the fact that the recorded EMG signals contain higher order nonlinear dynamics. The use of the third surrogate testing method (i.e. IAAFT algorithm) expands our understanding from the nature of the dynamical nonlinearity. In our investigation, the existence of deterministic patterns measured by % DET is a key to the SYNERGOS equation. Therefore the use of third surrogate algorithm, IAAFT was justified. In previous studies amplitude adjusted Fourier transform (AAFT) algorithms were used to investigate the nature of the EMG signal [40] however it has been argued that for short and highly correlated data series the AAFT algorithm may result in flatness of power spectrums [36] therefore IAAFT algorithm was introduced to overcome such a bias by iteratively correcting the deviations in the power spectrum. [36, 39].

[9] Del SantoF., et al., Recurrence quantification analysis of surface EMG detects changes in motor unit synchronization induced by recurrent inhibition. Exp Brain Res,

SYNERGOS: A Multiple Muscle Activation Index

http://dx.doi.org/10.5772/56168

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