Selection Procedures for the Largest Lyapunov Exponent in Gait Biomechanics

Abstract

The present study was aimed at investigating the effectiveness of the Wolf et al. (LyE_W) and Rosenstein et al. largest Lyapunov Exponent (LyE_R) algorithms to differentiate data sets with distinctly different temporal structures. The three-dimensional displacement of the sacrum was recorded from healthy subjects during walking and running at two speeds; one low speed close to the preferred walking speed and one high speed close to the preferred running speed. LyE_R and LyE_W were calculated using four different time series normalization procedures. The performance of the algorithms were evaluated based on their ability to return relative low values for slow walking and fast running and relative high values for fast walking and slow running. Neither of the two algorithms outperformed the other; however, the effectiveness of the two algorithms was highly dependent on the applied time series normalization procedure. Future studies using the LyE_R should normalize the time series to a fixed number of strides and a fixed number of data points per stride or data points per time series while the LyE_W should be applied to time series normalized to a fixed number of data points or a fixed number of strides.

Keywords: Dynamics; Locomotion; Nonlinear analysis; Variability; Walking.

Figure 1:

State space reconstruction of the anterior-posterior, mediolateral and vertical displacement of the sacrum marker for the four conditions. This representation illustrates that the three signals substantially changes dynamics when changing gait mode between walk and run and when increasing or decreasing speed. Qualitatively evaluated, the trajectory dynamics during running at the low speed and walking at the high speed appear more disorganized and random compared to walking at low speed and running at high speed. This confirms the validity of the raised criteria.

Figure 2:

Mean and standard deviation of the LyE calculated from the anterior-posterior (AP), mediolateral (ML) and vertical (Vert) displacement of the sacrum marker during the four conditions using the Rosenstein et al. algorithm and the four different normalization procedures. 1 above the bars indicates significantly higher LyE for the sacrum displacements during running at the low speed compared to walking at the low speed, 4 above the bars indicates significantly higher LyE for the sacrum displacements during walking at the high speed compared to walking at the low speed. N=10 except for walk 2.46 m/s where N=7.

Figure 3:

Mean and standard deviation of the LyE calculated from the anterior-posterior (AP), mediolateral (ML) and vertical (Vert) displacement of the sacrum marker during the four conditions using the Wolf et al. algorithm and the four different normalization procedures. 2 above the bars indicates significantly higher LyE for the sacrum displacements during walking at the high speed compared to running at the high speed, 4 above the bars indicates significantly higher LyE for the sacrum displacements during walking at the high speed compared to walking at the low speed. N=10 except for walk 2.46 m/s where N=7.