Time-varying delay estimators for measuring muscle fiber conduction velocity from the surface electromyogram
Graphical abstract
Introduction
The estimation of time delays is an important topic in several biomedical applications. For example, muscle fiber conduction velocity (MFCV) can be measured by estimating the time delay between surface EMG signals recorded by electrodes aligned with the fiber direction (for a recent review, see [1]). The measure of MFCV has been shown to be relevant in the diagnosis of pathologies [2], fatigue [3] and in the detection of neuromuscular system adjustments due to exercise [4]. A major objective of these studies is to understand the motor unit (MU) recruitment strategies of the motor control. Most of these studies were conducted during static activity. However, it has been demonstrated that the MU recruitment was task-dependent [5] suggesting that this recruitment was different between dynamic and static contractions. Any results extrapolation from static to dynamic studies remains de facto speculative. Therefore, there is a real necessity to estimate the MFCV during dynamic activities which compose a main part of common activities (walking, running, cycling or jumping). The most commonly applied methods for estimating MFCV are based on the assumption of signal stationarity and constant delay within the processing interval, which is usually in the range 250 ms–1 s [6]. This assumption is valid for static contractions of moderate force but not during dynamic tasks or contractions consisting in rapid changes in the force expressed [7]. Indeed, in dynamic conditions, the EMG signal is highly non-stationary and the time delay between recording sites may vary rapidly over time due to recruitment and/or derecruitment of motor units with different MFCVs. In order to properly analyze the MFCV evolutions during these tasks, it is necessary to develop processing techniques that are adapted to non-stationary conditions. In the non-stationary case, the MFCV estimation is a time-varying delay estimation problem. Only few previous studies addressed the problem of estimating time delays from surface EMG signals recorded during dynamic tasks. Farina et al. [8] adapted a maximum likelihood estimator to short analysis intervals for its use in dynamic contractions, however the resulting approach does not provide an instantaneous delay estimation. Non-stationarity implies time-varying statistics of the data. Consequently, the analysis methods should provide local estimates. Time-frequency representations (TFR) and adaptive filtering techniques are among those approaches which are suited for non-stationary signals analyses. For example, time-varying delay estimators have been developed for turbulent flow analysis based on the wavelet cross-power spectrum [9]. Adaptive filtering techniques for time-varying delay estimation have also been theoretically developed and investigated in terms of convergence rates and accuracy [10], [11]. In this paper, we address the problem of estimating the time-varying delay between two EMG signals with the purpose of measuring MFCV in non-static conditions. For that, we propose to develop a set of delay estimation methods based on TFRs that are the most original part of this study. These methods are theoretically analyzed and modified to be adjusted to the EMG signal characteristics. In order to evaluate the quality of the methods proposed, an alternative method based on an adaptive filtering procedure already used for time-varying delay measure [12] is presented and tested. The different approaches are compared through Monte–Carlo simulations in order to classify them and to determine if their performance statistics are sufficient for practical applications. Moreover, we confront these methods to experimental data collected in dynamic exercise conditions in order to test the ability of the methods to track the MFCV in the range of physiological acceptable values. The study presents for the first time a set of approaches for instantaneous delay estimation from EMG signals.
Section snippets
Problem definition
We adopt a surface EMG model that allows the generation of signals with a well defined time-varying delay. The EMG signal is the sum of motor unit action potentials and can be considered as a Gaussian process when a sufficient number of contributions are present [13]. The power spectral density of this signal can be modeled by an analytical parameterized shape [14]. Moreover, the signal is contaminated by instrumentation and recording noise sources, which are uncorrelated to the EMG. Thus, the
Simulated data
The methods have to be tested in realistic conditions regarding the SNR level. The selection of the noise level range has been achieved by comparison of an experimental raw surface EMG with simulated noisy EMGs. The experimental EMG is shown in Fig. 3 with simulated ones that have been generated at different SNR values. The level of noise in experimental signals may fit with the 20–40 dB cases in the simulations, which is the range of SNR investigated in the following.
Fig. 4, Fig. 5, Fig. 6,
Discussion
Time-varying delay has been well tracked using the methods developed in the present study. In optimal simulated conditions with the simple model, statistics showed low errors with respect to the chosen variation law pattern: 0.01 ± 0.05 samples averaged on the TFR/TSR methods, at 30 dB vs 3 samples averaged delay value on the 5 s variation law, with a range from 1.7 to 5.1 samples. For the more realistic model, this error grew to 0.04 ± 0.17 samples which was acceptable regarding the absolute
Conclusion
We have proposed and compared on simulated and experimental signals several methods for estimating time-varying delays from two-channel recordings. The methods have been applied to surface EMG signals during dynamic exercise for the purpose of measuring MFCV. This kind of exercise is a nonstationary condition for EMG signal. It is the reason why TFR, TSR, and adaptive filtering were proposed for the analysis of non-stationary signals and time-varying delays. In the present study, these
Acknowledgment
The present work is embedded in a French national project ECOTECH (www.echostechsan.org) and is supported by the French National Agency for research under the contract Nr ANR-12-TECS-0020.
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