EMG Probability Density Function: A New Way to Look at EMG Signal Filling From Single Motor Unit Potential to Full Interference Pattern

Abstract

An analytical derivation of the EMG signal's amplitude probability density function (EMG PDF) is presented and used to study how an EMG signal buildsup, or fills, as the degree of muscle contraction increases. The EMG PDF is found to change from a semi-degenerate distribution to a Laplacian-like distribution and finally to a Gaussian-like distribution. We present a measure, the EMG filling factor, to quantify the degree to which an EMG signal has been built-up. This factor is calculated from the ratio of two non-central moments of the rectified EMG signal. The curve of the EMG filling factor as a function of the mean rectified amplitude shows a progressive and mostly linear increase during early recruitment, and saturation is observed when the EMG signal distribution becomes approximately Gaussian. Having presented the analytical tools used to derive the EMG PDF, we demonstrate the usefulness of the EMG filling factor and curve in studies with both simulated signals and real signals obtained from the tibialis anterior muscle of 10 subjects. Both simulated and real EMG filling curves start within the 0.2 to 0.35 range and rapidly rise towards 0.5 (Laplacian) before stabilizing at around 0.637 (Gaussian). Filling curves for the real signals consistently followed this pattern (100% repeatability within trials in 100% of the subjects). The theory of EMG signal filling derived in this work provides (a) an analytically consistent derivation of the EMG PDF as a function of motor unit potentials and motor unit firing patterns; (b) an explanation of the change in the EMG PDF according to degree of muscle contraction; and (c) a way (the EMG filling factor) to quantify the degree to which an EMG signal has been built-up.

Validation of the Filling Factor Index to Study the Filling Process of the Semg Signal in the Quadriceps

“…1). In our preliminary experiments, we verified that the filling factor increases its value as the sEMG progressively fills up with MUPs (and MUP overlapping increases), and the PDF shape evolves towards a Gaussian (Navallas et al, 2023). To illustrate this process, three sEMG signals selected at different degrees of filling are shown in Fig.…”

“…Then, the EMG filling factor (FF) is calculated as the ratio between m 1 2 and m 2 as follows (Navallas et al, 2023), or, in other words, the filling factor is the ratio between the squared ARV and the RMS:…”

“…The process of sEMG filling was analyzed by calculating the filling factor of successive non-overlapping segments of the sEMG signal. The duration of the segments was 0.7 s, similar to that utilized by Navallas et al (2023). Then, the filling factor values were plotted (and studied) against the rectified sEMG mean (i.e., the above-defined m 1 ), as done previously by other authors (Nandedkar et al, 2020;Navallas et al, 2023).…”

“…The duration of the segments was 0.7 s, similar to that utilized by Navallas et al (2023). Then, the filling factor values were plotted (and studied) against the rectified sEMG mean (i.e., the above-defined m 1 ), as done previously by other authors (Nandedkar et al, 2020;Navallas et al, 2023). For various reasons the filling factor was studied against the rectified sEMG mean, instead of as a function of force.…”

“…In this respect, it has been observed that, at low contraction forces, the sEMG PDF approximates to a Laplacian due to the little overlapping between MUPs of different motor units, whereas, as force increases, overlapping also increases and the PDF converges to a Gaussian (Nazarpour et al, 2007(Nazarpour et al, , 2013Ayachi et al, 2014). In a recent study of our group we presented a new index, the EMG filling factor to quantify the degree to which an EMG signal has been filled (Navallas et al, 2023). The filling factor is grounded on an analytical derivation of the EMG PDF based on the MUP waveforms and firing rates, and thus this index informs about the shape of the sEMG PDF distribution, and also serves as a measure of the overlapping between MUPs (Fig.…”