Long-time Correlations in Electromyography Signals

We have previously reported that the mean-square displacement calculated from electromyography time series of low back muscles exhibit a plateau-like behavior for intermediate times [$50 \, \mbox{ms} < t < 0.5 \, \mbox{s}$], so that $\left< [x_ {t} - x_{0} ]^{2} \right> \sim t^{0}$. This behavior is unexpected, and indicates the presence of long-time correlations in the signal. For fractal Brownian motion, the Hurst exponent calculated from the mean-square displacement and the exponent from the spectral density $P ( f) \sim 1/f^{\alpha}$, $\alpha = 2 H + 1$. For the EMG time series $y^{0}_{i} = x_{i}$, we have generated iterated time series, $y_{i}^{n+1} = [y_{2 i }^{n} + y_{2i+1}]/2$, and have calculated the corresponding time correlation functions, $C^{n} ( t) = \left< x_{i+ t}^{n} x_{i}^{n}\right>/\left<(x_{i}^{n})^{2} \right>$. We find that the correlation functions converge to a simple limit, $C(0) = 1$, $C(1) = -0.5$ and $C(n) =0 $ for $n \geq 2$. This limit is consistent with the plateau behavior of the mean- square displacement. We discuss the connection between the behavior of the iterated correlation functions and the properties of the spectrum.