The effects of motor unit synchronization on the power spectrum of the electromyogram

A realistic model for two synchronized motor unit action potential trains (MUAPT) is presented in which the variability of the time difference between corresponding action potentials (hereafter denoted by delay) is taken into account. Specifically, this delay is modeled as a continuous random variable that may assume both positive and negative values. Expressions are derived for the auto- and cross-power spectra of two such trains using their relations with the auto- and cross-correlation functions, respectively, with which they form Fourier transform pairs. The results show that the auto- and the cross-power spectra of two such synchronized MUAPTs differ from the auto- and the cross-spectra of two independent MUAPTs. The contribution of the statistics of the interpulse intervals to one of the auto-power spectra is smaller and the cross-power spectra no longer reduce to a Dirac sigma-function at the origin but are now determined by the other auto-power spectrum and by the Fourier transform of the density function associated with the time difference between corresponding action potentials. As a consequence of this change in the cross-power spectra synchronization leads to an absolute increase of power at low frequencies and to a relative decrease of power at high frequencies. The results are then generalized to electromyograms (EMG) composed of more than just two MUAPTs and illustrated with simulated power spectra with which the theory shows excellent agreement.