Revisiting convolutive blind source separation for identifying spiking motor neuron activity: From theory to practice

Objective: Identifying the activity of motor neurons (MNs) non-invasively is possible by decomposing signals from muscles, e.g., surface electromyography (EMG) or ultrasound. The theoretical background of MN identification is convolutive blind source separation (cBSS), and different algorithms have been developed and validated. Yet, the existence and identifiability of inverse solutions and the corresponding estimation errors are not fully understood. Further, the guidelines for selecting appropriate parameters are often built on empirical observations, limiting the translation to clinical applications and other modalities. Approach: We revisited the cBSS model for MN identification, augmented it with new theoretical insights and derived a framework that can predict the existence of inverse solutions. This framework allows the quantification of estimation errors due to the imperfect inversion of the motor unit action potentials (MUAP), noise sources, and the ill-conditioning of the inverse problem. To bridge the gap between theory and practice, we used computer simulations. Main results: (1) Increasing the similarity of MUAPs or correlation between spike trains increases the bias for detecting high amplitude MUs. (2) The optimal objective function depends on the expected spike amplitude, spike amplitude statistics and the amplitude of background spikes. (3) There is some wiggle room for MN detection given non-stationary MUAPs. (4) There is no connection between MUAP duration and extension factor, in contrast to previous guidelines. (5) Source quality metrics like the silhouette score (SIL) or the pulse-to-noise ratio (PNR) are highly correlated with a source's objective function output. (6) SIL is superior to PNR. Significance: These findings will guide cBSS algorithm developments tailored to MN identification and clinical application translation.