Recently, high-density surface EMG electrode grids and multi-channel amplifiers became available for non-invasive recording of human motor units (MUs). We present a way to decompose surface EMG signals into MU firing patterns, whereby we concentrate on the importance of two-dimensional spatial differences between the MU action potentials (MUAPs). Our method is exemplified with high-density EMG data from the vastus lateralis muscle of a single subject. Bipolar and Laplacian spatial filtering was applied to the monopolar raw signals. From the single recording in this subject six different simultaneously active MUs could be distinguished using the spatial differences between MUAPs in the direction perpendicular to the muscle fiber direction. After spike-triggered averaging, 125-channel two-dimensional MUAP templates were obtained. Template-matching allowed tracking of all MU firings. The impact of spatial information was measured by using subsets of the MUAP templates, either in parallel or perpendicular to the muscle fiber direction. The use of one-dimensional spatial information perpendicular to the muscle fiber direction was superior to the use of a linear array electrode in the longitudinal direction. However, to detect the firing events of the MUs with a high accuracy, as needed for instance for estimation of firing synchrony, two-dimensional information from the complete grid electrode appears essential.