Spherical splines and average referencing in scalp electroencephalography

EEG analysis and interpretation are affected by the reference electrode. Average referenced potentials are used widely to approximate the potentials relative to infinity, but estimates of the average surface potential are prone to errors due to incomplete sampling of the scalp surface. Even if the electrode density is high, this arises by not sampling the inferior scalp surface. This paper shows analytically how the spherical splines represent the average surface potential. It also shows that, for spline orders m >= 3, the interpolating function is well approximated by its large-m limit, weighting near and distant electrodes with opposite signs. Together these motivate the hypothesis that spherical splines permit a better estimate of the potentials relative to infinity than the discrete average computed over superior scalp electrodes. it tests this hypothesis using numerical simulations in a four-sphere head model with single- and many-dipole sources, and variations in spline order, electrode number and head model parameters. The results confirm that the spherical splines yield a better estimate of the potentials relative to infinity, provided the electrode sampling density is adequate.