Summary
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arfken, G.B. and Weber, H.J. MathematicalMethods for Physicists. Academic Press, 1995.
Baumann, S.B., Wonzy, D.R., Kelly, S.K. and Meno, F.M. Theelectrical conductivity of human cerebrospinal fluid at bodytemperature. IEEE Trans. Biomed. Eng., 1997, 44(3): 220–223.
Bertrand, O., Perrin, F. and Pernier, J. A theoreticaljustification of the average reference in topographic evokedpotential studies. Electroenceph. Clin. Neurophysiol., 1985, 62(6): 462–464.
Burger, H.C. and van Milaan, J.B. Measurements of thespecific resistance of the human body to direct current. ActaMed. Scand. Fasc. VI, 1943, 114: 584–607.
Desmedt, J.E. and Tomberg, C. Topographic analysis inbrain mapping can be compromised by the average reference. BrainTopogr., 1990, 3(1): 35–42.
Duchon, J. Interpolation des fonctions de deuxvariables suivant le principe de la flexion des plaques minces. RAIRO Anal. Num., 1976, 10: 5–12.
Fletcher, E.M., Kussmaul, C.L. and Mangun, G.R. Estimation of interpolation errors in scalp topographic mapping. Electroenceph. Clin. Neurophysiol., 1996, 98: 422–434.
Gencer, N.G., Williamson, S.J., Gueziec, A. and Hummel, R. Optimal reference electrode selection for electric source imaging. Electroenceph. Clin. Neurophysiol., 1996, 99(2): 163–173.
Geselowitz, D.B. The zero of potential. IEEEEng. Med. Biol. Mag., 1998, 17(1): 128–132.
Hamalainen, M.S. and Sarvas, J. Realistic conductorgeometry model of the human head for interpretation ofneuromagnetic data. IEEE Trans. Biomed. Eng., 1989, 36:165–171.
Hjorth, B. An on-line transformation of EEG scalppotentials into orthogonal source derivations. Electroenceph.Clin. Neurophysiol., 1975, 39: 526–530.
Junghofer, M., Elbert, T., Tucker, D.M. and Braun, C. Thepolar average reference effect: A bias in estimating the headsurface integral in EEG recording. Clin. Neurophysiol., 1999, 110(6): 1149–1155.
Law, S.K., Nunez, P.L. and Wijesinghe, R.S. High-resolution EEG using spline generated surface Laplacians onspherical and ellipsoidal surfaces. IEEE Trans. Biomed. Eng., 1993, 40(2): 145–153.
Lehmann, D., Ozaki, H. and Pal, I. Averaging of spectralpower and phase via vector diagram best fits without referenceelectrode or reference channel. Electroencephalogr. Clin.Neurophysiol., 1986, 64(4): 350–363.
Nunez, P.L. Electric fields of the brain: Theneurophysics of EEG. New York, Oxford University Press, 1981.
Nunez, P.L. and Srinivasan, R. Electric fields ofthe brain: The neurophysics of EEG, 2nd edition. New York, OxfordUniversity Press, 2006.
Nunez, P.L., Srinivasan, R., Westdorp, A.F., Wijesinghe, R.S., Tucker, D.M., Silberstein, R.B. and Cadusch, P.J. EEG coherency.I: Statistics, reference electrode, volume conduction, Laplacians, cortical imaging, and interpretation at multiple scales. Electroencephalogr. Clin. Neurophysiol., 1997, 103(5): 499–515.
Orekhova, E.V., Wallin, B.G. and Hedstrom, A. Modificationof the average reference montage: Dynamic average reference. J.Clin. Neurophysiol., 2002, 19(3): 209–218.
Perrin, F., Pernier, J., Bertrand, O., Giard, M.H. and Echallier, J.F. Mapping of scalp potentials by surface spline interpolation. Electroenceph. Clin. Neurophysiol., 1987, 66: 75–81.
Perrin, F., Pernier, J., Bertrand, O. and Echallier, J.F. Spherical splines for scalp potential and current density mapping. Electroenceph. Clin. Neurophysiol., 1989, 72: 184–187.
Perrin, F., Pernier, J., Bertrand, O. and Echallier, J.F. Corrigenda: EEG 02274. Electroenceph. Clin. Neurophysiol., 1990, 76: 565.
Press, W.H., Tuekolsky, S.A., Vetterling, W.T. and Flannery, B.P. Numerical recipes in C. Cambridge University Press, 1992.
Rush, S. and Blanchard, R.R. Skull physical andelectrical characteristics relevant to the distribution fromelectrodes on the scalp. Proceedings of the 19th ACEMB, 1966.
Rush, S. and Driscoll, D.A. Current distribution in thebrain from surface electrodes. Anesthesia analgesia, 1968, 47(6): 717–723.
Salu, Y., Cohen, L.G., Rose, D., Sato, S., Kufta, C. and Hallett, M. An improved method for localizing electric brain dipoles. IEEE Trans. Biomed. Eng., 1990, 37: 699–705.
Soong, A.C.K., Lind, J.C., Shaw, G.R. and Koles, Z.J. Systematic comparisons of interpolation techniques in topographic brain mapping. Electroenceph. Clin. Neurophysiol., 1993, 87:185–195.
Soufflet, L., Toussaint, M., Luthringer, R., Gresser, J., Minot, R. and Macher, J.P. A statistical evaluation of the main interpolation methods applied to 3-dimensional EEG mapping. Electroenceph. Clin. Neurophys., 1991, 79: 393–402.
Srinivasan, R., Tucker, D.M. and Murias, M. Estimating thespatial Nyquist of the human EEG. Behav. Res. Methods, Instrum.Comput., 1998, 30: 8–19.
Stok, C.J. The influence of model parameters onEEG/MEG single dipole source estimation. IEEE Trans. Biomed.Eng., 1987, 34(4): 289–296.
Sun, M. An efficient algorithm for computingmultishell spherical volume conductor models in EEG dipole sourcelocalization. IEEE Trans. Biomed. Eng., 1997, 44(12): 1243–1252.
Tomberg, C., Neol, P., POzaki, I. and Desmedt, J.E. Inadequacy of the average reference for the topographic mapping offocal enhancements of brain potentials. Electroenceph. Clin.Neurophys., 1990, 77(4): 259–265.
Wahba, G. and Wendelberger, J. Some new mathematicalmethods for variational objective analysis using splines and crossvalidation. Mon. Weather Rev., 1980, 108: 1122–1143.
Wahba, G. Spline interpolation and smoothing onthe sphere. SIAM J. Sci. Stat. Comput., 1981, 2(1): 5–16.
Wahba, G. Erratum: Spline interpolation andsmoothing on the sphere. SIAM J. Sci. Stat. Comput. 1982, 3(3): 385–386.
Yao, D.A. method to standardize a reference ofscalp EEG recordings to a point at infinity. Physiol. Meas.,2001, 22(4): 693–711.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ferree, T.C. Spherical Splines and Average Referencing in Scalp Electroencephalography. Brain Topogr 19, 43–52 (2006). https://doi.org/10.1007/s10548-006-0011-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10548-006-0011-0