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Spherical Splines and Average Referencing in Scalp Electroencephalography

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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.

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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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Google Scholar 

  • Desmedt, J.E. and Tomberg, C. Topographic analysis inbrain mapping can be compromised by the average reference. BrainTopogr., 1990, 3(1): 35–42.

    CAS  Google Scholar 

  • Duchon, J. Interpolation des fonctions de deuxvariables suivant le principe de la flexion des plaques minces. RAIRO Anal. Num., 1976, 10: 5–12.

    Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • Geselowitz, D.B. The zero of potential. IEEEEng. Med. Biol. Mag., 1998, 17(1): 128–132.

    Article  CAS  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • Hjorth, B. An on-line transformation of EEG scalppotentials into orthogonal source derivations. Electroenceph.Clin. Neurophysiol., 1975, 39: 526–530.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • Nunez, P.L. Electric fields of the brain: Theneurophysics of EEG. New York, Oxford University Press, 1981.

    Google Scholar 

  • Nunez, P.L. and Srinivasan, R. Electric fields ofthe brain: The neurophysics of EEG, 2nd edition. New York, OxfordUniversity Press, 2006.

    Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • Perrin, F., Pernier, J., Bertrand, O. and Echallier, J.F. Corrigenda: EEG 02274. Electroenceph. Clin. Neurophysiol., 1990, 76: 565.

    Article  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  Google Scholar 

  • Srinivasan, R., Tucker, D.M. and Murias, M. Estimating thespatial Nyquist of the human EEG. Behav. Res. Methods, Instrum.Comput., 1998, 30: 8–19.

    Google Scholar 

  • Stok, C.J. The influence of model parameters onEEG/MEG single dipole source estimation. IEEE Trans. Biomed.Eng., 1987, 34(4): 289–296.

    CAS  PubMed  Google Scholar 

  • Sun, M. An efficient algorithm for computingmultishell spherical volume conductor models in EEG dipole sourcelocalization. IEEE Trans. Biomed. Eng., 1997, 44(12): 1243–1252.

    Article  CAS  PubMed  Google Scholar 

  • 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.

    Article  CAS  Google Scholar 

  • Wahba, G. and Wendelberger, J. Some new mathematicalmethods for variational objective analysis using splines and crossvalidation. Mon. Weather Rev., 1980, 108: 1122–1143.

    Article  Google Scholar 

  • Wahba, G. Spline interpolation and smoothing onthe sphere. SIAM J. Sci. Stat. Comput., 1981, 2(1): 5–16.

    Article  Google Scholar 

  • Wahba, G. Erratum: Spline interpolation andsmoothing on the sphere. SIAM J. Sci. Stat. Comput. 1982, 3(3): 385–386.

    Article  Google Scholar 

  • Yao, D.A. method to standardize a reference ofscalp EEG recordings to a point at infinity. Physiol. Meas.,2001, 22(4): 693–711.

    Article  CAS  PubMed  Google Scholar 

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  1. Center for Mind and Brain, University of California, Davis, CA, 95616, USA

    Thomas C. Ferree

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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

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