Shown first is the equivalence between the multiple expansion (ME) of the brain electrical generator and the spherical harmonic spectra (SHS) of the potential generated by the electrical generator in an infinite volume conductor. Based on the equivalence, the SHS and the spatial filters which connect the SHS with the ME are deduced, in a concentric 3 sphere conductor and for the 5 EEG source mappings. They are cortical potential mapping (CPM), scalp Laplacian mapping (LM), pseudo-cortical potential mapping (PCPM), equivalent dipole layer mapping (EDM) and equivalent charge layer mapping (ECM). The theoretical simulation study of the spatial filters and mappings indicate that all 5 mappings provide higher resolution imaging maps of brain electrical activity than the scalp potential map. In the inverse problem, a spherical spline fit algorithm is provided to reconstruct the SHS of the scalp recording potential, and then the SHS and maps of the 5 mappings are reconstructed by utilizing the spatial filters and the SHS of the scalp potential. The results indicate that the correlativity order between a reconstructed map and the actual cortical potential map is CPM greater than or equal to EDM >PCPM >LM >ECM. An empirical VEP data study shows that any one of the 5 mappings also provides higher spatial resolution than the scalp potential map. (C) 2000 Elsevier Science Ireland Ltd. All rights reserved.