A third-order approximate solution of the EEG forward problem in four-shell ellipsoidal geometry

We present a solution of the electroencephalographic (EEG) forward problem for the case when the head´s geometry is modeled using a four-shell ellipsoidal geometry and the source is a current dipole. The EEG potentials generated by this forward model have been previously approximated with elliptic integrals and harmonics up to second-order. Here, we evaluate the solution up to the third-order terms and compare the corresponding EEG against those generated using the second-order approximation and a realistic model solved by the boundary element method (BEM). A comparison is also performed in terms of the bias in estimating the location of brain sources when using the second and third-order forward solutions. Our simulations show that the third-order approximation achieves EEG magnitudes that are closer to realistic values (computed by the BEM model) in comparison to the second-order approximation. However, the third-order approximation did not offer a significant improvement in estimating the location of dipole sources, as the mean bias was the same as the one produced by the second-order approximation.