Revisiting an analytical solution for the three-shell spherical human head model in electroencephalography

Abstract

The complete three-shell spherical human head model in electroencephalography (EEG) is revisited and an analytical solution of the forward problem is derived. The introduced geometrical model involves four concentric spheres that represent the successive interfaces between the cerebrum, the cerebrospinal fluid, the skull and the skin, which are characterized by different conductivities, while the outer environment is evidently the non-conductive air. The neuronal operation of the brain is considered to be represented by an equivalent and arbitrarily orientated electric dipole that is located in the inner sphere. The dipole source produces a bipolar primary current and the electric activity is initiated by means of the generated electric field, which is associated with the corresponding potential functions within each one of the conductive compartments of the model, inferring crucial information about EEG effects outside the head. The potentials formulae are obtained in compact fashion via the solution of a sequence of interconnected elliptic-type boundary value problems with Dirichlet and Neumann transmission conditions, where the consistent behavior of the fields in the brain and far away from the system has been taken into account. The efficiency of the suggested mathematical model is numerically implemented and the impact of the brain electric response to the exterior measurable potential field is demonstrated, implying a solid and sufficient head representation for the EEG forward problem.