Power Dissipation and Surface Charge in EEG: Application to Eigenvalue Structure of Integral Operators

Objective: To demonstrate the role of surface charge and power dissipation in the analysis of EEG measurements. Methods: The forward EEG problem is formulated in terms of surface charge density. Using bounds based on power dissipation, the integral equations for forward solutions are shown to satisfy bounds on their eigenvalue structure. Results: We show that two physical variables, dissipated power and the accumulated charge at interfaces, can be used in formulating the forward problem. We derive the boundary integral equations satisfied by the charge and show their connection to the integral equations for the potential that are known from other approaches. We show how the dissipated power determines bounds on the range of eigenvalues of the integral operators that appear in EEG boundary element methods. Using the eigenvalue structure, we propose a new method for the solution of the forward problem, where the integral kernels are regularized by the exclusion of eigenvectors associated to a finite range of eigenvalues. We demonstrate the method on a head model with realistic shape. Conclusion: The eigenvalue analysis of the EEG forward problem is given a clear interpretation in terms of power dissipation and surface charge density. Significance: The use of these variables enhances our understanding of the structure of EEG, makes connection with other techniques and contributes to the development of new analysis algorithms.