The Localized Subtraction Approach For EEG and MEG Forward Modeling

In FEM-based EEG and MEG source analysis, the subtraction approach has been proposed to simulate sensor measurements generated by neural activity. While this approach possesses a rigorous foundation and produces accurate results, its major downside is that it is computationally prohibitively expensive in practical applications. To overcome this, we developed a new approach, called the localized subtraction approach. This approach is designed to preserve the mathematical foundation of the subtraction approach, while also leading to sparse right-hand sides in the FEM formulation, making it efficiently computable. We achieve this by introducing a cut-off into the subtraction, restricting its influence to the immediate neighborhood of the source. In this work, this approach will be presented, analyzed, and compared to other state-of-the-art FEM right-hand side approaches. Furthermore, we discuss how to arrive at an efficient and stable implementation. We perform validation in multi-layer sphere models where analytical solutions exist. There, we demonstrate that the localized subtraction approach is vastly more efficient than the subtraction approach. Moreover, we find that for the EEG forward problem, the localized subtraction approach is less dependent on the global structure of the FEM mesh when compared to the subtraction approach. Additionally, we show the localized subtraction approach to rival, and in many cases even surpass, the other investigated approaches in terms of accuracy. For the MEG forward problem, we show the localized subtraction approach and the subtraction approach to produce highly accurate approximations of the volume currents close to the source. The localized subtraction approach thus reduces the computational cost of the subtraction approach to an extent that makes it usable in practical applications without sacrificing rigorousness and accuracy.