Finite element representation of the EEG forward problem with multipole expansion
- Multipolentwicklung zur Lösung des EEG Vorwärtsproblems mittels Finiter Elemente
Hanrath, Anne; Grasedyck, Lars (Thesis advisor); Wolters, Carsten H. (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2019
Brain activity imaging using Electroencephalography (EEG) is a vital aspect of cognitive science and neurology. However, the reliability of brain activity localization based on EEG data has been a hard endeavor so far. In this dissertation, we propose a new, reliable mathematical algorithm for this purpose. First, we will provide an overview of the biological background of methodology in cognitive science. To localize brain activity from EEG data, we need to solve an elliptic partial differential equation (PDE) with a mathematical dipole as a right--hand side. For this, we use Finite Element Methods (FEM). However, FE cannot discretize a mathematical dipole. We solve this issue with regularization. Secondly, we list contemporary regularization approaches. The most successful one - the Venant Approach - has so far lacked a mathematical and theoretical foundation. This dissertation is expanding that approach both theoretically and practically. Thirdly, we formulate a new approach using the idea of the Venant Approach. It originates from the idea to substitute the mathematical dipole with a distribution of monopoles. Monopoles are placed exactly on FE nodes, using the corresponding monopole loads as FE values. The substitution of a mathematical dipole with a monopole distribution results in a new PDE. With assumptions for both monopole loads and electrical conductivity in the PDE, we prove that there is a unique solution for the PDE's ultra-weak formulation using the duality method. We can derive a more intuitive way to compute monopoles with the help of multipole expansion. This involves expanding the potential of the mathematical dipole and the potential of the monopole distribution and comparing the different terms. This new approach is called Multipole Approach. For the Multipole Approach, we can give an error bound with an error decay in the direction of the boundary for a small sphere of uniform conductivity around the dipole location. Fourthly and finally, we employ the most popular approaches, together with the new Multiple Approach in numerical computations. First, we test with unique conductivity in 2D. Second, we employ the approaches on spherical models with different layers of distinct conductivity in 3D. Finally, we compute a goal function scan on one 3D spherical model. We study the Multipole Approach in different configurations and find the optimal one. In this optimal case, the new approach outperforms the contemporary approaches effortlessly.