Uncertainty- and Sensitivity Analysis of Coupled Systems Composed of an Electromagnetic Field Problem and a Dynamic Nonlinear Network Using Spectral Methods

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The present project addresses the uncertainty and sensitivity analysis of coupled systems composed of an electromagnetic field problem and a complex dynamic and nonlinear network including lumped parameters. Especially in case of coupled systems with a high number of variables it is of great interest to determine the most important model parameters.It is the goal to address this question in the scope of transcranial magnetic stimulation (TMS) and neural mass modelling (NMM). The electric field induced during TMS can be predicted by means of numerical techniques like the finite element method but depends on the electrical conductivities assigned to the neural tissues, which are subject to uncertainties. Besides that, NMM enable to model neural activity in consequence of external stimulation. They are described in accordance to electrical circuits by systems of differential equations. Those models involve a high number of parameters and can be declared as high-dimensional. Due to individual differences and their intrinsic nature, the parameters are subject to high variabilities. Those variabilities propagate through the system under investigation and directly affect the predicted output of it. This fact becomes even more apparent when multiple parameters undergo a certain amount of uncertainty at the same time, which makes it difficult to distinguish between individual effects and how they were caused. In this project, it is intended to couple the electromagnetic field induced during TMS with an NMM, taking into account the uncertainties of the parameters in both systems. Therefore, a stochastic coupling model between both systems is developed. The methodological basis rests on the generalized polynomial chaos technique (gPC). Since NMM are multistable and high-dimensional, it is the goal to further develop the gPC method in these terms. The stochastic analysis of the overall system composed of the induction problem in TMS, the coupling model and the NMM necessitates efficient algorithms. In order to ensure computational efficiency, it is the goal to utilize model order reduction techniques together with sparse grids and reduced basis polynomials. The obtained methodology is of general use and can be applied for a variety of engineering problems, such as electrical circuit design and nondestructive testing, where uncertainties are involved and could harm safety. In times characterized by a strong demand on technology and automation, safety is of the utmost priority. This underpins the need of highly efficient methods to analyse the impact of uncertainties in complex and coupled systems.