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

本课题研究由电磁场问题和包含集总参数的复杂动态非线性网络组成的耦合系统的不确定性和灵敏度分析。特别是在具有大量变量的耦合系统中，确定最重要的模型参数是非常重要的。目的是在经颅磁刺激（TMS）和神经团块建模（NMM）的范围内解决这个问题。经颅磁刺激过程中产生的电场可以通过有限元等数值方法来预测，但这取决于分配给神经组织的电导率，这是不确定的。此外，NMM还可以模拟外部刺激导致的神经活动。它们是根据电路用微分方程组来描述的。这些模型涉及大量参数，可以声明为高维模型。由于个体差异及其内在性质，这些参数具有很高的可变性。这些变量在被研究的系统中传播，并直接影响系统的预测输出。当多个参数同时经历一定程度的不确定性时，这一事实变得更加明显，这使得很难区分个别影响及其引起方式。在这个项目中，考虑到两个系统参数的不确定性，打算将TMS过程中产生的电磁场与NMM耦合。因此，建立了两个系统之间的随机耦合模型。方法基础是广义多项式混沌技术（gPC）。由于NMM是多稳定和高维的，因此在这些方面进一步发展gPC方法是目标。由TMS中的感应问题、耦合模型和NMM组成的整个系统的随机分析需要高效的算法。为了保证计算效率，将模型降阶技术与稀疏网格和约简基多项式相结合是目标。所获得的方法是通用的，可以应用于各种工程问题，如电路设计和无损检测，其中涉及的不确定性和可能危害安全。在对技术和自动化有强烈需求的时代，安全是最重要的。这就需要高效的方法来分析复杂和耦合系统中不确定性的影响。

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.