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Review on solving the forward problem in EEG source analysis

机译：解决脑电信号源分析中正向问题的综述

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Background The aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes. Methods While other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field. Results It starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method. Conclusion Solving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.
机译：背景脑电图（EEG）源定位的目的是找到负责感兴趣的EEG波的大脑区域。它包括解决正向和反向问题。通过从给定的电源开始并计算电极上的电势可以解决正向问题。这些评估对于解决反问题是必要的，反问题定义为寻找脑电源，该脑源负责在EEG电极上测得的电势。方法尽管其他评论对正向和反向问题都进行了广泛的总结，但这篇评论文章侧重于解决正向问题的不同方面，旨在供该研究领域的新手使用。结果首先，它着重于脑电图的产生器：锥体神经元顶突的突触后电位。这些细胞产生细胞外电流，可以通过泊松微分方程以及Neumann和Dirichlet边界条件进行建模。这些电流在其中流动的隔室可以是各向异性的（例如头骨和白质）。在三壳球头模型中，存在解析表达式来解决正向问题。在过去的二十年中，研究人员试图在从3D医学图像获得的逼真的头部模型中求解泊松方程，这需要数值方法。相互比较了以下方法：边界元法（BEM），有限元法（FEM）和有限差分法（FDM）。在后两种方法中，可以方便地引入各向异性导电隔室。然后，重点将放在在脑电图源本地化中使用互惠性。引入它是为了加快向前计算，此处对每个电极位置而不是每个偶极位置进行计算。利用FEM和FDM求解泊松方程对应于求解大型稀疏线性系统。解决这些稀疏线性系统需要迭代方法。讨论了以下迭代方法：连续过度松弛，共轭梯度法和代数多重网格法。结论在过去的几十年中，解决前向问题的记录非常丰富。过去使用简化的球形头部模型，而如今使用成像模态的组合来准确描述头部模型的几何形状。在现实地描述头部模型的形状，组织类型的异质性以及现实地确定电导率方面已经做出了努力。然而，体内电导率值的确定和验证仍然是该领域中的重要课题。此外，必须对磁头模型的所有参数和数值技术对正解问题的影响进行更多的研究。

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