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.

翻译：在基于有限元法的脑电图和脑磁图源分析中，减法方法被提出用于模拟神经活动产生的传感器测量值。尽管该方法具有严谨的理论基础并能生成精确结果，其主要缺点是在实际应用中计算成本过高。为克服这一局限，我们开发了一种名为"局部化减法方法"的新方法。该方法旨在保留减法方法的数学基础，同时使有限元公式中的右端项保持稀疏性，从而实现高效计算。我们通过在减法过程中引入截断机制，将其影响限制在源的直接邻域内。本文将对这一方法进行阐述、分析，并与其他前沿的有限元法右端项处理方法进行比较。此外，我们讨论了如何实现高效稳定的实施方案。在存在解析解的多层球体模型中进行了验证，结果表明局部化减法方法的计算效率远高于传统减法方法。研究发现，对于脑电图正演问题，局部化减法方法相比减法方法对有限元网格整体结构的依赖性更弱。同时，局部化减法方法在精度方面可与研究中的其他方法相媲美，并在许多情况下甚至超越它们。针对脑磁图正演问题，局部化减法方法与减法方法均能对源附近体电流产生高精度的近似。因此，局部化减法方法在保持严谨性和精确性的前提下，将减法方法的计算成本降低至可实际应用的水平。