arXiv 1702.075852017-02-24math.NAcs.NA

一种用于含传输延迟的神经场方程的空间-时间有限元方法

A space-time finite element method for neural field equations with transmission delays

Bolyai Institute, University of Szeged（塞格德大学博莱伊研究所）；ELI-ALPS, ELI-HU Ltd（ELI-ALPS，ELI-HU有限公司）；Mathematics of Computational Science Group, Department of Applied Mathematics, University of Twente（特温特大学应用数学学院计算科学数学组）；Applied Analysis, Department of Applied Mathematics, University of Twente（特温特大学应用数学学院应用分析）

M. Polner, J. J. W. van der Vegt, S. A. van Gils

发表信息：: SIAM Journal on Scientific Computing 39(5), 2017
DOI：: 10.1137/16M1085024

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AI总结：

本文提出并分析了一种新的空间-时间有限元方法，用于求解含传输延迟的神经场方程，该方法通过不连续时间和连续空间基函数（dGcG-FEM）有效处理空间依赖延迟，提供了先验误差分析和多个神经场模型的应用演示。

摘要：

我们提出并分析了一种新的空间-时间有限元方法，用于求解含传输延迟的神经场方程。这些系统的数值处理在文献中很少见，目前在空间域和涉及的函数上存在诸多限制，如连通性和延迟函数。使用空间-时间离散化，基函数在时间上不连续而在空间上连续（dGcG-FEM），是处理空间依赖延迟的自然方法，这对许多神经场应用至关重要。本文详细描述了用于神经延迟方程的空间-时间dGcG-FEM算法，包括先验误差分析。我们展示了dGcG-FEM算法在多个神经场模型中的应用，包括具有非均匀核的问题。

英文摘要：

We present and analyze a new space-time finite element method for the solution of neural field equations with transmission delays. The numerical treatment of these systems is rare in the literature and currently has several restrictions on the spatial domain and the functions involved, such as connectivity and delay functions. The use of a space-time discretization, with basis functions that are discontinuous in time and continuous in space (dGcG-FEM), is a natural way to deal with space-dependent delays, which is important for many neural field applications. In this article we provide a detailed description of a space-time dGcG-FEM algorithm for neural delay equations, including an a-priori error analysis. We demonstrate the application of the dGcG-FEM algorithm on several neural field models, including problems with an inhomogeneous kernel.

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