In this paper, we consider a time-dependent singularly perturbed differential-difference equation with small shifts arising in the field of neuroscience. The terms containing the delay and advance parameters are approximated by using the Taylor's series expansion. The continuous problem is semi-discretized using the Crank–Nicolson finite difference method in the time direction on uniform mesh and quadratic B-spline collocation method in the space direction on exponentially graded mesh. The method is shown to be second-order uniformly convergent in space and time direction. Theoretical estimates are carried out which support the obtained numerical experiments.

中文翻译：

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神经元活动建模中出现的时间相关奇摄动微分-差分方程的二次 B 样条配置方法

在本文中，我们考虑了神经科学领域中出现的具有小偏移的时间相关的奇摄动微分-差分方程。包含延迟和提前参数的项通过使用泰勒级数展开来近似。连续问题在时间方向上使用 Crank-Nicolson 有限差分法在均匀网格上使用二次 B 样条配置方法在空间方向上对指数级网格进行半离散化。该方法被证明在空间和时间方向上是二阶一致收敛的。进行了支持所获得的数值实验的理论估计。