In this work, we consider the Fokker-Planck equation of the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model for neuron networks. Due to the firing events of neurons at the microscopic level, this Fokker-Planck equation contains dynamic boundary conditions involving specific internal points. To efficiently solve this problem and explore the properties of the unknown, we construct a flexible numerical scheme for the Fokker-Planck equation in the framework of spectral methods that can accurately handle the dynamic boundary condition. This numerical scheme is stable with suitable choices of test function spaces, and asymptotic preserving, and it is easily extendable to variant models with multiple time scales. We also present extensive numerical examples to verify the scheme properties, including order of convergence and time efficiency, and explore unique properties of the model, including blow-up phenomena for the NNLIF model and learning and discriminative properties for the NNLIF model with learning rules.

翻译：本文研究神经元网络非线性噪声漏电整合-发放（NNLIF）模型的Fokker-Planck方程。由于微观尺度上神经元的发放事件，该Fokker-Planck方程包含涉及特定内点的动态边界条件。为了有效求解该问题并探究未知量的性质，我们在谱方法框架下构建了一种灵活的数值格式，该格式能够精确处理动态边界条件。该数值格式在测试函数空间的适当选取下具有稳定性，且具有渐进保持特性，并易于扩展到含多时间尺度的变体模型。我们还给出了大量数值算例来验证格式的性质，包括收敛阶和时间效率，并探讨了模型的独特性质，包括NNLIF模型的爆破现象以及含学习规则NNLIF模型的学习与判别特性。