In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method.

翻译：本文聚焦于高效灵活地模拟与非线性噪声泄漏积分发放（NNLIF）模型相关的Fokker-Planck方程，该方程反映了神经元网络的动态行为。我们采用Galerkin谱方法，通过构建满足复杂边界条件的变分形式对空间域进行离散化。此外，该变分形式中的边界条件仅包含零阶项，一阶条件被自然地纳入其中。这使得数值格式可进一步扩展至包含突触延迟与不应期状态的兴奋-抑制群体模型。同时，我们证明了该数值格式的一致性。通过精度测试、爆发事件与周期性振荡等实验结果，验证了所提方法的特性。