In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. While the equations exhibit diversified solutions from various numerical observations, the properties of solutions are not yet completely understood, and by far there has been no rigorous numerical analysis work concerning such models. We propose a conservative and conditionally positivity preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models.

在这项工作中，我们关注与神经网络的非线性噪声泄漏积分与发射模型相关的Fokker-Planck方程。由于在微观层面上的跳跃机制，此类Fokker-Planck方程具有非常规的结构：将边界通量传输到特定的内部点。尽管方程式显示了来自各种数值观测的多种解，但解的性质尚未完全理解，到目前为止，尚未有关于此类模型的严格的数值分析工作。我们为这些Fokker-Planck方程提出了一个保守的，有条件的正性保留方案，并且证明了在线性情况下，半离散方案满足离散的相对熵估计，它基本上与唯一已知的长时间渐近解性质匹配。我们还提供了广泛的数值测试来验证方案特性，并进行了几组数值实验，包括有限时间爆炸，收敛到平衡以及捕获变体模型的时间周期解。