In this work, we are concerned with a Fokker-Planck equation related to the nonlinear noisy leaky integrate-and-fire model for biological neural networks which are structured by the synaptic weights and equipped with the Hebbian learning rule. The equation contains a small parameter $\varepsilon$ separating the time scales of learning and reacting behavior of the neural system, and an asymptotic limit model can be derived by letting $\varepsilon\to 0$, where the microscopic quasi-static states and the macroscopic evolution equation are coupled through the total firing rate. To handle the endowed flux-shift structure and the multi-scale dynamics in a unified framework, we propose a numerical scheme for this equation that is mass conservative, unconditionally positivity preserving, and asymptotic preserving. We provide extensive numerical tests to verify the schemes' properties and carry out a set of numerical experiments to investigate the model's learning ability, and explore the solution's behavior when the neural network is excitatory.

中文翻译：

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具有学习规则的结构化神经网络 Fokker-Planck 方程的结构保持数值方案

在这项工作中，我们关注与生物神经网络的非线性噪声泄漏积分和发射模型相关的 Fokker-Planck 方程，该模型由突触权重构成并配备 Hebbian 学习规则。该方程包含一个小参数 $\varepsilon$ 分隔神经系统的学习和反应行为的时间尺度，并且可以通过让 $\varepsilon\to 0$ 推导出渐近极限模型，其中微观准静态状态和宏观演化方程通过总射速耦合。为了在统一框架中处理固有的通量位移结构和多尺度动力学，我们为该方程提出了一种质量保守、无​​条件保持正性和渐进保持的数值方案。