A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration
A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration
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DOI：
10.1016/j.jcp.2021.110195
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发表时间：
2019-11
期刊：
影响因子：
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通讯作者：
Jingwei Hu;Jian‐Guo Liu;Yantong Xie;Zhennan Zhou
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文献类型：
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作者：
Jingwei Hu;Jian‐Guo Liu;Yantong Xie;Zhennan Zhou
在这项工作中，我们关注的是与非线性噪声泄漏积分点火模型相关的Fokker-Planck方程。由于微观层面的跳跃机制，这种Fokker-Planck方程被赋予了一种非常规的结构：将边界通量传输到特定的内部点。虽然这些方程从不同的数值观测中得到了不同的解，但解的性质还没有完全被理解，到目前为止，还没有关于这些模型的严格的数值分析工作。对于这些Fokker-Planck方程，我们提出了一种保守且有条件保正的格式，并证明了在线性情况下，半离散格式满足离散相对熵估计，这基本上符合已知的唯一长时间渐近解的性质。我们还提供了大量的数值测试来验证方案的性质，并进行了几组数值实验，包括有限时间爆破、收敛到平衡和捕获变量模型的时间段解。
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.