Mathematics > Numerical Analysis
[Submitted on 10 Sep 2021 (v1), last revised 27 Jun 2022 (this version, v2)]
Title:A structure preserving numerical scheme for Fokker-Planck equations of structured neural networks with learning rules
View PDFAbstract: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.
Submission history
From: Qing He [view email][v1] Fri, 10 Sep 2021 05:02:35 UTC (582 KB)
[v2] Mon, 27 Jun 2022 03:56:42 UTC (881 KB)
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