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arXiv · 2109.04667

A structure preserving numerical scheme for Fokker-Planck equations of structured neural networks with learning rules

Qing He·Jingwei Hu·Zhennan Zhou
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Abstract

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.

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Qing He, Jingwei Hu, Zhennan Zhou. 2021-09-10. A structure preserving numerical scheme for Fokker-Planck equations of structured neural networks with learning rules. https://doi.org/10.48550/arxiv.2109.04667

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