Weak formulation and spectral approximation of a Fokker-Planck equation for neural ensembles
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909428272332800 |
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| author | Yan, Ling Zhang, Pei Wang, Yanli Zhou, Zhennan |
| author_facet | Yan, Ling Zhang, Pei Wang, Yanli Zhou, Zhennan |
| contents | In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10676 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak formulation and spectral approximation of a Fokker-Planck equation for neural ensembles Yan, Ling Zhang, Pei Wang, Yanli Zhou, Zhennan Numerical Analysis 35Q92, 65M70, 92B20 In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method. |
| title | Weak formulation and spectral approximation of a Fokker-Planck equation for neural ensembles |
| topic | Numerical Analysis 35Q92, 65M70, 92B20 |
| url | https://arxiv.org/abs/2412.10676 |