A comprehensive numerical investigation of a coupled mathematical model of neuronal excitability
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| Format: | Preprint |
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2025
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| _version_ | 1866916590126104576 |
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| author | Gürbüz, Burcu Gökçe, Aytül Modanlı, Mahmut |
| author_facet | Gürbüz, Burcu Gökçe, Aytül Modanlı, Mahmut |
| contents | Being an example for a relaxation oscillator, the FitzHugh-Nagumo model has been widely employed for describing the generation of action potentials. In this paper, we begin with a biological interpretation of what the subsequent mathematical and numerical analyses of the model entail. The interaction between action potential variable and recovery variable is then revisited through linear stability analysis around the equilibrium and local stability conditions are determined. Analytical results are compared with numerical simulations. The study aims to show an alternative approach regarding Taylor polynomials and constructed difference scheme which play a key role in the numerical approach for the problem. The robustness of the schemes is investigated in terms of convergency and stability of the techniques. This systematic approach by the combination of numerical techniques provides beneficial results which are uniquely designed for the FitzHugh-Nagumo model. We describe the matrix representations with the collocation points. Then the method is applied in order to acquire a system of nonlinear algebraic equations. On the other hand, we apply finite difference scheme and its stability is also performed. Moreover, the numerical simulations are shown. Consequently, a comprehensive investigation of the related model is examined. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_18013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A comprehensive numerical investigation of a coupled mathematical model of neuronal excitability Gürbüz, Burcu Gökçe, Aytül Modanlı, Mahmut Numerical Analysis Dynamical Systems Neurons and Cognition 65L12, 65L60, 34D20 Being an example for a relaxation oscillator, the FitzHugh-Nagumo model has been widely employed for describing the generation of action potentials. In this paper, we begin with a biological interpretation of what the subsequent mathematical and numerical analyses of the model entail. The interaction between action potential variable and recovery variable is then revisited through linear stability analysis around the equilibrium and local stability conditions are determined. Analytical results are compared with numerical simulations. The study aims to show an alternative approach regarding Taylor polynomials and constructed difference scheme which play a key role in the numerical approach for the problem. The robustness of the schemes is investigated in terms of convergency and stability of the techniques. This systematic approach by the combination of numerical techniques provides beneficial results which are uniquely designed for the FitzHugh-Nagumo model. We describe the matrix representations with the collocation points. Then the method is applied in order to acquire a system of nonlinear algebraic equations. On the other hand, we apply finite difference scheme and its stability is also performed. Moreover, the numerical simulations are shown. Consequently, a comprehensive investigation of the related model is examined. |
| title | A comprehensive numerical investigation of a coupled mathematical model of neuronal excitability |
| topic | Numerical Analysis Dynamical Systems Neurons and Cognition 65L12, 65L60, 34D20 |
| url | https://arxiv.org/abs/2501.18013 |