Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo System
Creating adequate mathematical models of processes in living nature is an important task of modern biophysics. Blood clotting, nerve impulse propagation, reduction of the heart muscle, the pattern-formation in nature are auto-wave processes. FitzHugh–Nagumo system of equations is used to describe th...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Yaroslavl State University
2016-10-01
|
| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/389 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850024078307491840 |
|---|---|
| author | A. A. Melnikova R. L. Argun |
| author_facet | A. A. Melnikova R. L. Argun |
| author_sort | A. A. Melnikova |
| collection | DOAJ |
| description | Creating adequate mathematical models of processes in living nature is an important task of modern biophysics. Blood clotting, nerve impulse propagation, reduction of the heart muscle, the pattern-formation in nature are auto-wave processes. FitzHugh–Nagumo system of equations is used to describe the auto-wave processes in active media. Such math problems are usually solved by numerical methods. The use of resource-intensive algorithms is required in the case of auto-wave solutions with sharp gradients. Therefore, it is appropriate to use the analytical methods for this type of problems. In this paper, the asymptotic method of contrast structures theory is used to obtain an approximate solution of a singularly perturbed system of FitzHugh–Nagumo type. The method allows to reduce the non-linear system of equations to a number of problems that can be solved analytically or with a stable numerical algorithm. This study presents the asymptotic approximation of a stationary auto-wave solution of the considered system. Additionally, this paper provides a formula that specifies the location of internal transition layers. The results were compared with the numerical solution. The application of contrast structures theory to the study of active media models can be used for analytical studies of other such systems, improving existing models and increasing the efficiency of the numerical calculations. |
| format | Article |
| id | doaj-art-bf3d3aae581c468bb2e8f8d030ffa79b |
| institution | DOAJ |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2016-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-bf3d3aae581c468bb2e8f8d030ffa79b2025-08-20T03:01:13ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-10-0123555956710.18255/1818-1015-2016-5-559-567325Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo SystemA. A. Melnikova0R. L. Argun1Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow 119991, Russian FederationLomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow 119991, Russian FederationCreating adequate mathematical models of processes in living nature is an important task of modern biophysics. Blood clotting, nerve impulse propagation, reduction of the heart muscle, the pattern-formation in nature are auto-wave processes. FitzHugh–Nagumo system of equations is used to describe the auto-wave processes in active media. Such math problems are usually solved by numerical methods. The use of resource-intensive algorithms is required in the case of auto-wave solutions with sharp gradients. Therefore, it is appropriate to use the analytical methods for this type of problems. In this paper, the asymptotic method of contrast structures theory is used to obtain an approximate solution of a singularly perturbed system of FitzHugh–Nagumo type. The method allows to reduce the non-linear system of equations to a number of problems that can be solved analytically or with a stable numerical algorithm. This study presents the asymptotic approximation of a stationary auto-wave solution of the considered system. Additionally, this paper provides a formula that specifies the location of internal transition layers. The results were compared with the numerical solution. The application of contrast structures theory to the study of active media models can be used for analytical studies of other such systems, improving existing models and increasing the efficiency of the numerical calculations.https://www.mais-journal.ru/jour/article/view/389asymptotic approximationsmall parametersingular perturbationinner transition layeractivator-inhibitor system |
| spellingShingle | A. A. Melnikova R. L. Argun Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo System Моделирование и анализ информационных систем asymptotic approximation small parameter singular perturbation inner transition layer activator-inhibitor system |
| title | Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo System |
| title_full | Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo System |
| title_fullStr | Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo System |
| title_full_unstemmed | Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo System |
| title_short | Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo System |
| title_sort | asymptotic approximation of the stationary solution with internal layer for fitzhugh nagumo system |
| topic | asymptotic approximation small parameter singular perturbation inner transition layer activator-inhibitor system |
| url | https://www.mais-journal.ru/jour/article/view/389 |
| work_keys_str_mv | AT aamelnikova asymptoticapproximationofthestationarysolutionwithinternallayerforfitzhughnagumosystem AT rlargun asymptoticapproximationofthestationarysolutionwithinternallayerforfitzhughnagumosystem |