FrOsFHN Software Package for Quantitative and Qualitative Analysis of the FitzHugh-Nagumo Fractional Oscillator with Variable Memory
DOI:
https://doi.org/10.71310/pcam.6_70.2025.01Keywords:
FitzHugh-Nagumo fractional oscillator, oscillograms, phase trajectories, nonlocal explicit finite-difference scheme, algorithms, Python, PyCharm, bifurcation diagramAbstract
This article describes and provides examples of the FrOsFHN software package for the quantitative and qualitative analysis of a nonlinear FitzHugh-Nagumo oscillator with variable memory (the fractional FitzHugh-Nagumo oscillator (FrOsFHN)). Variable memory was taken into account in the FitzHugh-Nagumo oscillator model equation using fractional-order derivatives of the Gerasimov-Caputo type with respect to time. The quantitative analysis in the software package was implemented based on a numerical algorithm for a nonlocal explicit finite-difference scheme of first-order accuracy, and the qualitative analysis was implemented using algorithms for visualizing the numerical solution in the form of oscillograms, phase trajectories, and bifurcation diagrams. To conduct the quantitative analysis, the user was able to select the functional dependence of the fractional derivative orders on time: a linear decreasing function, a trigonometric function, or a decreasing exponential function. To construct bifurcation diagrams, we implemented the ability to consider a parallel version of the algorithm in addition to the sequential version, leveraging the computing resources of the computer’s central processor. The software package also features the ability to save calculations to a text file, as well as result graphs. The software package was written in Python using the PyCharm environment.
References
FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane // Biophysical Journal. – 1961. – Vol. 1, Issue 6. – P. 445-466. – doi: http://dx.doi.org/10.1016/S0006-3495(61)86902-6.
Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon // Proceedings of the IRE. – 1962. – Vol. 50, Issue 10. – P. 2061-2070. – doi: http://dx.doi.org/10.1109/JRPROC.1962.288235.
Hodgkin A.L., Huxley A.F. A quantitative description of membrane current and its application to conduction and excitation in nerve // J. Physiol.. – 1952. – Vol. 117, Issue 4. – P. 500-544. – doi: http://dx.doi.org/10.1113/jphysiol.1952.sp004764.
Kazarnikov A.V., Revina S.V. Monotonnaya neustoychivost' v sisteme FittsKH'yu-Nagumo s diffuziyey // Izvestiya vysshikh uchebnykh zavedeniy. Severo-Kavkazskiy region. Yestestvennyye nauki. – 2018. – №4(200). – C. 18-24
Nakhushev A.M. Drobnoye ischisleniye i yego primeneniye. – M.: Fizmatlit, 2003. – 272 s.
Lipko O.D. Mathematical model of propagation of nerve impulses with regard hereditarity // Vestnik KRAUNC. Fiziko-matematicheskie nauki. – 2017. – №1(17). – P. 33-43. – doi: http://dx.doi.org/10.18454/2079-6641-2017-17-1-33-43.
Lipko O.D., Parovik R.I. Some aspects of investigation of limit cycles of Fitzhugh-Nagumo oscillator with degree memory // Journal of Physics: Conference Series. – 2018. – Vol. 1141. – Art. 012125. – doi: http://dx.doi.org/10.1088/1742-6596/1141/1/012125.
Lipko O.D., Parovik R.I. The study of chaotic and regular regimes of the fractal oscillators FitzHugh-Nagumo // E3S Web of Conferences. – 2018. – Vol. 62. – Art. 02017. – doi: http://dx.doi.org/10.1051/e3sconf/20186202017.
Volterra V. Functional theory, integral and integro-differential equations. – New York: Dover Publications, 2005. – 288 p.
Parovik R.I. Khaoticheskiye i regulyarnyye rezhimy drobnykh ostsillyatorov. – Petropavlovsk-Kamchatskiy: KAMCHATPRESS, 2019. – 132 s.
Novozhenova O.G. Life and science of Alexey Gerasimov, one of the pioneers of fractional calculus in Soviet Union // Fractional Calculus and Applied Analysis. – 2017. – Vol. 20, Issue 3. – P. 790-809. – doi: http://dx.doi.org/10.1515/fca-2017-0040.
Caputo M., Fabrizio M. On the notion of fractional derivative and applications to the hysteresis phenomena // Meccanica. – 2017. – Vol. 52, Issue 13. – P. 3043-3052. – doi: http://dx.doi.org/10.1007/s11012-017-0652-y.
Alimova N.B. Matematicheskoye modelirovaniye avtokolebaniy neyrona v kletochnoy membrane s ispol'zovaniyem drobnoy modeli FittsKH'yu-Nagumo s funktsiyey intensivnosti razdrazhitelya // Vestnik KRAUNTS. Fiziko-matematicheskiye nauki. – 2024. – T. 48, №3. – C. 56-69. – doi: http://dx.doi.org/10.26117/2079-6641-2024-48-3-56-69.
Alimova N.B., Parovik R.I. Ostsillyator FittsKH'yu-Nagumo s peremennoy nasledstvennost'yu i vneshnim vozdeystviyem // Problemy vychislitel'noy i prikladnoy matematiki. – 2025. – №1(63). – C. 5-16. – doi: http://dx.doi.org/10.71310/pcam.1_63.2025.01.
Sun H., Chang A., Zhang Y., Chen W. A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications // Fractional Calculus and Applied Analysis. – 2019. – Vol. 22, Issue 1. – P. 27-59. – doi: http://dx.doi.org/10.1515/fca-2019-0003.
Patnaik S., Hollkamp J.P., Semperlotti F. Applications of variable-order fractional operators: a review // Proc. R. Soc. A. – 2020. – Vol. 476, Issue 2234. – Art. 20190498. – doi: http://dx.doi.org/10.1098/rspa.2019.0498.
Shaw Z.A. Learn Python the Hard Way. – 5th ed. – Boston: Addison-Wesley Professional, 2024. – 352 p.
Van Horn B.M. II, Nguyen Q. Hands-On Application Development with PyCharm: Build Applications like a Pro with the Ultimate Python Development Tool. – 2nd ed. – Birmingham: Packt Publishing, 2023. – 652 p.
Li´enard A. Etude des oscillations entretenues // Revue g´en´erale de l’´electricit´e. – 1928. – Vol. 23. – P. 901-912, 946-954.
Pskhu A.V., Rekhviashvili S.Sh. Analysis of forced oscillations of a fractional oscillator // Technical Physics Letters. – 2018. – Vol. 44, Issue 12. – P. 1218-1221. – doi: http://dx.doi.org/10.1134/S1063785019010164.
Parovik R.I. Analiz dobrotnosti vynuzhdennykh kolebaniy drobnogo lineynogo ostsillyatora // Zhurnal tekhnicheskoy fiziki. – 2020. – T. 90, №7. – C. 1059-1063. – doi: http://dx.doi.org/10.21883/JTF.2020.07.49436.233-19.
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