najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2019-10-6-623-626najo-755Research ArticleMATHEMATICSМАТЕМАТИКАMetric graph version of the FitzHugh–Nagumo modelMetric graph version of the FitzHugh–Nagumo modelFedorovЕ. G.FedorovE. G.

Kronverkskii, 49, St. Petersburg, 197101

fedevg95@mail.ruPopovА. I.PopovA. I.

Kronverkskii, 49, St. Petersburg, 197101

popov239@gmail.comPopovI. Y.PopovI. Y.

Kronverkskii, 49, St. Petersburg, 197101

popov1955@gmail.comITMO UniversityITMO University201913082025106623626Copyright © Fedorov E.G., Popov A.I., Popov I.Y., 20252025Fedorov Е.G., Popov А.I., Popov I.Y.Fedorov E.G., Popov A.I., Popov I.Y.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/755

The FitzHugh–Nagumo model on a metric graph is studied. System of delayed differential equations is used to model a pair of FitzHughNagumo excitable systems with time-delayed fast threshold modulation coupling. The model can be used for description of signal transmission in different nanostructures, microsystems or neural networks. The effect of time delay on the impulse transmission is studied.

The FitzHugh–Nagumo model on a metric graph is studied. System of delayed differential equations is used to model a pair of FitzHughNagumo excitable systems with time-delayed fast threshold modulation coupling. The model can be used for description of signal transmission in different nanostructures, microsystems or neural networks. The effect of time delay on the impulse transmission is studied.

FitzHugh–Nagumo systemtime-delayed couplingtravelling wavemetric graphneural networkFitzHugh–Nagumo systemtime-delayed couplingtravelling wavemetric graphneural networkThis work was partially financially supported by the Government of the Russian Federation (grant 08-08), grant 16-11-10330 of Russian Science FoundationThis work was partially financially supported by the Government of the Russian Federation (grant 08-08), grant 16-11-10330 of Russian Science FoundationReferencesRosenblum M.G., Pikovsky A.S. Controlling synchronization in an ensemble of globally coupled oscillators. Physical Review Letters, 2004, 92, P. 114102.Rosenblum M.G., Pikovsky A.S. Controlling synchronization in an ensemble of globally coupled oscillators. Physical Review Letters, 2004, 92, P. 114102.Gassel M., Glatt E., Kaiser F. Delay-sustained pattern formation in subexcitable media. Physical Review E, 2008, 77, P. 066220.Gassel M., Glatt E., Kaiser F. Delay-sustained pattern formation in subexcitable media. Physical Review E, 2008, 77, P. 066220.Rankovic D. Bifurcations of FitzHugh–Nagumo excitable systems with chemical delayed coupling. Matematicki Vesnik, 2011, 63, P. 103114.Rankovic D. Bifurcations of FitzHugh–Nagumo excitable systems with chemical delayed coupling. Matematicki Vesnik, 2011, 63, P. 103114.Buric N., Todorovic D. Bifurcations due to small time-lag in coupled excitable systems. International Journal of Bifurcation and Chaos, 2005, 15, P. 1775-1785.Buric N., Todorovic D. Bifurcations due to small time-lag in coupled excitable systems. International Journal of Bifurcation and Chaos, 2005, 15, P. 1775-1785.Domogo A.A., Collera J. Symmetric solutions to a system of mutually delay-coupled oscillators with conjugate coupling. Journal of Physics: Conference Series, 2018, 1123(1), P. 012028.Domogo A.A., Collera J. Symmetric solutions to a system of mutually delay-coupled oscillators with conjugate coupling. Journal of Physics: Conference Series, 2018, 1123(1), P. 012028.Hodgkin A.L., Huxley A.F. A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 1952, 117(4), P. 500–544.Hodgkin A.L., Huxley A.F. A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 1952, 117(4), P. 500–544.Izhikevich E.M. Dynamical systems in neuroscience, MIT Press, Cambridge, MA, 2007.Izhikevich E.M. Dynamical systems in neuroscience, MIT Press, Cambridge, MA, 2007.Murray J.D., Mathematical Biology. Biomathematics Vol. 19, Springer-Verlag, Berlin, 1989.Murray J.D., Mathematical Biology. Biomathematics Vol. 19, Springer-Verlag, Berlin, 1989.FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical journal, 1961, 1(6), P. 445–466.FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical journal, 1961, 1(6), P. 445–466.Nagumo J., Arimoto S. and Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 1962, 50(10), P. 2061–2070.Nagumo J., Arimoto S. and Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 1962, 50(10), P. 2061–2070.B. Van der Pol. On relaxation-oscillations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1926, 1(11), P. 978–992.B. Van der Pol. On relaxation-oscillations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1926, 1(11), P. 978–992.Postnikov E.B., Titkova O.V. A Correspondence between the Models of Hodgkin-Huxley and FitzHugh–Nagumo Revised. Eur. Phys. J. Plus, 2016, 131(11), P. 411.Postnikov E.B., Titkova O.V. A Correspondence between the Models of Hodgkin-Huxley and FitzHugh–Nagumo Revised. Eur. Phys. J. Plus, 2016, 131(11), P. 411.Saha A. and Feudel U. Extreme Events in FitzHugh–Nagumo Oscillators Coupled with Two Time Delays. Phys. Rev. E, 2017, 95(6), P. 062219.Saha A. and Feudel U. Extreme Events in FitzHugh–Nagumo Oscillators Coupled with Two Time Delays. Phys. Rev. E, 2017, 95(6), P. 062219.Kudryashov N.A. Asymptotic and Exact Solutions of the FitzHughNagumo Model. Regul. Chaotic Dyn., 2018, 23(2), P. 152-160.Kudryashov N.A. Asymptotic and Exact Solutions of the FitzHughNagumo Model. Regul. Chaotic Dyn., 2018, 23(2), P. 152-160.D.-P. Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete and Continuous Dynamical Systems B, 2016, 21(8), P. 2601–2614.D.-P. Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete and Continuous Dynamical Systems B, 2016, 21(8), P. 2601–2614.Gu A., Wang B. Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise. Discrete and Continuous Dynamical Systems B, 2018, 23(4), P. 1689–1720.Gu A., Wang B. Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise. Discrete and Continuous Dynamical Systems B, 2018, 23(4), P. 1689–1720.Bur N., Grozdanov I., and Vasov N. Type I vs. type II excitable systems with delayed coupling. Chaos, Solitons and Fractals, 2005, 23(4), P. 1221–1233.Bur N., Grozdanov I., and Vasov N. Type I vs. type II excitable systems with delayed coupling. Chaos, Solitons and Fractals, 2005, 23(4), P. 1221–1233.Campbell S.A. Time delays in neural systems in Handbook of brain connectivity, Springer, Berlin, Heidelberg, 2007, P. 65–90.Campbell S.A. Time delays in neural systems in Handbook of brain connectivity, Springer, Berlin, Heidelberg, 2007, P. 65–90.Rinzel J. Models in neurobiology in Nonlinear phenomena in physics and biology, Springer, Boston, MA, 1981, P. 345–367.Rinzel J. Models in neurobiology in Nonlinear phenomena in physics and biology, Springer, Boston, MA, 1981, P. 345–367.Bur N. and Todorov D. Dynamics of FitzHugh–Nagumo excitable systems with delayed coupling. Physical Review E, 2003, 67(6), P. 066222.Bur N. and Todorov D. Dynamics of FitzHugh–Nagumo excitable systems with delayed coupling. Physical Review E, 2003, 67(6), P. 066222.Berkolaiko G., Kuchment P. Introduction to Quantum Graphs. AMS, Providence, 2012.Berkolaiko G., Kuchment P. Introduction to Quantum Graphs. AMS, Providence, 2012.Eremin D.A., Grishanov E.N., Nikiforov D.S., Popov I.Y. Wave dynamics on time-depending graph with Aharonov-Bohm ring. Nanosystems: Physics, Chemistry, Mathematics, 2018, 9, P. 457–463.Eremin D.A., Grishanov E.N., Nikiforov D.S., Popov I.Y. Wave dynamics on time-depending graph with Aharonov-Bohm ring. Nanosystems: Physics, Chemistry, Mathematics, 2018, 9, P. 457–463.

The authors declare that there are no conflicts of interest present.