najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2021-12-5-553-562najo-508Research ArticleMATHEMATICSМАТЕМАТИКАProperties of an oriented ring of neurons with the FitzHugh-Nagumo modelСвойства ориентированного кольца нейронов в рамках модели Фитцхью-НагумоFedorovE. G.FedorovE. G.

Kronverkskiy, 49, St. Petersburg, 197101

FedorovEGe@gmail.comITMO UniversityRussian Federation202105082025125553562Copyright © Fedorov E.G., 20252025Fedorov E.G.Fedorov E.G.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/508

The transmission of an impulse through a neuron is provided by processes that occur at the nanoscale level. This paper will build a model for an oriented ring of connected neurons. To describe the process of impulse transmission through a neuron, the FitzHugh-Nagumo model is used, which allows one to set a higher abstraction level by simulating an impulse. In this case, when transmitting impulses between neurons, the delay is taken into account. For the constructed model, the dependence of the number of neurons on the dynamics of the network as a whole is studied, and local bifurcations are found. All results are verified numerically. It is shown that the period of self-oscillations of such a network depends on the number of neurons.

Прохождение импульса через нейрон обусловлено процессами на нономасштабном уровне. В статье строится модель ориентированного кольца связанных нейронов. Для описания процесса прохождения импульса через нейрон используется модель ФитцХью-Нагумо, которая позволяет моделировать импульс на достаточно абстрактном уровне. В этой модели при передаче импульса между нейронами принимается во внимание задержка.  Для построенной модели влияние числа нейронов на динамику сети изучено и найдены локальные бифуркации. Все результаты проверены численно. Показано, что период осцилляций в сети зависит от числа нейронов.

сеть нейроновбифуркация Хопфасистема ФитцХью-Нагумозадержка связиустойчивостьNeural networksHopf bifurcationsFitzHugh-Nagumo systemcommunication delaystabilityThis work was supported by the Russian Foundation for Basic Research (RFBR) project number 20-31-90036.ReferencesSch¨oll E., Hiller G., H¨ovel P., Dahlem M. Time-delayed feedback in neurosystems. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2009. 1891 (367). 1079–1096.Sch¨oll E., Hiller G., H¨ovel P., Dahlem M. Time-delayed feedback in neurosystems. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2009. 1891 (367). 1079–1096.Buri´c N., Todorovi´c D. Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling. Physical Review E. 2003. 6 (67). 066222 p.Buri´c N., Todorovi´c D. Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling. Physical Review E. 2003. 6 (67). 066222 p.Song Z., Xu J., Zhen B. Multitype activity coexistence in an inertial two-neuron system with multiple delays. International Journal of Bifurcation and Chaos. 2015. 13 (25). 1530040 p.Song Z., Xu J., Zhen B. Multitype activity coexistence in an inertial two-neuron system with multiple delays. International Journal of Bifurcation and Chaos. 2015. 13 (25). 1530040 p.Xu C., Zhang Q., Wu Y. Bifurcation analysis in a three-neuron artificial neural network model with distributed delays. Neural Processing Letters. 2016. 2 (44). 343–373.Xu C., Zhang Q., Wu Y. Bifurcation analysis in a three-neuron artificial neural network model with distributed delays. Neural Processing Letters. 2016. 2 (44). 343–373.Karaolu E., Yılmaz E., Merdan H. Stability and bifurcation analysis of two-neuron network with discrete and distributed delays. Neurocomputing. 2016. 182. 102–110.Karaolu E., Yılmaz E., Merdan H. Stability and bifurcation analysis of two-neuron network with discrete and distributed delays. Neurocomputing. 2016. 182. 102–110.Xu C. Local and global Hopf bifurcation analysis on simplified bidirectional associative memory neural networks with multiple delays. Mathematics and Computers in Simulation. 2018. 149. 69–90.Xu C. Local and global Hopf bifurcation analysis on simplified bidirectional associative memory neural networks with multiple delays. Mathematics and Computers in Simulation. 2018. 149. 69–90.FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical journal. 1961. 6 (1). 445–466.FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical journal. 1961. 6 (1). 445–466.Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE. 1962. 10 (50). 2061– 2070.Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE. 1962. 10 (50). 2061– 2070.Fedorov E.G., Popov A.I., Popov I.Y. Metric graph version of the FitzHugh-Nagumo model. Nanosystems: Physics, Chemistry, Mathematics. 2019. 10 (6). 623–626.Fedorov E.G., Popov A.I., Popov I.Y. Metric graph version of the FitzHugh-Nagumo model. Nanosystems: Physics, Chemistry, Mathematics. 2019. 10 (6). 623–626.Buri´c N., Grozdanovi´c I., Vasovi´c N. Type I vs. type II excitable systems with delayed coupling. Chaos, Solitons and Fractals. 2005. 5 (23). 1221–1233.Buri´c N., Grozdanovi´c I., Vasovi´c N. Type I vs. type II excitable systems with delayed coupling. Chaos, Solitons and Fractals. 2005. 5 (23). 1221–1233.

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