Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh-Rose and FitzHugh-Nagumo neurons with two time delays

doi:10.1088/1674-1056/aca601

Abstract A memristor-coupled heterogenous neural network consisting of two-dimensional (2D) FitzHugh-Nagumo (FHN) and Hindmarsh-Rose (HR) neurons with two time delays is established. Taking the time delays as the control parameters, the existence of Hopf bifurcation near the stable equilibrium point in four cases is derived theoretically, and the validity of the Hopf bifurcation condition is verified by numerical analysis. The results show that the two time delays can make the stable equilibrium point unstable, thus leading to periodic oscillations induced by Hopf bifurcation. Furthermore, the time delays in FHN and HR neurons have different effects on the firing activity of neural network. Complex firing patterns, such as quiescent state, chaotic spiking, and periodic spiking can be induced by the time delay in FHN neuron, while the neural network only exhibits quiescent state and periodic spiking with the change of the time delay in HR neuron. Especially, phase synchronization between the heterogeneous neurons is explored, and the results show that the time delay in HR neurons has a greater effect on blocking the synchronization than the time delay in FHN neuron. Finally, the theoretical analysis is verified by circuit simulations.

Keywords: memristor time delay heterogeneous neurons hopf bifurcation phase synchronization

Received: 02 September 2022
Revised: 24 October 2022
Accepted manuscript online: 25 November 2022

PACS:	87.19.ll	(Models of single neurons and networks)
	87.19.lj	(Neuronal network dynamics)
	05.45.Pq	(Numerical simulations of chaotic systems)

Fund: Project supported by the National Natural Science Foundations of China (Grant Nos. 62171401 and 62071411).

Corresponding Authors: Zhi-Jun Li
E-mail: lizhijun_320@163.com

Cite this article:

Zhan-Hong Guo(郭展宏), Zhi-Jun Li(李志军), Meng-Jiao Wang(王梦蛟), and Ming-Lin Ma(马铭磷) Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh-Rose and FitzHugh-Nagumo neurons with two time delays 2023 Chin. Phys. B 32 038701

[1] Yao Y, Yang L, Wang C and Liu Q 2018 Complexity 10 5632650
[2] Yao Y and Ma J 2018 Cognitive Neurodynamics 12 343
[3] Wu F, Ma J and Zhang G 2019 Applied Mathematic and Computation 347 590
[4] Hindmarsh J L and Rose R M 1982 Nature 296 162
[5] Hindmarsh J L and Rose R M 1984 Proc. Roy. Soc. Lond. Ser. B Biolog. Sci. 221 87
[6] Wang H, Lu Q and Wang Q 2008 Commun. Nonlinear Sci. Numeri. Simul. 13 1668
[7] Bao B, Yang Q, Zhu L and Bao H 2019 Int. J. Bifur. Chaos 29 1950134
[8] Ge P and Cao H 2021 Int. J. Bifur. Chaos 31 2150233
[9] Ma J, Yang Z Q, Yang L J and Tang J 2019 Journal of Zhejiang University-Science A 20 639
[10] Vaidyanathan S 2015 International Journal of PharmTech Research 8 117
[11] An X and Qiao S 2021 Chaos, Solitons & Fractals 143 110587
[12] Lv M, Wang C, Ren G, Ma J and Song X 2016 Nonlinear Dyn. 85 1479
[13] Song X, Wang H and Chen Y 2019 Nonlinear Dyn. 96 2341
[14] Bashkirtseva I, Nasyrova V and Ryashko L 2018 Chaos, Solitons & Fractals 110 76
[15] Wu J, Xu Y and Ma J 2017 PLoS One. 12 e0174330
[16] Wang Y, Ma J, Xu Y, Wu F and Zhou P 2017 Int. J. Bifur. Chaos 27 1750030
[17] Wu F, Wang C, Jin W and Ma J 2017 Physica A 469 81
[18] Ma J and Tang J 2017 Nonlinear Dyn. 89 1569
[19] Li Z and Zhou H 2021 Electron. Lett. 57 715
[20] Chua L 1971 IEEE Trans. Circuit Theory 18 507
[21] Ma M, Yang Y, Qiu Z, Peng Y and Sun Y 2022 Nonlinear Dyn. 107 2935
[22] Wen Z, Li Z and Li X 2019 Chaos Solitons & Fractals 128 58
[23] Shao N, Zhang S and Shao S 2019 Acta Phys. Sin. 68 198502 (in Chinese)
[24] Zhu L and Wang F 2019 Acta Phys. Sin. 68 198501 (in Chinese)
[25] Eccles J C 1982 Ann. Rev. Neurosci. 5 325
[26] Zhang P, Xia M, Zhuge F, Zhou Y and Wang Z 2019 Nano Lett. 19 4279
[27] Kazantsev V B and Asatryan S Y 2011 Phys. Rev. E 84 031913
[28] Tan Y and Wang C 2020 Chaos 30 053118
[29] Lin H, Wang C, Hong Q and Sun Y 2020 IEEE Transacti on Circuits and Systems II: Express Briefs 67 3472
[30] Wu F, Zhang Y and Zhang X 2019 Nonlinear Dyn. 98 971
[31] Li R, Wang Z and Dong E 2021 Nonlinear Dyn. 104 4459
[32] Xu Y, Jia Y, Ma J, Alsaedi A and Ahmad B 2017 Chaos, Solitons & Fractals 104 435
[33] Bao H, Zhang Y, Liu W and Bao B 2020 Nonlinear Dyn. 100 937
[34] Li Z, Zhou H, Wang M and Ma M 2021 Nonlinear Dyn. 104 1455
[35] Hu D and Cao H 2016 Int. J. Bifur. Chaos. 26 1650187
[36] Liao X, Wong K W, Leung C S and Wu Z 2001 Chaos, Solitons & Fractals 12 1535
[37] Xu Y and Ma J 2021 Chin. Phys. B 30 100501
[38] Wang Z, Wang X, Li Y and Huang X 2017 Int. J. Bifur. Chaos 27 1750209
[39] Wang C J, Yang K L and Qu S X 2014 Phys. Scr. 89 105001
[40] Li Z, Guo Z, Wang M and Ma, M 2021 AEU-International Journal of Electronics and Communications 142 153995
[41] Liu Q, Liao X, Liu Y, Zhou S and Guo S 2009 Nonlinear Dyn. 58 573
[42] Liao X, Guo S and Li C 2007 Nonlinear Dyn. 49 319
[43] He X, Li C, Huang T and Li C 2013 IEEE Transactions on Neural Networks and Learning Systems 24 1001
[44] Zhang W, Li C, Huang T and He X 2015 IEEE Transactions on Neural Networks and Learning Systems 26 3308
[45] Huang S, Zhang J, Wang M and Hu C K 2018 Physica A 499 88
[46] Xiao M, Zheng W X, Jiang G and Cao J 2020 IEEE Transactions on Neural Networks and Learning Systems 32 1974
[47] Li L, Zhao Z and Gu H 2019 Int. J. Bifur. Chaos 29 1950147
[48] Marsden J E and McCracken M 2012 Springer Science & Business Media 19 13542
[49] Walschap G 2012 Springer Science & Business Media 224 116
[50] Bao H, Hu A, Liu W and Bao B 2019 IEEE Transactions on Neural Networks and Learning Systems 31 502
[51] Bao B, Qian H, Xu Q, Chen M and Wang J 2017 Frontiers in Computational Neuroscience 11 81
[52] Ma M, Xie X and Li Z 2023 Chin. Phys. B 32 058701
[53] Ma M, Lu Y and Li Z 2023 Fractal and Fractional. 7 82
[54] Ma M, Xiong K and Li Z 2023 Mathematics 11 375
[55] Ma M, Yang Y and Qiu Z 2022 Nonlinear Dyn. 107 2935

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Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh-Rose and FitzHugh-Nagumo neurons with two time delays

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