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The dynamic behaviors of coupled neurons with different mathematical representations have received more and more attention in recent years. The coupling among heterogeneous neurons can show richer dynamic phenomena, which is of great significance in understanding the function of the human brain. In this paper, we present a fraction-order heterogeneous network with three neurons, which is built by coupling an FN neuron with two HR neurons. Complex electromagnetic surroundings have meaningful physical influence on the electrical activities of neurons. To imitate the effects of electromagnetic induction on the three-neuron heterogeneous network, we introduce a fraction-order locally active memristor in the neural network. The characteristics of this memristor are carefully analyzed by pinched hysteresis loops and its locally active characteristic is proved by the power-off plot and the DC v-i plot. Then, the parameter-dependent dynamic activities are investigated numerically by using several dynamical analysis methods, such as the phase diagrams, bifurcation diagrams, Lyapunov exponent spectra, and attraction basins. In addition, the network also reveals rich dynamic behaviors, including coexisting activities, anti-monotonicity phenomena, transient chaos and firing patterns, providing support for further investigating the firing patterns of the human brain. In particular, complex dynamics, including coexisting attractors, anti-monotonicity, and firing patterns, can be influenced by the order and strength of electrical synaptic coupling and electromagnetic induction. The control of the bistable state can be realized through the time feedback control method, so that the bistable state can be transformed into an ideal monostable state. The study of the fraction-order memristive neural network may expand the field of view for understanding the collective behaviors of neurons. Finally, based on the ARM platform, we give a digital implementation of the fraction-order memristive neural network, which can verify the consistency with the numerical simulation results. In the future, we will explore more interesting memristive neural networks and study different types of methods to control the firing behaviors of the networks.
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Keywords:
- memristive electromagnetic induction /
- fraction-order neural network /
- attractor coexistence /
- anti-monotonicity
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Google Scholar
Zhou X R, Luo X S, Jing P Q, Yuan W J 2007 Acta Phys. Sin. 56 5679
Google Scholar
[2] Xu Y, Jia Y, Ge M Y, Lu L L, Yang L J, Zhan X 2018 Neurocomputing 283 196
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[3] Izhikevich E M 2003 IEEE Trans. Neural Netw. 14 1569
Google Scholar
[4] Bao B C, Yang Q, Zhu L, Bao H 2019 Int. J. Bifurc. Chaos 29 1930010
Google Scholar
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[7] Izhikevich E M, Fitzhugh R 2006 Scholarpedia 1 1349
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[8] Chay T R 1985 Physica D 16 233
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[10] Li Z J, Zhou H Y, Wang M J, Ma M L 2021 Nonlinear Dyn. 104 1455
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[11] Ding D W, Jiang L, Hu Y B, Yang Z L, Li Q 2021 Chaos 31 083107
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Google Scholar
Sun X J, Yang B H, Wu Y, Xiao J H 2014 Acta Phys. Sin. 63 120502
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Google Scholar
Ding X L, Gu H G, Jia B, Li Y Y 2021 Acta Phys. Sin. 70 218701
Google Scholar
[16] 吴莹, 徐健学, 何岱海, 靳伍银 2005 物理学报 54 3457
Google Scholar
Wu Y, Xu J X, He D H, Jin W Y 2005 Acta Phys. Sin. 54 3457
Google Scholar
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Google Scholar
[18] Hu X Y, Liu C X, Liu L, Ni J K, Yao Y P 2018 Nonlinear Dyn. 91 1541
Google Scholar
[19] Wan Q Z, Yan Z D, Li F, Chen S M, Liu J 2022 Chaos 32 073107
Google Scholar
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[31] Chen L, He Z L, Li C D, Wen S P, Chen Y R 2020 Int. J. Bifurcat. Chaos 30 2050172
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 分数阶忆阻器的磁滞回线 (a)不同频率; (b)不同振幅; (c)不同阶数
Figure 1. Hysteresis loops of fraction-order memristor: (a) Different frequencies; (b) different amplitudes; (c) different orders.
图 2 (a) 忆阻器的断电图; (b) 忆阻器的直流v-i图
Figure 2. (a) POP plot of the memristor; (b) DC v-i plot of the memristor.
图 3 耦合神经元的拓扑结构
Figure 3. Topology of coupled neurons.
图 4 网络的平衡点
Figure 4. Equilibrium point of the network.
图 5 (a) q的分岔图; (b)对应的前3个Lyapunov指数谱
Figure 5. (a) Bifurcation diagram of q; (b) the corresponding Lyapunov exponent spectrum.
图 6 (a)双参数分岔图; (b) SE复杂度图
Figure 6. (a) Two-parameter bifurcation diagram; (b) SE complexity diagram.
图 7 (a)迭代序列图; (b)瞬态混沌图
Figure 7. (a) Iterative sequences; (b) phases of transient chaotic.
图 8 (a) 共存分岔图; (b) 最大Lyapunov指数; (c) 周期与混沌共存吸引子; (d)不同周期共存吸引子
Figure 8. (a) Coexisting bifurcation diagrams; (b) the largest Lyapunov exponents; (c) coexisting attractors of periodic and chaotic states; (d) coexisting attractors of different periodic states.
图 9 (a) $k$的分岔图; (b) $k{\text{ - }}{m_{23}}$平面的双参数分岔图
Figure 9. (a) Bifurcation diagram of the $k$; (b) the two-parameter bifurcation diagrams in the $k{\text{ - }}{m_{23}}$ plane.
图 10 电磁感应强度不同时, ${m_{23}}$的分岔图 (a) $k = - 0.3$; (b) $k = 0$; (c) $k = 0.2$; (d) $k = 0.9$
Figure 10. Bifurcation diagrams of the ${m_{23}}$ for different $k$: (a) $k = - 0.3$; (b) $k = 0$; (c) $k = 0.2$; (d) $k = 0.9$.
图 11 电磁感应相关的${x_1}$, ${x_2}$, ${x_3}$和$\varphi $放电行为 (a1)—(a4) $k = 0.3$; (b1)—(b4) $k = 0.5$; (c1)—(c4) $k = 0.7$
Figure 11. Induction strength-relied firing patterns for ${x_1}$, ${x_2}$, ${x_3}$ and $\varphi $: (a1)–(a4) $k = 0.3$; (b1)–(b4) $k = 0.5$; (c1)–(c4) $k = 0.7$
图 12 (a) ${x_{10}}{\text{ - }}{\varphi _0} $平面上的吸引盆; (b) x1-y1平面上的相图
Figure 12. (a) Basin of attraction of ${x_{10}}{\text{ - }}{\varphi _0} $ plane; (b) phases of x1-y1 plane.
图 13 (a) 原分岔图; (b) 时间反馈控制后的分岔图
Figure 13. (a) Bifurcation diagram; (b) the desired distribution diagram controlled via temporal feedback.
图 14 设计网络的硬件框图
Figure 14. Hardware block diagram of the designed network.
图 15 基于ARM平台的硬件实现 (a)硬件连接图; (b), (c) MATLAB仿真图; (d), (e)单片机实验结果
Figure 15. Hardware implementation based on ARM platform: (a) Hardware connection plot; (b), (c) MATLAB simulation diagram; (d), (e) ARM implementation of the attractors.
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[1] 周小荣, 罗晓曙, 蒋品群, 袁五届 2007 物理学报 56 5679
Google Scholar
Zhou X R, Luo X S, Jing P Q, Yuan W J 2007 Acta Phys. Sin. 56 5679
Google Scholar
[2] Xu Y, Jia Y, Ge M Y, Lu L L, Yang L J, Zhan X 2018 Neurocomputing 283 196
Google Scholar
[3] Izhikevich E M 2003 IEEE Trans. Neural Netw. 14 1569
Google Scholar
[4] Bao B C, Yang Q, Zhu L, Bao H 2019 Int. J. Bifurc. Chaos 29 1930010
Google Scholar
[5] Hindmarsh J L, Rose R M 1982 Nature 296 5853
Google Scholar
[6] Hindmarsh J L, Rose R M 1984 Proc. R. Soc. Lond, Ser B: Biol. Sci. 221 87
Google Scholar
[7] Izhikevich E M, Fitzhugh R 2006 Scholarpedia 1 1349
Google Scholar
[8] Chay T R 1985 Physica D 16 233
Google Scholar
[9] Xu Q, Liu T, Feng C T, Bao H, Wu H G, Bao B C 2021 Chin. Phys. B 30 128702
Google Scholar
[10] Li Z J, Zhou H Y, Wang M J, Ma M L 2021 Nonlinear Dyn. 104 1455
Google Scholar
[11] Ding D W, Jiang L, Hu Y B, Yang Z L, Li Q 2021 Chaos 31 083107
Google Scholar
[12] Njitacke Z T, Awrejcewicz J, Ramakrishnan B, Rajagopal K, Kengne J 2022 Nonlinear Dyn. 107 2867
Google Scholar
[13] De S, Balakrishnan J 2020 Commun. Nonlinear Sci. Numer. Simul. 90 105391
Google Scholar
[14] 孙晓娟, 杨白桦, 吴晔, 肖井华 2014 物理学报 63 120502
Google Scholar
Sun X J, Yang B H, Wu Y, Xiao J H 2014 Acta Phys. Sin. 63 120502
Google Scholar
[15] 丁学利, 古华光, 贾冰, 李玉叶 2021 物理学报 70 218701
Google Scholar
Ding X L, Gu H G, Jia B, Li Y Y 2021 Acta Phys. Sin. 70 218701
Google Scholar
[16] 吴莹, 徐健学, 何岱海, 靳伍银 2005 物理学报 54 3457
Google Scholar
Wu Y, Xu J X, He D H, Jin W Y 2005 Acta Phys. Sin. 54 3457
Google Scholar
[17] Lv M, Ma J, Yao Y G, Alzahrani F 2019 Sci. China-Technol. Sci. 62 448
Google Scholar
[18] Hu X Y, Liu C X, Liu L, Ni J K, Yao Y P 2018 Nonlinear Dyn. 91 1541
Google Scholar
[19] Wan Q Z, Yan Z D, Li F, Chen S M, Liu J 2022 Chaos 32 073107
Google Scholar
[20] Lin H R, Wang C H, Hong Q H, Sun Y C 2020 IEEE Trans. Circuits Syst. II-Express Briefs 67 3472
Google Scholar
[21] Li C L, Li H D, Xie W W, Du J R 2021 Nonlinear Dyn. 106 1041
Google Scholar
[22] Yu F, Zhang Z N, Shen H, Huang Y Y, Cai S, Du S C 2022 Chin. Phys. B 31 020505
Google Scholar
[23] 罗佳, 孙亮, 乔印虎 2022 计算物理 39 109
Google Scholar
Luo J, Sun L, Qiao Y H 2022 Chin. J. Comput. Phys. 39 109
Google Scholar
[24] Xu Q, Ju Z T, Ding S K, Feng C T, Chen M, Bao B C 2022 Cogn. Neurodyn. 16 1221
Google Scholar
[25] Bao H, Hu A H, Liu W B, Bao B C 2020 IEEE Trans. Neural Netw. Learn. Syst. 31 502
Google Scholar
[26] Bao H, Liu W B, Hu A H 2019 Nonlinear Dyn. 95 43
Google Scholar
[27] Tenreiro Machado J, Duarte F B, Duarte G M, 2012 Int. J. Bifurcat. Chaos 22 1250249
Google Scholar
[28] Tenreiro Machado J A, Lopes A M 2016 Nonlinear Dyn. 86 1613
Google Scholar
[29] Tzounas G, Dassios I, Murad M A A, Milano F 2020 IEEE Trans. Power Syst. 35 4622
Google Scholar
[30] Kërçi T, Giraldo J, Milano F 2020 Int. J. Elec. Power 119 105819
Google Scholar
[31] Chen L, He Z L, Li C D, Wen S P, Chen Y R 2020 Int. J. Bifurcat. Chaos 30 2050172
Google Scholar
[32] Jahanshahi H, Yousefpour A, Munoz-Pacheco J M, Kacar S, Viet-Thanh P, Alsaadi F E 2020 Appl. Math. Comput. 383 125310
Google Scholar
[33] Dong J, Zhang G J, Xie Y, Yao H, Wang J 2014 Cogn. Neurodyn. 8 167
Google Scholar
[34] Wu G C, Baleanu D 2015 Commun. Nonlinear Sci. Numer. Simul. 22 95
Google Scholar
[35] Lin H R, Wang C H, Sun Y C, Yao W 2020 Nonlinear Dyn. 100 3667
Google Scholar
[36] Chua L 2013 Nanotechnology 24 383001
Google Scholar
[37] Chua L 2015 Radioengineering 24 319
Google Scholar
[38] Weiher M, Herzig M, Tetzlaff R, Ascoli A, Mikolajick T, Slesazeck S 2019 IEEE Trans. Circuits Syst. I-Regul 66 2627
Google Scholar
[39] Yu H, Du S Z, Dong E Z, Tong J G 2022 Chaos Solitons Fractals 160 112220
Google Scholar
[40] Li C B, Wang X, Chen G R 2017 Nonlinear Dyn. 90 1335
Google Scholar
[41] Li C B, Sprott J C, Hu W, Xu Y J 2017 Int. J. Bifurcat. Chaos 27 1750160
Google Scholar
[42] Park J H, Huh S H, Kim S H, Seo S J, Park G T 2005 IEEE Trans. Neural Netw. 16 414
Google Scholar
[43] Yadav K, Prasad A, Shrimali M D 2018 Phys. Lett. A 382 2127
Google Scholar
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