Coexisting discharge and synchronization of heterogeneous discrete neural network with crosstalk memristor synapses

Wang Xuan; Du Jian-Rong; Li Zhi-Jun; Ma Ming-Lin; Li Chun-Lai

doi:10.7498/aps.73.20231972

Abstract

Synaptic crosstalk, which occurs due to the overflow of neurotransmitters between neighboring synapses, holds a crucial position in shaping the discharge characteristics and signal transmission within nervous systems. In this work, two memristors are employed to simulate biological neural synapses and bidirectionally coupled Chialvo discrete neuron and Rulkov discrete neuron. Thus, a heterogeneous discrete neural network with memristor-synapse coupling is constructed, with the crosstalk behavior between memristor synapses in the coupled state taken into account. The analysis demonstrates that the quantity and stability of fixed points within this neural network greatly depend on the strength of synaptic crosstalk. Additionally, through a thorough investigation of bifurcation diagrams, phase diagrams, Lyapunov exponents, and time sequences, we uncover the multi-stable state property exhibited by the neural network. This characteristic manifests as the coexistence of diverse discharge behaviors, which significantly change with the intensity of synaptic crosstalk. Interestingly, the introduction of control parameter into state variables can lead the bias to increase, and also the infinite stable states to occur in the neural network. Furthermore, we comprehensively study the influence of synaptic crosstalk strength on the synchronization behavior of the neural network, with consideration of various coupling strengths, initial conditions, and parameters. Our analysis, which is based on the phase difference and synchronization factor of neuronal discharge sequences, reveales that the neural network maintains phase synchronization despite the variations of the two crosstalk strengths. The insights gained from this work provide important support for elucidating the electrophysiological mechanisms behind the processing and transmission of biological neural information. Especially, the coexisting discharge phenomenon in the neural network provides an electrophysiological theoretical foundation for the clinical symptoms and diagnosis of the same neurological disease among different individuals or at different stages. And the doctors can predict the progression and prognosis of neurological disease based on the patterns and characteristics of coexisting discharge in patients, enabling them to adopt appropriate intervention measures and monitoring plans. Therefore, the research on coexisting discharge in the neural system contributes to the comprehensive treatment of nervous system disease.

Keywords:

Authors and contacts

1.
Key Laboratory of Hunan Province on Information Photonics and Freespace Optical Communications, Hunan Institute of Science and Technology, Yueyang 414006, China
2.
School of Computer Science & School of Cyberspace Science, Xiangtan University, Xiangtan 411105, China
3.
School of Automation and Electronic Information, Xiangtan University, Xiangtan 411105, China

Corresponding author: Li Chun-Lai, lichunlai33@126.com

Funds: Project supported by the Hunan Provincial Natural Science Foundation of China (Grant No. 2020JJ4337) and the National Natural Science Foundation of China (Grant No. 62171401).

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References

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[2]	Stein R B, Gossen E R, Jones K E 2005 Nat. Rev. Neurosci. 6 389
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[4]	Tan F, Zhou L L, Lu J W, Quan, H Z, Liu K Y 2022 Eur. J. Control 70 100764
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[12]	Li C L, Yang Y Y, Yang X B, Zi X Y, Xiao F L 2022 Nonlinear Dyn. 108 1697 Google Scholar
[13]	Li C L, Wang X, Du J R, Li Z J 2023 Nonlinear Dyn. 111 21333 Google Scholar
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[19]	Rosenblatt F 1963 Am. J. Psychol. 76 705 Google Scholar
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[25]	Chua L O 2005 Int. J. Bifurcation Chaos 15 3435 Google Scholar
[26]	Jin P P, Wang G Y, Liang Y, Iu H H, Chua L O 2021 IEEE Trans. Circuits Syst. Regul. Pap. 68 4419 Google Scholar
[27]	Lai Q, Yang L 2022 Chaos, Solitons Fractals 165 112781 Google Scholar
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[29]	Lu Y M, Wang C H, Deng Q L, Xu C 2022 Chin. Phys. B 31 060502 Google Scholar
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[33]	Ding D W, Chen X Y, Yang Z L, Hu Y B, Wang M Y, Zhang H W, Zhang X 2022 Chaos, Solitons Fractals 158 112014 Google Scholar
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[36]	Pool R 1989 Science 243 604 Google Scholar
[37]	Adhikari S P, Sah M P, Kim H, Chua L O 2013 IEEE Trans. Circuits Syst. Regul. Pap. 60 3008 Google Scholar
[38]	Ren L J, Mou J, Banerjee S, Zhang Y S 2023 Chaos, Solitons Fractals 167 113024 Google Scholar
[39]	Li C, Yi C, Li Y, Mitro S, Wang Z 2024 Chaos 34 031102 Google Scholar
[40]	Ma C G, Mou J, Xiong L, Banerjee S, Liu T M, Han X T 2021 Nonlinear Dyn. 103 2867 Google Scholar
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Cited By

图 1 忆阻器的电压-电流特性曲线　(a) $\varphi $(0) = 0.1, ω = 0.1, A = 0.2, 0.3, 0.4; (b) $\varphi $(0) = 0.1, A = 0.5, ω = 0.2, 0.4, 0.8; (c) A = 0.5, ω = 0.2, $\varphi $(0) = –3.0, 0.1, 3.0

Figure 1. Hysteresis loop for discrete memristor: (a) $\varphi $(0) = 0.1, ω = 0.1 and A = 0.2, 0.3, 0.4; (b) $\varphi $(0) = 0.1, A = 0.5 and ω = 0.2, 0.4, 0.8; (c) A = 0.5, ω = 0.2 and $\varphi $(0) = –3.0, 0.1, 3.0.

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图 2 忆阻器的(a) POP和(b) DC V-I图

Figure 2. (a) POP and (b) DC V-I plot of memristor.

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图 3 异质离散神经网络的拓扑结构

Figure 3. Topology of heterogeneous discrete neural network.

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图 4 不同初始条件时, 状态变量x₁(n)关于p₁的分岔图　(a) (1, 1, 1, –1, 1, 0); (b) (1, 1, 1, –2.4, 1, 0); (c) (1, 1, 1, 1, 1, 0); (d) (1, 1, 1, 2, 1, 0)

Figure 4. Bifurcation diagrams of state variable x₁(n) versus p₁ under different initial conditions: (a) (1, 1, 1, –1, 1, 0); (b) (1, 1, 1, –2.4, 1, 0); (c) (1, 1, 1, 1, 1, 0); (d) (1, 1, 1, 2, 1, 0).

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图 5 p₁取不同值时, 神经网络的共存吸引子及共存放电序列　(a)—(c) p₁ = –0.8; (d)—(f) p₁ = –0.4; (g)—(i) p₁ = 0.2; (j)—(l) p₁ = 0.8

Figure 5. Coexisting attractors and coexisting discharge sequences under different p₁ value: (a)–(c) p₁ = –0.8; (d)–(f) p₁ = –0.4; (g)–(i) p₁ = 0.2; (j)–(l) p₁ = 0.8.

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图 6 不同初始条件时状态变量x₁(n)关于p₂的分岔图　(a) (1, 1, 1, –5, 1, 0); (b) (1, 1, 1, 5, 1, 0); (c) (1, 1, 1, –1, 1, 0); (d) (1, 1, 1, –3, 1, 0)

Figure 6. Bifurcation diagrams of state variable x₁(n) versus p₂ under different initial conditions: (a) (1, 1, 1, –5, 1, 0); (b) (1, 1, 1, 5, 1, 0); (c) (1, 1, 1, –1, 1, 0); (d) (1, 1, 1, –3, 1, 0).

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图 7 不同p₂值时神经网络的共存吸引子及共存放电序列　(a)—(c) p₂ = –1.1; (d)—(f) p₂ = –0.4; (g)—(i) p₂ = 0.2; (j)—(l) p₂ = 0.8

Figure 7. Coexisting attractors and coexisting discharge sequences under different p₂ value: (a)–(c) p₂ = –0.7; (d)–(f) p₂ = –0.4; (g)–(i) p₂ = 0.2; (j)–(l) p₂ = 0.8.

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图 8 (a), (b) 状态变量x₂, y₂关于参数m的分岔图; (c)关于参数m的李雅普诺夫指数图

Figure 8. (a), (b) Bifurcation diagram and (c) Lyapunov exponent versus m.

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图 9 不同m取值时 (a) x₂-y₂的相图; (b), (c)状态变量x₂, y₂的放电序列

Figure 9. For different values of m: (a) x₂-y₂ phase diagrams; (b), (c) discharge sequences for state variables x₂, y₂.

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图 10 不同耦合强度k　(a) p₁对同步因子的影响; (b) p₁对相位差的影响

Figure 10. For different k: (a) Dependence of synchronization factor on p₁; (b) dependence of phase difference on p₁.

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图 11 不同初始值y₂(0)　(a) p₁对同步因子的影响; (b) p₁对相位差的影响

Figure 11. For different y₂(0): (a) Dependence of synchronization factor on p₁; (b) dependence of phase difference on p₁.

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图 12 不同系统参数a₂　(a) p₁对同步因子的影响; (b) p₁对相位差的影响

Figure 12. For different a₂: (a) Dependence of synchronization factor on p₁; (b) dependence of phase difference on p₁.

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图 13 (a) p₁ = –5时的时间序列; (b) p₁ = –5时的相位差; (c) p₁ = 5时的时间序列; (d) p₁ = 5时的相位差

Figure 13. (a) Time series when p₁ = –5; (b) phase difference when p₁ = –5; (d) time series when p₁ = 5; (d) phase difference when p₁ = 5.

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图 14 不同耦合强度k　(a) p₂对同步因子的影响; (b) p₂对相位差的影响

Figure 14. For different k: (a) Dependence of synchronization factor on p₂; (b) dependence of phase difference on p₂.

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图 15 不同初始值y₂(0)　(a) p₂对同步因子的影响; (b) p₂对相位差的影响

Figure 15. For different y₂(0): (a) Dependence of synchronization factor on p₂; (b) dependence of phase difference on p₂.

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图 16 不同参数a₂　(a) p₂对同步因子的影响; (b) p₂对相位差的影响

Figure 16. For different a₂: (a) Dependence of synchronization factor on p₂; (b) dependence of phase difference on p₂.

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图 17 (a) p₂ = –5时的时间序列; (b) p₂ = –5时的相位差; (c) p₂ = 5时的时间序列; (d) p₂ = 5时的相位差.

Figure 17. (a) Time series when p₂ = –5; (b) phase difference when p₂ = –5; (d) time series when p₂ = 5; (d) phase difference when p₂ = 5.

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表 1 神经网络的平衡点、特征值和稳定性

Table 1. Equilibrium points, eigenvalues, and stability of neural network.

(p₁, p₂)	(x₁(n), y₂(n), $\varphi $₁(n), $\varphi $₂(n))	特征值	稳定性
(0.1, 0.1)	(0.0342, 0.001, 0.0323, 0.0086)	–0.5593, 0.8269, 0.8889, 0.9994±0.0022i, 1	不稳定鞍焦点
	(0.0352, 0.001, 0.036, –0.3372)	–0.5593, 0.8509, 0.8291, 0.8881, 0.9990, 0.9996	不稳定鞍结点
	(0.0392, 0.0021, –0.2489, –0.3372)	–0.5587, 0.8296, 0.9425, 0.9017, 0.8926, 0.9994	不稳定鞍结点
	(168.8416, 0.0715, –0.7368, –0.3416)	0, 0.8900, –0.5580, 0.1857, 0.8236, 0.9994	不稳定鞍结点
(1, 1)	(0.0339, 0.0009, 0.0973, –0.0618)	–0.5589, 0.8097, 0.8892, 0.9953±0.0015i, 0.9995	不稳定鞍焦点
	(0.0347, 0.0018, 0.1062, –0.3375)	–0.5586, 0.8277±0.0047i, 0.8890, 0.9955, 0.9994	不稳定鞍焦点
	(0.0383, 0.0031–0.224, –0.3365)	–0.5569, 0.8305, 0.8838±0.0020i, 0.9645, 0.9994	不稳定鞍焦点
	(168.8425, 0.1017, –0.7368, –0.3416)	0, 0.89, –0.5457, 0.1857, 0.8113, 0.9994	不稳定鞍结点

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[1]	白婧, 关富荣, 唐国宁 2021 物理学报 70 170502 Google Scholar Bai J, Guan F R, Tang G N 2021 Acta Phys. Sin. 70 170502 Google Scholar
[2]	Stein R B, Gossen E R, Jones K E 2005 Nat. Rev. Neurosci. 6 389
[3]	安新磊, 乔帅, 张莉 2021 物理学报 70 050501 Google Scholar An X L, Qiao S, Zhang L 2021 Acta Phys. Sin. 70 050501 Google Scholar
[4]	Tan F, Zhou L L, Lu J W, Quan, H Z, Liu K Y 2022 Eur. J. Control 70 100764
[5]	徐泠风, 李传东, 陈玲 2016 物理学报 65 240701 Google Scholar Xu L F, Li C D, Chen L 2016 Acta Phys. Sin. 65 240701 Google Scholar
[6]	Liu B, Peng X N, Li C L 2024 Int. J. Electron. Commun. 178 155283
[7]	Guo M, Zhu Y L, Liu R Y, Zhao K X, Dou G 2021 Neurocomputing 472 12
[8]	徐聪, 王春华, 孙晶茹 2023 中国科学: 信息科学 53 164 Google Scholar Xu C, Wang C H, Sun J R 2023 Sci. Sin. Inf. 53 164 Google Scholar
[9]	张美娇, 张建刚, 魏立祥, 南梦冉 2021 中国医学物理学杂志 38 1273 Zhang M J, Zhang J G, Wei L X, Nan M R 2021 Chin. J. Med. Phys. 38 1273
[10]	Xu Q, Zhu D 2020 IETE Tech. Rev. 38 563
[11]	Lu Y C, Li H M, Li C L 2023 Neurocomputing 544 126246 Google Scholar
[12]	Li C L, Yang Y Y, Yang X B, Zi X Y, Xiao F L 2022 Nonlinear Dyn. 108 1697 Google Scholar
[13]	Li C L, Wang X, Du J R, Li Z J 2023 Nonlinear Dyn. 111 21333 Google Scholar
[14]	Tian Z K, Zhou D 2019 Front. Comput. Neurosci. 14 40
[15]	Ma M L, Xiong K L, Li Z J Sun Y C 2023 Mathematics 11 375 Google Scholar
[16]	Bashkirtseva I A, Nasyrova V, Ryashko L B 2018 Chaos, Solitons Fractals 110 76 Google Scholar
[17]	Kinouchi O, Tragtenberg M H R 1996 Int. J. Bifurcation Chaos 6 2343 Google Scholar
[18]	Girardi-Schappo M, Tragtenberg M H, Kinouchi O 2013 J. Neurosci. Methods 220 116 Google Scholar
[19]	Rosenblatt F 1963 Am. J. Psychol. 76 705 Google Scholar
[20]	Li H D, Li C L, He S B 2023 Int. J. Bifurcation Chaos 33 2350032 Google Scholar
[21]	郭慧朦, 梁燕, 董玉姣, 王光义 2023 物理学报 72 070501 Google Scholar Guo H M, Liang Y, Dong Y J, Wang G Y 2023 Acta Phys. Sin. 72 070501 Google Scholar
[22]	Zhou L L, Lin H C, Tan F 2023 Chaos, Solitons Fractals 173 113643 Google Scholar
[23]	秦铭宏, 赖强, 吴永红 2022 物理学报 71 160502 Google Scholar Qin M H, Lai Q, Wu Y H 2022 Acta Phys. Sin. 71 160502 Google Scholar
[24]	吴朝俊, 方礼熠, 杨宁宁 2024 物理学报 73 010501 Google Scholar Wu C J, Fang L Y, Yang N N 2024 Acta Phys. Sin. 73 010501 Google Scholar
[25]	Chua L O 2005 Int. J. Bifurcation Chaos 15 3435 Google Scholar
[26]	Jin P P, Wang G Y, Liang Y, Iu H H, Chua L O 2021 IEEE Trans. Circuits Syst. Regul. Pap. 68 4419 Google Scholar
[27]	Lai Q, Yang L 2022 Chaos, Solitons Fractals 165 112781 Google Scholar
[28]	Li K X, Bao H, Li H Z, Ma J, Hua Z Y, Bao B C 2021 IEEE Trans. Ind. Inf. 18 1726
[29]	Lu Y M, Wang C H, Deng Q L, Xu C 2022 Chin. Phys. B 31 060502 Google Scholar
[30]	Kawahara M, Kato‐Negishi M, Tanaka K 2017 Metallomics 9 619 Google Scholar
[31]	A N Shrivastava, A Triller, W Sieghart 2011 Front. Cell. Neurosci. 5 7
[32]	Li Z J, Yi Z 2022 Electron. Lett. 58 539 Google Scholar
[33]	Ding D W, Chen X Y, Yang Z L, Hu Y B, Wang M Y, Zhang H W, Zhang X 2022 Chaos, Solitons Fractals 158 112014 Google Scholar
[34]	Li Z J, Peng C, Wang M J, Ma M L 2024 Indian J. Phys. 98 1043 Google Scholar
[35]	Ma M L, Xiong K L, Li Z J, He S B 2024 Chin. Phys. B 33 028706 Google Scholar
[36]	Pool R 1989 Science 243 604 Google Scholar
[37]	Adhikari S P, Sah M P, Kim H, Chua L O 2013 IEEE Trans. Circuits Syst. Regul. Pap. 60 3008 Google Scholar
[38]	Ren L J, Mou J, Banerjee S, Zhang Y S 2023 Chaos, Solitons Fractals 167 113024 Google Scholar
[39]	Li C, Yi C, Li Y, Mitro S, Wang Z 2024 Chaos 34 031102 Google Scholar
[40]	Ma C G, Mou J, Xiong L, Banerjee S, Liu T M, Han X T 2021 Nonlinear Dyn. 103 2867 Google Scholar
[41]	Li C, Wang X, Chen G 2017 Nonlinear Dyn. 90 1335 Google Scholar
[42]	Li C, Gao Y, Lei T, Li RY, Xu Y 2024 Int. J. Bifurcation Chaos 34 2450008 Google Scholar

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Vol. 73, Issue 11 - 2024-06-05

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