November  2021, 29(5): 3323-3340. doi: 10.3934/era.2021041

Synchronization for a class of complex-valued memristor-based competitive neural networks(CMCNNs) with different time scales

1. 

School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, China

2. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China

* Corresponding author: Yong Zhao

Received  December 2020 Revised  March 2021 Published  November 2021 Early access  May 2021

Fund Project: The first author is supported by NSF grant No.12062004 and No.11972115

In this paper, the synchronization problem of complex-valued memristive competitive neural networks(CMCNNs) with different time scales is investigated. Based on differential inclusions and inequality techniques, some novel sufficient conditions are derived to ensure synchronization of the drive-response systems by designing a proper controller. Finally, a numerical example is provided to illustrate the usefulness and feasibility of our results.

Citation: Yong Zhao, Shanshan Ren. Synchronization for a class of complex-valued memristor-based competitive neural networks(CMCNNs) with different time scales. Electronic Research Archive, 2021, 29 (5) : 3323-3340. doi: 10.3934/era.2021041
References:
[1]

A. AscoliV. LanzaF. Corinto and R. Tetzlaff, Synchronization conditions in simple memristor neural networks, Journal of the Franklin Institute, 352 (2015), 3196-3220.  doi: 10.1016/j.jfranklin.2015.06.003.  Google Scholar

[2]

J. L. BonaS. Vento and F. B. Weissler, Singularity formation and blowup of complex-valued solutions of the modified KdV equation, Discrete Cont. Dyn., 33 (2013), 4811-4840.  doi: 10.3934/dcds.2013.33.4811.  Google Scholar

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D. FanY. ZhengZ. Yang and Q. Wang, Improving control effects of absence seizures using single-pulse alternately resetting stimulation (SARS) of corticothalamic circuit, Appl. Math. Mech. -Engl. Ed., 41 (2020), 1287-1302.  doi: 10.1007/s10483-020-2644-8.  Google Scholar

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S. GongS. YangZ. Guo and T. Huang, Global exponential synchronization of memristive competitive neural networks with time-varying delay via nonlinear control, Neural Process. Lett., 49 (2018), 103-119.  doi: 10.1007/s11063-017-9777-1.  Google Scholar

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F. HanM. WiercigrochJ.-A. Fang and Z. Wang, Excitement and synchronization of small-world neuronal networks with short-term synaptic plasticity, Int. J. Neural Syst., 21 (2011), 415-425.  doi: 10.1142/S0129065711002924.  Google Scholar

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F. HanB. ZhenY. DuY. Zheng and M. Wiercigroch, Global Hopf bifurcation analysis on a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme, Discrete Cont. Dyn. Syst.-B, 16 (2011), 457-474.  doi: 10.3934/dcdsb.2011.16.457.  Google Scholar

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D. LiuS. Zhu and K. Sun, Global anti-synchronization of complex-valued memristive neural networks with time delays, IEEE T. Cybernetics, 49 (2019), 1735-1747.  doi: 10.1109/TCYB.2018.2812708.  Google Scholar

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A. Meyer-BäseF. Ohl and H. Scheich, Singular perturbation analysis of competitive neural networks with different time scales, Neural. Comput., 8 (1996), 1731-1742.  doi: 10.1162/neco.1996.8.8.1731.  Google Scholar

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A. Meyer-Bäse and V. Thummler, Local and global stability analysis of an unsupervised competitive neural network, IEEE T. Neural Networ., 19 (2008), 346-351.  doi: 10.1109/TNN.2007.908626.  Google Scholar

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A. Meyer-BäseP. SergeiA. Wismueller and F. Simon, Local exponential stability of competitive neural networks with different time scales, Eng. Appl. Artif. Intell., 17 (2004), 227-232.  doi: 10.1016/j.engappai.2004.02.010.  Google Scholar

[13]

S. RenY. Zhao and Y. Xia, Anti-synchronization of a class of fuzzy memristive competitive neural networks with different time scales, Neural Process. Lett., 52 (2020), 647-661.  doi: 10.1007/s11063-020-10269-w.  Google Scholar

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A. RiehleS. Grun and M. Diesmann, Spike synchronization and rate modulation differentially involved in motor cortical function, Science, 278 (1997), 1950-1953.  doi: 10.1126/science.278.5345.1950.  Google Scholar

[15]

Y. Shi and P. Zhu, Synchronization of memristive competitive neural networks with different time scales, Neural. Comput. and Applic., 25 (2014), 1163-1168.  doi: 10.1007/s00521-014-1598-9.  Google Scholar

[16]

J. SunG. HanY. WangH. Zhang and L. Wu, Hybrid memristor chaotic system, J. Nanorlectron. Optoe., 13 (2018), 812-818.  doi: 10.1166/jno.2018.2326.  Google Scholar

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J. M. Tour and T. He, Electronics: The fourth element, Nature, 453 (2008), 42-43.  doi: 10.1038/453042a.  Google Scholar

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M. E. Valle, Complex-valued recurrent correlation neural networks, IEEE Trans. Neural. Netw. Learn. Syst., 25 (2014), 1600-1612.  doi: 10.1109/TNNLS.2014.2341013.  Google Scholar

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M. Di VentraY. V. Pershin and L. O. Chua, Circuit elements with memory: Memristors, memcapacitors, and meminductors, P. IEEE, 97 (2009), 1717-1724.  doi: 10.1109/JPROC.2009.2021077.  Google Scholar

[20]

D. Wang, L. Huang and L. Tang, New results for global exponential synchronization in neural networks via functional differential inclusions, Chaos, 25 (2015), 083103, 11 pp. doi: 10.1063/1.4927737.  Google Scholar

[21]

Y. YuX. M. WangQ. S. Wang and Q. Y. Wang, A review of computational modeling and deep brain stimulation: Applications to Parkinson's disease, Appl. Math. Mech. -Engl. Ed., 41 (2020), 1747-1768.   Google Scholar

[22]

J. Zhang and X. Liao, Effects of initial conditions on the synchronization of the coupled memristor neural circuits, Nonlinear Dynamics, 95 (2019), 1269-1282.  doi: 10.1007/s11071-018-4628-9.  Google Scholar

[23]

W. Zhang, C. Li and T. Huang,, Global robust stability of complex-valued recurrent neural networks with time-delays and uncertainties, Int. J. Biomath., 7 (2014), 1450016. doi: 10.1142/S1793524514500168.  Google Scholar

[24]

Y. ZhaoJ. Kurths and L. Duan, Input-to-state stability analysis for memristive Cohen-Grossberg-type neural networks with variable time delays, Chaos Soliton. Fract., 114 (2018), 364-369.  doi: 10.1016/j.chaos.2018.07.021.  Google Scholar

[25]

L. Zhou and Z. Zhao, Exponential stability of a class of competitive neural networks with multi-proportional delays, Neural Process. Lett., 44 (2016), 651-663.  doi: 10.1007/s11063-015-9486-6.  Google Scholar

[26]

S. ZhuD. LiuC. Yang and J. Fu, Synchronization of memristive complex-valued neural networks with time delays via pinning control method, IEEE T. Cybernetics, 50 (2020), 3806-3815.  doi: 10.1109/TCYB.2019.2946703.  Google Scholar

[27]

J. Zhuang, Y. Zhou and Y. Xia, Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations, Discrete Cont. Dyn.-S., 14 (2021), 1607–-1629. doi: 10.3934/dcdss.2020279.  Google Scholar

show all references

References:
[1]

A. AscoliV. LanzaF. Corinto and R. Tetzlaff, Synchronization conditions in simple memristor neural networks, Journal of the Franklin Institute, 352 (2015), 3196-3220.  doi: 10.1016/j.jfranklin.2015.06.003.  Google Scholar

[2]

J. L. BonaS. Vento and F. B. Weissler, Singularity formation and blowup of complex-valued solutions of the modified KdV equation, Discrete Cont. Dyn., 33 (2013), 4811-4840.  doi: 10.3934/dcds.2013.33.4811.  Google Scholar

[3]

L. Chua, Memristor–The missing circuit element, IEEE Trans. Circ. Theor., 18 (1971), 507-519.  doi: 10.1109/TCT.1971.1083337.  Google Scholar

[4]

D. FanY. ZhengZ. Yang and Q. Wang, Improving control effects of absence seizures using single-pulse alternately resetting stimulation (SARS) of corticothalamic circuit, Appl. Math. Mech. -Engl. Ed., 41 (2020), 1287-1302.  doi: 10.1007/s10483-020-2644-8.  Google Scholar

[5]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht, 18 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[6]

S. GongS. YangZ. Guo and T. Huang, Global exponential synchronization of memristive competitive neural networks with time-varying delay via nonlinear control, Neural Process. Lett., 49 (2018), 103-119.  doi: 10.1007/s11063-017-9777-1.  Google Scholar

[7]

F. HanM. WiercigrochJ.-A. Fang and Z. Wang, Excitement and synchronization of small-world neuronal networks with short-term synaptic plasticity, Int. J. Neural Syst., 21 (2011), 415-425.  doi: 10.1142/S0129065711002924.  Google Scholar

[8]

F. HanB. ZhenY. DuY. Zheng and M. Wiercigroch, Global Hopf bifurcation analysis on a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme, Discrete Cont. Dyn. Syst.-B, 16 (2011), 457-474.  doi: 10.3934/dcdsb.2011.16.457.  Google Scholar

[9]

D. LiuS. Zhu and K. Sun, Global anti-synchronization of complex-valued memristive neural networks with time delays, IEEE T. Cybernetics, 49 (2019), 1735-1747.  doi: 10.1109/TCYB.2018.2812708.  Google Scholar

[10]

A. Meyer-BäseF. Ohl and H. Scheich, Singular perturbation analysis of competitive neural networks with different time scales, Neural. Comput., 8 (1996), 1731-1742.  doi: 10.1162/neco.1996.8.8.1731.  Google Scholar

[11]

A. Meyer-Bäse and V. Thummler, Local and global stability analysis of an unsupervised competitive neural network, IEEE T. Neural Networ., 19 (2008), 346-351.  doi: 10.1109/TNN.2007.908626.  Google Scholar

[12]

A. Meyer-BäseP. SergeiA. Wismueller and F. Simon, Local exponential stability of competitive neural networks with different time scales, Eng. Appl. Artif. Intell., 17 (2004), 227-232.  doi: 10.1016/j.engappai.2004.02.010.  Google Scholar

[13]

S. RenY. Zhao and Y. Xia, Anti-synchronization of a class of fuzzy memristive competitive neural networks with different time scales, Neural Process. Lett., 52 (2020), 647-661.  doi: 10.1007/s11063-020-10269-w.  Google Scholar

[14]

A. RiehleS. Grun and M. Diesmann, Spike synchronization and rate modulation differentially involved in motor cortical function, Science, 278 (1997), 1950-1953.  doi: 10.1126/science.278.5345.1950.  Google Scholar

[15]

Y. Shi and P. Zhu, Synchronization of memristive competitive neural networks with different time scales, Neural. Comput. and Applic., 25 (2014), 1163-1168.  doi: 10.1007/s00521-014-1598-9.  Google Scholar

[16]

J. SunG. HanY. WangH. Zhang and L. Wu, Hybrid memristor chaotic system, J. Nanorlectron. Optoe., 13 (2018), 812-818.  doi: 10.1166/jno.2018.2326.  Google Scholar

[17]

J. M. Tour and T. He, Electronics: The fourth element, Nature, 453 (2008), 42-43.  doi: 10.1038/453042a.  Google Scholar

[18]

M. E. Valle, Complex-valued recurrent correlation neural networks, IEEE Trans. Neural. Netw. Learn. Syst., 25 (2014), 1600-1612.  doi: 10.1109/TNNLS.2014.2341013.  Google Scholar

[19]

M. Di VentraY. V. Pershin and L. O. Chua, Circuit elements with memory: Memristors, memcapacitors, and meminductors, P. IEEE, 97 (2009), 1717-1724.  doi: 10.1109/JPROC.2009.2021077.  Google Scholar

[20]

D. Wang, L. Huang and L. Tang, New results for global exponential synchronization in neural networks via functional differential inclusions, Chaos, 25 (2015), 083103, 11 pp. doi: 10.1063/1.4927737.  Google Scholar

[21]

Y. YuX. M. WangQ. S. Wang and Q. Y. Wang, A review of computational modeling and deep brain stimulation: Applications to Parkinson's disease, Appl. Math. Mech. -Engl. Ed., 41 (2020), 1747-1768.   Google Scholar

[22]

J. Zhang and X. Liao, Effects of initial conditions on the synchronization of the coupled memristor neural circuits, Nonlinear Dynamics, 95 (2019), 1269-1282.  doi: 10.1007/s11071-018-4628-9.  Google Scholar

[23]

W. Zhang, C. Li and T. Huang,, Global robust stability of complex-valued recurrent neural networks with time-delays and uncertainties, Int. J. Biomath., 7 (2014), 1450016. doi: 10.1142/S1793524514500168.  Google Scholar

[24]

Y. ZhaoJ. Kurths and L. Duan, Input-to-state stability analysis for memristive Cohen-Grossberg-type neural networks with variable time delays, Chaos Soliton. Fract., 114 (2018), 364-369.  doi: 10.1016/j.chaos.2018.07.021.  Google Scholar

[25]

L. Zhou and Z. Zhao, Exponential stability of a class of competitive neural networks with multi-proportional delays, Neural Process. Lett., 44 (2016), 651-663.  doi: 10.1007/s11063-015-9486-6.  Google Scholar

[26]

S. ZhuD. LiuC. Yang and J. Fu, Synchronization of memristive complex-valued neural networks with time delays via pinning control method, IEEE T. Cybernetics, 50 (2020), 3806-3815.  doi: 10.1109/TCYB.2019.2946703.  Google Scholar

[27]

J. Zhuang, Y. Zhou and Y. Xia, Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations, Discrete Cont. Dyn.-S., 14 (2021), 1607–-1629. doi: 10.3934/dcdss.2020279.  Google Scholar

Figure 1.  Synchronization errors of $ e^R_1,e^R_2,h^R_1,h^R_2 $ of system (45) and (46) with the controllers (20)
Figure 2.  Synchronization errors of $ e^I_1,e^I_2,h^I_1,h^I_2 $ of system (45) and (46) with the controllers (20)
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