Abstract
In this paper, a class of complex-valued neural networks (CVNNs) with stochastic parameters and mixed time delays are proposed. The random fluctuation of system parameters is considered in order to describe the implementation of CVNNs more practically. Mixed time delays including distributed delays and time-varying delays are also taken into account in order to reflect the influence of network loads and communication constraints. Firstly, the stability problem is investigated for the CVNNs. In virtue of Lyapunov stability theory, a sufficient condition is deduced to ensure that CVNNs are asymptotically stable in the mean square. Then, for an array of coupled identical CVNNs with stochastic parameters and mixed time delays, synchronization issue is investigated. A set of matrix inequalities are obtained by using Lyapunov stability theory and Kronecker product and if these matrix inequalities are feasible, the addressed CVNNs are synchronized. Finally, the effectiveness of the obtained theoretical results is demonstrated by two numerical examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing not applicable to this paper as no datasets were generated or analysed during the current study.
References
Aizenberg I (2011) Complex-valued neural networks with multi-valued neurons. Springer, Berlin
Cao Y, Sriraman R, Shyamsundarraj N, Samidurai R (2020) Robust stability of uncertain stochastic complex-valued neural networks with additive time-varying delays. Math Comput Simul 171:207–220. https://doi.org/10.1016/j.matcom.2019.05.011
Cao Y, Ramajayam S, Sriraman R, Samidurai R (2021) Leakage delay on stabilization of finite-time complex-valued BAM neural network: decomposition approach. Neurocomputing 463:505–513. https://doi.org/10.1016/j.neucom.2021.08.056
Chen J, Chen X (2001) Special matrices. Tsinghua Publisher, Beijing, China
Chen S, Song Q, Zhao Z, Liu Y, Alsaadi FE (2021) Global asymptotic stability of fractional-order complex-valued neural networks with probabilistic time-varying delays. Neurocomputing 450:311–318. https://doi.org/10.1016/j.neucom.2021.04.043
Ding D, Wang Z, Han QL, Wei G (2018) Security control for discrete-time stochastic nonlinear systems subject to deception attacks. IEEE Trans Syst Man Cybern Syst 48(5):779–789. https://doi.org/10.1109/TSMC.2016.2616544
Fan D, Wang Q (2017) Synchronization and bursting transition of the coupled Hindmarsh–Rose systems with asymmetrical time-delays. Sci China Technol Sci 60(7):1019–1031. https://doi.org/10.1007/s11431-016-0169-8
Feng L, Hu C, Yu J, Jiang H, Wen S (2021) Fixed-time synchronization of coupled memristive complex-valued neural networks. Chaos Solit Fractals 148:110993. https://doi.org/10.1016/j.chaos.2021.110993
Ge X, Han QL, Wang Z (2019) A threshold-parameter-dependent approach to designing distributed event-triggered \(H_{\infty }\) consensus filters over sensor networks. IEEE Trans Cybern 49(4):1148–1159. https://doi.org/10.1109/TCYB.2017.2789296
Gong W, Liang J, Cao J (2015) Matrix measure method for global exponential stability of complex-valued recurrent neural networks with time-varying delays. Neural Netw 70:81–89. https://doi.org/10.1016/j.neunet.2015.07.003
Gürüler H (2017) A novel diagnosis system for Parkinson’s disease using complex-valued artificial neural network with k-means clustering feature weighting method. Neural Comput Appl 28(7):1657–1666. https://doi.org/10.1007/s00521-015-2142-2
Hirose A (2012) Complex-valued neural networks. Springer, Berlin, Heidelberg
Hu J, Wang J (2012) Global stability of complex-valued recurrent neural networks with time-delays. IEEE Trans Neural Netw Learn Syst 23(6):853–865. https://doi.org/10.1109/TNNLS.2012.2195028
Hu J, Cao J, Alofi A, Al-Mazrooei A, Elaiw A (2015) Pinning synchronization of coupled inertial delayed neural networks. Cogn Neurodyn 9:341–350. https://doi.org/10.1007/s11571-014-9322-0
Huang Y, Wu F (2021) Finite-time passivity and synchronization of coupled complex-valued memristive neural networks. Inf Sci 580:775–800. https://doi.org/10.1016/j.ins.2021.09.050
Huang T, Li C, Duan S, Starzyk JA (2012) Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects. IEEE Trans Neural Netw Learn Syst 23(6):866–875. https://doi.org/10.1109/TNNLS.2012.2192135
Huang YB, He Y, An J, Wu M (2021) Polynomial-type Lyapunov–Krasovskii functional and Jacobi-Bessel inequality: Further results on stability analysis of time-delay systems. IEEE Trans Autom Control 66(6):2905–2912. https://doi.org/10.1109/TAC.2020.3013930
Kim SY, Lim W (2020) Effect of interpopulation spike-timing-dependent plasticity on synchronized rhythms in neuronal networks with inhibitory and excitatory populations. Cogn Neurodyn 14:535–567. https://doi.org/10.1007/s11571-020-09580-y
Li X, Ho DWC, Cao J (2019) Finite-time stability and settling-time estimation of nonlinear impulsive systems. Automatica 99:361–368. https://doi.org/10.1016/j.automatica.2018.10.024
Li L, Shi X, Liang J (2019) Synchronization of impulsive coupled complex-valued neural networks with delay: The matrix measure method. Neural Netw 117:285–294. https://doi.org/10.1016/j.neunet.2019.05.024
Li Q, Liang J, Qu H (2021) \(H_\infty\) estimation for stochastic semi-Markovian switching CVNNs with missing measurements and mode-dependent delays. Neural Netw 141:281–293. https://doi.org/10.1016/j.neunet.2021.04.022
Liu X, Chen T (2016) Global exponential stability for complex-valued recurrent neural networks with asynchronous time delays. IEEE Trans Neural Netw Learn Syst 27(3):593–606. https://doi.org/10.1109/TNNLS.2015.2415496
Liu Y, Wang Z, Liang J, Liu X (2009) Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays. IEEE Trans Neural Netw 20(7):1102–1116. https://doi.org/10.1109/TNN.2009.2016210
Lu J, Ho DWC, Cao J, Kurths J (2011) Exponential synchronization of linearly coupled neural networks with impulsive disturbances. IEEE Trans Neural Netw 22(2):329–335. https://doi.org/10.1109/TNN.2010.2101081
Ma L, Wang Z, Liu Y, Alsaadi FE (2019) Distributed filtering for nonlinear time-delay systems over sensor networks subject to multiplicative link noises and switching topology. Int J Robust Nonlinear Control 29(10):2941–2959. https://doi.org/10.1002/rnc.4535
Nitta T (2003) Solving the XOR problem and the detection of symmetry using a single complex-valued neuron. Neural Netw 16(8):1101–1105. https://doi.org/10.1016/S0893-6080(03)00168-0
Peker M, Sen B, Delen D (2016) A novel method for automated diagnosis of epilepsy using complex-valued classifiers. IEEE J Biomed Health Inform 20(1):108–118. https://doi.org/10.1109/JBHI.2014.2387795
Qi J, Li C, Huang T (2014) Stability of delayed memristive neural networks with time-varying impulses. Cogn Neurodyn 8:429–436. https://doi.org/10.1007/s11571-014-9286-0
Rakkiyappan R, Cao J, Velmurugan G (2015) Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans Neural Netw Learn Syst 26(1):84–97. https://doi.org/10.1109/TNNLS.2014.2311099
Slotine JE, Li W (1991) Applied nonlinear control. Englewood Cliffs, New Jersey
Solomon O, Fridman E (2013) New stability conditions for systems with distributed delays. Automatica 49(11):3467–3475. https://doi.org/10.1016/j.automatica.2013.08.025
Sriraman R, Cao Y, Samidurai R (2020) Global asymptotic stability of stochastic complex-valued neural networks with probabilistic time-varying delays. Math Comput Simul 171:103–118. https://doi.org/10.1016/j.matcom.2019.04.001
Subramanian K, Muthukumar P (2017) Global asymptotic stability of complex-valued neural networks with additive time-varying delays. Cogn Neurodyn 11:293–306. https://doi.org/10.1007/s11571-017-9429-1
Sunaga Y, Natsuaki R, Hirose A (2019) Land form classification and similar land-shape discovery by using complex-valued convolutional neural networks. IEEE Trans Geosci Remote Sens 57(10):7907–7917. https://doi.org/10.1109/TGRS.2019.2917214
Uhlhaas PJ, Singer W (2006) Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron 52(1):155–168. https://doi.org/10.1016/j.neuron.2006.09.020
Wang B, Zhu Q (2018) Stability analysis of semi-Markov switched stochastic systems. Automatica 94:72–80. https://doi.org/10.1016/j.automatica.2018.04.016
Wang H, Duan S, Huang T, Wang L, Li C (2017) Exponential stability of complex-valued memristive recurrent neural networks. IEEE Trans Neural Netw Learn Syst 28(3):766–771. https://doi.org/10.1109/TNNLS.2015.2513001
Wang S, Guo Z, Wen S, Huang T (2020) Global synchronization of coupled delayed memristive reaction-diffusion neural networks. Neural Netw 123:362–371. https://doi.org/10.1016/j.neunet.2019.12.016
Wang M, Wang Z, Dong H, Han QL (2021) A novel framework for backstepping-based control of discrete-time strict-feedback nonlinear systems with multiplicative noises. IEEE Trans Autom Control 66(4):1484–1496. https://doi.org/10.1109/TAC.2020.2995576
Wen G, Wang P, Yu X, Yu W, Cao J (2019) Pinning Synchronization of complex switching networks with a leader of nonzero control inputs. IEEE Trans Circuits Syst I Regul Pap 66(8):3100–3112. https://doi.org/10.1109/TCSI.2019.2904946
Wu A, Chen Y, Zeng Z (2021) Quantization synchronization of chaotic neural networks with time delay under event-triggered strategy. Cogn Neurodyn 15:897–914. https://doi.org/10.1007/s11571-021-09667-0
Yu Y, Hao Y, Wang Q (2020) Model-based optimized phase-deviation deep brain stimulation for Parkinson’s disease. Neural Netw 122:308–319. https://doi.org/10.1016/j.neunet.2019.11.001
Zhang L, Song Q, Zhao Z (2017) Stability analysis of fractional-order complex-valued neural networks with both leakage and discrete delays. Appl Math Comput 298:296–309. https://doi.org/10.1016/j.amc.2016.11.027
Zhang XM, Han QL, Wang Z, Zhang BL (2017) Neuronal state estimation for neural networks with two additive time-varying delay components. IEEE Trans Cybernet 47(10):3184–3194. https://doi.org/10.1109/TCYB.2017.2690676
Zhang Z, Guo R, Liu X, Lin C (2020) Lagrange exponential stability of complex-valued BAM neural networks with time-varying delays. IEEE Trans Syst Man Cybernet Syst 50(8):3072–3085. https://doi.org/10.1109/TSMC.2018.2840091
Zou L, Wang Z, Gao H, Liu X (2017) State estimation for discrete-time dynamical networks with time-varying delays and stochastic disturbances under the Round-Robin protocol. IEEE Trans Neural Netw Learn Syst 28(5):1139–1151. https://doi.org/10.1109/TNNLS.2016.2524621
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grants 61873059 and 61922024, the Program of Shanghai Academic/Technology Research Leader under Grant 20XD1420100, and the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University of China under Grants CUSF-DH-D-2020085 and CUSF-DH-D-2021056.
Author information
Authors and Affiliations
Contributions
YL, BS and JS contributed equally to the background analysis, literature search, theoretical derivation, theoretical verification, and writing. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
Ethics approval was not required for this research.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Liu, Y., Shen, B. & Sun, J. Stability and synchronization for complex-valued neural networks with stochastic parameters and mixed time delays. Cogn Neurodyn 17, 1213–1227 (2023). https://doi.org/10.1007/s11571-022-09823-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11571-022-09823-0