Skip to main content
SpringerLink
Log in

Synchronization of Hypercomplex Neural Networks with Mixed Time-Varying Delays

  • Published:
Cognitive Computation Aims and scope Submit manuscript

Abstract

This article discusses the fixed-time synchronization (FTS) of hypercomplex neural networks (HCNNs) with mixed time-varying delays. Unlike finite-time synchronization (FNTS) based on initial conditions, the settling time of FTS can be adjusted to meet the needs. The state vector, weight matrices, activation functions, and input vectors of HCNNs are all hypercomplex numbers. The techniques used in complex-valued neural networks (CVNNs) and quaternion-valued neural networks (QVNNs) cannot be used directly with HCNNs because they do not work with eight or more dimensions. To begin with, the decomposition method is used to split the HCNNs into \((n+1)\) real-valued neural networks (RVNNs) applying distributive law to handle non-commutativity and non-associativity. A nonlinear controller is constructed to synchronize the master-response systems of the HCNNs. Lyapunov-based method is used to prove the stability of an error system. The FTS of mixed time-varying delayed HCNNs is achieved using a suitable lemma, Lipschitz condition, appropriate Lyapunov functional construction, and designing suitable controllers. Two different algebraic criteria for settling time have been achieved by employing two distinct lemmas. It is demonstrated that the settling time derived from Lemma 1 produces a more precise result than that obtained from Lemma 2. Three numerical examples for CVNNs, QVNNs, and octonions-valued neural networks (OVNNs) are provided to demonstrate the efficacy and effectiveness of the proposed theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Cao Y, Samidurai R, Sriraman R. Robust passivity analysis for uncertain neural networks with leakage delay and additive time-varying delays by using general activation function. Math Comput Simul. 2019;155:57–77.

    Article  MathSciNet  Google Scholar 

  2. Wei X, Zhang Z, Lin C, Chen J. Synchronization and anti-synchronization for complex-valued inertial neural networks with time-varying delays. Appl Math Comput. 2021;403:126194.

    MathSciNet  Google Scholar 

  3. Aizenberg NN, Ivaskiv YL, Pospelov, DA. A certain generalization of threshold functions. In: Doklady Akademii Nauk (vol. 196, no. 6). Russian Academy of Sciences; 1971. pp. 1287–90.

  4. Noest AJ. Associative memory in sparse phasor neural networks. EPL (Europhysics Letters). 1988;6(5):469.

  5. Aizenberg NN, Aizenberg IN. CNN based on multi-valued neuron as a model of associative memory for grey scale images. In: CNNA’92 Proceedings Second International Workshop on Cellular Neural Networks and their Applications. IEEE; 1992. p. 36–41.

    Chapter  Google Scholar 

  6. Kantor IL, Solodovnikov AS, Shenitzer A. Hypercomplex numbers: An elementary introduction to algebras, vol. 302. Springer; 1989.

    Book  Google Scholar 

  7. Hitzer E, Nitta T, Kuroe Y. Applications of Clifford’s geometric algebra. Adv Appl Clifford Algebras. 2013;23(2):377–404.

    Article  MathSciNet  Google Scholar 

  8. Aizenberg I. Complex-valued neural networks with multi-valued neurons, vol. 353. Springer; 2011.

    Book  Google Scholar 

  9. Nitta T. Complex-valued neural networks: Utilizing high-dimensional parameters: Utilizing high-dimensional parameters. IGI Global; 2009.

  10. Hirose A. Complex-valued neural networks, vol. 400. Springer Science & Business Media; 2012.

    Google Scholar 

  11. Jankowski S, Lozowski A, Zurada JM. Complex-valued multistate neural associative memory. IEEE Trans Neural Networks. 1996;7(6):1491–6.

    Article  CAS  PubMed  Google Scholar 

  12. Kobayashi M. Symmetric complex-valued Hopfield neural networks. IEEE Trans Neural Netw Learn Syst. 2016;28(4):1011–5.

    Article  PubMed  Google Scholar 

  13. Isokawa T, Yamamoto H, Nishimura H, Yumoto T, Kamiura N, Matsui N. Complex-valued associative memories with projection and iterative learning rules. J Artif Intell Soft Comput Res. 2018;8(3):237–49.

    Article  Google Scholar 

  14. Nitta T, Kuroe Y. Hyperbolic gradient operator and hyperbolic back-propagation learning algorithms. IEEE Trans Neural Netw Learn Syst. 2017;29(5):1689–702.

    Article  MathSciNet  PubMed  Google Scholar 

  15. Isokawa T, Nishimura H, Kamiura N, Matsui N. Associative memory in quaternionic Hopfield neural network. Int J Neural Syst. 2008;18(02):135–45.

    Article  PubMed  Google Scholar 

  16. Parcollet T, Morchid M, Linarès G. A survey of quaternion neural networks. Artif Intell Rev. 2020;53(4):2957–82.

    Article  Google Scholar 

  17. Kuroe Y. Models of recurrent Clifford neural networks and their dynamics. Complex-Valued Neural Netw: Adv Appl. 2013;133–51. Wiley Online Library.

  18. Peng W, Varanka T, Mostafa A, Shi H, Zhao G. Hyperbolic deep neural networks: a survey. IEEE Trans Pattern Anal Mach Intell. 2021;44(12):10023–44.

    Article  Google Scholar 

  19. Kuroe Y, Iima H. A model of hopfield-type octonion neural networks and existing conditions of energy functions. In: 2016 International Joint Conference on Neural Networks (IJCNN). IEEE; 201+. p. 4426–30.

  20. Liu Y, Zhang D, Lu J, Cao J. Global \(\mu \)-stability criteria for quaternion-valued neural networks with unbounded time-varying delays. Inf Sci. 2016;360:273–88.

    Article  ADS  Google Scholar 

  21. Baluni S, Das S, Yadav VK, Cao J. Lagrange \(\alpha \)-exponential synchronization of non-identical fractional-order complex-valued neural networks. Circuits Systems Signal Process. 2022;41(10):5632–52.

    Article  Google Scholar 

  22. Popa C-A. Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays. Neurocomputing. 2018;309:117–33.

    Article  Google Scholar 

  23. Huang C, Zhao X, Cao J, Alsaadi FE. Global dynamics of neoclassical growth model with multiple pairs of variable delays. Nonlinearity. 2020;33(12):6819.

    Article  ADS  MathSciNet  Google Scholar 

  24. Song C, Fei S, Cao J, Huang C. Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control. Mathematics. 2019;7(7):599.

    Article  Google Scholar 

  25. Gao J, Dai L, Jiang H. Stability analysis of pseudo almost periodic solutions for octonion-valued recurrent neural networks with proportional delay. Chaos Solitons Fractals. 2023;175:114061.

    Article  MathSciNet  Google Scholar 

  26. Yang D, Li X, Qiu J. Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback. Nonlinear Anal Hybrid Syst. 2019;32:294–305.

    Article  MathSciNet  Google Scholar 

  27. Yang X, Li X, Xi Q, Duan P. Review of stability and stabilization for impulsive delayed systems. Math Biosci Eng. 2018;15(6):1495.

  28. Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett. 1990;64(8):821.

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  29. He W, Cao J. Exponential synchronization of chaotic neural networks: a matrix measure approach. Nonlinear Dyn. 2009;55(1):55–65.

    Article  MathSciNet  Google Scholar 

  30. Das S, Srivastava M, Leung A. Hybrid phase synchronization between identical and nonidentical three-dimensional chaotic systems using the active control method. Nonlinear Dyn. 2013;73:2261–72.

    Article  MathSciNet  Google Scholar 

  31. Wei R, Cao J, Gorbachev S. Fixed-time control for memristor-based quaternion-valued neural networks with discontinuous activation functions. Cogn Comput. 2023;15(1):50–60.

    Article  Google Scholar 

  32. Liu Y, Wang Z, Liu X. Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw. 2006;19(5):667–75.

    Article  PubMed  Google Scholar 

  33. Kumar U, Das S, Huang C, Cao J. Fixed-time synchronization of quaternion-valued neural networks with time-varying delay. Proceedings of the Royal Society A. 2020;476(2241):20200324.

    Article  ADS  MathSciNet  Google Scholar 

  34. Cai Z, Huang L, Zhu M, Wang D. Finite-time stabilization control of memristor-based neural networks. Nonlinear Anal Hybrid Syst. 2016;20:37–54.

    Article  MathSciNet  Google Scholar 

  35. Polyakov A. Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans Autom Control. 2011;57(8):2106–10.

    Article  MathSciNet  Google Scholar 

  36. Cao J, Li R. Fixed-time synchronization of delayed memristor-based recurrent neural networks. Science China Inf Sci. 2017;60(3):1–15.

    Article  ADS  Google Scholar 

  37. Deng H, Bao H. Fixed-time synchronization of quaternion-valued neural networks. Physica A. 2019;527:121351.

  38. Singh S, Kumar U, Das S, Alsaadi F, Cao J. Synchronization of quaternion valued neural networks with mixed time delays using Lyapunov function method. Neural Process Lett. 2021;1–17. Springer.

  39. Zhou C, Zhang W, Yang X, Xu C, Feng J. Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations. Neural Process Lett. 2017;46(1):271–91.

    Article  Google Scholar 

  40. Lu W, Chen T. Synchronization of coupled connected neural networks with delays. IEEE Trans Circuits Syst I Regul Pap. 2004;51(12):2491–503.

    Article  MathSciNet  Google Scholar 

  41. Chen C, Li L, Peng H, Yang Y, Mi L, Zhao H. A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks. Neural Netw. 2020;123:412–9.

    Article  PubMed  Google Scholar 

  42. Valle ME, Lobo RA. Hypercomplex-valued recurrent correlation neural networks. Neurocomputing. 2021;432:111–23.

    Article  Google Scholar 

Download references

Acknowledgements

The authors are extending their heartfelt thanks to the revered reviewers for their valuable comments towards the upgradation of the article.

Funding

The work is supported by the MATRICS scheme, SERB, Govt. of India (File No.: MTR/2020/000053); and DST, Govt. of India INSPIRE Program (IF-180997).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Sapna Baluni & Subir Das

  2. Department of Mathematics, Amity School of Applied Sciences, Amity University, Gurugram, Haryana, 122413, India

    Vijay K. Yadav

  3. School of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

Authors
  1. Sapna Baluni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vijay K. Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Subir Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jinde Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinde Cao.

Ethics declarations

Human and Animal Rights

This article does not contain any studies with human or animal subjects performed by any of the authors.

Informed Consent

Informed consent was obtained from all individual participants included in the study.

Conflict of Interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baluni, S., K. Yadav, V., Das, S. et al. Synchronization of Hypercomplex Neural Networks with Mixed Time-Varying Delays. Cogn Comput (2024). https://doi.org/10.1007/s12559-024-10253-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12559-024-10253-9

Keywords

Search

Navigation