Skip to main content
SpringerLink
Log in

Chaos and firing patterns in a discrete fractional Hopfield neural network model

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This article explores the importance of neuronal firing patterns in transmitting information within the human brain. These patterns are unique to each group of neurons and play a crucial role in understanding their diverse behavior. The article introduces various activation functions for neurons and develops a 4D discrete fractional order Hopfield neural network model to showcase the complex dynamics involved. Additionally, the neurons exhibit nonlinear behavior in their self-synaptic weight functions due to external stimuli. The article examines the dynamics of the network with and without external stimulus, presenting bifurcation diagrams that illustrate the transition between chaotic and stable states. The research also investigates the region of chaos in relation to the fractional order and the nonlinear synaptic function. The largest Lyapunov exponents are used to illustrate this chaotic region. It also demonstrates the network’s sensitivity to even the smallest changes in parameter values, visualizing different firing patterns. This research emphasizes how the choice of activation functions, fractional order, and external input greatly influence the equilibrium and behavior of the network’s state variables. By analyzing the system’s dynamics and changes in equilibrium states over time, the study sheds light on the diverse dynamical characteristics exhibited by the system.

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29

Similar content being viewed by others

Data Availability

No datasets are generated or analyzed during the current study.

References

  1. Abbes, A., Ouannas, A., Shawagfeh, N., Khennaoui, A.A.: Incommensurate fractional discrete neural network: chaos and complexity. Eur. Phys. J. Plus 137(2), 235 (2022)

    Google Scholar 

  2. Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62(3), 1602–1611 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Abdeljawad, T., Banerjee, S., Wu, G.C.: Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption. Optik 218, 163698 (2020)

    Google Scholar 

  4. Adeli, H.: Neural networks in civil engineering: 1989–2000. Comput. Aided Civil Infrastruct. Eng. 16(2), 126–142 (2001)

    Google Scholar 

  5. Aguilar, C.Z., Gómez-Aguilar, J., Alvarado-Martínez, V., Romero-Ugalde, H.: Fractional order neural networks for system identification. Chaos Solitons Fractals 130, 109444 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Aihara, K.: Chaos engineering and its application to parallel distributed processing with chaotic neural networks. Proc. IEEE 90(5), 919–930 (2002)

    Google Scholar 

  7. Alzabut, J., Tyagi, S., Abbas, S.: Discrete fractional-order bam neural networks with leakage delay: existence and stability results. Asian J. Control 22(1), 143–155 (2020)

    MathSciNet  Google Scholar 

  8. Atici, F., Eloe, P.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137(3), 981–989 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Bi, P., Hu, Z.: Hopf bifurcation and stability for a neural network model with mixed delays. Appl. Math. Comput. 218(12), 6748–6761 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Bishop, C.M.: Neural networks and their applications. Rev. Sci. Instrum. 65(6), 1803–1832 (1994)

    Google Scholar 

  11. Čermák, J., Győri, I., Nechvátal, L.: On explicit stability conditions for a linear fractional difference system. Fract. Calc. Appl. Anal. 18(3), 651–672 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Chen, J., Zeng, Z., Jiang, P.: Global mittag-leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw. 51, 1–8 (2014)

    MATH  Google Scholar 

  13. Cheng, Z., Xie, K., Wang, T., Cao, J.: Stability and hopf bifurcation of three-triangle neural networks with delays. Neurocomputing 322, 206–215 (2018)

    Google Scholar 

  14. Ge, J., Xu, J.: Stability and hopf bifurcation on four-neuron neural networks with inertia and multiple delays. Neurocomputing 287, 34–44 (2018)

    Google Scholar 

  15. Goodrich, C., Peterson, A.C.: Discrete fractional calculus, vol. 10. Springer, Berlin (2015)

    MATH  Google Scholar 

  16. Hioual, A., Ouannas, A., Oussaeif, T.E., Grassi, G., Batiha, I.M., Momani, S.: On variable-order fractional discrete neural networks: solvability and stability. Fractal Fract. 6(2), 119 (2022)

    Google Scholar 

  17. Huang, C., Wang, H., Cao, J.: Fractional order-induced bifurcations in a delayed neural network with three neurons. Chaos Interdiscip. J. Nonlinear Sci. 33(3), 033143 (2023)

    MathSciNet  Google Scholar 

  18. Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw. 32, 245–256 (2012)

    MATH  Google Scholar 

  19. Li, L., Wang, Z., Li, Y., Shen, H., Lu, J.: Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays. Appl. Math. Comput. 330, 152–169 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Lin, H., Wang, C., Deng, Q., Xu, C., Deng, Z., Zhou, C.: Review on chaotic dynamics of memristive neuron and neural network. Nonlinear Dyn. 106(1), 959–973 (2021)

    Google Scholar 

  21. Ma, T., Mou, J., Li, B., Banerjee, S., Yan, H.: Study on the complex dynamical behavior of the fractional-order hopfield neural network system and its implementation. Fractal Fract. 6(11), 637 (2022)

    Google Scholar 

  22. Magin, R.: Fractional calculus in bioengineering, part 1. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)

    Google Scholar 

  23. Odom, M.D., Sharda, R.: A neural network model for bankruptcy prediction. In: 1990 IJCNN International Joint Conference on neural networks, 163–168. IEEE (1990)

  24. Ostalczyk, P.: Discrete fractional calculus: applications in control and image processing, vol. 4. World scientific, Singapore (2015)

    MATH  Google Scholar 

  25. Ouannas, A., Khennaoui, A.A., Momani, S., Grassi, G., Pham, V.T.: Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization. AIP Adv. 10(4), 045310 (2020)

    Google Scholar 

  26. Petras, I.: A note on the fractional-order cellular neural networks. In: The 2006 IEEE International Joint Conference on Neural Network Proceedings, 1021–1024. IEEE (2006)

  27. Selvam, A.G.M., Baleanu, D., Alzabut, J., Vignesh, D., Abbas, S.: On Hyers-Ulam Mittag-Leffler stability of discrete fractional Duffing equation with application on inverted pendulum. Adv. Differ. Equ. 2020(456), 1–15 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Shammakh, W., Selvam, A.G.M., Dhakshinamoorthy, V., Alzabut, J.: A study of generalized hybrid discrete pantograph equation via hilfer fractional operator. Fractal Fract. 6(3), 152 (2022)

    Google Scholar 

  29. Shida, L., Shaoying, S., Shishi, L., Fuming, L.: The bridge between weather and climate-fractional derivatives. Weather Technol. 35(1), 15–19 (2007)

    Google Scholar 

  30. Sierociuk, D., Petráš, I.: Modeling of heat transfer process by using discrete fractional-order neural networks. In: 2011 16th International Conference on Methods & Models in Automation & Robotics, 146–150. IEEE (2011)

  31. Sierociuk, D., Sarwas, G., Dzieliński, A.: Discrete fractional order artificial neural network. Acta Mech. et Automatica 5(2), 128–132 (2011)

    MATH  Google Scholar 

  32. Song, C., Cao, J.: Dynamics in fractional-order neural networks. Neurocomputing 142, 494–498 (2014)

    Google Scholar 

  33. Udhayakumar, K., Rihan, F.A., Rakkiyappan, R., Cao, J.: Fractional-order discontinuous systems with indefinite lkfs: an application to fractional-order neural networks with time delays. Neural Netw. 145, 319–330 (2022)

    MATH  Google Scholar 

  34. Vignesh, D., Banerjee, S.: Dynamical analysis of a fractional discrete-time vocal system. Nonlinear Dyn. 111, 4501–4515 (2023)

    MATH  Google Scholar 

  35. Vignesh, D., Banerjee, S.: Reversible chemical reactions model with fractional difference operator: dynamical analysis and synchronization. Chaos Interdiscip. J. Nonlinear Sci. 33(3), 033126 (2023)

    MathSciNet  Google Scholar 

  36. Vignesh, D., He, S., Banerjee, S.: Modelling discrete time fractional rucklidge system with complex state variables and its synchronization. Appl. Math. Comput. 455, 128111 (2023)

    MathSciNet  MATH  Google Scholar 

  37. Wu, G.C., Abdeljawad, T., Liu, J., Baleanu, D., Wu, K.T.: Mittag-leffler stability analysis of fractional discrete-time neural networks via fixed point technique. Nonlinear Anal. Model. Control 24(6), 919–936 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Wu, G.C., Baleanu, D.: Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 95–100 (2015)

    MathSciNet  MATH  Google Scholar 

  39. Xu, C., Mu, D., Liu, Z., Pang, Y., Liao, M., Li, P., Yao, L., Qin, Q.: Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed bam neural networks. Nonlinear Anal. Model. Control 27(6), 1030–1053 (2022)

    MathSciNet  MATH  Google Scholar 

  40. Yan, X.P.: Hopf bifurcation and stability for a delayed tri-neuron network model. J. Comput. Appl. Math. 196(2), 579–595 (2006)

    MathSciNet  MATH  Google Scholar 

  41. Yang, Q., Chen, D., Zhao, T., Chen, Y.: Fractional calculus in image processing: a review. Fract. Calcul. Appl. Anal. 19(5), 1222–1249 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Yang, X.S., Yuan, Q.: Chaos and transient chaos in simple hopfield neural networks. Neurocomputing 69(1–3), 232–241 (2005)

  43. Yao, W., Wang, C., Sun, Y., Gong, S., Lin, H.: Event-triggered control for robust exponential synchronization of inertial memristive neural networks under parameter disturbance. Neural Netw. 164, 67–80 (2023)

    Google Scholar 

  44. Yao, W., Wang, C., Sun, Y., Zhou, C.: Robust multimode function synchronization of memristive neural networks with parameter perturbations and time-varying delays. IEEE Trans. Syst. Man Cybernet. Syst. 52(1), 260–274 (2020)

  45. Yao, W., Wang, C., Sun, Y., Zhou, C., Lin, H.: Exponential multistability of memristive cohen-grossberg neural networks with stochastic parameter perturbations. Appl. Math. Comput. 386, 125483 (2020)

    MathSciNet  MATH  Google Scholar 

  46. Zhang, L., Song, Q., Zhao, Z.: Stability analysis of fractional-order complex-valued neural networks with both leakage and discrete delays. Appl. Math. Comput. 298, 296–309 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by the Natural Science Foundation of China (Nos. 61901530, 62061008), the Natural Science Foundation of Hunan Province (No.2020JJ5767).

Author information

Authors and Affiliations

  1. School of Automation and Electronic Information, Xiangtan University, Xiangtan, 411105, China

    Shaobo He

  2. Department of Mathematics, School of Engineering and Technology, CMR University, Chagalahatti, Bangalore, Karnataka, 562149, India

    D. Vignesh

  3. Department of Mathematical Sciences, Giuseppe Luigi Lagrange, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy

    Lamberto Rondoni & Santo Banerjee

Authors
  1. Shaobo He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Vignesh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lamberto Rondoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Santo Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to the study conception and design. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Santo Banerjee.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose. The authors declare that they have no conflict of interest.

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

He, S., Vignesh, D., Rondoni, L. et al. Chaos and firing patterns in a discrete fractional Hopfield neural network model. Nonlinear Dyn 111, 21307–21332 (2023). https://doi.org/10.1007/s11071-023-08972-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08972-z

Keywords

Search

Navigation