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A practical synthesis and analysis of the fractional-order FitzHugh-Nagumo neuronal model

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Abstract

This work focuses on the practical and reasonable synthesis of the fractional-order FitzHugh-Nagumo (FHN) neuronal model. First of all, the descriptive equations of the fractional FHN neuronal system have been given, and then the system stability has been analyzed according to these equations. Secondly, the Laplace-Adomian-decomposition-method is introduced for the numerical solution of the fractional-order FHN neuron model. By means of this method, rapid convergence can be achieved as well as advantages in terms of low hardware cost and uncomplicated computation. In numerical analysis, different situations have been evaluated in detail, depending on the values of fractional-order parameter and external stimulation. Third, the coupling status of fractional-order FHN neuron models is discussed. Finally, experimental validation of the numerical results obtained for the fractional-order single and coupled FHN neurons has been performed by means of the digital signal processor control card F28335 Delfino. Thus, the efficiency of the introduced method for synthesizing the fractional FHN neuronal model in a fast, low cost and simple way has been demonstrated.

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References

  1. Dayan, P., Abbott, L.F.: Theoretical neuroscience: computational and mathematical modeling of neural systems. J. Cogn. Neurosci. 15(1), 154–155 (2003)

    Google Scholar 

  2. Barranco, B.L., Sinencio, E.S., Vazquez, A.R., Huertas, J.L.: A CMOS implementation of FitzHugh-Nagumo neuron model. IEEE J. Solid-State Circuits 26(7), 956–965 (1991)

    Article  ADS  Google Scholar 

  3. Kandel, E.R., Schwartz, J.H., Jessell, T.M.: Principles of neural science, (Vol. 4), McGraw-Hill New York, (2000).

  4. Nouri, M., Karimi, Gh.R., Ahmadi, A., Abbott, D.: Digital multiplierless implementation of the biological FitzHugh–Nagumo model. Neurocomputing 165, 468–476 (2015). https://doi.org/10.1016/j.neucom.2015.03.084

    Article  Google Scholar 

  5. Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952). https://doi.org/10.1113/jphysiol.1952.sp004764

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. FitzHugh, R.: Mathematical models for excitation and propagation in nerve. In: Schawn, H.P. (ed.) Biological Engineering, vol. 1, pp. 1–85. McGraw-Hill, New York (1969)

    Google Scholar 

  7. Hindmarsh, J.L., Rose, R.M.: A model of neural bursting using three couple first order differential equations. Proc. R. Soc. Lond. Biol. Sci. 221(1222), 87–102 (1984). https://doi.org/10.1098/rspb.1984.0024

    Article  ADS  CAS  Google Scholar 

  8. Izhikevich, E.M.: Simple model of spiking neurons. IEEE Trans. Neural Netw. 14(6), 1569–1572 (2003). https://doi.org/10.1109/tnn.2003.820440

    Article  MathSciNet  CAS  PubMed  Google Scholar 

  9. Karaca, Z., Korkmaz, N., Altuncu, Y., Kılıç, R.: An extensive FPGA-based realization study about the Izhikevich neurons and their bio-inspired applications. Nonlinear Dyn. 105, 3529–3549 (2021). https://doi.org/10.1007/s11071-021-06647-1

    Article  Google Scholar 

  10. Rocsoreanu, C., Georgescu, A., Giurgiteanu, N.: The FitzHugh-Nagumo model: bifurcation and dynamics. Springer Sci. Business Media (2012). https://doi.org/10.1007/978-94-015-9548-3

    Article  Google Scholar 

  11. Bisquert, J.: A frequency domain analysis of the excitability and bifurcations of the FitzHugh–Nagumo neuron model. J. Phys. Chem. Lett. 12, 11005–11013 (2021). https://doi.org/10.1021/acs.jpclett.1c03406

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Danca, M.F., Wang, Q.: Synthesizing attractors of Hindmarsh-rose neuronal systems. Nonlinear Dyn. 62, 437–446 (2010). https://doi.org/10.1007/s11071-010-9730-6

    Article  MathSciNet  Google Scholar 

  13. Storace, M., Linaro, D., de Lange, E.: The Hindmarsh—rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos Interdiscip. J. Nonlinear Sci. 18(3), 033128 (2008)

    Article  MathSciNet  Google Scholar 

  14. Rajagopal, K., Khalaf, A.J.M., Parastesh, F., Moroz, I., Karthikeyan, A., Jafari, S.: Dynamical behavior and network analysis of an extended Hindmarsh-Rose neuron model. Nonlinear Dyn. 98(1), 477–487 (2019). https://doi.org/10.1007/s11071-019-05205-0

    Article  Google Scholar 

  15. Usha, K., Subha, P.A.: Hindmarsh-rose neuron model with memristors. Biosystems 178, 1–9 (2019). https://doi.org/10.1016/j.biosystems.2019.01.005

    Article  Google Scholar 

  16. Tamura, A., Ueta, T., Tsuji, S.: Bifurcation analysis of Izhikevich neuron model. Dyn. Cont. Discret. Impuls. Syst. Ser. A Math. Anal. 16(6), 759–775 (2009)

    MathSciNet  Google Scholar 

  17. Kafraj, M.S., Parastesh, F., Jafari, S.: Firing patterns of an improved Izhikevich neuron model under the effect of electromagnetic induction and noise. Chaos Solitons Fract. 137, 109782 (2020). https://doi.org/10.1016/j.chaos.2020.109782

    Article  MathSciNet  Google Scholar 

  18. Wouapi, M.K., Fotsin, B.H., Ngouonkadi, E.B.M., Kemwoue, F.F., Njitacke, Z.T.: Complex bifurcation analysis and synchronization optimal control for Hindmarsh-rose neuron model under magnetic flow effect. Cogn. Neurodyn. 15(2), 315–347 (2021). https://doi.org/10.1007/s11571-020-09606-5

    Article  PubMed  Google Scholar 

  19. Drapaca, C.: Fractional calculus in neuronal electromechanics. J. Mech. Mater. Struct. 12(1), 35–55 (2016). https://doi.org/10.2140/jomms.2017.12.35

    Article  MathSciNet  Google Scholar 

  20. Jun, D., Guang-Jun, Z., Yong, X., Hong, Y., Jue, W.: Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model. Cogn. Neurodyn. 8(2), 167–175 (2014). https://doi.org/10.1007/s11571-013-9273-x

    Article  PubMed  Google Scholar 

  21. Kaslik, E.: Analysis of two-and three-dimensional fractional-order Hindmarsh-rose type neuronal models. Fract. Calculus Appl. Anal. 20(3), 623–645 (2017). https://doi.org/10.1515/fca-2017-0033

    Article  MathSciNet  Google Scholar 

  22. Yu, Y., Shi, M., Kang, H., Chen, M., Bao, B.: Hidden dynamics in a fractional-order memristive Hindmarsh-rose model. Nonlinear Dyn. 100(1), 891–906 (2020). https://doi.org/10.1007/s11071-020-05495-9

    Article  Google Scholar 

  23. Mondal, A., Sharma, S.K., Upadhyay, R.K., Mondal, A.: Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics. Sci. Rep. 9(1), 1–11 (2019). https://doi.org/10.1038/s41598-019-52061-4

    Article  CAS  Google Scholar 

  24. Teka, W.W., Upadhyay, R.K., Mondal, A.: Spiking and bursting patterns of fractional-order Izhikevich model. Commun. Nonlinear Sci. Numer. Simul. 56, 161–176 (2018). https://doi.org/10.1016/j.cnsns.2017.07.026

    Article  ADS  MathSciNet  Google Scholar 

  25. Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Numerical simulations of fractional systems: an overview of existing methods and improvements. Nonlinear Dyn. 38(1), 117–131 (2004). https://doi.org/10.1007/s11071-004-3750-z

    Article  Google Scholar 

  26. Jia, H., Guo, Z., Qi, G., Chen, Z.: Analysis of a four-wing fractional-order chaotic system via frequency-domain and time-domain approaches and circuit implementation for secure communication. Optik 155, 233–241 (2018). https://doi.org/10.1016/j.ijleo.2017.10.076

    Article  ADS  Google Scholar 

  27. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1), 3–22 (2002). https://doi.org/10.1023/A:1016592219341

    Article  MathSciNet  Google Scholar 

  28. Scherer, R., Kalla, S.L., Tang, Y., Huang, J.: The Grünwald-Letnikov method for fractional differential equations. Comput. Math. Appl. 62(3), 902–917 (2011). https://doi.org/10.1016/j.camwa.2011.03.054

    Article  MathSciNet  Google Scholar 

  29. Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier (1998).

  30. Milici, C., Drăgănescu, G., Machado, J.T.: Introduction to fractional differential equations (Vol. 25). Springer (2018). https://doi.org/10.1007/978-3-030-00895-6

  31. Daftardar-Gejji, V., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301(2), 508–518 (2005). https://doi.org/10.1016/j.jmaa.2004.07.039

    Article  MathSciNet  Google Scholar 

  32. Li, C., Wang, Y.: Numerical algorithm based on Adomian decomposition for fractional differential equations. Comput. Math. Appl. 57(10), 1672–1681 (2009). https://doi.org/10.1016/j.camwa.2009.03.079

    Article  MathSciNet  Google Scholar 

  33. Duan, J.S., Rach, R., Baleanu, D., Wazwaz, A.M.: A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Fract. Calc. 3(2), 73–99 (2012)

    Google Scholar 

  34. Mohammed, P.O., Machado, J.A.T., Guirao, J.L., Agarwal, R.P.: Adomian decomposition and fractional power series solution of a class of nonlinear fractional differential equations. Mathematics 9(9), 1070 (2021). https://doi.org/10.3390/math9091070

    Article  Google Scholar 

  35. Tavazoei, M.S., Haeri, M.: Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Proc. 1(4), 171–181 (2007). https://doi.org/10.1049/iet-spr:20070053

    Article  Google Scholar 

  36. Tavazoei, M.S., Haeri, M.: Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal. Theor. Methods Appl. 69(4), 1299–1320 (2008). https://doi.org/10.1016/j.na.2007.06.030

    Article  MathSciNet  Google Scholar 

  37. Haq, F., Shah, K., Ur Rahman, G., Shahzad, M.: Numerical solution of fractional order smoking model via laplace Adomian decomposition method. Alex. Eng. J. 57(2), 1061–1069 (2018). https://doi.org/10.1016/j.aej.2017.02.015

    Article  Google Scholar 

  38. Mahmood, S., Shah, R., Arif, M.: Laplace adomian decomposition method for multi dimensional time fractional model of Navier-stokes equation. Symmetry 11(2), 149 (2019). https://doi.org/10.3390/sym11020149

    Article  ADS  Google Scholar 

  39. Shah, R., Khan, H., Arif, M., Kumam, P.: Application of Laplace-Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations. Entropy 21(4), 335 (2019). https://doi.org/10.3390/e21040335

    Article  ADS  MathSciNet  PubMed  PubMed Central  Google Scholar 

  40. Silva-Juárez, A., Tlelo-Cuautle, E., de la Fraga, L.G., Li, R.: FPAA-based implementation of fractional-order chaotic oscillators using first-order active filter blocks. J. Adv. Res. 25, 77–85 (2020). https://doi.org/10.1016/j.jare.2020.05.014

    Article  PubMed  PubMed Central  Google Scholar 

  41. Malik, S.A., Mir, A.H.: FPGA realization of fractional order neuron. Appl. Math. Model. 81, 372–385 (2020). https://doi.org/10.1016/j.apm.2019.12.008

    Article  MathSciNet  Google Scholar 

  42. Malik, S.A., Mir, A.H.: Discrete multiplierless implementation of fractional order Hindmarsh-rose model. IEEE Trans. Emerg. Topics Comput. (2020). https://doi.org/10.1109/TETCI.2020.2979462

    Article  Google Scholar 

  43. Tolba, M.F., Elsafty, A.H., Armanyos, M., Said, L.A., Madian, A.H., Radwan, A.G.: Synchronization and FPGA realization of fractional-order Izhikevich neuron model. Microelectron. J. 89, 56–69 (2019). https://doi.org/10.1016/j.mejo.2019.05.003

    Article  Google Scholar 

  44. Khanday, F.A., Kant, N.A., Dar, M.R., Zulkifli, T.Z.A., Psychalinos, C.: Low-voltage low-power integrable CMOS circuit implementation of integer-and fractional–order FitzHugh–Nagumo neuron model. IEEE Trans. Neural Netw. Learn. Syst. 30(7), 2108–2122 (2018). https://doi.org/10.1109/TNNLS.2018.2877454

    Article  MathSciNet  PubMed  Google Scholar 

  45. Liu, Y., Xie, Y., Kang, Y., Tan, N., Jiang, J., Xu, J.X.: Dynamical characteristics of the fractional-order FitzHugh-Nagumo model neuron. In: Advances in Cognitive Neurodynamics (II), Springer, Dordrecht, pp. 253–258, (2011). https://doi.org/10.1007/978-90-481-9695-1_39

  46. Armanyos M., Radwan A.G.: Fractional-order Fitzhugh-Nagumo and Izhikevich neuron models. In: IEEE 13th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), Chiang Mai, Thailand, pp. 1–5 (2016). https://doi.org/10.1109/ECTICon.2016.7561406

  47. Alidousti, J., Ghaziani, R.K.: Spiking and bursting of a fractional order of the modified FitzHugh-Nagumo neuron model. Math. Models Comput. Simul. 9(3), 390–403 (2017). https://doi.org/10.1134/S2070048217030036

    Article  MathSciNet  Google Scholar 

  48. Shatnawi, M.T., Abbes, A., Ouannas, A., Batiha, I.M.: Hidden multistability of fractional discrete non-equilibrium point memristor based map. Phys. Scr. 98(3), 035213 (2023). https://doi.org/10.1088/1402-4896/acafac

    Article  ADS  Google Scholar 

  49. Al-Qurashi, M., Rashid, S., Jarad, F., Ali, E., Egami, R.H.: Dynamic prediction modelling and equilibrium stability of a fractional discrete biophysical neuron model. Results Phys. 48, 106405 (2023). https://doi.org/10.1016/j.rinp.2023.106405

    Article  Google Scholar 

  50. Ouannas, A., Mesdoui, F., Momani, S., Batiha, I., Grassi, G.: Synchronization of FitzHugh-Nagumo reaction-diffusion systems via one-dimensional linear control law. Archiv. Control Sci. 31(2), 333–345 (2021)

    MathSciNet  Google Scholar 

  51. Shatnawi, M.T., Khennaoui, A.A., Ouannas, A., Grassi, G., Radogna, A.V., Bataihah, A., Batiha, I.M.: A multistable discrete memristor and its application to discrete-time FitzHugh–Nagumo model. Electronics 12(13), 2929 (2023). https://doi.org/10.3390/electronics12132929

    Article  Google Scholar 

  52. Xu, Q., Chen, X., Chen, B., Wu, H., Li, Z., Bao, H.: Dynamical analysis of an improved FitzHugh-Nagumo neuron model with multiplier-free implementation. Nonlinear Dyn. 111, 8737–8749 (2023). https://doi.org/10.1007/s11071-023-08274-4

    Article  Google Scholar 

  53. Fang, X., Duan, S., Wang, L.: Memristive FHN spiking neuron model and brain-inspired threshold logic computing. Neurocomputing 517, 93–105 (2023). https://doi.org/10.1016/j.neucom.2022.08.056

    Article  Google Scholar 

  54. Yao, Z., Sun, K., He, S.: Firing patterns in a fractional-order FithzHugh–Nagumo neuron model. Nonlinear Dyn. 110, 1807–1822 (2022). https://doi.org/10.1007/s11071-022-07690-2

    Article  Google Scholar 

  55. Chen, G., Zheng, Y., Zeng, Q., Yi, D.: Stability and Hopf bifurcation of FHN neuron model with time delay under magnetic flow. Int. J. Dynam. Control 11, 985–994 (2023). https://doi.org/10.1007/s40435-022-01048-7

    Article  MathSciNet  Google Scholar 

  56. Tavazoei, M.S., Haeri, M.: Chaotic attractors in incommensurate fractional order systems. Physica D 237(20), 2628–2637 (2008). https://doi.org/10.1016/j.physd.2008.03.037

    Article  ADS  MathSciNet  Google Scholar 

  57. Tavazoei, M.S., Haeri, M.: A necessary condition for double scroll attractor existence in fractional-order systems. Phys. Lett. A 367(1–2), 102–113 (2007). https://doi.org/10.1016/j.physleta.2007.05.081

    Article  ADS  CAS  Google Scholar 

  58. Cafagna, D., Grassi, G.: Fractional-order Chua’s circuit: time-domain analysis, bifurcation, chaotic behavior and test for chaos. Int. J. Bifurc. Chaos 18(3), 615–639 (2008). https://doi.org/10.1142/S0218127408020550

    Article  MathSciNet  Google Scholar 

  59. Liao, H., Ding, Y., Wang, L.: Adomian decomposition algorithm for studying incommensurate fractional-order memristor-based Chua’s system. Int. J. Bifurcation Chaos 28(11), 1850134 (2018). https://doi.org/10.1142/S0218127418501341

    Article  ADS  MathSciNet  Google Scholar 

  60. Liu, T., Yan, H., Banerjee, S., Mou, J.: A fractional-order chaotic system with hidden attractor and self-excited attractor and its DSP implementation. Chaos Solitons Fract. 145, 110791 (2021). https://doi.org/10.1016/j.chaos.2021.110791

    Article  MathSciNet  Google Scholar 

  61. Wazwaz, A.M.: A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. Math. Comput. 111(1), 33–51 (2000). https://doi.org/10.1016/S0096-3003(99)00063-6

    Article  MathSciNet  Google Scholar 

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  1. Clinical Engineering Research and Implementation Center (ERKAM), Erciyes University, 38030, Kayseri, Turkey

    İbrahim Ethem Saçu

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Saçu, İ.E. A practical synthesis and analysis of the fractional-order FitzHugh-Nagumo neuronal model. J Comput Electron 23, 188–207 (2024). https://doi.org/10.1007/s10825-023-02120-x

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