Skip to main content
SpringerLink
Log in

Firing activity in a simplified Hodgkin–Huxley circuit with memristive sodium and potassium ion channels

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

Sodium and potassium ion channels are significant for generating spiking behaviors in excitable neurons. The marvelous Hodgkin–Huxley (HH) circuit employs time-varying resistors to describe electrophysiological properties of the ion channels and to constrict the relation between membrane potential and ion currents. It is a difficult task to analog implement the marvelous HH circuit, since it contains mixed exponential function for describing the ion currents. To solve and mitigate this issue, a simplified HH circuit with memristive sodium and potassium ion channels is constructed. In the simplified memristive Hodgkin–Huxley (mHH) circuit, a second-order (2nd-order) locally active memristor (LAM) for characterizing sodium ion channel and a first-order (1st-order) LAM for characterizing the potassium ion channel are employed. Numerical simulations employing several numerical tools are utilized to offer unique insight into exploring the dynamical behavior and firing activity of the simplified mHH circuit, which delight that LAMs- and stimulus-related parameters can be used to regulate the generation of various spiking behaviors. Moreover, a PCB-based analog circuit is made by using discrete circuit components and hardware experiment is performed. The experimentally measured results well verify the numerically simulated spiking behaviors. The numerical simulations and experimental confirmations display that the simplified mHH circuit is feasible in producing various neuron firing activities and benefit for developing spike-based applications.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical research work, so no additional data are associated with this work.]

References

  1. A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952). https://doi.org/10.1113/jphysiol.1952.sp004764

    Article  Google Scholar 

  2. Q. Xu, X. Tan, D. Zhu, H. Bao, Y.H. Hu, B.C. Bao, Bifurcations to bursting and spiking in the chay neuron and their validation in a digital circuit. Chaos Solit. Fract. 141, 110353 (2020). https://doi.org/10.1016/j.chaos.2020.110353

    Article  MathSciNet  Google Scholar 

  3. H. Bao, Z.H. Yu, Q. Xu, H.G. Wu, B.C. Bao, Three-dimensional memristive morris-lecar model with magnetic induction effects and its fpga implementation. Cogn. Neurodyn. 17, 1079–1092 (2023). https://doi.org/10.1007/s11571-022-09871-6

    Article  Google Scholar 

  4. H.R. Wilson, Simplified dynamics of human and mammalian neocortical neurons. J. Theor. Biol. 200(4), 375–388 (1999). https://doi.org/10.1006/jtbi.1999.1002

    Article  ADS  Google Scholar 

  5. M. Nouri, M. Hayati, T. Serrano-Gotarredona, D. Abbott, A digital neuromorphic realization of the 2D Wilson neuron model. IEEE Trans. Circuits Syst. II 66(1), 136–140 (2019). https://doi.org/10.1109/TCSI.2021.3126555

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. Q. Xu, X. Chen, B. Chen, H.G. Wu, Z. Li, H. Bao, Dynamical analysis of an improved FitzHugh–Nagumo neuron mode with multiplier-free implementation. Nonlinear Dyn. 111(9), 8737–8749 (2023). https://doi.org/10.1007/s11071-023-08274-4

    Article  Google Scholar 

  8. K.M. Wouapi, B.H. Fotsin, F.P. Louodop, K.F. Feudjio, Z.T. Njitacke, T.H. Djeudjo, Various firing activities and finite-time synchronization of an improved Hindmarsh-Rose neuron model under electric field effect. Cogn. Neurodyn. 14, 375–397 (2020). https://doi.org/10.1007/s11571-020-09570-0

    Article  Google Scholar 

  9. X.Y. Zhou, Y. Xu, G.W. Wang, Y. Jia, Ionic channel blockage in stochastic Hodgkin-Huxley neuronal model driven by multiple oscillatory signals. Cogn. Neurodyn. 14, 569–578 (2020). https://doi.org/10.1007/s11571-020-09570-0

    Article  Google Scholar 

  10. Y. Xu, J. Ma, X. Zhan, L.J. Yang, Y. Ja, Temperature effect on memristive ion channels. Cogn. Neurodyn. 13, 601–611 (2019). https://doi.org/10.1007/s11571-019-09547-8

    Article  Google Scholar 

  11. A. Basu, P.E. Hasler, Nullcline-based design of a silicon neuron. IEEE Trans. Circuits Syst. I 57(11), 2938–2947 (2010). https://doi.org/10.1109/TCSI.2010.2048772

    Article  MathSciNet  Google Scholar 

  12. S. Haghiri, A. Naderi, B. Ghanbari, A. Ahmadi, High speed and low digital resources implementation of Hodgkin-Huxley neuronal model using base-2 functions. IEEE Trans. Circuits Syst. I 68(1), 275–287 (2021). https://doi.org/10.1109/TCSI.2020.3026076

    Article  MathSciNet  Google Scholar 

  13. L.O. Chua, Local activity if the origin of complexity. Int. J. Bifurc. Chaos 15(11), 3435–3456 (2005)

    Article  MATH  Google Scholar 

  14. L.O. Chua, V. Sbitnev, H. Kim, Neurons are poised near the edge of chaos. Int. J. Bifurc. Chaos 22(4), 1250098 (2012). https://doi.org/10.1142/S0218127412500988

    Article  Google Scholar 

  15. C.X. Pan, Q.H. Hong, X.P. Wang, A novel memristive chaotic neuron circuit and its application in chaotic neural networks for associative memory, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 40(3), 521–532 (2021). https://doi.org/10.1109/TCAD.2020.3002568

    Article  Google Scholar 

  16. P.P. Jin, G.Y. Wang, Y. Liang, H.H. Iu, L.O. Chua, Neuromorphic dynamics of Chua corsage memristor. IEEE Trans. Circuits Syst. I 68(11), 4419–4432 (2021). https://doi.org/10.1109/TCSI.2021.3121676

    Article  Google Scholar 

  17. S. Kim, C. Du, P. Sheridan, W. Ma, S. Choi, W.D. Lu, Experimental demonstration of a second-order memristor and its ability to biorealistically implement synaptic plasticity. ACS Nano 15(3), 2203–2211 (2015). https://doi.org/10.1021/acs.nanolett.5b00697

    Article  ADS  Google Scholar 

  18. Q. Xu, Y.T. Wang, H.H.C. Iu, N. Wang, H. Bao, Locally active memristor based neuromorphic circuit: Firing pattern and hardware experiment. IEEE Trans. Circuits Syst. I 70(8), 3130–3141 (2023). https://doi.org/10.1109/TCSI.2023.3276983

    Article  Google Scholar 

  19. Q. Xu, Y.T. Wang, B. Chen, Z. Li, N. Wang, Firing pattern in a memristive Hodgkin–Huxley circuit: numerical simulation and analog circuit validation. Chaos Solitons Fract. 172, 113627 (2023). https://doi.org/10.1016/j.chaos.2023.113627

    Article  MathSciNet  Google Scholar 

  20. X.Y. Hu, C.X. Liu, Dynamic property analysis and circuit implementation of simplified memristive Hodgkin–Huxley model. Nonlinear Dyn. 97, 1721–1733 (2019). https://doi.org/10.1007/s11071-019-05100-8

    Article  MATH  Google Scholar 

  21. F.F. Yang, Y. Xu, J. Ma, A memristive neuron and its adaptability to external electric field. Chaos 33(2), 023110 (2023). https://doi.org/10.1063/5.0136195

    Article  ADS  MathSciNet  Google Scholar 

  22. F.Q. Wu, Y.T. Guo, J. Ma, Reproduce the biophysical function of chemical synapse by using a memristive synapse. Nonlinear Dyn. 109, 2063–2084 (2022). https://doi.org/10.1007/s11071-022-07533-0

    Article  Google Scholar 

  23. J. Ma, Biophysical neurons, energy, and synapse controllability: a review. J. Zhejiang Univ.-Sci. A (Appl. Phys. Eng.) 24, 109–129 (2023). https://doi.org/10.1631/jzus.A2200469

    Article  Google Scholar 

  24. A.L. Hodgkin, A.F. Huxley, B. Katz, Ionic currents underlying activities in the giant axon of the squid. Arch. Sci. Physiol. 3, 129–150 (1949)

    Google Scholar 

  25. X.Y. Hu, S. Wang, C.X. Liu, Hidden coexisting firing patterns and bubble-like bifurcation in HR neuron model under electromagnetic induction, Chin. J. Phys. 77, 2541–2549 (2022). https://doi.org/10.1016/j.cjph.2022.04.016

    Article  MathSciNet  Google Scholar 

  26. W. Zhang, B.Q. Fan, D. Agarwal, T. Li, Y.G. Yu, Axonal sodium and potassium conductance density determines spiking dynamical properties of regular- and fast-spiking neurons. Nonlinear Dyn. 95, 1035–1052 (2019). https://doi.org/10.1007/s11071-018-4613-3

    Article  Google Scholar 

  27. Z.T. Ju, Y. Lin, B. Chen, H.G. Wu, M. Chen, Q. Xu, Electromagnetic radiation induced non-chaotic behaviors in a Wilson neuron model, Chin. J. Phys. 77, 214–222 (2022). https://doi.org/10.1016/j.cjph.2022.03.012

    Article  MathSciNet  Google Scholar 

  28. S.K. Ding, N. Wang, H. Bao, B. Chen, H.G. Wu, Q. Xu, Memristor synapse-coupled piecewise-linear simplified Hopfield neural network: dynamics analysis and circuit implementation. Chaos Solitons Fract. 166, 112899 (2023). https://doi.org/10.1016/j.chaos.2022.112899

    Article  MathSciNet  Google Scholar 

  29. D.S. Yu, H.H.C. Iu, A.L. Fitch, Y. Liang, A floating memristor emulator based relaxation oscillator. IEEE Trans. Circuits Syst. I 61(10), 2888–2896 (2014). https://doi.org/10.1109/TCSI.2014.2333687

    Article  Google Scholar 

  30. X.J. Chen, N. Wang, Y.T. Wang, H.G. Wu, Q. Xu, Memristor initial-offset boosting and its bifurcation mechanism in a memristive Fitzhugh-Nagumo neuron model with hidden dynamics. Chaos Solitons Fract. 174, 113836 (2023). https://doi.org/10.1016/j.chaos.2023.113836

    Article  MathSciNet  Google Scholar 

  31. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining lyapunov exponents from a time series. Phys. D 16(3), 285–317 (1985). https://doi.org/10.1016/0167-2789(85)90011-9

    Article  MathSciNet  MATH  Google Scholar 

  32. X.L. An, S. Qiao, The hidden, period-adding, mixed-mode oscillations and control in a hr neuron under electromagnetic induction. Chaos Solitons Fract. 143, 110587 (2021). https://doi.org/10.1016/j.chaos.2020.110587

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Panahi, S. Jafari, A.J.M. Khalaf, K. Rajagopal, V.T. Pham, F.E. Alsaadi, Complete dynamical analysis of a neuron under magnetic flow effect, Chin. J. Phys. 56, 2254–2264 (2018). https://doi.org/10.1016/j.cjph.2018.08.008

    Article  Google Scholar 

  34. Y.Z. Cheng, F.H. Min, Z. Rui, Y.P. Dou, Firing multistability, symmetry, bubbles of a Shinriki oscillator with mem-elements, Chin. J. Phys. 74, 157–174 (2021). https://doi.org/10.1016/j.cjph.2021.09.002

    Article  Google Scholar 

  35. M.L. Ma, Y. Yang, Z.C. Qiu, Y.X. Peng, Y.C. Sun, Z.J. Li, M.J. Wang, A locally active discrete memristor model and its application in a hyperchaotic map. Nonlinear Dyn. 107, 2935–2949 (2022). https://doi.org/10.1007/s11071-021-07132-5

    Article  Google Scholar 

  36. J.H. Kim, J.K. Lee, H.G. Kim, K.B. Kim, H.R. Kim, Possible effects of radio frequency electromagnetic field exposure on central nerve system. Biomol. Ther. 27(3), 265–275 (2019). https://doi.org/10.4062/biomolther.2018.152

    Article  Google Scholar 

  37. Y. Yao, J. Ma, Weak periodic signal detection by sine- Wiener-noise-induced resonance in the FitzHugh-Nagumo neuron. Cogn. Neurodyn. 12, 343–349 (2018). https://doi.org/10.1007/s11571-018-9475-3

    Article  Google Scholar 

  38. P.S. Sachdeva, J.A. Livezey, M.R. DeWeese, Heterogeneous synaptic weighting improves neural coding in the presence of common noise. Neural Comput. 32, 1239–1276 (2020). https://doi.org/10.1162/neco_a_01287

    Article  MathSciNet  MATH  Google Scholar 

  39. F. Yu, H. Shen, Q. Yu, X. Kong, P.K. Sharma, S. Cai, Privacy protection of medical data based on multi-scroll memristive Hopfield neural network. IEEE Trans. Netw. Sci. Eng. 10, 845–858 (2023). https://doi.org/10.1109/TNSE.2022.3223930

    Article  Google Scholar 

  40. A. Basu, S. Ramakrishnan, C. Petre, S. Koziol, S. Brink, P.E. Hasler, Neural dynamics in reconfigurable silicon. IEEE Trans. Biomed. Circuits Syst. 4(5), 311–319 (2010). https://doi.org/10.1109/TBCAS.2010.2055157

    Article  Google Scholar 

  41. Q. Xu, Z.T. Ju, S.K. Ding, C.T. Feng, M. Chen, B.C. Bao, Electromagnetic induction effects on electrical activity within a memristive Wilson neuron model. Cogn. Neurodyn. 16, 1221–1231 (2022). https://doi.org/10.1007/s11571-021-09764-0

    Article  Google Scholar 

  42. M.Y. Ge, Y. Jia, Y. Xu, L.J. Yang, Mode transition in electrical activities of neuron driven by high and low frequency stimulus in the presence of electromagnetic induction and radiation. Nonlinear Dyn. 91, 515–523 (2018). https://doi.org/10.1007/s11071-017-3886-2

    Article  Google Scholar 

  43. A. Rao, P. Plank, A. Wild, W. Maass, A long short-term memory for AI applications in spike-based neuromorphic hardware. Nat. Mach. Intell. 4, 467–479 (2022). https://doi.org/10.1007/s11071-017-3886-2

    Article  Google Scholar 

  44. M.D. Pickett, G. Medeiros-Ribeiro, R.S. Williams, A scalable neuristor built with Mott memristors. Nature Mater. 12(2), 114–117 (2013). https://doi.org/10.1038/nmat3510

    Article  ADS  Google Scholar 

  45. S. Saïghi, Y. Bornat, J. Tomas, G.L. Masson, S. Renaud, A library of analog operators based on the Hodgkin-Huxley formalism for the design of tunable, real-time, silicon neurons. IEEE Trans. Biomed. Circuits Syst. 5(1), 3–19 (2011). https://doi.org/10.1109/TBCAS.2010.2078816

    Article  Google Scholar 

  46. J.M. Cai, H. Bao, Q. Xu, Z.Y. Hua, B.C. Bao, Smooth nonlinear fitting scheme for analog multiplierless implementation of Hindmarsh-rose neuron model. Nonlinear Dyn. 104(4), 4379–4389 (2021). https://doi.org/10.1007/s11071-021-06453-9

    Article  Google Scholar 

  47. G. Dou, K.X. Zhao, M. Guo, J. Mou, Memristor-based lstm network for text classification. Fractals 2023, 2340040 (2023). https://doi.org/10.1142/S0218348X23400406

    Article  Google Scholar 

  48. X.Y. Gao, B. Sun, Y.H. Cao, S. Banerjee, J. Mou, A color image encryption algorithm based on hyperchaotic map and dna mutation. Chin. Phys. B 32, 030501 (2023). https://doi.org/10.1088/1674-1056/ac8cdf

    Article  ADS  Google Scholar 

  49. J.N. Teramae, Y. Tsubo, T. Fukai, Optimal spike-based communication in excitable networks with strong-sparse and weak-dense links. Sci. Rep. 2, 485 (2012). https://doi.org/10.1038/srep00485

    Article  Google Scholar 

  50. B. Singh, C. Jain, A. Bansal, An improved adjustable step adaptive neuron-based control approach for the grid-supportive spv system. IEEE Trans. Ind. Appl. 54, 563–570 (2018). https://doi.org/10.1109/TIA.2017.2732338

    Article  Google Scholar 

  51. Y.M. Yang, J. Ma, Y. Xu, Y. Jia, Energy dependence on discharge model of izhikevich neuron driven by external stimulus under electromagnetic induction. Cogn. Neurodyn. 15, 265–277 (2021). https://doi.org/10.1007/s11571-020-09596-4

    Article  Google Scholar 

  52. F.Q. Wu, J. Ma, G. Zhang, Energy estimation and coupling synchronization between biological neurons. Sci. China Tech. Sci. 63, 625–636 (2020). https://doi.org/10.1007/s11431-019-9670-1

    Article  Google Scholar 

Download references

Funding

This work was supported by the grants from the National Natural Science Foundations of China under 12172066 and 61801054, the Natural Science Foundation of Jiangsu Province, China, under BK20160282, the Project 333 of Jiangsu Province, the Postgraduate Research and Practice Innovation Program of Jiangsu Province, China under Grant No. KYCX23_3168, and the College Students’ Innovation and Entrepreneurship Training Program of Changzhou Jiangsu Province, China Under Grant No. 202310292042Z. The authors acknowledge the anonymous referees for their valuable comments.

Author information

Authors and Affiliations

  1. School of Microelectronics and Control Engineering, Changzhou University, Changzhou, 213159, China

    Weiwei Fan, Yiteng Wang, Ning Wang, Yufan Shan & Quan Xu

Authors
  1. Weiwei Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yiteng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ning Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yufan Shan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Quan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quan Xu.

Ethics declarations

Conflict of interest

No conflicts of interest about the publication by all authors.

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

Fan, W., Wang, Y., Wang, N. et al. Firing activity in a simplified Hodgkin–Huxley circuit with memristive sodium and potassium ion channels. Eur. Phys. J. Plus 138, 834 (2023). https://doi.org/10.1140/epjp/s13360-023-04472-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-023-04472-6

Search

Navigation