Skip to main content
SpringerLink
Log in

Chaotic resonance in Hodgkin–Huxley neuron

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

Abstract

Chaotic Resonance (CR), whereby the response of a nonlinear system to a weak signal can be enhanced by the assistance of chaotic activities that can be intrinsic or extrinsic, has recently been studied widely. In this paper, the effects of extrinsic chaotic signal on the weak signal detection performance of the Hodgkin–Huxley neuron are examined via numerical simulation. The chaotic signal has been derived from Lorenz system and is injected to neuron as a current. Obtained results have revealed that the H–H neuron exhibits CR phenomenon depending on the chaotic current intensity. Also, we have found an optimal chaotic current intensity ensuring the best detection of the weak signal in H–H neuron via CR. In addition, we have calculated the maximal Lyapunov exponent to determine whether the H–H neuron is in chaotic regime. After determining the state of the neuron, we have shown that the H–H neuron can be able to detect the weak signal even if it is in the chaotic regime. Finally, we have investigated the effects of chaotic activity on the collective behavior of H–H neurons in small-world networks and have concluded that CR effect is a robust phenomenon which can be observed both in single neurons and neuronal networks.

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

Similar content being viewed by others

References

  1. Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70(1), 223–287 (1998)

    Article  Google Scholar 

  2. Russell, D.F., Wilkens, L.A., Moss, F.: Use of behavioural stochastic resonance by paddle fish for feeding. Nature 402(6759), 291–294 (1999)

    Article  Google Scholar 

  3. Douglass, J.K., Wilkens, L., Pantazelou, E., Moss, F.: Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature 365(6444), 337–340 (1993)

    Article  Google Scholar 

  4. Anishchenko, V.S., Neiman, A.B., Moss, F., Schimansky-Geier, L.: Uspekhi Fizicheskih Nauk. Sov. Phys. Usp. 42, 7 (1999)

    Article  Google Scholar 

  5. McNamara, B., Wiesenfeld, K.: Theory of stochastic resonance. Phys. Rev. A 39(9), 4854–4869 (1989)

    Article  Google Scholar 

  6. Palonpon, A., Amistoso, J., Holdsworth, J., Garcia, W., Saloma, C.: Measurement of weak transmittances by stochastic resonance. Opt. Lett. 23(18), 1480–1482 (1998)

    Article  Google Scholar 

  7. Hänggi, P.: Stochastic resonance in biology how noise can enhance detection of weak signals and help improve biological information processing. ChemPhysChem 3(3), 285–290 (2002)

    Article  Google Scholar 

  8. Wiesenfeld, K., Moss, F.: Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs. Nature 373(6509), 33–36 (1995)

    Article  Google Scholar 

  9. Wiesenfeld, K., Jaramillo, F.: Minireview of stochastic resonance. Chaos 8(3), 539–548 (1998)

    Article  Google Scholar 

  10. Longtin, A.: Stochastic resonance in neuron models. J. Stat. Phys. 70(1–2), 309–327 (1993)

    Article  MATH  Google Scholar 

  11. Moss, F., Ward, L.M., Sannita, W.G.: Stochastic resonance and sensory information processing: a tutorial and review of application. Clin. Neurophysiol. 115(2), 267–281 (2004)

    Article  Google Scholar 

  12. Yasuda, H., Miyaoka, T., Horiguchi, J., Yasuda, A., Hänggi, P., Yamamoto, Y.: Novel class of neural stochastic resonance and error-free information transfer. Phys. Rev. Lett. 100(11), 118103–118106 (2008)

    Article  Google Scholar 

  13. Yilmaz, E., Uzuntarla, M., Ozer, M., Perc, M.: Stochastic resonance in hybrid scale-free neuronal networks. Physica A 392(22), 5735–5741 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Collins, J.J., Imhoff, T.T., Grigg, P.: Noise-enhanced tactile sensation. Nature 383(6603), 770 (1996)

    Article  Google Scholar 

  15. Russell, D.F., Wilkens, L.A., Moss, F.: Use of behavioural stochastic resonance by paddle fish for feeding. Nature 402(6759), 291–294 (1999)

    Article  Google Scholar 

  16. Perc, M.: Stochastic resonance on excitable small-world networks via a pacemaker. Phys. Rev. E 76(6), 066203–6 (2007)

    Article  MathSciNet  Google Scholar 

  17. Bezrukov, S.M., Vodyanoy, I.: Noise-induced enhancement of signal transduction across voltage-dependent ion channels. Nature 378(6555), 362–364 (1995)

    Article  Google Scholar 

  18. Schmid, G., Goychuk, I., Hänggi, P.: Stochastic resonance as a collective property of ion channel assemblies. EPL (Europhysics Letters) 56(1), 22–28 (2001)

    Article  Google Scholar 

  19. Guo, D., Li, C.: Stochastic resonance in Hodgkin-Huxley neuron induced by unreliable synaptic transmission. J. Theor. Biol. 308, 105–114 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Izhikevich, E.M.: Simple model of spiking neurons. IEEE Trans. Neural Netw. 14(6), 1569–1572 (2003)

    Article  MathSciNet  Google Scholar 

  21. Landa, P.S., McClintock, P.V.: Vibrational resonance. J. Phys. A: Math. Gen. 33(45), L433 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ullner, E., Zaikin, A., Garcıa-Ojalvo, J., Bascones, R., Kurths, J.: Vibrational resonance and vibrational propagation in excitable systems. Phys. Lett. A 312(5–6), 348–354 (2003)

    Article  MathSciNet  Google Scholar 

  23. Deng, B., Wang, J., Wei, X., Tsang, K.M., Chan, W.L.: Vibrational resonance in neuron populations. Chaos 20(1), 013113 (2010)

    Article  Google Scholar 

  24. Yu, H., Wang, J., Liu, C., Deng, B., Wei, X.: Vibrational resonance in excitable neuronal systems. Chaos 21(4), 043101 (2011)

    Article  Google Scholar 

  25. Stiefel, K.M., Englitz, B., Sejnowski, T.J.: Origin of intrinsic irregular firing in cortical interneurons. Proc. Natl. Acad. Sci. 110(19), 7886–7891 (2013)

    Article  MATH  Google Scholar 

  26. Fellous, J.M., Rudolph, M., Destexhe, A., Sejnowski, T.J.: Synaptic background noise controls the input/output characteristics of single cells in an in vitro model of in vivo activity. Neuroscience 122(3), 811–829 (2003)

    Article  Google Scholar 

  27. Destexhe, A., Rudolph, M., Paré, D.: The high-conductance state of neocortical neurons in vivo. Nat. Rev. Neurosci. 4(9), 739–751 (2003)

    Article  Google Scholar 

  28. Hansel, D., Mato, G.: Short-term plasticity explains irregular persistent activity in working memory tasks. J. Neurosci. 33(1), 133–149 (2013)

    Article  Google Scholar 

  29. Ardid, S., Wang, X.J., Gomez-Cabrero, D., Compte, A.: Reconciling coherent oscillation with modulationof irregular spiking activity in selective attention: gamma-range synchronization between sensoryand executive cortical areas. J. Neurosci. 30(8), 2856–2870 (2010)

    Article  Google Scholar 

  30. Doron, G., von Heimendahl, M., Schlattmann, P., Houweling, A.R., Brecht, M.: Spiking irregularity and frequency modulate the behavioral report of single-neuron stimulation. Neuron 81(3), 653–663 (2014)

    Article  Google Scholar 

  31. Arbib, M.A., Fellous, J.M.: Emotions: from brain to robot. Trends. Cogn. Sci. 8(12), 554–561 (2004)

    Article  Google Scholar 

  32. Freeman, W.J.: Evidence from human scalp electroencephalograms of global chaotic itinerancy. Chaos 13(3), 1067–1077 (2003)

    Article  MathSciNet  Google Scholar 

  33. Korn, H., Faure, P.: Is there chaos in the brain? II. Experimental evidence and related models. C. R. Biol. 326(9), 787–840 (2003)

    Article  Google Scholar 

  34. El Boustani, S., Destexhe, A.: Brain dynamics at multiple scales: can one reconcile the apparent low-dimensional chaos of macroscopic variables with the seemingly stochastic behavior of single neurons? Int. J. Bifurc. Chaos 20(06), 1687–1702 (2010)

    Article  MathSciNet  Google Scholar 

  35. Hayashi, H., Ishizuka, S., Hirakawa, K.: Chaotic response of the pacemaker neuron. J. Phys. Soc. Jpn. 54(6), 2337–2346 (1985)

    Article  Google Scholar 

  36. Freeman, W. J.: On the problem of anomalous dispersion in chaoto-chaotic phase transitions of neural masses, and its significance for the management of perceptual information in brains. In: Synergetics of cognition (pp. 126-143). Springer, Berlin (1990)

  37. Freeman, W.J.: A proposed name for aperiodic brain activity: stochastic chaos. Neural Netw. 13(1), 11–13 (2000)

    Article  Google Scholar 

  38. Freeman, W.J., Burke, B.C., Holmes, M.D.: Aperiodic phase re-setting in scalp EEG of beta-gamma oscillations by state transitions at alpha-theta rates. Hum. Brain Mapp. 19(4), 248–272 (2003)

    Article  Google Scholar 

  39. Paul, K., Cauller, L.J., Llano, D.A.: Presence of a chaotic region at the sleep-wake transition in a simplified thalamocortical circuit model. Front. Comput. Neurosci. 10, 91 (2016)

    Article  Google Scholar 

  40. Carroll, T.L., Pecora, L.M.: Stochastic resonance and crises. Phys. Rev. Lett. 70(5), 576–579 (1993)

    Article  Google Scholar 

  41. Carroll, T.L., Pecora, L.M.: Stochastic resonance as a crisis in a period-doubled circuit. Phys. Rev. E 47(6), 3941–3949 (1993)

    Article  Google Scholar 

  42. Nobukawa, S., Nishimura, H., Yamanishi, T., Liu, J.Q.: Analysis of chaotic resonance in Izhikevich neuron model. PloS ONE 10(9), e0138919 (2015)

    Article  Google Scholar 

  43. Nobukawa, S., Nishimura, H., Yamanishi, T., Liu, J.Q.: Chaotic states induced by resetting process in Izhikevich neuron model. JAISCR 5(2), 109–119 (2015)

    Google Scholar 

  44. Schweighofer, N., Doya, K., Fukai, H., Chiron, J.V., Furukawa, T., Kawato, M.: Chaos may enhance information transmission in the inferior olive. Proc. Natl. Acad. Sci. 101(13), 4655–4660 (2004)

    Article  Google Scholar 

  45. Hodgkin, A. L., Huxley, A. F.: Movement of sodium and potassium ions during nervous activity. In: Cold Spring Harbor symposia on quantitative biology (Vol. 17, pp. 43–52). Cold Spring Harbor Laboratory Press (1952)

  46. Ma, J., Ying, H.P., Pu, Z.S.: An anti-control scheme for spiral under Lorenz chaotic signals. Chin. Phys. Lett. 22(5), 1065–1068 (2005)

    Article  Google Scholar 

  47. Wilson, H. R.: Spikes, decisions, and actions: the dynamical foundations of neurosciences. (1999)

  48. Levin, J.E., Miller, J.P.: Broadband neural encoding in the cricket cereal sensory system enhanced by stochastic resonance. Nature 380(6570), 165 (1996)

    Article  Google Scholar 

  49. Pankratova, E.V., Polovinkin, A.V., Mosekilde, E.: Resonant activation in a stochastic Hodgkin–Huxley model: interplay between noise and suprathreshold driving effects. Eur. Phys. J. B 45(3), 391–397 (2005)

    Article  Google Scholar 

  50. Yu, Y., Liu, F., Wang, W.: Frequency sensitivity in Hodgkin–Huxley systems. Biol. Cybern. 84(3), 227–235 (2001)

    Article  Google Scholar 

  51. Wang, W., Wang, Y., Wang, Z.D.: Firing and signal transduction associated with an intrinsic oscillation in neuronal systems. Phys. Rev. E 57(3), R2527 (1998)

    Article  Google Scholar 

  52. Xie, Y., Chen, L., Kang, Y.M., Aihara, K.: Controlling the onset of Hopf bifurcation in the Hodgkin–Huxley model. Phys. Rev. E 77(6), 061921 (2008)

    Article  MathSciNet  Google Scholar 

  53. Kaplan, D.T., Clay, J.R., Manning, T., Glass, L., Guevara, M.R., Shrier, A.: Subthreshold dynamics in periodically stimulated squid giant axons. Phys. Rev. Lett. 76(21), 4074 (1996)

    Article  Google Scholar 

  54. Matsumoto, G.: Periodic and Nonperiodic responces of membrane potentials in squid giant axons during sinusoidal current stimulation. J. Theor. Neurobiol. 3, 1–14 (1984)

    Google Scholar 

  55. Aihara, K.: Chaos in neurons. Scholarpedia 3(5), 1786 (2008)

    Article  Google Scholar 

  56. Guttman, R., Feldman, L., Jakbsson, E.: Frequency entrainment of squid axon membrane. J. Membr. Biol. 56(1), 9–18 (1980)

    Article  Google Scholar 

  57. Lee, S.G., Kim, S.: Bifurcation analysis of mode-locking structure in a Hodgkin–Huxley neuron under sinusoidal current. Phys. Rev. E 73(4), 041924 (2006)

    Article  MathSciNet  Google Scholar 

  58. Borkowski, L.S.: Bistability and resonance in the periodically stimulated Hodgkin–Huxley model with noise. Phys. Rev. E 83(5), 051901 (2011)

    Article  Google Scholar 

  59. Borkowski, L.S.: Response of a Hodgkin–Huxley neuron to a high-frequency input. Phys. Rev. E 80(5), 051914 (2009)

    Article  Google Scholar 

  60. Parmananda, P., Mena, C.H., Baier, G.: Resonant forcing of a silent Hodgkin–Huxley neuron. Phys. Rev. E 66(4), 047202 (2002)

    Article  MathSciNet  Google Scholar 

  61. Bassett, D.S., Meyer-Lindenberg, A., Achard, S., Duke, T., Bullmore, E.: Adaptive reconfiguration of fractal small-world human brain functional networks. Proc. Natl. Acad. Sci. 103(51), 19518–19523 (2006)

    Article  Google Scholar 

  62. Kitzbichler, M.G., Smith, M.L., Christensen, S.R., Bullmore, E.: Broadband criticality of human brain network synchronization. PLoS Comp. Biol. 5(3), e1000314 (2009)

    Article  MathSciNet  Google Scholar 

  63. Lü, J., Chen, G.: Generating multiscroll chaotic attractors: theories, methods and applications. Int. J. Bifurcat. Chaos 16(04), 775–858 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  64. Chua, L.E.O.N.O., Komuro, M., Matsumoto, T.: The double scroll family. IEEE Trans. Circuits Syst. 33(11), 1072–1118 (1986)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Biomedical Engineering, Bülent Ecevit University, 67100, Zonguldak, Turkey

    Veli Baysal & Ergin Yilmaz

  2. Department of Electrical-Electronics Engineering, Bülent Ecevit University, 67100, Zonguldak, Turkey

    Zehra Saraç

Authors
  1. Veli Baysal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zehra Saraç
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ergin Yilmaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ergin Yilmaz.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baysal, V., Saraç, Z. & Yilmaz, E. Chaotic resonance in Hodgkin–Huxley neuron. Nonlinear Dyn 97, 1275–1285 (2019). https://doi.org/10.1007/s11071-019-05047-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05047-w

Keywords

Search

Navigation