Multiple coherence resonances evoked from bursting and the underlying bifurcation mechanism


Coherence resonance (CR), which is typical nonlinear phenomenon that noise plays construct roles, has been widely investigated in single neurons at resting state instead of bursting behavior when noise free. In the present paper, multiple CRs evoked from bursting instead of resting state and the underlying bifurcation mechanism are studied in a theoretical neuron model composed of fast subsystem and slow variable, which is very important for the bursting neurons receiving strong synaptic noise in the central nervous system. With increasing noise intensity within a range, the bursting period decreases to induce the burst number increases firstly and then becomes irregular, which results in the increase firstly and then decrease in the corresponding peak of the power spectrum of membrane potentials, i.e., CR is evoked from bursting instead of resting state. With further increasing noise intensity, two other peaks exhibit CR phenomenon, one at middle frequency and middle noise intensity and the other at large frequency and strong noise intensity, which shows that multiple CRs are evoked from bursting. Furthermore, the bifurcation mechanism of the latter two resonant peaks is acquired with fast–slow variable dissection method. With increasing the value of slow variable, the fast subsystem exhibits a fold bifurcation of limit cycle and a subcritical Hopf bifurcation to a stable focus, and the imaginary part of the characteristic value of the focus increases, which implies that spikes evoked from the focus exhibit increasing frequency. The trajectory of the deterministic bursting in phase space locates between the two bifurcations and alternates between the coexisting limit cycle and focus of the fast subsystem. With increasing noise intensity, the value of slow variable of the stochastic bursting increases and moves to a range of the focus of the fast subsystem, and the spikes of the stochastic bursting are evoked from the focus by noise. Therefore, the stronger the noise intensity, the larger the value of slow variable, and the higher frequency the spikes, due to the changes of characteristic values, which is the cause that the peaks with middle and high frequency appear at middle and strong noise intensity, respectively. The results of the present paper extend the concept of CR to multiple CRs evoked from bursting, provide the bifurcation mechanism underlying the multiple CRs, and imply multiple chances to utilize noise to enhance information of bursting neurons.

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

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


  1. 1.

    Benzi, R., Sutera, A., Vulpiani, A.: Stochastic resonance in climatic change. J. Phys. A 14, L453–L457 (1981)

    MATH  Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 3.

    Lindnera, B., García-Ojalvob, J., Neiman, A., Schimansky-Geiere, L.: Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)

    Google Scholar 

  4. 4.

    Simakov, D.S., Pérez-Mercader, J.: Noise induced oscillations and coherence resonance in a generic model of the nonisothermal chemical oscillator. Sci. Rep. 3, 2404 (2013)

    Google Scholar 

  5. 5.

    McDonnell, M.D., Iannella, N., To, M.S., Tuckwell, H.C., Jost, J., Gutkin, B.S., Ward, L.M.: A review of methods for identifying stochastic resonance in simulations of single neuron models. Network 26(2), 35–71 (2015)

    Google Scholar 

  6. 6.

    Sun, G., Jusup, M., Jin, Z., Wang, Y., Wang, Z.: Pattern transitions in spatial epidemics: mechanisms and emergent properties. Phys. Life Rev. 19, 43–73 (2016)

    Google Scholar 

  7. 7.

    Longtin, A., Bulsara, A., Moss, F.: Time-interval sequences in bistable systems and the noise-induced transmission of information by sensory neurons. Phys. Rev. Lett. 67(5), 656–659 (1991)

    Google Scholar 

  8. 8.

    Braun, H.A., Wissing, H., Schäfer, K., Hirsch, M.C.: Oscillation and noise determine signal transduction in shark multimodal sensory cells. Nature 367, 270–273 (1994)

    Google Scholar 

  9. 9.

    McDonnell, M.D., Abbott, D.: What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLoS Comput. Biol. 5(5), e1000348 (2009)

    MathSciNet  Google Scholar 

  10. 10.

    Pikovsky, A.S., Kurths, J.: Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett. 78, 775–778 (1997)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Longtin, A.: Autonomous stochastic resonance in bursting neurons. Phys. Rev. E 55(1), 868–876 (1997)

    Google Scholar 

  12. 12.

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

    Google Scholar 

  13. 13.

    Levin, J.E., Miller, J.P.: Broadband neural encoding in the cricket cereal sensory system enhanced by stochastic resonance. Nature 380, 165–168 (1996)

    Google Scholar 

  14. 14.

    Gu, H., Zhao, Z., Jia, B., Chen, S.: Dynamics of on-off neural firing patterns and stochastic effects near a sub-critical Hopf bifurcation. PLoS One 10(4), e0121028 (2015)

    Google Scholar 

  15. 15.

    Jia, B., Gu, H.: Dynamics and physiological roles of stochastic neural firing patterns near bifurcation points. Int. J. Bifurc. Chaos 27(7), 1750113 (2017)

    MATH  Google Scholar 

  16. 16.

    Méndez-Balbuena, I., Huidobro, N., Silva, M., Flores, A., Trenado, C., Quintanar, L., Arias-Carrión, O., Kristeva, R., Manjarrez, E.: Effect of mechanical tactile noise on amplitude of visual evoked potentials: multisensory stochastic resonance. J. Neurophysiol. 114(4), 2132–2143 (2015)

    Google Scholar 

  17. 17.

    van der Groen, O., Wenderoth, N.: Transcranial random noise stimulation of visual cortex: stochastic resonance enhances central mechanisms of perception. J. Neurosci. 36(19), 5289–5298 (2016)

    Google Scholar 

  18. 18.

    Wuehr, M., Boerner, J.C., Pradhan, C., Decker, J., Jahn, K., Brandt, T., Schniepp, R.: Stochastic resonance in the human vestibular system\(-\)Noise\(-\)induced facilitation of vestibulospinal reflexes. Brain Stimul. 11(2), 261–263 (2018)

    Google Scholar 

  19. 19.

    Angwin, A.J., Wilson, W.J., Arnott, W.L., Signorini, A., Barry, R.J., Copland, D.A.: White noise enhances new-word learning in healthy adults. Sci. Rep. 7(1), 13045 (2017)

    Google Scholar 

  20. 20.

    Itzcovich, E., Riani, M., Sannita, W.G.: Stochastic resonance improves vision in the severely impaired. Sci. Rep. 7(1), 12840 (2017)

    Google Scholar 

  21. 21.

    Hilliard, D., Passow, S., Thurm, F., Schuck, N.W., Garthe, A., Kempermann, G., Li, S.C.: Noisy galvanic vestibular stimulation modulates spatial memory in young healthy adults. Sci. Rep. 9(1), 9310 (2019)

    Google Scholar 

  22. 22.

    Krauss, P., Metzner, C., Schilling, A., Schutz, C., Tziridis, K., Fabry, B., Schulze, H.: Adaptive stochastic resonance for unknown and variable input signals. Sci. Rep. 7(1), 2450 (2017)

    Google Scholar 

  23. 23.

    Nobusako, S., Osumi, M., Matsuo, A., Fukuchi, T., Nakai, A., Zama, T., Shimada, S., Morioka, S.: Stochastic resonance improves visuomotor temporal integration in healthy young adults. PLoS One 13(12), e0209382 (2018)

    Google Scholar 

  24. 24.

    Wu, S., Ren, W., He, K., Huang, Z.: Burst and coherence resonance in Rose-Hindmarsh model induced by additive noise. Phys. Lett. A 279(5–6), 347–354 (2001)

    MATH  Google Scholar 

  25. 25.

    Gu, H., Yang, M., Li, L., Liu, Z., Ren, W.: Experimental observation of the stochastic bursting caused by coherence resonance in a neural pacemaker. NeuroReport 13(13), 1657–1660 (2002)

    Google Scholar 

  26. 26.

    Gu, H., Yang, M., Li, L., Liu, Z., Ren, W.: Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios. Phys. Lett. A 319(1–2), 89–96 (2003)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Yao, Y., Ma, J.: Weak periodic signal detection by sine-Wiener-noise-induced resonance in the FitzHugh-Nagumo neuron. Cogn. Neurodyn. 12(3), 343–349 (2018)

    MathSciNet  Google Scholar 

  28. 28.

    Zhou, C., Kurths, J., Hu, B.: Array-enhanced coherence resonance: nontrivial effects of heterogeneity and spatial independence of noise. Phys. Rev. Lett. 87(9), 098101 (2001)

    Google Scholar 

  29. 29.

    Perc, M.: Spatial coherence resonance in excitable media. Phys. Rev. E 72, 016207 (2005)

    MathSciNet  Google Scholar 

  30. 30.

    Sun, X., Perc, M., Lu, Q., Kurths, J.: Spatial coherence resonance on diffusive and small-world networks of Hodgkin–Huxley neurons. Chaos 18(2), 023102 (2008)

    MathSciNet  Google Scholar 

  31. 31.

    Guo, D., Li, C.: Stochastic and coherence resonance in feed-forward-loop neuronal network motifs. Phys. Rev. E. 79, 051921 (2009)

    Google Scholar 

  32. 32.

    Wang, Y., Ma, J., Xu, Y., Wu, F., Zhou, P.: The electrical activity of neurons subject to electromagnetic induction and gaussian white noise. Int. J. Bifurc. Chaos 27(02), 1750030 (2017)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Jia, Y., Gu, H.: Transition from double coherence resonances to single coherence resonance in a neuronal network with phase noise. Chaos 25(12), 123124 (2015)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Vilar, J.M.G., Rubí, J.M.: Stochastic multiresonance. Phys. Rev. Lett. 78(15), 2882–2885 (1997)

    Google Scholar 

  35. 35.

    Jiang, Y.: Multiple dynamical resonances in a discrete neuronal model. Phys. Rev. E 71, 057103 (2005)

    Google Scholar 

  36. 36.

    Liang, G., Cao, L., Wang, J., Wu, D.: Modulated stochastic multiresonance in a single-mode laser system driven by colored additive and multiplicative noises without external periodic force. Physica A 327(3–4), 304–312 (2003)

    Google Scholar 

  37. 37.

    Yilmaz, E., Ozer, M., Baysal, V., Perc, M.: Autapse-induced multiple coherence resonance in single neurons and neuronal networks. Sci. Rep. 6, 30914 (2016)

    Google Scholar 

  38. 38.

    Yang, X., Yu, Y., Sun, Z.: Autapse-induced multiple stochastic resonances in a modular neuronal network. Chaos 27(8), 083117 (2017)

    MathSciNet  Google Scholar 

  39. 39.

    Wang, Q., Perc, M., Duan, Z., Chen, G.: Delay-induced multiple stochastic resonances on scale-free neuronal networks. Chaos 19(2), 023112 (2009)

    Google Scholar 

  40. 40.

    Wang, Q., Zhang, H., Perc, M., Chen, G.: Multiple firing coherence resonances in excitatory and inhibitory coupled neurons. Commun. Nonlinear Sci. Numer. Simul. 17(10), 3979–3988 (2012)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Perc, M.: Noise-induced spatial periodicity in excitable chemical media. Chem. Phys. Lett. 410(1–3), 49–53 (2005)

    Google Scholar 

  42. 42.

    Perc, M.: Spatial decoherence induced by small-world connectivity in excitable media. New J. Phys. 7, 252–252 (2005)

    Google Scholar 

  43. 43.

    Jung, P., Mayer-Kress, G.: Spatiotemporal stochastic resonance in excitable media. Phys. Rev. Lett. 74(11), 2130–2133 (1995)

    Google Scholar 

  44. 44.

    Gu, H., Jia, B., Li, Y., Chen, G.: White noise-induced spiral waves and multiple spatial coherence resonances in a neuronal network with type I excitability. Physica A 392(6), 1361–1374 (2013)

    MathSciNet  Google Scholar 

  45. 45.

    Li, Y., Gu, H.: The influence of initial values on spatial coherence resonance in neuronal networks. Int. J. Bifurc. Chaos 25(8), 1550104 (2015)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Tateno, T., Pakdaman, K.: Random dynamics of the Morris–Lecar neural model. Chaos 14, 511–530 (2004)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Gu, H., Zhang, H., Wei, C., Yang, M., Liu, Z., Ren, W.: Coherence resonance induced stochastic neural firing at a saddle-node bifurcation. Int. J. Mod. Phys. B 25(29), 3977–3986 (2011)

    Google Scholar 

  48. 48.

    Jia, B., Gu, H., Li, L., Zhao, X.: Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns. Cogn. Neurodyn. 6(1), 89–106 (2012)

    Google Scholar 

  49. 49.

    Gu, H., Ren, W., Lu, Q., Wu, S., Chen, W.: Integer multiple spiking in neuronal pacemakers without external periodic stimulation. Phys. Lett. A 285, 63–68 (2001)

    MathSciNet  Google Scholar 

  50. 50.

    Kim, J.H., Lee, H.J., Min, C.H., Lee, K.J.: Coherence resonance in bursting neural networks. Phys. Rev. E 92(4), 042701 (2015)

    Google Scholar 

  51. 51.

    Lisman, J.E.: Bursts as a unit of neural information: making unreliable synapses reliable. Trends Neurosci. 20(1), 38–43 (1997)

    Google Scholar 

  52. 52.

    Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10(6), 1171–1266 (2000)

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge (2007)

    Google Scholar 

  54. 54.

    Del Negro, C.A., Hsiao, C.F., Chandler, S.H., Garfinkel, A.: Evidence for a novel bursting mechanism in rodent trigeminal neurons. Biophys. J. 75, 174–182 (1998)

    Google Scholar 

  55. 55.

    Li, Y., Gu, H.: The distinct stochastic and deterministic dynamics between period-adding and period-doubling bifurcations of neural bursting patterns. Nonlinear Dyn. 87(4), 2541–2562 (2017)

    Google Scholar 

  56. 56.

    Wu, F., Gu, H., Li, Y.: Inhibitory electromagnetic induction current induced enhancement instead of reduction of neural bursting activities. Commun. Nonlinear Sci. Numer. Simul. 79, 104924 (2019)

    MathSciNet  Google Scholar 

  57. 57.

    Li, Y., Gu, H., Ding, X.: Bifurcations of enhanced neuronal bursting activities induced by the negative current mediated by inhibitory autapse. Nonlinear Dyn. 97(4), 2091–2105 (2019)

    Google Scholar 

  58. 58.

    Cao, B., Guan, L., Gu, H.: Bifurcation mechanism of not increase but decrease of spike numbers within a neural burst induced by excitatory effect. Acta Phys. Sin. 67(24), 240502 (2018). (in chinese)

    Google Scholar 

  59. 59.

    Wu, F., Gu, H.: Bifurcations of negative responses to positive feedback current mediated by memristor in neuron model with bursting patterns. Int. J. Bifurc. Chaos 30(4), 2030009 (2020)

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Mysin, I.E., Kitchigina, V.F., Kazanovich, Y.: Modeling synchronous theta activity in the medial septum: key role of local communications between different cell populations. J. Comput. Neurosci. 39, 1–16 (2015)

    Google Scholar 

  61. 61.

    Wang, X.: Pacemaker neurons for the theta rhythm and their synchronization in the septohippocampal reciprocal loop. J. Neurophysiol. 87, 889–900 (2002)

    Google Scholar 

  62. 62.

    Colom, L.V., Castaneda, M.T., Reyna, T., Hernandez, S., Garrido-Sanabria, E.: Characterization of medial septal glutamatergic neurons and their projection to the hippocampus. Synapse 58(3), 151–164 (2005)

    Google Scholar 

  63. 63.

    Hangya, B., Borhegyi, Z., Szilágyi, N., Freund, T.F., Varga, V.: GABAergic neurons of the medial septum lead the hippocampal network during theta activity. J. Neurosci. 29(25), 8094–8102 (2009)

    Google Scholar 

  64. 64.

    Mannella, R., Palleschi, V.: Fast and precise algorithm for compute simulation of stochastic differential equations. Phys. Rev. A 40, 3381–3386 (1989)

    Google Scholar 

  65. 65.

    Ermentrout, B.: Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. SIAM, Philadelphia (2002)

    Google Scholar 

  66. 66.

    Hu, G., Ditzinger, T., Ning, C.Z., Haken, H.: Stochastic resonance without external periodic force. Phys. Rev. Lett. 71(6), 807 (1993)

    Google Scholar 

  67. 67.

    Semenova, N., Zakharova, A., Anishchenko, V., Schöll, E.: Coherence-resonance chimeras in a network of excitable elements. Phys. Rev. Lett. 117(1), 014102 (2016)

    Google Scholar 

  68. 68.

    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)

    Google Scholar 

  69. 69.

    McDonnell, M.D., Ward, L.M.: The benefits of noise in neural systems: bridging theory and experiment. Nat. Rev. Neurosci. 12(7), 415–426 (2011)

    Google Scholar 

  70. 70.

    Faisal, A.A., Selen, L.P., Wolpert, D.M.: Noise in the nervous system. Nat. Rev. Neurosci. 9(4), 292–303 (2008)

    Google Scholar 

  71. 71.

    Yu, H., Zhang, L., Guo, X., Wang, J., Cao, Y., Liu, J.: Effect of inhibitory firing pattern on coherence resonance in random neural networks. Phys. A 490, 1201–1210 (2018)

    MathSciNet  Google Scholar 

  72. 72.

    Zhang, X., Gu, H., Guan, L.: Stochastic dynamics of conduction failure of action potential along nerve fiber with Hopf bifurcation. Sci. China Technol. Sci. 62(9), 1502–1511 (2019)

    Google Scholar 

  73. 73.

    Ma, J., Wang, C., Jin, W., Wu, Y.: Transition from spiral wave to target wave and other coherent structures in the networks of Hodgkin–Huxley neurons. Appl. Math. Comput. 217(8), 3844–3852 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Huaguang Gu.

Ethics declarations

Conflict of interest

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.

This work was sponsored by the National Natural Science Foundation of China (Grant Numbers: 11872276, 11572225, 11802086)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guan, L., Gu, H. & Jia, Y. Multiple coherence resonances evoked from bursting and the underlying bifurcation mechanism. Nonlinear Dyn 100, 3645–3666 (2020).

Download citation


  • Coherence resonance
  • Bifurcation
  • Bursting
  • Neural firing pattern