Nonlinear Dynamics

, Volume 98, Issue 2, pp 1029–1039 | Cite as

Extended analysis of stochastic resonance in a modular neuronal network at different scales

  • XiaoLi YangEmail author
  • Na Li
  • ZhongKui Sun
Original paper


Based on the firing dynamics at three different levels of microscale, mesoscale and macroscale, this study presents an extended analysis of stochastic resonance in a modular neuronal network in the spatially correlated white noise environment. Two well-defined modules of small-world subnetwork and scale-free subnetwork constitute the modular neuronal network in a hierarchical way. When a subthreshold periodic input is incorporated into this network, numerical results illustrate that a collective pattern of stochastic resonance emerges at macroscopic scale when the intensity of the correlated noise is appropriately tuned. Through extended analysis, one can detect that the firing rhythms of individual neurons gradually follow those of the periodic input at microscale. In addition, the occurrence of stochastic resonance at mesoscale in the small-world subnetwork is earlier than that in the scale-free subnetwork, and the peak height of resonance curve in the former subnetwork is remarkably higher than that in the latter one. These combined results indicate that the small-world subnetwork is more favorable than the scale-free subnetwork to induce stochastic resonance in this constructed modular network. The robustness of the extended analysis of stochastic resonance against variations in noise correlation coefficient and intra-module probability is also unveiled. This study provides a new perspective and tool to understand the collective phenomenon of stochastic resonance in realistic neuronal systems.


Stochastic resonance Modular neuronal network Macroscale Mesoscale Microscale 



This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11572180, 11972217, 11671243), the Fundamental Funds Research for the Central Universities (Grant Nos. GK201901008, GK201701001).

Compliance with ethical standards

Conflict of interest

We declare that we have no conflict of interest.


  1. 1.
    Pakdamana, K., Mestivier, D.: Noise induced synchronization in a neuronal oscillator. Physica D 192, 123–137 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Zhang, J.T., Sun, Z.K., Yang, X.L., Xu, W.: Controlling bifurcation in fractional-delay systems with colored noise. Int. J. Bifurc. Chaos 28, 1850137 (2018)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Sagues, F., Sancho, J.M., Ojalvo, J.G.: Spatiotemporal order out of noise. Rev. Mod. Phys. 79, 829–884 (2007)Google Scholar
  4. 4.
    Guo, Q., Sun, Z.K., Xu, W.: Stochastic bifurcations in a birhythmic biological model with time-delayed feedbacks. Int. J. Bifurc. Chaos 28, 1850048 (2018)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Guo, D.Q., Perc, M., Liu, T.J., Yao, D.Z.: Functional importance of noise in neuronal information processing. EPL 124, 50001 (2018)Google Scholar
  6. 6.
    Lindner, B., Ojalvo, J.G., Neiman, A.L., Geier, S.: Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)Google Scholar
  7. 7.
    Collins, J.J., Chow, C.C., Imhoff, T.T.: Stochastic resonance without tuning. Nature 376, 236–238 (1995)Google Scholar
  8. 8.
    Guo, D.Q., Li, C.G.: Stochastic resonance in Hodgkin–Huxley neuron induced by unreliable synaptic transmission. J. Theor. Biol. 308, 104 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Pikovsky, A.S., Kurths, J.: Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett. 78, 775 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Wang, Q.Y., Zhang, H.H., Chen, G.R.: Effect of the heterogeneous neuron and information transmission delay on stochastic resonance of neuronal networks. Chaos 22, 043123 (2012)MathSciNetGoogle Scholar
  11. 11.
    Wang, Q.Y., Zhang, H.H., Perc, M., Chen, G.R.: Multiple firing coherence resonances in excitatory and inhibitory coupled neurons. Commun. Nonlinear Sci. 17, 3979–3988 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hu, B., Zhou, C.S.: Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance. Phys. Rev. E 61, R1001–R1004 (2000)Google Scholar
  13. 13.
    Yang, X.L., Senthilkumar, D.V., Kurths, J.: Impact of connection delays on noise-induced spatiotemporal patterns in neuronal networks. Chaos 22, 043150 (2012)MathSciNetGoogle Scholar
  14. 14.
    Yilmaz, E., Baysal, V., Perc, M., Ozer, M.: Enhancement of pacemaker induced stochastic resonance by an autapse in a scale-free neuronal network. Sci. China Technol. Sci. 59, 364–370 (2016)Google Scholar
  15. 15.
    Yang, X.L., Jia, Y.B., Zhang, L.: Impact of bounded noise and shortcuts on the spatiotemporal dynamics of neuronal networks. Physica A 393, 617–623 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guo, D.Q., Li, C.G.: Stochastic and coherence resonance in feed-forward-loop neuronal network motifs. Phys. Rev. E 79, 051921 (2009)Google Scholar
  17. 17.
    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
  18. 18.
    Yilmaz, E., Baysal, V., Ozer, M., Perc, M.: Autaptic pacemaker mediated propagation of weak rhythmic activity across small-world neuronal networks. Physica A 444, 538–546 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Wang, Q.Y., Perc, M., Duan, Z.S., Chen, G.R.: Delay-induced multiple stochastic resonances on scale-free neuronal networks. Chaos 19, 023112 (2009)Google Scholar
  20. 20.
    Sun, X.J., Liu, Z.F.: Combined effects of time delay and noise on the ability of neuronal network to detect the subthreshold signal. Nonlinear Dyn. 92, 1707–1717 (2018)Google Scholar
  21. 21.
    Perc, M.: Stochastic resonance on excitable small-world networks via a pacemaker. Phys. Rev. E 76, 066203 (2007)Google Scholar
  22. 22.
    Guo, D.Q., Perc, M., Zhang, Y.S., Xu, P., Yao, D.Z.: Frequency-difference-dependent stochastic resonance in neural systems. Phys. Rev. E 96, 022415 (2017)Google Scholar
  23. 23.
    Sun, X.J., Liu, Z.F., Perc, M.: Effects of coupling strength and network topology on signal detection in small-world neuronal networks. Nonlinear Dyn. 96, 2145–2155 (2019)Google Scholar
  24. 24.
    Li, H.Y., Sun, X.J., Xiao, J.H.: Stochastic multiresonance in coupled excitable FHN neurons. Chaos 28, 043113 (2018)MathSciNetGoogle Scholar
  25. 25.
    Gu, H.G., Jia, B., Li, Y.Y., Chen, G.R.: White noise-induced spiral waves and multiple spatial coherence resonances in a neuronal network with type I excitability. Physica A 392, 1361–1374 (2013)MathSciNetGoogle Scholar
  26. 26.
    Peter, J., Gottfried, M.K.: Spatiotemporal stochastic resonance in excitable media. Phys. Rev. Lett. 74, 2130 (1995)Google Scholar
  27. 27.
    Wang, Q.Y., Lu, Q.S., Chen, G.R.: Spatio-temporal patterns in a square-lattice Hodgkin–Huxley neural network. Eur. Phys. J. B 54, 255–261 (2006)Google Scholar
  28. 28.
    Ma, J., Zhang, A.H., Tang, J., Jin, W.Y.: Collective behaviors of spiral wave in the networks of Hodgkin–Huxley neurons in presence of channel noise. J. Biol. Syst. 18, 243–259 (2010)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ma, J., Wu, Y., Ying, H.P., Jia, Y.: Channel noise-induced phase transition of spiral wave in networks of Hodgkin–Huxley neurons. Chin. Sci. Bull. 56, 151–157 (2011)Google Scholar
  30. 30.
    Zhou, C.S., Kurths, J., Hu, B.: Array-enhanced coherence resonance: nontrivial effects of heterogeneity and spatial independence of noise. Phys. Rev. Lett. 87, 098101 (2001)Google Scholar
  31. 31.
    Doiron, B., Lindner, B., Longtin, A., Maler, L., Bastian, J.: Oscillatory activity in electrosensory neurons increases with the spatial correlation of the stochastic input stimulus. Phys. Rev. Lett. 93, 048101 (2004)Google Scholar
  32. 32.
    Sun, X.J., Lu, Q.S., Kurths, J.: Correlated noise induced spatiotemporal coherence resonance in a square lattice network. Physica A 387, 6679 (2008)Google Scholar
  33. 33.
    Hilgetag, C.C., Burns, G.A., O’neill, M.A., Scannell, J.W., Young, M.P.: Anatomical connectivity defines the organization of clusters of cortical areas in the macaque monkey and the cat. Philos. Trans. R. Soc. Lond. Ser B. 355, 91 (2000)Google Scholar
  34. 34.
    Hilgetag, C.C., Kaiser, M.: Clustered organization of cortical connectivity. Neuroinformatics 2, 353 (2004)Google Scholar
  35. 35.
    Jia, Y.B., Yang, X.L., Kurths, J.: Diversity and time delays induce resonance in a modular neuronal network. Chaos 24, 043140 (2014)Google Scholar
  36. 36.
    Yang, X.L., Li, H.D., Sun, Z.K.: Partial coupling delay induced multiple spatiotemporal orders in a modular neuronal network. PLoS ONE 12, e0177918 (2017)Google Scholar
  37. 37.
    Yang, X.L., Yu, Y.H., Sun, Z.K.: Autapse-induced multiple stochastic resonances in a modular neuronal network. Chaos 27, 083117 (2017)MathSciNetGoogle Scholar
  38. 38.
    Yu, H.T., Wang, J., Liu, Q.X., Wen, J.X., Deng, B., Wei, X.L.: Chaotic phase synchronization in a modular neuronal network of small-world subnetworks. Chaos 21, 043125 (2011)zbMATHGoogle Scholar
  39. 39.
    Zamora-López, G., Zhou, C.S., Kurths, J.: Graph analysis of cortical networks reveals complex anatomical communication substrate. Chaos 19, 015117 (2009)Google Scholar
  40. 40.
    Belykh, I., Hasler, M.: Mesoscale and clusters of synchrony in networks of bursting neurons. Chaos 21, 016106 (2011)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Arenas, A., Guilera, A.D., Perez-Vicente, C.J.: Synchronization processes in complex networks. Physica D 224, 27–34 (2006)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Ma, J., Tang, J.: A review for dynamics of collective behaviors of network of neurons. Sci. China Technol. Sci. 58, 2038–2045 (2015)Google Scholar
  43. 43.
    Guo, D.Q., Li, C.C.: Self-sustained irregular activity in 2-D small-world networks of excitatory and inhibitory neurons. IEEE Trans. Neural Netw. 21, 895–905 (2010)Google Scholar
  44. 44.
    Eguíluz, V.M., Chialvo, D.R., Cecchi, G.A., Baliki, M., Apkarian, A.V.: Scale-free brain functional networks. Phys. Rev. Lett. 94, 018102 (2005)Google Scholar
  45. 45.
    Bassett, D., Bullmore, E.: Small-world brain networks. Neuroscientist 12, 512–523 (2006)Google Scholar
  46. 46.
    Newman, M.E.J., Watts, D.J.: Renormalization group analysis of the small-world network. Phys. Lett. A 263, 341 (1999)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509 (1999)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China
  2. 2.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

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