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The probability density function of interspike intervals in an FHN model with α-stable noise

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Abstract

In this paper, we expand the paradigmatic Fitzhugh–Nagumo (FHN) model by including α-stable noise and study the firing rate and coherence resonance via probability density function (PDF) of the interspike intervals (ISIs). In the previous work, different indicators and measurements are given to illustrate the firing rate and coherence resonance. However, we find that they can be described simply by the PDF of ISIs, which contains the underlying information of spiking neuron signals. For example, the width of the peak of the PDF of ISIs can characterize the phenomenon of coherence resonance. The narrower width of peak stands for the better coherence resonance.

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References

  1. W. Gerstner, W. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity (Cambridge University Press, Cambridge, 2002).

    Book  MATH  Google Scholar 

  2. D. Andrieux, T. Monnai, Firing rate of noisy integrate-and-fire neurons with synaptic current dynamics. Phys. Rev. E 80, 021933 (2009)

    Article  ADS  Google Scholar 

  3. M. Yi, L. Yang, Propagation of firing rate by synchronization and coherence of firing pattern in a feed-forward multilayer neural network. Phys. Rev. E 81, 061924 (2010)

    Article  ADS  Google Scholar 

  4. S. Liu, Z. He, M. Zhan, Firing rates of coupled noisy excitable elements. Front. Phys. 9, 120–127 (2014)

    Article  ADS  Google Scholar 

  5. M. Levakova, M. Tamborrino, L. Kostal et al., Accuracy of rate coding: when shorter time window and higher spontaneous activity help. Phys. Rev. E 95, 022310 (2017)

    Article  ADS  Google Scholar 

  6. M. Richardson, Spike shape and synaptic-amplitude distribution interact to set the high-frequency firing-rate response of neuronal populations. Phys. Rev. E 98, 042405 (2018)

    Article  ADS  Google Scholar 

  7. F. Han, X. Gu, Z. Wang et al., Global firing rate contrast enhancement in E/I neuronal networks by recurrent synchronized inhibition. Chaos: Interdiscip. J. Nonlinear Sci. 28, 106324 (2018)

    Article  MathSciNet  Google Scholar 

  8. F. Gabbiani, C. Koch, Principles of spike train analysis. Methods Neuronal Model. 12, 313–360 (1998)

    Google Scholar 

  9. M. Forrest, Intracellular calcium dynamics permit a Purkinje neuron model to perform toggle and gain computations upon its inputs. Front. Comput. Neurosci. 8, 86 (2014)

    Article  Google Scholar 

  10. G.D. Scholes, G.R. Fleming, L.X. Chen et al., Using coherence to enhance function in chemical and biophysical systems. Nature 543, 647–656 (2017)

    Article  ADS  Google Scholar 

  11. V. Djokic, G. Enzian, F. Vewinger et al., Resonance retrieval of stored coherence in an rf-optical double-resonance experiment. Phys. Rev. A 92, 063802 (2015)

    Article  ADS  Google Scholar 

  12. G. Hu, T. Ditzinger, C. Ning et al., Stochastic resonance without external periodic force. Phys. Rev. Lett. 71, 807–810 (1993)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. S. Biswas, D. Das, P. Parmananda et al., Predicting the coherence resonance curve using a semianalytical treatment. Phys. Rev. E 80, 046220 (2009)

    Article  ADS  Google Scholar 

  15. O. Rosso, C. Masoller, Detecting and quantifying stochastic and coherence resonances via information-theory complexity measurements. Phys. Rev. E 79, 040106 (2009)

    Article  ADS  Google Scholar 

  16. N. Kouvaris, L. Schimansky-Geier, E. Schöll, Control of coherence in excitable systems by the interplay of noise and time-delay. Eur. Phys. J.-Spec. Top. 191, 29–51 (2010)

    Article  Google Scholar 

  17. P. Borowski, R. Kuske, Y. Li et al., Characterizing mixed mode oscillations shaped by noise and bifurcation structure. Chaos: Interdiscip. J. Nonlinear Sci. 20, 043117 (2010)

    Article  MathSciNet  Google Scholar 

  18. P. Shaw, D. Saha, S. Ghosh et al., Intrinsic noise induced coherence resonance in a glow discharge plasma. Chaos: Interdiscip. J. Nonlinear Sci. 25, 043101 (2015)

    Article  Google Scholar 

  19. M. Calderón Ramírez, R. Rico Martínez, E. Ramírez Álvarez et al., Tracking stochastic resonance curves using an assisted reference model. Chaos: Interdiscip. J. Nonlinear Sci. 25, 063107 (2015)

    Article  MathSciNet  Google Scholar 

  20. R. Fitzhugh, Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J. Gen. Physiol. 43, 867–896 (1960)

    Article  Google Scholar 

  21. J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)

    Article  Google Scholar 

  22. C. Takembo, A. Mvogo, H. Fouda et al., Effect of electromagnetic radiation on the dynamics of spatiotemporal patterns in memristor-based neuronal network. Nonlinear Dyn. 95, 1067–1078 (2019)

    Article  MATH  Google Scholar 

  23. Z. Luo, L. Song, Research on the new dynamics properties for a noise-induced excited system. Neural Comput. Appl. 24, 521–529 (2014)

    Article  Google Scholar 

  24. H. Li, X. Sun, J. Xiao, Stochastic multiresonance in coupled excitable FHN neurons. Chaos: Interdiscip. J. Nonlinear Sci. 28, 043113 (2018)

    Article  MathSciNet  Google Scholar 

  25. I. Franović, K. Todorović, N. Vasović et al., Cluster synchronization of spiking induced by noise and interaction delays in homogenous neuronal ensembles. Chaos: Interdiscip. J. Nonlinear Sci. 22, 033147 (2012)

    Article  MathSciNet  Google Scholar 

  26. X. Sun, Z. Liu, M. Perc, Effects of coupling strength and network topology on signal detection in small-world neuronal networks. Nonlinear Dyn. 96, 2145–2155 (2019)

    Article  MATH  Google Scholar 

  27. Y. Xu, B. Pei, G. Guo, Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise. Appl. Math. Comput. 263, 398–409 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Y. Li, Y. Xu, J. Kurths et al., Lévy-noise-induced transport in a rough triple-well potential. Phys. Rev. E 94, 042222 (2016)

    Article  ADS  Google Scholar 

  29. D. Roberts, A. Kalloniatis, Synchronisation under shocks: the Lévy Kuramoto model. Physica D 368, 10–21 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Y. Li, Y. Xu, J. Kurths, Roughness-enhanced transport in a tilted ratchet driven by Lévy noise. Phys. Rev. E 96, 052121 (2017)

    Article  Google Scholar 

  31. R. Cai, X. Chen, J. Duan et al., Lévy noise-induced escape in an excitable system. J. Stat. Mech. 2017, 063503 (2017)

    Article  MATH  Google Scholar 

  32. Z. Wang, Y. Xu, H. Yang, Lévy noise induced stochastic resonance in an FHN model. Sci. China Technol. Sci. 59, 371–375 (2016)

    Article  ADS  Google Scholar 

  33. Z. Wang, Y. Xu, Y. Li et al., α-stable noise-induced coherence on a spatially extended Fitzhugh–Nagumo system. J. Stat. Mech: Theory Exp. 2019, 103501 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Ma, Y. Xu, Y. Li et al., Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions. Appl. Math. Mech. 42, 65–84 (2021)

    Article  Google Scholar 

  35. X. Zhang, Y. Xu, Q. Liu et al., Rate-dependent tipping-delay phenomenon in a thermoacoustic system with colored noise. Sci. China Technol. Sci. 63, 2315–2327 (2020)

    Article  ADS  Google Scholar 

  36. H. Li, Y. Xu, Y. Li et al., Transition path dynamics across rough inverted parabolic potential barrier. Eur. Phys. J. Plus 135, 1–22 (2020)

    Article  ADS  Google Scholar 

  37. R. Mei, Y. Xu, Y. Li et al., The steady current analysis in a periodic channel driven by correlated noises. Chaos Solitons Fractals 135, 109766 (2020)

    Article  MathSciNet  Google Scholar 

  38. J. Ma, Y. Xu, W. Xu et al., Slowing down critical transitions via Gaussian white noise and periodic force. Sci. China Technol. Sci. 62, 2144–2152 (2019)

    Article  ADS  Google Scholar 

  39. Y. Li, R. Mei et al., Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity. New J. Phys. 22, 053016 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  40. H. Kitano, Towards a theory of biological robustness. Mol. Syst. Biol. 3, 137 (2007)

    Article  Google Scholar 

  41. F.S. Borges, P.R. Protachevicz, E.L. Lameu et al., Synchronised firing patterns in a random network of adaptive exponential integrate-and-fire neuron model. Neural Netw. 90, 1–7 (2017)

    Article  MATH  Google Scholar 

  42. J.S. Seeler, A. Dejean, SUMO and the robustness of cancer. Nat. Rev. Cancer 17, 184–197 (2017)

    Article  Google Scholar 

  43. J. Chambers, C. Mallows, B. Stuck, A method for simulating stable random variables. J. Am. Stat. Assoc. 71, 340–344 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Chambers, C. Mallows, B. Stuck, Correction to: “A method for simulating stable random variables". J. Am. Stat. Assoc. 82, 704 (1987)

    Article  Google Scholar 

  45. D. Fulger, E. Scalas, G. Germano, Random numbers from the tails of probability distributions using the transformation method. Fract. Calc. Appl. Anal. 16, 332–353 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This paper was supported by the National Natural Science Foundation of China under Grant Nos. 11772255 and 11902118, the Fundamental Research Funds for the Central Universities, the Research Funds for Interdisciplinary Subject of Northwestern Polytechnical University, the Shaanxi Project for Distinguished Young Scholars, Shaanxi Provincial Key R&D Program 2020KW-013 and 2019TD-010, and National Key Research and Development Program of China under Grant No. 2018AAA0102201. JK acknowledges support from Russian Ministry of Science and Education “Digital biodesign and personalised healthcare”.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China

    Zhanqing Wang, Yong Xu & Yongge Li

  2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an, 710072, China

    Yong Xu

  3. Potsdam Institute for Climate Impact Research, 14412, Potsdam, Germany

    Jürgen Kurths

  4. Centre for Analysis of Complex Systems, Sechenov First Moscow State Medical University, Moscow, Russia

    Jürgen Kurths

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  1. Zhanqing Wang
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  2. Yong Xu
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  3. Yongge Li
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  4. Jürgen Kurths
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Correspondence to Yong Xu or Yongge Li.

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Guest editors: J. C. Cortés, T. Caraballo, C. Pinto.

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Wang, Z., Xu, Y., Li, Y. et al. The probability density function of interspike intervals in an FHN model with α-stable noise. Eur. Phys. J. Plus 136, 299 (2021). https://doi.org/10.1140/epjp/s13360-021-01245-x

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