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Autapse-induced firing patterns transitions in the Morris–Lecar neuron model

  • Xinlin Song
  • Hengtong Wang
  • Yong ChenEmail author
Original Paper
  • 65 Downloads

Abstract

In 1948, Hodgkin identified three firing patterns of a single neuron in response to increasing the external DC input. In this work, we investigated the responses of a single neuron with an autapse based on a modified Morris–Lecar neuron model, which can exhibit the three types of firing patterns by changing only one parameter. An excitatory autapse was found to enhance the firing frequency, but an inhibitory autapse suppressed neuron firing. With excitatory autaptic feedback, the firing of a Class-1 neuron could be switched to that of a Class-2 neuron, and a Class-3 neuron could exhibit a Class-2 firing pattern. The sustained response frequency of the Class-2 neuron transferred from that of Class-3 is only dependent on the autaptic time delay, and the frequency decays gradually with increased the delay time.

Keywords

Autapse Morris–Lecar neuron Firing pattern Time delay 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China with Grant No. 11675008 (YC), Grant No. 21434001 (YC) and Grant No. 11447027 (HTW). HTW acknowledges in addition supports from Natural Science Basic Research Plan in Shaanxi Province of China with Grant No. 2016JQ1037.

Compliance with ethical standards

Conflicts of interest

All the authors declare that there are no any conflict with the publication of this work.

References

  1. 1.
    Abbott, L.F.: Lapicque’s introduction of the integrate-and-fire model neuron (1907). Brain Res. Bull. 50, 303–304 (1999)CrossRefGoogle Scholar
  2. 2.
    Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35(1), 193–213 (1981)CrossRefGoogle Scholar
  3. 3.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952)CrossRefGoogle Scholar
  4. 4.
    Suresh, R., Senthilkumar, D.V., Lakshmanan, M., et al.: Emergence of a common generalized synchronization manifold in network motifs of structurally different time-delay systems. Chaos Solitons Fractals 93, 235–245 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hettiarachchi, I.T., Lakshmanan, S., Bhatti, A., et al.: Chaotic synchronization of time-delay coupled Hindmarsh–Rose neurons via nonlinear control. Nonlinear Dyn. 86, 1249–1262 (2016)CrossRefzbMATHGoogle Scholar
  6. 6.
    Yamakou, E.M., Inack, E.M., Kakmeni, F.M.M.: Ratcheting and energetic aspects of synchronization in coupled bursting neurons. Nonlinear Dyn. 83, 541–554 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kim, S.Y., Lim, W.: Dynamical responses to external stimuli for both cases of excitatory and inhibitory synchronization in a complex neuronal network. Cogn. Neurodyn. 11, 395–413 (2017)CrossRefGoogle Scholar
  8. 8.
    Panaggio, M.J., Abrams, D.M.: Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28(3), R67 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hizanidis, J., Kanas, V.G., Bezerianos, A., et al.: Chimera states in networks of nonlocally coupled Hindmarsh–Rose neuron models. Int. J. Bifurc. Chaos 24(03), 1450030 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Vogels, T.P., Abbott, L.F.: Gating multiple signals through detailed balance of excitation and inhibition in spiking networks. Nat. Neurosci. 12(4), 483 (2009)CrossRefGoogle Scholar
  11. 11.
    Wang, L.F., Jia, F., Liu, X.Z., et al.: Temperature effects on information capacity and energy efficiency of Hodgkin–Huxley neuron. Chin. Phys. Lett. 32(10), 108701 (2015)CrossRefGoogle Scholar
  12. 12.
    Ju, H., Hines, M.L., Yu, Y.: Cable energy function of cortical axons. Sci. Rep. 6, 29686 (2016)CrossRefGoogle Scholar
  13. 13.
    Christoph, J., Chebbok, M., Richter, C., et al.: Electromechanical vortex filaments during cardiac fibrillation. Nature 555(7698), 667 (2018)CrossRefGoogle Scholar
  14. 14.
    Jalife, J.: The tornadoes of sudden cardiac arrest. Nature 555, 597–598 (2018)CrossRefGoogle Scholar
  15. 15.
    Tateno, T., Harsch, A., Robinson, H.P.C.: Threshold firing frequency–current relationships of neurons in rat somatosensory cortex: type 1 and type 2 dynamics. J. Neurophysiol. 92(4), 2283–2294 (2004)CrossRefGoogle Scholar
  16. 16.
    Yu, L.C., Ma, J., Zhang, G.Y., et al.: Suppression of spiral waves by voltage clamp techniques in a conductance-based cardiac tissue model. Chin. Phys. Lett. 25(7), 2706 (2008)CrossRefGoogle Scholar
  17. 17.
    Wu, X.Y., Ma, J.: The formation mechanism of defects, spiral wave in the network of neurons. PLoS ONE 8(1), e55403 (2013)CrossRefGoogle Scholar
  18. 18.
    Zhang, J., Tang, J., Ma, J., et al.: The dynamics of spiral tip adjacent to inhomogeneity in cardiac tissue. Physica A 491, 340–346 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Prescott, S.A., De Koninck, Y., Sejnowski, T.J.: Biophysical basis for three distinct dynamical mechanisms of action potential initiation. PLoS Comput. Biol. 4(10), e1000198 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Huang, C., Sun, W., Zheng, Z., et al.: Hopf bifurcation control of the M–L neuron model with type I. Nonlinear Dyn. 87(2), 755–766 (2017)CrossRefGoogle Scholar
  21. 21.
    Hodgkin, A.L.: The local electric changes associated with repetitive action in a non-medullated axon. J. Physiol. 107(2), 165–181 (1948)CrossRefGoogle Scholar
  22. 22.
    Izhikevich, E.M.: Dynamical Systems in Neuroscience. The MIT Press, Cambridge, MA (2007)Google Scholar
  23. 23.
    Tateno, T., Robinson, H.P.C.: Rate coding and spike-time variability in cortical neurons with two types of threshold dynamics. J. Neurophysiol. 95(4), 2650–2663 (2006)CrossRefGoogle Scholar
  24. 24.
    Sadeghi, S., Valizadeh, A.: Synchronization of delayed coupled neurons in presence of inhomogeneity. J. Comput. Neurosci. 36(1), 55–66 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Esfahani, Z.G., Valizadeh, A.: Zero-lag synchronization despite inhomogeneities in a relay system. PLoS ONE 9(12), e112688 (2014)CrossRefGoogle Scholar
  26. 26.
    Wang, H.T., Wang, L.F., Yu, L.C., et al.: Response of Morris–Lecar neurons to various stimuli. Phys. Rev. E 83(2), 021915 (2011)CrossRefGoogle Scholar
  27. 27.
    Wang, H.T., Chen, Y., Chen, Y.: First-spike latency in Hodgkin’s three classes of neurons. J. Theor. Biol. 328, 19–25 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wang, C.N., Guo, S.L., Xu, Y., et al.: Formation of autapse connected to neuron and its biological function. Complexity 2017, 5436737 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Xu, Y., Ying, H.P., Jia, Y., et al.: Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci. Rep. 7, 43452 (2017)CrossRefGoogle Scholar
  30. 30.
    Qin, H.X., Ma, J., Wang, C.N., et al.: Autapse-induced target wave, spiral wave in regular network of neurons. Sci. China Phys. Mech. Astron. 57(10), 1918–1926 (2014)CrossRefGoogle Scholar
  31. 31.
    Ma, J., Song, X.L., Tang, J., et al.: Wave emitting and propagation induced by autapse in a forward feedback neuronal network. Neurocomputing 167, 378–389 (2015)CrossRefGoogle Scholar
  32. 32.
    Wang, L., Wang, H., Yu, L., et al.: Role of axonal sodium-channel band in neuronal excitability. Phys. Rev. E 85(5), 052901 (2011)CrossRefGoogle Scholar
  33. 33.
    Kole, M.H.P., Stuart, G.J.: Signal processing in the axon initial segment. Neuron 73(2), 235–247 (2012)CrossRefGoogle Scholar
  34. 34.
    Van der Loos, H.: Neuronal circuitry and its development. Perspect. Brain Res. 45, 259–278 (1976)CrossRefGoogle Scholar
  35. 35.
    Yin, L., Zheng, R., Ke, W., et al.: Autapses enhance bursting and coincidence detection in neocortical pyramidal cells. Nat. Commun. 9(1), 4890 (2018)CrossRefGoogle Scholar
  36. 36.
    Song, S., Miller, K.D., Abbott, L.F.: Competitive Hebbian learning through spike-timing-dependent synaptic plasticity. Nat. Neurosci. 3(9), 919 (2000)CrossRefGoogle Scholar
  37. 37.
    Song, X., Wang, H., Chen, Y.: Coherence resonance in an autaptic Hodgkin–Huxley neuron with time delay. Nonlinear Dyn. 94(1), 141 (2018)CrossRefGoogle Scholar
  38. 38.
    Wang, H.T., Wang, L.F., Chen, Y., et al.: Effect of autaptic activity on the response of a Hodgkin–Huxley neuron. Chaos 24(3), 033122 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yilmaz, E., Ozer, M., Baysal, V., et al.: Autapse-induced multiple coherence resonance in single neurons and neuronal networks. Sci. Rep. 6, 30914 (2016)CrossRefGoogle Scholar
  40. 40.
    Yilmaz, E., Ozer, M.: Delayed feedback and detection of weak periodic signals in a stochastic Hodgkin–Huxley neuron. Physica A 421, 455–462 (2015)CrossRefGoogle Scholar
  41. 41.
    Wang, H.T., Chen, Y.: Response of autaptic Hodgkin–Huxley neuron with noise to subthreshold sinusoidal signals. Physica A 462, 321–329 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wang, H.T., Sun, Y.J., Li, Y.C., et al.: Influence of autapse on mode-locking structure of a Hodgkin–Huxley neuron under sinusoidal stimulus. J. Theor. Biol. 358, 25–30 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Wang, H.T., Ma, J., Chen, Y., et al.: Effect of an autapse on the firing pattern transition in a bursting neuron. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3242–3254 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wang, H.T., Chen, Y.: Firing dynamics of an autaptic neuron. Chin. Phys. B 24(12), 128709 (2015)CrossRefGoogle Scholar
  45. 45.
    Guo, D., Chen, M., Perc, M., et al.: Firing regulation of fast-spiking interneurons by autaptic inhibition. EPL (Europhys. Lett.) 114(3), 30001 (2016)CrossRefGoogle Scholar
  46. 46.
    Zhao, Z., Jia, B., Gu, H.G.: Bifurcations and enhancement of neuronal firing induced by negative feedback. Nonlinear Dyn. 86(3), 1549–1560 (2016)CrossRefGoogle Scholar
  47. 47.
    Zhao, Z., Gu, H.G.: Transitions between classes of neuronal excitability and bifurcations induced by autapse. Sci. Rep. 7(1), 6760 (2017)CrossRefGoogle Scholar
  48. 48.
    Connelly, W.M., Lees, G.: Modulation and function of the autaptic connections of layer V fast spiking interneurons in the rat neocortex. J. Physiol. 588, 2047–2063 (2009)CrossRefGoogle Scholar
  49. 49.
    Ermentrout, G.B., Terman, D.H.: Mathematical Foundations of Neuroscience. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  50. 50.
    Sterratt, D., Graham, B., Gillies, A., et al.: Principles of Computational Modelling in Neuroscience. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  51. 51.
    Smeal, R.M., Ermentrout, G.B., White, J.A.: Phase-response curves and synchronized neural networks. Philos. Trans. R. Soc. B 365(1551), 2407–2422 (2010)CrossRefGoogle Scholar
  52. 52.
    Wang, L.F., Wang, H.T., Yu, L.C., et al.: Spike-threshold variability originated from separatrix-crossing in neuronal dynamics. Sci. Rep. 6, 31719 (2016)CrossRefGoogle Scholar
  53. 53.
    Wu, S.D., Zhang, Y.S., Cui, Y., et al.: Heterogeneity of synaptic input connectivity regulates spike-based neuronal avalanches. Neural Netw. 110, 95–103 (2019).  https://doi.org/10.1016/j.neunet.2018.10.017 CrossRefGoogle Scholar
  54. 54.
    Guo, D.Q., Wu, S.D., Chen, M.M., et al.: Regulation of irregular neuronal firing by autaptic transmission. Sci. Rep. 6, 2609 (2016)Google Scholar
  55. 55.
    Hashemi, M., Valizadeh, A., Azizi, Y.: Effect of duration of synaptic activity on spike rate of a Hodgkin–Huxley neuron with delayed feedback. Phys. Rev. E 85(2), 021917 (2012)CrossRefGoogle Scholar
  56. 56.
    Kalat, J.W.: Biological Psychology. Cengage Learning, Wadsworth (2009)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center of Soft Matter Physics and Its ApplicationsBeihang UniversityBeijingChina
  2. 2.School of Physics and Nuclear Energy EngineeringBeihang UniversityBeijingChina
  3. 3.School of Physics and Information TechnologyShaanxi Normal UniversityXi’anChina

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