Advertisement

Nonlinear Dynamics

, Volume 97, Issue 4, pp 2091–2105 | Cite as

Bifurcations of enhanced neuronal bursting activities induced by the negative current mediated by inhibitory autapse

  • Yuye Li
  • Huaguang GuEmail author
  • Xueli Ding
Original paper
  • 83 Downloads

Abstract

In contrast to the traditional viewpoint that inhibitory effect can often induce the reduction of neural firing activities, negative self-feedback current mediated by inhibitory autapse is identified to enhance neuronal bursting activities in the Chay model composed of fast and slow subsystems. With increasing autaptic strength of the autapse, bursting patterns exhibit period-adding bifurcations, which leads to the increase in the mean firing frequency. Such a phenomenon can be well interpreted with fast–slow variable dissection and bifurcation analysis to the bursting patterns. The initial and termination phases of the burst correspond to a saddle-node bifurcation and a saddle-homoclinic (SH) bifurcation of the fast subsystem, respectively. With increasing the autaptic strength, the termination phase of the burst delays and the initial phase remains nearly unchanged. Therefore, the width of the burst becomes wider to contain more spikes, which leads to the period-adding bifurcations and the increases of mean firing frequency. The detailed relationship between the delay of termination phase (SH point) and the characteristic of the inhibitory autaptic current is identified. Moreover, the distinction between the present paper and the previous investigations of the enhancement of bursting activities induced by inhibitory current with time delay is discussed. Based on the results that inhibitory autapse without time delay can enhance bursting activities via bifurcation mechanism, a novel case of nonlinear dynamics in contrast to the traditional viewpoint of the inhibitory effect, a possible function of the autapse, a novel potential measure to modulate bursting activities are present.

Keywords

Bifurcation Neural firing Inhibitory effect Autapse Fast–slow variable dissection 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Braun, H.A., Wissing, H., Schäfer, K., Hirsch, M.C.: Oscillation and noise determine signal transduction in shark multimodal sensory cells. Nature 367(6460), 270–273 (1994)Google Scholar
  2. 2.
    Jia, B., Gu, H.G., Xue, L.: A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space. Cognit. Neurodyn. 11(2), 189–200 (2017)Google Scholar
  3. 3.
    Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurcat. Chaos 10(06), 1171–1266 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Terman, D.: The transition from bursting to continuous spiking in excitable membrane models. J. Nonlinear Sci. 2(2), 135–182 (1992)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Glass, L.: Synchronization and rhythmic processes in physiology. Nature 410(6825), 277–284 (2001)Google Scholar
  6. 6.
    Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dyn. 89(3), 1569–1578 (2017)MathSciNetGoogle Scholar
  7. 7.
    Yao, C.G., Ma, J., He, Z.W., Qian, Y., Liu, L.P.: Transmission and detection of biharmonic envelope signal in a feed-forward multilayer neural network. Physica A 523, 797–806 (2019)MathSciNetGoogle Scholar
  8. 8.
    Gu, H.G.: Different bifurcation scenarios of neural firing patterns observed in the biological experiment on identical pacemakers. Int. J. Bifurcat. Chaos 23(12), 1350195 (2013)MathSciNetGoogle Scholar
  9. 9.
    Gu, H.G., Pan, B.B., Chen, G.R., Duan, L.X.: Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn. 78(1), 391–407 (2014)MathSciNetGoogle Scholar
  10. 10.
    Gu, H.G.: Experimental observation of transition from chaotic bursting to chaotic spiking in a neural pacemaker. Chaos 23(2), 023126 (2013)Google Scholar
  11. 11.
    Dodla, R., Rinzel, J.: Enhanced neuronal response induced by fast inhibition. Phys. Rev. E 73(1), 010903 (2006)Google Scholar
  12. 12.
    Beiderbeck, B., Myoga, M.H., Müller, N., Callan, A.R., Friauf, E., Grothe, B., Pecka, M.: Precisely timed inhibition facilitates action potential firing for spatial coding in the auditory brainstem. Nat. Commun. 9(1), 1771 (2018)Google Scholar
  13. 13.
    Zhao, Z.G., Jia, B., Gu, H.G.: Bifurcations and enhancement of neuronal firing induced by negative feedback. Nonlinear Dyn. 86(3), 1549–1560 (2016)Google Scholar
  14. 14.
    Jia, B.: Negative feedback mediated by fast inhibitory autapse enhances neuronal oscillations near a Hopf bifurcation point. Int. J. Bifurcat. Chaos 28(2), 1850030 (2018)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Duan, L.X., Liu, J., Chen, X., Xiao, P.C., Zhao, Y.: Dynamics of in-phase and anti-phase bursting in the coupled pre-Bötzinger complex cells. Cognit. Neurodyn. 11(1), 91–97 (2017)Google Scholar
  16. 16.
    Mondal, A., Upadhyay, R.K., Ma, J., Yadav, B.K., Sharma, S.K., Mondal, A.: Bifurcation analysis and diverse firing activities of a modified excitable neuron model. Cognit. Neurodyn. (2019).  https://doi.org/10.1007/s11571-019-09526-z Google Scholar
  17. 17.
    Grace, A.A., Bunney, B.S., Moore, H., Todd, C.L.: Dopamine-cell depolarization block as a model for the therapeutic actions of antipsychotic drugs. Trends Neurosci. 20(1), 31–37 (1997)Google Scholar
  18. 18.
    Valenti, O., Cifelli, P., Gill, K.M., Grace, A.A.: Antipsychotic drugs rapidly induce dopamine neuron depolarization block in a developmental rat model of schizophrenia. J. Neurosci. 31(34), 12330–12338 (2011)Google Scholar
  19. 19.
    Bang, S., Lee, B.J., Lee, S.R., Na, S., Jang, J.M., Kang, M., Kim, S., Min, D., Song, J.M., Ho, W., Jeon, N.: Reliable autapse formation using the single-cell patterning method. Biofabrication 11(1), 015008 (2018)Google Scholar
  20. 20.
    Saada, R., Miller, N., Hurwitz, I., Susswein, A.J.: Autaptic excitation elicits persistent activity and a plateau potential in a neuron of known behavioral function. Curr. Biol. 19(6), 479–684 (2009)Google Scholar
  21. 21.
    Bacci, A., Huguenard, J.R.: Enhancement of spike-timing precision by autaptic transmission in neocortical inhibitory interneurons. Neuron 49(1), 119–130 (2006)Google Scholar
  22. 22.
    Jiang, M., Zhu, J., Liu, Y.P., Yang, M.P., Tian, C.P., Jiang, S., Wang, Y., Guo, H., Wang, K., Shu, Y.: Enhancement of asynchronous release from fast-spiking interneuron in human and rat epileptic neocortex. PLoS Biol. 10(5), e1001324 (2012)Google Scholar
  23. 23.
    Bacci, A., Huguenard, J.R., Prince, D.A.: Functional autaptic neurotransmission in fast-spiking interneurons: a novel form of feedback inhibition in the neocortex. J. Neurosci. 23(3), 859–866 (2003)Google Scholar
  24. 24.
    Loos, H.V.D., Glaser, E.M.: Autapses in neocortex cerebri: synapses between a pyramidal cell’s axon and its own dendrites. Brain Res. 48(12), 355–360 (1972)Google Scholar
  25. 25.
    Cobb, S.R., Halasy, K., Vida, I., Nyiri, G., Tamas, G., Buhl, E.H., Somogyi, P.: Synaptic effects of identified interneurons innervating both interneurons and pyramidal cells in the rat hippocampus. Neuroscience 79(3), 629–648 (1997)Google Scholar
  26. 26.
    Pouzat, C., Marty, A.: Autaptic inhibitory currents recorded from interneurones in rat cerebellar slices. J. Physiol. 509(3), 777–783 (1998)Google Scholar
  27. 27.
    Tamás, G., Buhl, E.H., Somogyi, P.: Massive autaptic self-innervation of GABAergic neurons in cat visual cortex. J. Neurosci. 17(16), 6352–6364 (1997)Google Scholar
  28. 28.
    Yin, L.P., Zheng, R., Ke, W., He, Q.S., Zhang, Y., Li, J.L., Wang, B., Mi, Z., Long, Y.S., Rasch, M.J., Li, T.F., Luan, G.M., Shu, Y.S.: Autapses enhance bursting and coincidence detection in neocortical pyramidal cells. Nat. Commun. 9(1), 4890 (2018)Google Scholar
  29. 29.
    Song, X.L., Wang, H.T., Chen, Y.: Autapse-induced firing patterns transitions in the Morris-Lecar neuron model. Nonlinear Dyn. (2019).  https://doi.org/10.1007/s11071-019-04925-7 Google Scholar
  30. 30.
    Cao, B., Guan, L.N., Gu, H.G.: 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
  31. 31.
    Wang, H.T., Wang, L.F., Chen, Y.L., Chen, Y.: Effect of autaptic activity on the response of a Hodgkin-Huxley neuron. Chaos 24(3), 033122 (2014)MathSciNetGoogle Scholar
  32. 32.
    Wang, H.T., Ma, J., Chen, Y.L., Chen, Y.: Effect of an autapse on the firing pattern transition in a bursting neuron. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3242–3254 (2014)MathSciNetGoogle Scholar
  33. 33.
    Guo, D.Q., Chen, M.M., Perc, M., Wu, S.D., Xia, C., Zhang, Y.S., Xu, P., Xia, Y., Yao, D.Z.: Firing regulation of fast-spiking interneurons by autaptic inhibition. EPL (Europhys. Lett.) 114(3), 30001 (2016)Google Scholar
  34. 34.
    Zhao, Z.G., Gu, H.G.: Transitions between classes of neuronal excitability and bifurcations induced by autapse. Sci. Rep. 7(1), 6760 (2017)Google Scholar
  35. 35.
    Xu, Y., Ying, H.P., Jia, Y., Ma, J., Hayat, T.: Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci. Rep. 7, 43452 (2017)Google Scholar
  36. 36.
    Qin, H.X., Ma, J., Jin, W.Y., Wang, C.N.: Dynamics of electric activities in neuron and neurons of network induced by autapses. Sci. China Technol. Sci. 57(5), 936–946 (2014)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.
    Qin, H.X., Ma, J., Wang, C.N., Wu, Y.: Autapse-induced spiral wave in network of neurons under noise. PLoS ONE 9(6), e100849 (2014)Google Scholar
  39. 39.
    Wu, Y.N., Gong, Y.B., Wang, Q.: Autaptic activity-induced synchronization transitions in Newman–Watts network of Hodgkin–Huxley neurons. Chaos 25(4), 043113 (2015)MathSciNetzbMATHGoogle Scholar
  40. 40.
    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
  41. 41.
    Yang, X.L., Yu, Y.H., Sun, Z.K.: Autapse-induced multiple stochastic resonances in a modular neuronal network. Chaos 27(8), 083117 (2017)MathSciNetGoogle Scholar
  42. 42.
    Ding, X.L., Li, Y.Y.: Period-adding bifurcation of neural firings induced by inhibitory autapses with time-delay. Acta Phys. Sin. 65(21), 210502 (2016). (in chinese)Google Scholar
  43. 43.
    Gu, H.G., Pan, B.B.: A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model. Nonlinear Dyn. 81(4), 2107–2126 (2015)MathSciNetGoogle Scholar
  44. 44.
    González-Miranda, J.M.: Block structured dynamics and neuronal coding. Phys. Rev. E 72(5), 051922 (2005)MathSciNetGoogle Scholar
  45. 45.
    Gu, H.G., Zhao, Z.G.: Dynamics of time delay-induced multiple synchronous behaviors in inhibitory coupled neurons. PLoS ONE 10(9), e0138593 (2015)Google Scholar
  46. 46.
    Jia, B., Wu, Y.C., He, D., Guo, B.H., Xue, L.: Dynamics of transitions from anti-phase to multiple in-phase synchronizations in inhibitory coupled bursting neurons. Nonlinear Dyn. 93(3), 1599–1618 (2018)Google Scholar
  47. 47.
    Zhao, Z.G., Gu, H.G.: The influence of single neuron dynamics and network topology on time delay-induced multiple synchronous behaviors in inhibitory coupled network. Chaos Soliton Fract. 80, 96–108 (2015)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Elson, R.C., Selverston, A.I., Abarbanel, H.D.I., Rabinovich, M.I.: Inhibitory synchronization of bursting in biological neurons: dependence on synaptic time constant. J. Neurophysiol. 88(3), 1166–1176 (2002)Google Scholar
  49. 49.
    González-Miranda, J.M.: Nonlinear dynamics of the membrane potential of a bursting pacemaker cell. Chaos 22(1), 013123 (2012)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Barrio, R., Shilnikov, A.: Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh–Rose model. J. Math. Neurosci. 1(1), 1–6 (2011)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Lü, Z.S., Chen, L.N., Duan, L.X.: Bifurcation analysis of mixed bursting in the pre-Bötzinger complex. Appl. Math. Model. 67, 234–251 (2019)MathSciNetGoogle Scholar
  52. 52.
    Duan, L.X., Cao, Q.Y., Wang, Z.J., Su, J.W.: Dynamics of neurons in the pre-Bötzinger complex under magnetic flow effect. Nonlinear Dyn. 94(3), 1961–1971 (2018)Google Scholar
  53. 53.
    Chay, T.R.: Chaos in a three-variable model of an excitable cell. Physica D 16(2), 233–242 (1985)zbMATHGoogle Scholar
  54. 54.
    Fan, Y.S., Chay, T.R.: Generation of periodic and chaotic bursting in an excitable cell model. Biol. Cybern. 71(5), 417–431 (1994)zbMATHGoogle Scholar
  55. 55.
    Li, L., Gu, H.G., Liu, Z.Q., Yang, M.H., Ren, W.: A series of bifurcation scenarios in the firing pattern transitions in an experimental neural pacemaker. Int. J. Bifurcat. Chaos 14, 1813–1817 (2004)zbMATHGoogle Scholar
  56. 56.
    Wang, X.J., Rinzel, J.: Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput. 4(1), 84–97 (1992)Google Scholar
  57. 57.
    Wang, H.T., Chen, Y.: Firing dynamics of an autaptic neuron. Chin. Phys. B 24(12), 128709 (2015)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsChifeng UniversityChifengChina
  2. 2.Institute of Applied MathematicsChifeng UniversityChifengChina
  3. 3.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina
  4. 4.Department of Basic EducationFuyang Institute of TechnologyFuyangChina

Personalised recommendations