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Science China Technological Sciences

, Volume 62, Issue 9, pp 1502–1511 | Cite as

Stochastic dynamics of conduction failure of action potential along nerve fiber with Hopf bifurcation

  • XinJing Zhang
  • HuaGuang GuEmail author
  • LiNan Guan
Article
  • 9 Downloads

Abstract

Action potentials can be induced by external electronic impulsive stimulations applied at one end of the unmyelinated fibers (C-fibers), while some action potentials fail to conduct to the other end of the fiber when the stimulation frequency becomes high. Such a phenomenon is called as conduction failure, which was observed in the biological experiments and related to the painful diabetic neuropathy, inflammation, and trauma in the previous studies. On-off firing pattern was recorded from the fiber when conduction failure happened. In the present study, the diffusion Hodgkin-Huxley (HH) model with resting state near a Hopf bifurcation is adopted to simulate the experimental observations. When the periodic electrical pulses with high frequency are applied to one end of the fiber described by the deterministic HH model, conduction failure and the corresponding firing patterns different from the on-off firing pattern are simulated. When noise is introduced to form the stochastic HH model, the firing pattern corresponding to conduction failure becomes the on-off firing pattern, which is characterised by transition behaviors between on-phase (continuous action potentials) and off-phase (a long quiescent state) and large variations in the durations of both phases. Furthermore, the increase of potassium conductance can enhance the conduction failure degree, which closely matches those observed in the experiment and is suggested to be related to the reduction of pain signals. The results show that noise is an important factor to evoke the on-off firing pattern, reveal the functional capability in the pain signals propagation along C-fiber, and present a possible measure for the treatment of chronic pain.

Keywords

conduction failure Hopf bifurcation periodic pulse stimulus C-fiber 

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References

  1. 1.
    Bucher D, Goaillard J M. Beyond faithful conduction: Short-term dynamics, neuromodulation, and long-term regulation of spike propagation in the axon. Prog NeuroBiol, 2011, 94: 307–346CrossRefGoogle Scholar
  2. 2.
    Debanne D. Information processing in the axon. Nat Rev Neurosci, 2004, 5: 304–316; Pauli W. Die allgemeinen Prinzipien der Wellenmechanik. Berlin, Heidelberg: Springer, 1933CrossRefGoogle Scholar
  3. 3.
    van Beek M, van Kleef M, Linderoth B, et al. Spinal cord stimulation in experimental chronic painful diabetic polyneuropathy: Delayed effect of High-frequency stimulation. Eur J Pain, 2017, 21: 795–803CrossRefGoogle Scholar
  4. 4.
    Zhang Y, Bucher D, Nadim F. Ionic mechanisms underlying history-dependence of conduction delay in an unmyelinated axon. Elife, 2017, 6: e25386Google Scholar
  5. 5.
    Ramón F, Joyner R W, Moore J W. Propagation of action potentials in inhomogeneous axon regions. Fed Proc, 1975, 34: 1357–1363Google Scholar
  6. 6.
    Moore J W, Westerfield M. Action potential propagation and threshold parameters in inhomogeneous regions of squid axons. J Physiol, 1983, 336: 285–300CrossRefGoogle Scholar
  7. 7.
    Lambert L A, Lambert D H, Strichartz G R. Irreversible conduction block in isolated nerve by high concentrations of local anesthetics. Anesthesiology, 1994, 80: 1082–1093CrossRefGoogle Scholar
  8. 8.
    Babbs C F, Shi R. Subtle paranodal injury slows impulse conduction in a mathematical model of myelinated axons. PLoS ONE, 2013, 8: e67767CrossRefGoogle Scholar
  9. 9.
    Zhao S, Yang G, Wang J, et al. Effect of non-symmetric waveform on conduction block induced by high-frequency (kHz) biphasic stimulation in unmyelinated axon. J Comput Neurosci, 2014, 37: 377–386MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hamada M S, Popovic M A, Kole M H P. Loss of saltation and presynaptic action potential failure in demyelinated axons. Front Cell Neurosci, 2017, 11Google Scholar
  11. 11.
    Barron D H, Matthews B H. Intermittent conduction in the spinal cord. J Physiol, 1935, 85: 73–103CrossRefGoogle Scholar
  12. 12.
    Deschenes M, Landry P. Axonal branch diameter and spacing of nodes in the terminal arborization of identified thalamic and cortical neurons. Brain Res, 1980, 191: 538–544CrossRefGoogle Scholar
  13. 13.
    Soleng A F, Chiu K, Raastad M. Unmyelinated axons in the rat hippocampus hyperpolarize and activate an H current when spike frequency exceeds 1 Hz. J Physiol, 2003, 552: 459–470CrossRefGoogle Scholar
  14. 14.
    Meeks J P, Mennerick S. Selective effects of potassium elevations on glutamate signaling and action potential conduction in hippocampus. J Neuroscience, 2004, 24: 197–206CrossRefGoogle Scholar
  15. 15.
    Ducreux C, Reynaud J C, Puizillout J J. Spike conduction properties of T-shaped C neurons in the rabbit nodose ganglion. Pflugers Arch, 1993, 424: 238–244CrossRefGoogle Scholar
  16. 16.
    Zhu Z R, Tang X W, Wang W T, et al. Conduction failures in rabbit saphenous nerve unmyelinated fibers. Neurosignals, 2009, 17: 181–195CrossRefGoogle Scholar
  17. 17.
    Lüscher H R, Shiner J S. Computation of action potential propagation and presynaptic bouton bouton activation in terminal arborizations of different geometries. BioPhys J, 1990, 58: 1389–1399CrossRefGoogle Scholar
  18. 19.
    Sun W, Miao B, Wang X C, et al. Reduced conduction failure of the main axon of polymodal nociceptive C-fibres contributes to painful diabetic neuropathy in rats. Brain, 2012, 135: 359–375CrossRefGoogle Scholar
  19. 20.
    Wang X C,Wang S, Zhang M, et al. α-dendrotoxin-sensitive Kv1 channels contribute to conduction failure of polymodal nociceptive C-fibers from rat coccygeal nerve.. J NeuroPhysiol, 2016, 115: 947–957CrossRefGoogle Scholar
  20. 21.
    Bourque C W. Intraterminal recordings from the rat neurohypophysis in vitro. J Physiol, 1990, 421: 247–262CrossRefGoogle Scholar
  21. 22.
    Jackson M B, Zhang S J. Action potential propagation and propagation block by GABA in rat posterior pituitary nerve terminals. J Physiol, 1995, 483: 597–611CrossRefGoogle Scholar
  22. 23.
    Antic S, Wuskell J P, Loew L, et al. Functional profile of the giant metacerebral neuron of Helix aspersa: Temporal and spatial dynamics of electrical activity in situ. J Physiol, 2000, 527: 55–69CrossRefGoogle Scholar
  23. 24.
    Evans C G, Ludwar B C, Cropper E C. Mechanoafferent neuron with an inexcitable somatic region: Consequences for the regulation of spike propagation and afferent transmission. J NeuroPhysiol, 2007, 97: 3126–3130CrossRefGoogle Scholar
  24. 25.
    Debanne D, Boudkkazi S. New insights in information processing in the axon. In: New Aspects of Axonal Structure and Function. Boston: Springer, 2010. 55–83CrossRefGoogle Scholar
  25. 26.
    Tang J, Zhang J, Ma J, et al. Astrocyte calcium wave induces seizure-like behavior in neuron network. Sci China Tech Sci, 2017, 60: 1011–1018CrossRefGoogle Scholar
  26. 27.
    Gu H G, Chen S G. Potassium-induced bifurcations and chaos of firing patterns observed from biological experiment on a neural pacemaker. Sci China Tech Sci, 2014, 57: 864–871CrossRefGoogle Scholar
  27. 28.
    Sun X J, Shi X. Effects of channel blocks on the spiking regularity in clustered neuronal networks. Sci China Tech Sci, 2014, 57: 879–884CrossRefGoogle Scholar
  28. 29.
    Sun X J, Lei J Z, Perc M, et al. Effects of channel noise on firing coherence of small-world Hodgkin-Huxley neuronal networks. Eur Phys J B, 2011, 79: 61–66CrossRefGoogle Scholar
  29. 30.
    Braun H A, Wissing H, Schäfer K, et al. Oscillation and noise determine signal transduction in shark multimodal sensory cells. Nature, 1994, 367: 270–273CrossRefGoogle Scholar
  30. 31.
    Gu H G, Pan B B. Identification of neural firing patterns, frequency and temporal coding mechanisms in individual aortic baroreceptors. Front Comput Neurosci, 2015, 9: 00108CrossRefGoogle Scholar
  31. 32.
    Gu H G, Zhao Z G, Jia B,et al. Dynamics of on-off neural firing patterns and stochastic effects near a sub-critical Hopf bifurcation. PLoS ONE, 2015, 10: e0121028CrossRefGoogle Scholar
  32. 33.
    Jia B, Gu H G. Dynamics andphysiological roles ofstochastic firing patterns near bifurcation points. Int J Bifurcation Chaos, 2017, 27: 1750113CrossRefzbMATHGoogle Scholar
  33. 34.
    Gu H, Pan 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, 2015, 81: 2107–2126MathSciNetCrossRefGoogle Scholar
  34. 35.
    Paydarfar D, Forger D B, Clay J R. Noisy inputs and the induction of on-off switching behavior in a neuronal pacemaker. J NeuroPhysiol, 2006, 96: 3338–3348CrossRefGoogle Scholar
  35. 36.
    Xing J L, Hu S J, Long K P. Subthreshold membrane potential oscillations of type A neurons in injured DRG. Brain Res, 2001, 901: 128–136CrossRefGoogle Scholar
  36. 37.
    Huang C, Zhao X, Wang X, et al. Disparate delays-induced bifurcations in a fractional-order neural network. J Franklin Institute, 2019, 356: 2825–2846MathSciNetCrossRefzbMATHGoogle Scholar
  37. 38.
    Huang C D, Li H, Cao J D. A novel strategy of bifurcation control for a delayed fractional predator-prey model. Appl Math Comput, 2019, 347: 808–838MathSciNetCrossRefGoogle Scholar
  38. 39.
    Huang C D, Cao J D. Impact of leakage delay on bifurcation in highorder fractional BAM neural networks. Neural Networks, 2018, 98: 223–235CrossRefGoogle Scholar
  39. 40.
    Yanagita T, Nishiura Y, Kobayashi R. Signal propagation and failure in one-dimensional FitzHugh-Nagumo equations with periodic stimuli. Phys Rev E, 2005, 71: 036226CrossRefGoogle Scholar
  40. 41.
    Guo S L, Wang C N, Ma J, et al. Transmission of blocked electric pulses in a cable neuron model by using an electric field. Neurocomputing, 2016, 216: 627–637CrossRefGoogle Scholar
  41. 42.
    Maia P D, Kutz J N. Identifying critical regions for spike propagation in axon segments. J Comput Neurosci, 2014, 36: 141–155MathSciNetCrossRefGoogle Scholar
  42. 43.
    Ma J, Tang J. A review for dynamics in neuron and neuronal network. Nonlinear Dyn, 2017, 89: 1569–1578MathSciNetCrossRefGoogle Scholar
  43. 44.
    Ma J, Tang J. A review for dynamics of collective behaviors of network of neurons. Sci China Tech Sci, 2015, 58: 2038–2045CrossRefGoogle Scholar
  44. 45.
    Ma J, Song X, Tang J, et al. Wave emitting and propagation induced by autapse in a forward feedback neuronal network. Neurocomputing, 2015, 167: 378–389CrossRefGoogle Scholar
  45. 46.
    Zhang X, Roppolo J R, de Groat WC, et al. Simulation analysis of conduction block in myelinated axons induced by highfrequency biphasic rectangular pulses. IEEE Trans Biomed Eng, 2006, 53: 1433–1436CrossRefGoogle Scholar
  46. 47.
    Tai C F, de Groat W C, Roppolo J R. Simulation analysis of conduction block in unmyelinated axons induced by high-frequency biphasic rectangular pulses. IEEE Trans Biomed Eng, 2005, 52: 1323–1332CrossRefGoogle Scholar
  47. 48.
    Tai C F, Wang J C, Roppolo J R, et al. Relationship between temperature and stimulation frequency in conduction block of amphibian myelinated axon. J Comput Neurosci, 2009, 26: 331–338CrossRefGoogle Scholar
  48. 49.
    George S, Foster J M, Richardson G. Modelling in vivo action potential propagation along a giant axon. J Math Biol, 2015, 70: 237–263MathSciNetCrossRefzbMATHGoogle Scholar
  49. 50.
    Feng Z Y, Wang Z X, Guo Z S, et al. High frequency stimulation of afferent fibers generates asynchronous firing in the downstream neurons in hippocampus through partial block of axonal conduction. Brain Res, 2017, 1661: 67–78CrossRefGoogle Scholar
  50. 51.
    Hodgkin A L, Huxley A F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 1952, 117: 500–544CrossRefGoogle Scholar
  51. 52.
    Hassard B. Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon. J Theor Biol, 1978, 71: 401–420MathSciNetCrossRefGoogle Scholar
  52. 53.
    Ermentrout G B, Galán R F, Urban N N. Reliability, synchrony and noise. Trends Neurosci, 2008, 31: 428–434CrossRefGoogle Scholar
  53. 54.
    Wang Y, Ma J, Xu Y, et al. The electrical activity of neurons subject to electromagnetic induction and gaussian white noise. Int J Bifurcation Chaos, 2017, 27: 1750030MathSciNetCrossRefzbMATHGoogle Scholar
  54. 55.
    Thomas J W. Numerical Partial Differential Equations (Finite Difference Methods). New York: Springer, 1995CrossRefzbMATHGoogle Scholar
  55. 56.
    Wang X C, Wang S, Wang WT, et al. A novel intrinsic analgesic mechanism: the enhancement of the conduction failure along polymodal nociceptive C-fibers.. PAIN, 2016, 157: 2235–2247CrossRefGoogle Scholar
  56. 57.
    Tuckwell H C, Jost J. Weak noise in neurons may powerfully inhibit the generation of repetitive spiking but not its propagation. PLoS Comput Biol, 2010, 6: e1000794MathSciNetCrossRefGoogle Scholar
  57. 58.
    Tuckwell H C, Jost J. The effects of various spatial distributions of weak noise on rhythmic spiking. J Comput Neurosci, 2011, 30: 361–371MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina

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