Skip to main content
SpringerLink
Log in

Spike initiating dynamics at axonal afterpotentials: model-based mechanisms of the recovery cycle

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The axons exhibit depolarizing and hyperpolarizing afterpotentials, which result in complicated patterns of recovery cycle and influence the activation of subsequent action potential (AP) with deep brain stimulation (DBS). Our objective is to examine the spike initiation at axonal afterpotentials. We use biophysical models to simulate the afterpotentials and apply two-pulse conditioning-test paradigm to measure the stimulus threshold in recovery cycle. We analyze the phase plane portraits and interactions of ionic currents at spike threshold to determine the spike initiating dynamics associated with the recovery cycle. We show that the afterpotentials alter the net current at voltage threshold of subsequent AP, which results in the changes in spike threshold. The difference between spike threshold and afterpotentials determines the stimulus threshold for evoking subsequent AP, which governs the recovery cycle pattern. Our simulations provide a biophysical basis of the spike initiation at the afterpotentials, which is important for interpreting the activity-dependent modulations of axonal excitability. The predictions should be considered when understanding the frequency-dependent firing patterns in the axon with DBS.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included in this article.

References

  1. Bower, K.L., McIntyre, C.C.: Deep brain stimulation of terminating axons. Brain Stimul. 13(6), 1863–1870 (2020). https://doi.org/10.1016/j.brs.2020.09.001

    Article  Google Scholar 

  2. Lozano, A.M., Lipsman, N., Bergman, H., Brown, P., et al.: Deep brain stimulation: current challenges and future directions. Nat. Rev. Neurol. 15(3), 148–160 (2019). https://doi.org/10.1038/s41582-018-0128-2

    Article  Google Scholar 

  3. Yi, G.S., Wang, J., Wei, X.L., Che, Y.Q.: Energy cost of action potential generation and propagation in thalamocortical relay neurons during deep brain stimulation. IEEE Trans. Biomed. Eng. 66(12), 3457–3471 (2019). https://doi.org/10.1109/TBME.2019.2906114

    Article  Google Scholar 

  4. Kamiya, H.: Excitability tuning of axons by afterdepolarization. Front. Cell. Neurosci. 13, 407 (2019). https://doi.org/10.3389/fncel.2019.00407

    Article  Google Scholar 

  5. Stys, P.K., Waxman, S.G.: Activity-dependent modulation of excitability: implications for axonal physiology and pathophysiology. Muscle Nerve. 17(9), 969–974 (1994). https://doi.org/10.1002/mus.880170902

    Article  Google Scholar 

  6. George, A., Serra, J., Navarro, X., Bostock, H.: Velocity recovery cycles of single C fibres innervating rat skin. J. Physiol. 578(Pt 1), 213–232 (2007). https://doi.org/10.1113/jphysiol.2006.116129

    Article  Google Scholar 

  7. McIntyre, C.C., Grill, W.M.: Extracellular stimulation of central neurons: influence of stimulus waveform and frequency on neuronal output. J. Neurophysiol. 88(4), 1592–1604 (2002). https://doi.org/10.1152/jn.2002.88.4.1592

    Article  Google Scholar 

  8. David, G., Modney, B., Scappaticci, K.A., Barrett, J.N., et al.: Electrical and morphological factors influencing the depolarizing after-potential in rat and lizard myelinated axons. J. Physiol. 489(Pt 1), 141–157 (1995). https://doi.org/10.1113/jphysiol.1995.sp021037

    Article  Google Scholar 

  9. Barrett, E.F., Barrett, J.N.: Intracellular recording from vertebrate myelinated axons: mechanism of the depolarizing afterpotential. J. Physiol. 323, 117–144 (1982). https://doi.org/10.1113/jphysiol.1982.sp014064

    Article  Google Scholar 

  10. Kiernan, M.C., Mogyoros, I.: Burke, D: Differences in the recovery of excitability in sensory and motor axons of human median nerve. Brain 119(Pt 4), 1099–1105 (1996). https://doi.org/10.1093/brain/119.4.1099

    Article  Google Scholar 

  11. Meeks, J.P., Mennerick, S.: Selective effects of potassium elevations on glutamate signaling and action potential conduction in hippocampus. J. Neurosci. 24(1), 197–206 (2004). https://doi.org/10.1523/jneurosci.4845-03.2004

    Article  Google Scholar 

  12. Kocsis, J.D., Malenka, R.C., Waxman, S.G.: Effects of extracellular potassium concentration on the excitability of the parallel fibres of the rat cerebellum. J. Physiol. 334(1), 225–244 (1983). https://doi.org/10.1113/jphysiol.1983.sp014491

    Article  Google Scholar 

  13. Stephanova, D.I., Daskalova, M., Mladenov, M.: Conducting processes in simulated chronic inflammatory demyelinating polyneuropathy at 20°C-42°C. J. Integr. Neurosci. 14(1), 19–30 (2015). https://doi.org/10.1142/S0219635215500065

    Article  Google Scholar 

  14. Chen, Y.J., Chen, Y.C., Chan, P., Lin, C.I., et al.: Temperature regulates the arrhythmogenic activity of pulmonary vein cardiomyocytes. J. Biomed. Sci. 10(5), 535–543 (2003). https://doi.org/10.1007/BF02256115

    Article  Google Scholar 

  15. Bostock, H., Baker, M.: Evidence for two types of potassium channel in human motor axons in vivo. Brain Res. 462(2), 354–358 (1988). https://doi.org/10.1016/0006-8993(88)90564-1

    Article  Google Scholar 

  16. Burke, D., Kiernan, M.C., Bostock, H.: Excitability of human axons. Clin. Neurophysiol. 112(9), 1575–1585 (2001). https://doi.org/10.1016/s1388-2457(01)00595-8

    Article  Google Scholar 

  17. Chiu, S.Y., Ritchie, J.M., Rogart, R.B., Stagg, D.: A quantitative description of membrane currents in rabbit myelinated nerve. J. Physiol. 292, 149–166 (1979). https://doi.org/10.1113/jphysiol.1979.sp012843

    Article  Google Scholar 

  18. Robert, A., Jirounek, P.: Uptake of potassium by nonmyelinating Schwann cells induced by axonal activity. J. Neurophysiol. 72(6), 2570–2579 (1994). https://doi.org/10.1152/jn.1994.72.6.2570

    Article  Google Scholar 

  19. Burke, D., Howells, J., Trevillion, L., McNulty, P.A., et al.: Threshold behaviour of human axons explored using subthreshold perturbations to membrane potential. J. Physiol. 587(2), 491–504 (2009). https://doi.org/10.1113/jphysiol.2008.163170

    Article  Google Scholar 

  20. Tarnaud, T., Joseph, W., Martens, L., Tanghe, E.: Dependence of excitability indices on membrane channel dynamics, myelin impedance, electrode location and stimulus waveforms in myelinated and unmyelinated fibre models. Med. Biol. Eng. Comput. 56(9), 1595–1613 (2018). https://doi.org/10.1007/s11517-018-1799-y

    Article  Google Scholar 

  21. Tai, C.F., de Groat, W.C., Roppolo, J.R.: Simulation analysis of conduction block in unmyelinated axons induced by high-frequency biphasic electrical currents. IEEE Trans. Biomed. Eng. 52(7), 1323–1332 (2005). https://doi.org/10.1109/tbme.2005.847561

    Article  Google Scholar 

  22. Dekker, D.M., Briaire, J.J., Frijns, J.H.: The impact of internodal segmentation in biophysical nerve fiber models. J. Comput. Neurosci. 37(2), 307–315 (2014). https://doi.org/10.1007/s10827-014-0503-y

    Article  MathSciNet  MATH  Google Scholar 

  23. Yi, G.S., Wang, J., Deng, B., Wei, X.L.: Spike initiating dynamics of the neuron with different adaptation mechanisms to extracellular electric fields. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 574–586 (2015). https://doi.org/10.1016/j.cnsns.2014.07.020

    Article  MathSciNet  Google Scholar 

  24. Yi, G.S., Wang, J., Wei, X.L., Tsang, K.M., et al.: Neuronal spike initiation modulated by extracellular electric fields. PLoS One. 9(5), e97481 (2014). https://doi.org/10.1371/journal.pone.0097481

    Article  Google Scholar 

  25. Tonnelier, A.: Threshold curve for the excitability of bidimensional spiking neurons. Phys. Rev. E 90(2), 022701 (2014). https://doi.org/10.1103/PhysRevE.90.022701

    Article  MathSciNet  Google Scholar 

  26. Sekerli, M., Del Negro, C.A., Lee, R.H., Butera, R.J.: Estimating action potential thresholds from neuronal time-series: new metrics and evaluation of methodologies. IEEE Trans. Biomed. Eng. 51(9), 1665–1672 (2004). https://doi.org/10.1109/TBME.2004.827531

    Article  Google Scholar 

  27. 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). https://doi.org/10.1113/jphysiol.1952.sp004764

    Article  Google Scholar 

  28. Liu, H.L., Roppolo, J.R., de Groat, W.C., Tai, C.F.: The role of slow potassium current in nerve conduction block induced by high-frequency biphasic electrical current. IEEE Trans. Biomed. Eng. 56(1), 137–146 (2009). https://doi.org/10.1109/TBME.2008.2006013

    Article  Google Scholar 

  29. Smit, J.E., Hanekom, T., Hanekom, J.J.: Modelled temperature-dependent excitability behaviour of a generalised human peripheral sensory nerve fibre. Biol. Cybern. 101(2), 115–130 (2009). https://doi.org/10.1007/s00422-009-0324-7

    Article  MATH  Google Scholar 

  30. Smit, J.E., Hanekom, T., Wieringen, A., et al.: Threshold predictions of different pulse shapes using a human auditory nerve fibre model containing persistent sodium and slow potassium currents. Hear. Res. 269(1–2), 12–22 (2010). https://doi.org/10.1016/j.heares.2010.08.004

    Article  Google Scholar 

  31. McIntyre, C.C., Richardson, A.G., Grill, W.M.: Modeling the excitability of mammalian nerve fibers: influence of afterpotentials on the recovery cycle. J. Neurophysiol. 87(2), 995–1006 (2002). https://doi.org/10.1152/jn.00353.2001

    Article  Google Scholar 

  32. Tsuboi, R., Mochizuki, T., Ito, H., Kawano, S., et al.: Validation of a lateral flow immunochromatographic assay for tinea unguium diagnosis. J. Dermatol. 48(5), 633–637 (2021). https://doi.org/10.1111/1346-8138.15838

    Article  Google Scholar 

  33. FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17, 257–278 (1955). https://doi.org/10.1007/BF02477753

    Article  Google Scholar 

  34. Wang, L.F., Wang, H.T., Yu, L.C., Chen, Y.: Spike-threshold variability originated from separatrix-crossing in neuronal dynamics. Sci. Rep. 6, 31719 (2016). https://doi.org/10.1038/srep31719

    Article  Google Scholar 

  35. Prescott, S.A., Ratté, S., Koninck, Y.D., Sejnowski, T.J.: Nonlinear interaction between shunting and adaptation controls a switch between integration and coincidence detection in pyramidal neurons. J. Neurosci. 26(36), 9084–9097 (2006). https://doi.org/10.1523/JNEUROSCI.1388-06

    Article  Google Scholar 

  36. Coggan, J.S., Prescott, S.A., Bartol, T.M., Sejnowski, T.J.: Imbalance of ionic conductances contributes to diverse symptoms of demyelination. Proc. Natl. Acad. Sci. USA 107(48), 20602–20609 (2010). https://doi.org/10.1073/pnas.1013798107

    Article  Google Scholar 

  37. Platkiewicz, J., Brette, R.: A threshold equation for action potential initiation. PLoS Comput. Biol. 6(7), e1000850 (2010). https://doi.org/10.1371/journal.pcbi.1000850

    Article  MathSciNet  Google Scholar 

  38. Ermentrout, B.: Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. Pennsylvania (2002)

  39. Hines, M.L., Carnevale, N.T.: The NEURON simulation environment. Neural Comput. 9(6), 1179–1209 (1997). https://doi.org/10.1162/neco.1997.9.6.1179

    Article  Google Scholar 

  40. Kiernan, M.C., Mogyoros, I., Burke, D.: Differences in the recovery of excitability in sensory and motor axons of human median nerve. Brain 119(Pt 4), 1099–1105 (1996). https://doi.org/10.1093/brain/119.4.1099

    Article  Google Scholar 

  41. Gaines, J.L., Finn, K.E., Slopsema, J.P., Heyboer, L.A., et al.: A model of motor and sensory axon activation in the median nerve using surface electrical stimulation. J. Comput. Neurosci. 45(1), 29–43 (2018). https://doi.org/10.1007/s10827-018-0689-5

    Article  Google Scholar 

  42. Bowe, C.M., Kocsis, J.D., Waxman, S.G.: The association of the supernormal period and the depolarizing afterpotential in myelinated frog and rat sciatic nerve. Neuroscience 21(2), 585–593 (1987). https://doi.org/10.1016/0306-4522(87)90144-8

    Article  Google Scholar 

  43. Hedrich, U.B., Liautard, C., Kirschenbaum, D., Pofahl, M., et al.: Impaired action potential initiation in GABAergic interneurons causes hyperexcitable networks in an epileptic mouse model carrying a human Na(V)1.1 mutation. J. Neurosci. 34(45), 14874–14889 (2014). https://doi.org/10.1523/JNEUROSCI.0721-14.2014

    Article  Google Scholar 

  44. Akgul, G., Wollmuth, L.P.: Synapse-associated protein 97 regulates the membrane properties of fast-spiking parvalbumin interneurons in the visual cortex. J. Neurosci. 33(31), 12739–12750 (2013). https://doi.org/10.1523/JNEUROSCI.0040-13.2013

    Article  Google Scholar 

  45. Helm, J., Akgul, G., Wollmuth, L.P.: Subgroups of parvalbumin-expressing interneurons in layers 2/3 of the visual cortex. J. Neurophysiol. 109(6), 1600–1613 (2013). https://doi.org/10.1152/jn.00782.2012

    Article  Google Scholar 

  46. Guan, D.X., Armstrong, W.E., Foehring, R.C.: Electrophysiological properties of genetically identified subtypes of layer 5 neocortical pyramidal neurons: Ca2+ dependence and differential modulation by norepinephrine. J. Neurophysiol. 113(7), 2014–2032 (2015). https://doi.org/10.1152/jn.00524.2014

    Article  Google Scholar 

  47. Ma, C., Shu, Y.S., Zheng, Z., Chen, Y., et al.: Similar electrophysiological changes in axotomized and neighboring intact dorsal root ganglion neurons. J. Neurophysiol. 89(3), 1588–1602 (2003). https://doi.org/10.1152/jn.00855.2002

    Article  Google Scholar 

  48. Kim, C.H., Oh, S.H., Lee, J.H., Chang, S.O., et al.: Lobule-specific membrane excitability of cerebellar Purkinje cells. J. Physiol. 590(2), 273–288 (2012). https://doi.org/10.1113/jphysiol.2011.221846

    Article  Google Scholar 

  49. Jhangiani-Jashanmal, I.T., Yamamoto, R., Gungor, N.Z., Paré, D.: Electroresponsive properties of rat central medial thalamic neurons. J. Neurophysiol. 115(3), 1533–1541 (2016). https://doi.org/10.1152/jn.00982.2015

    Article  Google Scholar 

  50. Farries, M.A., Perkel, D.J.: Electrophysiological properties of avian basal ganglia neurons recorded in vitro. J. Neurophysiol. 84(5), 2502–2513 (2000). https://doi.org/10.1152/jn.2000.84.5.2502

    Article  Google Scholar 

  51. Gonzalez-Burgos, G., Miyamae, T., Pafundo, D.E., Yoshino, H., et al.: Functional maturation of GABA synapses during postnatal development of the monkey dorsolateral prefrontal cortex. Cereb. Cortex 25(11), 4076–4093 (2015). https://doi.org/10.1093/cercor/bhu122

    Article  Google Scholar 

  52. Higgs, M.H., Spain, W.J.: Kv1 channels control spike threshold dynamics and spike timing in cortical pyramidal neurones. J. Physiol. 589(21), 5125–5142 (2011). https://doi.org/10.1113/jphysiol.2011.216721

    Article  Google Scholar 

  53. Lubejko, S.T., Fontaine, B., Soueidan, S.E., MacLeod, K.M.: Spike threshold adaptation diversifies neuronal operating modes in the auditory brain stem. J. Neurophysiol. 122(6), 2576–2590 (2019). https://doi.org/10.1152/jn.00234.2019

    Article  Google Scholar 

  54. Wester, J.C., Contreras, D.: Biophysical mechanism of spike threshold dependence on the rate of rise of the membrane potential by sodium channel inactivation or subthreshold axonal potassium current. J. Comput. Neurosci. 35(1), 1–17 (2013). https://doi.org/10.1007/s10827-012-0436-2

    Article  MathSciNet  Google Scholar 

  55. Shvartsman, A., Kotler, O., Stoler, O., Khrapunsky, Y., et al.: Subcellular distribution of persistent sodium conductance in cortical pyramidal neurons. J. Neurosci. 41(29), 6190–6201 (2021). https://doi.org/10.1523/JNEUROSCI.2989-20.2021

    Article  Google Scholar 

  56. Magistretti, J., Castelli, L., Forti, L., D’Angelo, E.: Kinetic and functional analysis of transient, persistent and resurgent sodium currents in rat cerebellar granule cells in situ: an electrophysiological and modelling study. J. Physiol. 573(Pt 1), 83–106 (2006). https://doi.org/10.1113/jphysiol.2006.106682

    Article  Google Scholar 

  57. Xie, R.G., Zheng, D.W., Xing, J.L., Zhang, X.J., et al.: Blockade of persistent sodium currents contributes to the riluzole-induced inhibition of spontaneous activity and oscillations in injured DRG neurons. PLoS One 6(4), e18681 (2011). https://doi.org/10.1371/journal.pone.0018681

    Article  Google Scholar 

  58. Wu, S.N., Lo, Y.C., Chen, B.S., Cheung So, E., et al.: Contribution of blocked potassium current conductance and increased conductance of persistent sodium current to the afterdischarge in myelinated neuron. Muscle Nerve. 46(2), 297–299 (2012). https://doi.org/10.1002/mus.23400

    Article  Google Scholar 

  59. Baker, M., Bostock, H., Grafe, P., Martius, P.: Function and distribution of three types of rectifying channel in rat spinal root myelinated axons. J. Physiol. 383, 45–67 (1987). https://doi.org/10.1113/jphysiol.1987.sp016395

    Article  Google Scholar 

  60. Stys, P.K., Ashby, P.: An automated technique for measuring the recovery cycle of human nerves. Muscle Nerve. 13(8), 750–758 (1990). https://doi.org/10.1002/mus.880130814

    Article  Google Scholar 

  61. Hodgkin, A.L., Huxley, A.F., Katz, B.: Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J. Physiol. 116(4), 424–448 (1952). https://doi.org/10.1113/jphysiol.1952.sp004716

    Article  Google Scholar 

  62. Hille, B.: Ion channels of excitable membranes. Massachusetts (2001)

  63. Rinberg, A., Taylor, A.L., Marder, E.: The effects of temperature on the stability of a neuronal oscillator. PLoS Comput. Biol. 9(1), e1002857 (2013). https://doi.org/10.1371/journal.pcbi.1002857

    Article  Google Scholar 

  64. Bostock, H., Cikurel, K., Burke, D.: Threshold tracking techniques in the study of human peripheral nerve. Muscle Nerve. 21(2), 137–158 (1998). https://doi.org/10.1002/(sici)1097-4598(199802)21:2%3c137::aid-mus1%3e3.0.co;2-c

    Article  Google Scholar 

  65. Chiken, S., Nambu, A.: High-frequency pallidal stimulation disrupts information flow through the pallidum by GABAergic inhibition. J. Neurosci. 33(6), 2268–2280 (2013). https://doi.org/10.1523/JNEUROSCI.4144-11.2013

    Article  Google Scholar 

  66. Kuncel, A.M., Cooper, S.E., Wolgamuth, B.R., Grill, W.M.: Amplitude- and frequency-dependent changes in neuronal regularity parallel changes in tremor with thalamic deep brain stimulation. IEEE Trans. Neural. Syst. Rehabil. Eng. 15(2), 190–197 (2017). https://doi.org/10.1109/TNSRE.2007.897004

    Article  Google Scholar 

  67. Yi, G.S., Wang, J.: Frequency-dependent energy demand of dendritic responses to deep brain stimulation in thalamic neurons: a model-based study. IEEE Trans. Neural Netw. Learn. Syst. 32(7), 3056–3068 (2021). https://doi.org/10.1109/TNNLS.2020.3009293

    Article  MathSciNet  Google Scholar 

  68. McIntyre, C.C., Grill, W.M., Sherman, D.L., Thakor, N.V.: Cellular effects of deep brain stimulation: modelbased analysis of activation and inhibition. J. Neurophysiol. 91(4), 1457–1469 (2004). https://doi.org/10.1152/jn.00989.2003

    Article  Google Scholar 

  69. Yi, G.S., Grill, W.M.: Frequency-dependent antidromic activation in thalamocortical relay neurons: effects of synaptic inputs. J. Neural. Eng. 15(5), 0556001 (2018). https://doi.org/10.1088/1741-2552/aacbff

    Article  Google Scholar 

Download references

Acknowledgements

This work is funded by grants from the National Natural Science Foundation of China (62071324).

Author information

Authors and Affiliations

  1. School of Electrical and Information Engineering, Tianjin University, Tianjin, 300072, China

    Guosheng Yi & Qiang Zhao

Authors
  1. Guosheng Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guosheng Yi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yi, G., Zhao, Q. Spike initiating dynamics at axonal afterpotentials: model-based mechanisms of the recovery cycle. Nonlinear Dyn 111, 10487–10504 (2023). https://doi.org/10.1007/s11071-023-08362-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08362-5

Keywords

Search

Navigation