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Altered neuronal excitability in a Hodgkin-Huxley model incorporating channelopathies of the delayed rectifier potassium channel

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Abstract

Channelopathies involving acquired or genetic modifications of the delayed rectifier K+ channel Kv1.1 include phenotypes characterized by enhanced neuronal excitability. Affected Kv1.1 channels exhibit combinations of altered expression, voltage sensitivity, and rates of activation and deactivation. Computational modeling and analysis can reveal the potential of particular channelopathies to alter neuronal excitability. A dynamical systems approach was taken to study the excitability and underlying dynamical structure of the Hodgkin-Huxley (HH) model of neural excitation as properties of the delayed rectifier K+ channel were altered. Bifurcation patterns of the HH model were determined as the amplitude of steady injection current was varied simultaneously with single parameters describing the delayed rectifier rates of activation and deactivation, maximal conductance, and voltage sensitivity. Relatively modest changes in the properties of the delayed rectifier K+ channel analogous to what is described for its channelopathies alter the bifurcation structure of the HH model and profoundly modify excitability of the HH model. Channelopathies associated with Kv1.1 can reduce the threshold for onset of neural activity. These studies also demonstrate how pathological delayed rectifier K+ channels could lead to the observation of the generalized Hopf bifurcation and, perhaps, other variants of the Hopf bifurcation. The observed bifurcation patterns collectively demonstrate that properties of the nominal delayed rectifier in the HH model appear optimized to permit activation of the HH model over the broadest possible range of input currents.

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References

  • Adelman, J. P., Bond, C. T., Pessia, M., & Maylie, J. (1995). Episodic ataxia results from voltage-dependent potassium channels with altered functions. Neuron, 15(6), 1449–1454.

    Article  CAS  Google Scholar 

  • Berecki, G., Verkerk, A. O., van Ginneken, A. C. G., & Wilders, R. (2014). Dynamic clamp as a tool to study the functional effects of individual membrane currents. In M. Martina & S. Taverna (Eds.), Patch-Clamp Methods and Protocols (pp. 309–326). New York: Humana Press.

  • Champneys, A. R., & Kirk, V. (2004). The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities. Physica D: Nonlinear Phenomena, 195(1–2), 77–105.

    Article  Google Scholar 

  • Destexhe, A., Contreras, D., Sejnowski, T. J., & Steriade, M. (1994). A model of spindle rhythmicity in the isolated thalamic reticular nucleus. Journal of Neurophysiology, 72(2), 803–818.

    Article  CAS  Google Scholar 

  • Dhooge, A., Govaerts, W., Kuznetsov, Y. A., Meijer, H. G. E., & Sautois, B. (2008). New features of the software MatCont for bifurcation analysis of dynamical systems. Mathematical and Computer Modelling of Dynamical Systems, 14(2), 147–175.

    Article  Google Scholar 

  • Eunson, L. H., Rea, R., Zuberi, S. M., Youroukos, S., Panayiotopoulos, C. P., Liguori, R., et al. (2000). Clinical, genetic, and expression studies of mutations in the potassium channel gene KCNA1 reveal new phenotypic variability. Annals of Neurology, 48(4), 647–656.

    Article  CAS  Google Scholar 

  • Fukai, H., Doi, S., Nomura, T., & Sato, S. (2000). Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I. Global organization of bistable periodic solutions. Biological Cybernetics, 82(3), 215–222.

    Article  CAS  Google Scholar 

  • Gaspard, P. (1993). Local birth of homoclinic chaos. Physica D: Nonlinear Phenomena, 62(1–4), 94–122.

    Article  Google Scholar 

  • Guckenheimer, J. (1981). On a codimension two bifurcation. In D. A. Rand & L. S. Young (Eds.), Dynamical Systems and Turbulence, Warwick 1980 (vol 898, pp. 99–142, Lecture Notes in Mathematics). Berlin Heidelberg: Springer-Verlag.

  • Guckenheimer, J., & Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences, vol 42). New York: Springer-Verlag.

  • Guckenheimer, J., & Labouriau, I. S. (1993). Bifurcation of the Hodgkin and Huxley equations: A new twist. Bulletin of Mathematical Biology, 55(5), 937–952.

    Article  Google Scholar 

  • Guckenheimer, J., & Myers, M. (1996). Computing Hopf bifurcations II: Three examples from neurophysiology. SIAM Journal on Scientific Computing, 17(6), 1275–1301.

    Article  Google Scholar 

  • Guckenheimer, J., & Oliva, R. A. (2002). Chaos in the Hodgkin-Huxley model. SIAM Journal on Applied Dynamical Systems, 1(1), 105–114.

    Article  Google Scholar 

  • Hodgkin, A. L. (1948). The local electric changes associated with repetitive action in a non-medullated axon. The Journal of Physiology, 107(2), 165–181.

    Article  CAS  Google Scholar 

  • Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117(4), 500–544.

    Article  CAS  Google Scholar 

  • Izhikevich, E. M. (2000). Neural Excitability, Spiking and Bursting. International Journal of Bifurcation and Chaos, 10(06), 1171–1266.

    Article  Google Scholar 

  • Izhikevich, E. M. (2007). Dynamical systems in neuroscience: the geometry of excitability and bursting. Cambridge: MIT Press.

    Google Scholar 

  • Kuznetsov, Y. A. (2013). Elements of applied bifurcation theory. New York.

  • Pinsky, P. F., & Rinzel, J. (1994). Intrinsic and network rhythmogenesis in a reduced traub model for CA3 neurons. Journal of Computational Neuroscience, 1, 39–60.

    Article  CAS  Google Scholar 

  • Rinzel, J., & Ermentrout, G. B. (1998). Analysis of neural excitability and oscillations. In C. Koch & I. Segev (Eds.), Methods in Neuronal Modeling: From Ions to Networks (2nd ed., pp. 251–292). Cambridge: MIT Press.

    Google Scholar 

  • Rinzel, J., & Miller, R. N. (1980). Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations. Mathematical Biosciences, 49(1–2), 27–59.

    Article  Google Scholar 

  • Sharp, A. A., O'Neil, M. B., Abbott, L. F., & Marder, E. (1993a). The dynamic clamp: artificial conductances in biological neurons. Trends in Neurosciences, 16(10), 389–394.

    Article  CAS  Google Scholar 

  • Sharp, A. A., O'Neil, M. B., Abbott, L. F., & Marder, E. (1993b). Dynamic clamp: computer-generated conductances in real neurons. Journal of Neurophysiology, 69(3), 992–995.

    Article  CAS  Google Scholar 

  • Sutachan, J. J., Watanabe, I., Zhu, J., Gottschalk, A., Recio-Pinto, E., & Thornhill, W. B. (2005). Effects of Kv1.1 channel glycosylation on C-type inactivation and simulated action potentials. Brain Research, 1058(1–2), 30–43.

    Article  CAS  Google Scholar 

  • Wang, J., Geng, J., & Fei, X. (2005). Two-parameters Hopf bifurcation in the Hodgkin–Huxley model. Chaos, Solitons & Fractals, 23(3), 973–980.

    Article  Google Scholar 

  • Wang, X.-J., & Buzsáki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. The Journal of Neuroscience, 16(20), 6402–6413.

    Article  CAS  Google Scholar 

  • Watanabe, I., Zhu, J., Sutachan, J. J., Gottschalk, A., Recio-Pinto, E., & Thornhill, W. B. (2007). The glycosylation state of Kv1.2 potassium channels affects trafficking, gating, and simulated action potentials. Brain Research, 1144, 1–18.

    Article  CAS  Google Scholar 

  • Zerr, P., Adelman, J. P., & Maylie, J. (1998a). Characterization of three episodic ataxia mutations in the human Kv1.1 potassium channel. FEBS Letters, 431(3), 461–464.

    Article  CAS  Google Scholar 

  • Zerr, P., Adelman, J. P., & Maylie, J. (1998b). Episodic ataxia mutations in Kv1.1 alter potassium channel function by dominant negative effects or haploinsufficiency. Journal of Neuroscience, 18(8), 2842–2848.

    Article  CAS  Google Scholar 

  • Zuberi, S. M., Eunson, L. H., Spauschus, A., De Silva, R., Tolmie, J., Wood, N. W., et al. (1999). A novel mutation in the human voltage-gated potassium channel gene (Kv1.1) associates with episodic ataxia type 1 and sometimes with partial epilepsy. Brain, 122(5), 817–825.

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful for the assistance of William B. Thornhill, Ph.D. (Department of Biological Sciences, Fordham University, Bronx, NY, USA) in cataloging the representative sample of Kv1.1 channelopathies of Table 1.

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Authors and Affiliations

  1. Zanvyl Krieger School of Arts and Sciences, Johns Hopkins University, Baltimore, MD, USA

    Omar A. Hafez

  2. Department of Anesthesiology and Critical Care Medicine, Johns Hopkins Hospital, Baltimore, MD, USA

    Allan Gottschalk

  3. Department of Neurosurgery, Johns Hopkins Hospital, Baltimore, MD, USA

    Allan Gottschalk

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  1. Omar A. Hafez
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  2. Allan Gottschalk
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Correspondence to Allan Gottschalk.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Action Editor: J. Rinzel

Electronic supplementary material

Model code is provided in the accompanying files ESM_1.nb and ESM_2.txt. The former contains the modified Hodgkin-Huxley model as implemented in Mathematica along with tools to reproduce Fig. 1 and simulate the system given any injection current and delayed rectifier parameterization. ESM_2.txt contains the necessary information to input the model into MATCONT’s graphical user interface for bifurcation analysis (Figs. 2, 3, 4, and 5).

ESM 1

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Hafez, O.A., Gottschalk, A. Altered neuronal excitability in a Hodgkin-Huxley model incorporating channelopathies of the delayed rectifier potassium channel. J Comput Neurosci 48, 377–386 (2020). https://doi.org/10.1007/s10827-020-00766-1

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  • DOI: https://doi.org/10.1007/s10827-020-00766-1

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