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The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models

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Abstract

It is widely believed, following the work of Connor and Stevens (1971,J. Physiol. Lond. 214, 31–53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (I A). Using a reduction of the classical Hodgkin-Huxley model, we study the effects ofI A on steady firing rate, especially in the near-threshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based genericI A parameters, the model does not fire at arbitrarily low rates, although it can for the more atypicalI A parameters given by Connor and Stevens for the crab axon. When theI A inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases asg A decreases and as inactivation rate decreases. We also study howI A affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.

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References

  • Awiszus, F. 1992. Reduction of a Hodgkin-Huxley-type model for a mammalian neuron at body temperature.Biol. Cybern. 67, 427–432.

    Article  Google Scholar 

  • Baer, S. M., T. Erneux and J. Rinzel. 1989. The slow passage through a Hopf-bifurcation: delay, memory effects and resonance.SIAM J. Appl. Math. 49, 55–71.

    Article  MathSciNet  MATH  Google Scholar 

  • Bargas, J., E. Galarraga and J. Aceves. 1989. An early outward conductance modulates the firing latency and frequency of neostriatal neurons of the rat brain.Exp. Brain Res. 75, 146–156.

    Article  Google Scholar 

  • Bogdanov, R. I. 1975. Versal deformations of a singular point on the plane in the case of zero eigenvalues.Funct. Anal. Appl. 9, 144–145.

    Article  MathSciNet  MATH  Google Scholar 

  • Buchholtz, F., J. Golowasch, I. R. Epstein and E. Marder. 1992. Mathematical model of an identified stomatiogastric ganglion neuron.J. Neurophysiol. 67, 332–340.

    Google Scholar 

  • Byrne, J. H. 1980. Quantitative aspects of ionic conductance mechanisms of contributing to firing patterns of motor cells mediating inking behavior inaplysia california.J. Neurophysiol. 43, 651–668.

    Google Scholar 

  • Connor, J. A. and C. F. Stevens. 1971. Prediction of repetitive firing behavior from voltage-clamp data on an isolated neurone soma.J. Physiol. Lond. 214, 31–53.

    Article  Google Scholar 

  • Connor, J. A., D. Walter and R. McKown. 1977. Neural repetitive firing: modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons.Biophys. J. 18, 81–102.

    Article  Google Scholar 

  • Dekin, M. S. and P. A. Getting. 1987. In vitro characterization of neurons in the ventral part of the nucleus tractus solitarius. III. ionic basis for repetitive firing patterns.J. Neurophysiol. 58, 215–229.

    Google Scholar 

  • Dodge, F. A. 1972. On the transduction of visual, mechanical, and chemical stimuli.Int. J. Neurosc. 3, 5–14.

    Article  Google Scholar 

  • Doedel, E. J. 1981. AUTO: A program for the automatic bifurcation analysis of autonomous systems.Congr. Numer. 30, 265–284.

    MathSciNet  Google Scholar 

  • FitzHugh, R. 1976. Anodal excitation in the Hodgkin-Huxley nerve model.Biophy. J. 16, 209–226.

    Article  Google Scholar 

  • Gerber, B. and E. Jakobsson. 1993. The functional significance of the A-current.Biol. Cybernetics. 70, 109–114.

    Article  Google Scholar 

  • Getting, P. 1983. Mechanisms of pattern generating underlying swimming inTritonia. III. intrinsic and synaptic mechanisms for delayed excitation.J. Neurophysiol. 49, 1036–1050.

    Google Scholar 

  • Golowasch, J. and E. Marder. 1992. Ionic currents of the lateral pyloric neuron of the stomatogastric ganglion of the crab.J. Neurophysiol. 66, 318–331.

    Google Scholar 

  • Graubard, K. and D. K. Hartline. 1991. Voltage clamp analysis of intact stomatogastric neurons.Brain Res. 557, 241–254.

    Article  Google Scholar 

  • Guckenheimer, J., S. Gueron and R. M. Harris-Warrick. 1992. Mapping the dynamics of a bursting neuron.Phil. Trans. Roy. Soc. Lond. Ser. B 341, 345–359.

    Article  Google Scholar 

  • Hassard, B. D. and L. J. Shiau. 1991. Degenerate Hopf bifurcation and isolated periodic solutions of the Hodgkin-Huxley model with varying sodium ion concentration.J. Theor. Biol. 148, 157–173.

    Article  Google Scholar 

  • Hodgkin, A. L. and A. F. Huxley. 1952. A quantitative description of membrane current and its application to conduction in nerve.J. Physiol. Lond. 117, 500–544.

    Article  Google Scholar 

  • Huguenard, J. R. and D. A. McCormick. 1992. Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons.J. Neurophysiol. 68, 1373–1383.

    Google Scholar 

  • Kaczmarek, L. K. and F. Strumwasser. 1984. A voltage-clamp analysis of currents underlying cyclic amp-induced membrane modulation in isolated peptidergic neurons ofAplysia.J. Neurophysiol. 52, 340–349.

    Google Scholar 

  • Kernell, D. and H. Sjoholm. 1973. Repetitive impulse firing: comparison between neurone models based on voltage clamp equations and spinal motoneurones.Acta Physiol. Scand. 87, 40–56.

    Article  Google Scholar 

  • Labouriau, I. S. 1989. Degenerate Hopf bifurcation and nerve impulse. Part II.SIAM J. Math. Anal. 20, 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  • McCormick, D. A. 1991. Functional properties of a slowly inactivating potassium current in guinea pig dorsal lateral geniculate relay neurons.J. Neurophysiol. 66, 1176–1189.

    Google Scholar 

  • Neher, E. 1971. Two fast transient current components during voltage clamp on snail neurons.J. Gen. Physiol. 58, 36–53.

    Article  Google Scholar 

  • Rinzel, J. 1978. On repetitive activity in nerve.Federation Proc. 37, 2793–2802.

    Google Scholar 

  • Rinzel, J. 1985. Excitation dynamics: insights from simplified membrane models.Federation Proc. 44, 2944–2946.

    Google Scholar 

  • Rinzel J. 1987. A formal classification of bursting mechanisms in excitable systems. InMathematical Topics in Population Biology Morphogensis and Neurosciences. Lecture Notes in Biomathematics T. E. Yamaguti and M. Yamaguti (Eds), Vol. 71, pp. 267–281. New York: Springer-Verlag.

    Chapter  Google Scholar 

  • Rinzel, J. and G. B. Ermentrout. 1989. Analysis of neural excitability and oscillations. InMethods in Neuronal Modeling: From Synapses to Networks C. Koch and I. Seger (Eds), pp. 135–169. Cambridge MA: MIT Press.

    Google Scholar 

  • Rogawski, M. A. 1985. The A-current: how ubiquitous a feature of excitable cells is it?Trends Neurosci. 8, 214–219.

    Article  Google Scholar 

  • Rose, R. M. and J. L. Hindmarsh. 1985. A model of a thalamic neuron.Proc. Roy. Soc. Lond. Ser. B 225, 161–193.

    Article  Google Scholar 

  • Shapiro, B. I. and F. K. Lenherr. 1972. Increased modulation and linearity to response to constant current stimulus.Biophys. J. 12, 1145–1158.

    Article  Google Scholar 

  • Storm, J. F. 1988. Temporal integration by a slowly inactivating K+ current in hippocampal neurons.Nature 336, 379–381.

    Article  Google Scholar 

  • Surmeier, D. J., J. Bargas and S. T. Kitai. 1988. Voltage-clamp analysis of a transient potassium current in rat neostriatal neurons.Brain Res. 473, 187–192.

    Article  Google Scholar 

  • Surmeier, D. J., J. Bargus, and S. T. Kitai. 1989. Two types of A-current differing in voltage-dependence are expressed by neurons of the rat neostriatum.Neurosci. Lett. 103, 331–337.

    Article  Google Scholar 

  • Takens, F. 1974. Singularities of vector fields.Publ. Math. IHES 43, 47–100.

    Article  MathSciNet  Google Scholar 

  • Thompson, S. 1977. Three pharmacologically distinct potassium channels in molluscan neurones.J. Physiol. 265, 465–488.

    Article  Google Scholar 

  • Thompson, S. 1982. Aminopyridine block of transient potassium current.J. Gen. Physiol. 80, 1–18.

    Article  Google Scholar 

  • Wang, X. J. 1993. Ionic basis for intrinsic 40 Hz neuronal oscillations.NeuroReport 5, 221–224.

    Article  MATH  Google Scholar 

  • Williams, J. T., R. A. North, S. A. Shefner, S. Nishi and T. M. Egan. 1984. Membrane properties of rat locus coeruleus neurons.Neuroscience 13, 137–156.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, California State University, 93311, Bakersfield, California, USA

    Maureen E. Rush

  2. Mathematical Research Branch, NIDDK, National Institutes of Health, BSA Bldg., Suite 350, 20892, Bethesda, Maryland, USA

    John Rinzel

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  1. Maureen E. Rush
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Rush, M.E., Rinzel, J. The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models. Bltn Mathcal Biology 57, 899–929 (1995). https://doi.org/10.1007/BF02458299

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