Abstract
The excitation equation for an excitable membrane dV/dt=F(V) may have multiple equilibria where F(V)=0, and these may be stable or unstable. We demonstrate multiple equilibria in the Hodgkin-Huxley equations when either gK or [Ca2+]0 is lowered in the presence of a hyperpolarising current density. Under these conditions molluscan somata exhibit exotic behaviours-endogenous paroxysmal depolarising shifts and complex multiple spikes reminiscent of the normal complex activity of some mammalian central neurones. Complex discharge waveforms can be an expression of membrane (differential) properties, rather than electrotonic, geometric (partial differential) behaviour.
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References
Adams, D.J., Smith, S.J., Thompson, S.H.: Ionic currents in molluscan soma. Ann. Rev. Neurosci. 3, 141–167 (1980)
Aldrich, R.W., Jr., Getting, P.A., Thompson, S.H.: Mechanism of frequency-dependent broadening of molluscan neurone soma spikes. J. Physiol. 291, 531–544 (1979)
Armstrong, C.M.: Interaction of tetraethylammonium ion derivatives with the potassium ion channel of giant axons. J. Gen. Physiol. 58, 413–443 (1971)
Begenisich, T., Stevens, C.F.: How many conductance states do potassium channels have? Biophys. J. 15, 843–846 (1975)
Brown, R.H.: Membrane surface charge: discrete and uniform modelling. Progr. Biophys. Molec. Biol. 28, 341–370 (1974)
Carpenter, G.A.: A mathematical analysis of excitable membrane phenomena. Progr. Cybern. Syst. Res. 3, 505–514 (1978)
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)
Cohen, H.: Mathematical developments in Hodgkin-Huxley theory and its approximations. Lect. Math. Life Sci. 8, 89–123 (1976)
Conti, F., DeFelice, L.J., Wanke, E.: Potassium and sodium ion current noise in the membrane of the squid. J. Physiol. 248, 45–82 (1975)
Conti, F., Neher, E.: Single channel recordings of K+ currents in squid axons. Nature 285, 140–145 (1980)
Dodge, F.: A study of the ionic permeability changes underlying excitation in myelinated nerve fibres in the frog. Thesis: Rockefeller University. University Microfilms, Ann Arbor, Michigan, 64-7333 (1963)
Fitzhugh, R.: Mathematical models of excitation and propagation in nerve. In: Biological engineering, Schwan, H.P., ed. New York: McGraw-Hill 1969
Frankenheuser, B., Hodgkin, A.L.: The action of calcium on the electrical properties of squid axons. J. Physiol. 137, 218–244 (1957)
Frankenheuser, B., Huxley, A.F.: The action potential in the myelinated nerve fibre of Xenopus laevis as computed on the basis of voltage-clamp data. J. Physiol. 171, 302–315 (1964)
Grantyn, A., Grantyn, R., Holden, A.V., Schierwagen, A.: Double discharges produced in a giant neurone of the pond-snail and superior collicular neurones of the anaesthetised cat. J. Physiol. 324, 61P (1982)
Hassard, B.: Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon. J. Theor. Biol. 71, 401–420 (1978)
Haydon, P.G., Winlow, W., Holden, A.V.: The effects of menthol on central neurones of the pond-snail, Lymnaea stagnalis (L.). Comp. Biochem. Physiol. 73C, 95–100 (1982)
Herman, A., Gorman, A.L.F.: Effects of tetraethylammonium on potassium currents in a molluscan neurone. J. Gen. Physiol. 78, 87–110 (1981)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Holden, A.V.: Membrane current fluctuations and neuronal information processing. Advances in Physiological Science 30, 23–42 (1981)
Holden, A.V.: The mathematics of excitation. In: Biomathematics in 1980, pp. 15–47. Ricciardi, L., Scott, A.C., eds. Amsterdam: North-Holland 1982
Holden, A.V., Knapp, D.G., Yoda, M.: The periodic activity of excitable membranes develops from equilibria via Hopf bifurcations. J. Physiol. 330, 83–84 P (1983)
Holden, A.V., Winlow, W.: Bifurcation of periodic activity from periodic activity in a molluscan neurone. Biol. Cybern. 42, 189–194 (1981)
Holden, A.V., Winlow, W.: Neuronal activity as the behaviour of a differential system. IEEE Trans. SMC (submitted)
Holden, A.V., Winlow, W., Haydon, P.G.: Effects of tetraethylammonium and 4-aminopyridine on the somatic potentials of an identified molluscan neurone. Comp. Biochem. Physiol. 73A, 303–310 (1982a)
Holden, A.V., Winlow, W., Haydon, P.G.: The induction of periodic and chaotic activity in a molluscan neurone. Biol. Cybern 43, 169–173 (1982b)
Holden, A.V., Yoda, M.: Ionic channel density of excitable membranes can act as a bifurcation parameter. Biol. Cybern. 42, 29–38 (1981a)
Holden, A.V., Yoda, M.: The effects of ionic channel density in neuronal function. J. Theor. Neurobiol. 1, 60–81 (1981b)
Holden, A.V., Yoda, M.: Bifurcation theory and autorhythmicity of the excitable membrane of nerve cells. Proc. 2nd World Congress on Mathematics at the Service of Man, Las Palmas pp. 355–360 (1982)
Huxley, A.F.: Ion movements during nerve activity. Ann. N.Y. Acad. Sci. 81, 221–246 (1957)
Koestler, J., Byrne, J.H. (eds.): Molluscan nerve cells: from biophysics to behaviour. Cold Spring Harbor Laboratory (1980)
Llinas, R., Sugimori, M.: Electrophysiological properties of in vitro Purkinje cell somata in mammalian cerebellar slices. J. Physiol. 305, 171–198 (1980)
Rinzel, J.: On repetitive activity in nerve. Fed. Proc. Fed. Am. Soc. Exp. Biol. 37, 2793–2802 (1978)
Thompson, S.H.: Three pharmacologically distinct potassium channels in molluscan neurones. J. Physiol. 265, 465–488 (1977)
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Holden, A.V., Haydon, P.G. & Winlow, W. Multiple equilibria and exotic behaviour in excitable membranes. Biol. Cybern. 46, 167–172 (1983). https://doi.org/10.1007/BF00336798
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DOI: https://doi.org/10.1007/BF00336798