Abstract
We consider the problem of the existence of a negative slope region (NSR) in the voltage-current curve of the neuronal membrane and the relationship between this phenomenon and the membrane parameters. For the Hodgkin-Huxley model it is proposed to determine the dependence of the number of NSR on the values of the maximal sodium (gNa) and potassium (gK) conductances. The method is suggested for constructing the boundaries on the (gNa, gK) plane, in passing through which the number of NSR changes to 1. Using the method we partition the (gNa, gK) plane into the regions corresponding to the curves with the different number of NSR. This number can be changed from 0 to 2 in changing the values of gNa and gK over the physiologically possible range.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aihara K, Matsumoto G (1983) Two stable steady states in the Hodgkin-Huxley axons. Biophys J 41:87–89
Bruce JW, Giblin PG (1984) Curves and singularities. A geometrical introduction to singularity theory. Cambridge University Press, Cambridge, UK
Fields RD, Guthrie PB, Kater SB (1990) Calcium homeostatic capacity is regulated by patterned activity in the growth cones of mouse DRG neurons. Soc Neurosci Abstr 16:475
Guckenheimer J, Gueron S, Harris-Warrick R (1993) Mapping, the dynamics of bursting neuron. Philos Trans R Soc Lond [Biol] 341:345–359
Guthrie PB, Mattson M, Mills LR, Kater SB (1988) Calcium homeostasis, in molluscan and mammalian neurons: neuron-selective set-point of calcium rest concentrations. Soc Neurosci Abstr 14:582
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol (Lond) 117:500–544
Holden AV, Muhamad MA, Schierwagen AK (1985) Repolarizing currents and periodic activity in nerve membrane. J Theor Neurobiol 4:61–71
Ivanizky GP, Krinsky VI, Selkov EE (1978) Mathematical biophysics of a cell. Nauka, Moscow
Mills LR (1991) Neuron-specific and state-specific differences in calcium regulation. Ann NY Acad Sci 639:312–324
Moore JW (1959) Excitation of the squid giant axon membrane in potassium chloride. Nature 183:265–266
Plant RE (1981) Bifurcation and resonance in a model for bursting nerve cells. J Math Biol 11:15–32
Rinzel J, Lee YS (1987) Dissection of a model for neuronal parabolic bursting. J Math Biol 25:653–675
Smith TG, Barker JL, Gainer H (1975) Requirements for bursting pacemaker potential activity in molluscan neurones. Nature 253:450–452
Stampfli R (1958) Die Strom-Spannungscharakteristik der erregbaren Membrane eines einzelnen Schnurrings und ihre Abhangigkeit von der Ionenkonzentration. Helv Physiol Acta 16:127–146
Tasaki I, Hagivara S (1957) Demonstration of two stable potential states in the squid giant axon under tetraethylammonium chloride. J Gen Physiol 40:859–885
Wilson WA, Wachtel H (1974) Negative resistance characteristic essential for the maintenance of slow oscillations in bursting neurons. Science 186:932–934
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bedrov, Y.A., Akoev, G.N. & Dick, O.E. On the relationship between the number of negative slope regions in the voltage-current curve of the Hodgkin-Huxley model and its parameter values. Biol. Cybern. 73, 149–154 (1995). https://doi.org/10.1007/BF00204053
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00204053