On the relationship between the number of negative slope regions in the voltage-current curve of the Hodgkin-Huxley model and its parameter values

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.

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I. Global organization of bistable periodic solutions

“…Diverse behaviors of the HH, depending on the parameter values, have been studied for a long time (Hassard 1978;Rinzel 1978;Troy 1978;Bedrov et al 1992Bedrov et al , 1995Guckenheimer and Labouriau 1993). The dierence between our study and these studies is that we changed many parameters over wide ranges and investigated the global structure of bifurcations integrating several studies on the HH.…”

“…Guckenheimer and Labouriau (1993) reported the detailed bifurcation diagrams in the parameter plane I ext -V K , where the HH possess three equilibrium points. Bedrov et al (1992Bedrov et al ( , 1995 changed the parameter values representing the maximal conductances of Na and K currents (" g Na and "…”

Two-parameters Hopf bifurcation in the Hodgkin–Huxley model

“…Rinzel and Miller computed the stable and unstable solutions of Hodgkin-Huxley model with membrane injection current I ext and the influence of temperature on the bifurcation point [7]. Bedrov studied the possible bifurcation with sodium conductance g Na and potassium conductance g K [10,11]. Current research on bifurcation in Hodgkin-Huxley model focuses on both the dynamical behaviors in different regions of physiological parameter space, and applications of singularity theory by means of numerical methods.…”

Effects of Maximal Sodium and Potassium Conductance on the Stability of Hodgkin-Huxley Model

“…Guckenheimer and Labouriau investigated the influence of i ext and potassium ion potential V K on the bifurcations of the model [18]. Bedrov et al gave the relationship between the numbers of negative slope regions and presented some results about the possible bifurcation giving rise to maximal sodium conductance ${\stackrel{-}{g}}_{\text{Na}}$ and maximal potassium conductance ${\stackrel{-}{g}}_{\text{K}}$ [19, 20]. Fukai and his fellows examined the structure of the model's bifurcations produced by i ext and one of the other parameters [21].…”