Partition of the Hodgkin-Huxley type model parameter space into the regions of qualitatively different solutions

Abstract

We have examined the problem of obtaining relationships between the type of stable solutions of the Hodgkin-Huxley type system, the values of its parameters and a constant applied current (I). As variable parameters of the system the maximal Na+(-gNa), K+(-gK) conductances and shifts (Gm, Gh, Gn) of the voltage-dependences have been chosen. To solve this problem it is sufficient to find points belonging to the boundary, partitioning the parameter space of the system into the regions of the qualitatively different types of stable solutions (steady states and stable periodic oscillations). Almost all over the physiological range of I, a type of stable solution is determined by the type of steady state (stable or unstable). Using this fact, the approximate solution of this problem could be obtained by analyzing the spectrum of eigenvalues of the Jacobian matrix for the linearized system. The families of the plan sections of the boundary have been constructed in the three-parameter spaces (I, -gNa, -gK), (I, Gm, Gh), (I, Gm, Gn).

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 "…”

Neural Excitability, Spiking and Bursting

“…As a result, this bifurcation has been scrutinized numerically and analytically by many researches (see e.g. [Hassard, 1978;Troy, 1978;Rinzel & Miller, 1980;Hassard et al, 1981;Holden et al, 1991;Bedrov et al, 1992]). A remarkable historical fact is that many important neuroscience properties, such as all-or-none response, threshold, and integration, have been introduced or illustrated using classical HodgkinHuxley model despite the fact that the model does not exhibit any of these properties, as we see below.…”

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.…”