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⁺(g_Na),K⁺(g_K) 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 plane sections of the boundary have been constructed in the three-parameter spaces (I, g_Na,g_K), (I, Gm, Gh), (I, Gm, Gn).