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Partition of the Hodgkin-Huxley type model parameter space into the regions of qualitatively different solutions. (English) Zbl 0745.92004

Summary: 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 \(\hbox{Na}^ +(\bar g_{\hbox{Na}})\), \(\hbox{K}^ +(\bar g_{\hbox{K}})\) 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 \((I,\bar g_{\hbox{Na}},\bar g_{\hbox{K}})\), \((I,Gm,Gh)\), \((I,Gm,Gn)\).

MSC:

92C20 Neural biology
34C99 Qualitative theory for ordinary differential equations
34D99 Stability theory for ordinary differential equations
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