Altered neuronal excitability in a Hodgkin-Huxley model incorporating channelopathies of the delayed rectifier potassium channel

Channelopathies involving acquired or genetic modifications of the delayed rectifier K+ channel Kv1.1 include phenotypes characterized by enhanced neuronal excitability. Affected Kv1.1 channels exhibit combinations of altered expression, voltage sensitivity, and rates of activation and deactivation. Computational modeling and analysis can reveal the potential of particular channelopathies to alter neuronal excitability. A dynamical systems approach was taken to study the excitability and underlying dynamical structure of the Hodgkin-Huxley (HH) model of neural excitation as properties of the delayed rectifier K+ channel were altered. Bifurcation patterns of the HH model were determined as the amplitude of steady injection current was varied simultaneously with single parameters describing the delayed rectifier rates of activation and deactivation, maximal conductance, and voltage sensitivity. Relatively modest changes in the properties of the delayed rectifier K+ channel analogous to what is described for its channelopathies alter the bifurcation structure of the HH model and profoundly modify excitability of the HH model. Channelopathies associated with Kv1.1 can reduce the threshold for onset of neural activity. These studies also demonstrate how pathological delayed rectifier K+ channels could lead to the observation of the generalized Hopf bifurcation and, perhaps, other variants of the Hopf bifurcation. The observed bifurcation patterns collectively demonstrate that properties of the nominal delayed rectifier in the HH model appear optimized to permit activation of the HH model over the broadest possible range of input currents.