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

Hodgkin-Huxley (HH) equation is the first cell computing model in the world and pioneered the use of model to study electrophysiological problems. The model consists of four differential equations which are based on the experimental data of ion channels. Maximal conductance is an important characteristic of different channels. In this study, mathematical method is used to investigate the importance of maximal sodium conductance gNA and maximal potassium conductance gK. Applying stability theory, and taking gNA and gK as variables, we analyze the stability and bifurcations of the model. Bifurcations are found when the variables change, and bifurcation points and boundary are also calculated. There is only one bifurcation point when gNA is the variable, while there are two points when gK is variable. The (gNA,  gK) plane is partitioned into two regions and the upper bifurcation boundary is similar to a line when both gNA and gK are variables. Numerical simulations illustrate the validity of the analysis. The results obtained could be helpful in studying relevant diseases caused by maximal conductance anomaly.