^{1}

^{1}

^{1}

^{1}

^{1, 2}

^{1}

^{2}

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

Hodgkin-Huxley (HH) equation is created on the foundation of huge experimental data of sodium and potassium channels by Hodgkin and Huxley who are both excellent biology scientists and had long engaged in nerve conduction research. In about 1952, they took squid giant axon as experiment subject and continuously published four papers describing the electrical excitation of this kind of cell [

The work of Hodgkin and Huxley was recognized as excellent achievement and with significant contribution to the development of electrophysiology. It is the basis of the subsequent models of ion channels. Not only was the HH model consistent with the obtained experiment data accurately, but also it could precisely simulate the change of action potential. The model discovered the relationship of transmembrane potential and current and maximum conductance of ions. This made it possible to research the character of ion channel with mathematical methods. In 1960, Professor Nobel who pioneered the cardiac electrophysiology simulations applied HH model to myocardial cell and got the famous Purkinje fiber cell model [

Because of the importance of HH model, the stability has long attracted the researcher’s attention. Hassard et al. were the earlier researchers caring about the bifurcation phenomenon of HH model. And they indicated that bifurcation would occur at the equilibrium points when the external current

Wang et al. [

Bifurcation means qualitative changes in the solution structure of a dynamic system when the parameters vary. From analyzing the bifurcation, we can get the effects of the parameters. Further, changing the corresponding parameters, we could make the solution into an ideal condition. Bifurcation is an important branch in mathematics and applied to much different field [

In the past, for HH model, external current

The rest of the paper is organized as follows. The HH equations are introduced in detail in Section

HH model was proposed on the foundation of ion channels. The electrophysiological activities of a cell are shown in Figure ^{+} flow inward, forming current ^{+}, creating the current

The electrophysiological process and equivalent circuit of neuron.

Electrophysiological activities of cell

Electrical equivalent circuit of HH model

The equations were obtained according to electrical formulas and experimental data, which are shown as follows:

In these equations, ^{2} is membrane capacitance. ^{2}, ^{2}, and ^{2}, which are the ideal experimental data.

Stability is one of a model’s important properties. If the model is stable, it will reach a rest state at last. Otherwise, periodic phenomenon or chaos may appear. To analyze an ordinary differential system, equilibrium points are one of its most important aspects, which may be the final state of the system. Suppose (

We can get the eigenmatrix of (

According to Routh-Hurwitz criterion, if

In this section, we will investigate the influence of ^{2},

The relationship between

From Figure

Applying bifurcation theory and using the method of bisection, we can get one bifurcation point

Substituting

Here, we regard

Figure

The response of

The

The trajectory of

The

The trajectory of

Figures

In this part, we choose

The relationship between

From Figure

Using the method of bisection to calculate the eigenvalues, we can find two bifurcation points

Here,

The response of

The

The trajectory of

The

The trajectory of

The

The trajectory of

Figure

Figures

Figures

Both

The

From Figure

The effects of

In our analysis, when ^{+} current and the transient outward K^{+} current and block the acetylcholine-activated K^{+} current; however, it has no effect on Na^{+} current, L-type Ca^{2+} current, or even inward-rectifier K^{+} current [

Stable states indicate that the electrophysiological activity of cell will get to corresponding resting state at last, while periodic phenomenon looks like response of pathological cell’s action potentials caused by cardiac arrhythmias [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant no. 61173086 and no. 61179009.

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