We present the equilibrium point bifurcation and singularity analysis of HH model with constraints. We investigate the effect of constraints and parameters on the type of equilibrium point bifurcation. HH model with constraints has more transition sets. The Matcont toolbox software environment was used for analysis of the bifurcation points in conjunction with Matlab. We also illustrate the stability of the equilibrium points.
The HodgkinHuxley nonlinear model (HH) [
The HH comprises the following differential equations:
For the following bifurcation equation:
The mathematical expression with singlesided constraint is
The external current
Constraint transition set.
From Figure
The bifurcation diagram for
In order to show the bifurcation characteristics of
The bifurcation diagram and stability of equilibrium points for
Bifurcation diagram for
Stability of equilibrium points
Beginning from the left side of the abscissa of Figure
The limit cycle emerging from sH at
The limit cycle emerging from sH at
The limit cycle emerging from uH at
The limit cycle emerging from sH at
In Table
Bifurcation analysis results derived by the Matcont software.
Parameter  Equilibrium points 

Type of condition 




uH 


Neutral saddle  


Neutral saddle  


Limit point  


Limit point  


sH 
The bifurcation diagrams and the stability of equilibrium points are obtained in Figure
Bifurcation analysis results derived by the Matcont software.
Parameter  Equilibrium points 

Type of condition 




Limit point 


Neutral saddle  


Neutral saddle  


Limit point  


Neutral saddle  


uH 
The bifurcation diagram and stability of equilibrium points for
Bifurcation diagram for
Stability of equilibrium points
The limit cycle emerging from uH at
From Table
The bifurcation diagrams and the stability of equilibrium points are obtained in Figure
Bifurcation analysis results derived by the Matcont software.
Parameter  Equilibrium points 

Type of condition 




uH 


Neutral saddle  


Limit point  


Limit point  


Neutral saddle  


uH 
The bifurcation diagram and stability of equilibrium for
Bifurcation diagram for
Stability of equilibrium points
The limit cycle emerging from uH at
The limit cycle emerging from uH at
The bifurcation diagrams and the stability of equilibrium points are obtained in Figure
Bifurcation analysis results derived by the Matcont software.
Parameter  Equilibrium points 

Type of condition 




uH 


sH 
The bifurcation diagram and stability of equilibrium point for
Bifurcation diagram for
Stability of equilibrium
The limit cycle emerging from uH at
In this paper we present the equilibrium point bifurcation and singularity analysis of HH model with constraints. We investigate the effect of constraints and parameters on the type of equilibrium point bifurcation. We find that if we restrict
The authors declare that there is no conflict of interests regarding the publication of this paper.
This study is supported by the National Science Foundation of China (Grant no. 11172198) and the National Basic Research Program of China (Grant no. 11372211).