# Equilibrium Point Bifurcation and Singularity Analysis of HH Model with Constraint

Authors: Xinhe Zhu; Zhiqiang Wu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

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.


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## Body

## 1. Introduction

The Hodgkin-Huxley nonlinear model (HH) [1] is one of the biggest challenges in the life science in the near history. HH quantitatively describes the electrical excitations of squid giant axon. Under the HH formalism, many mathematical models (HH-type) for diverse neurons are established [2–5]. A bifurcation is a qualitative change in the behavior of a nonlinear dynamical system as its parameters pass through critical values [6]. The study of bifurcations in neural models is important to understand the dynamical origin of many neurons and the organization of behavior. Many studies have been done on the bifurcation analysis of HH model. Guckenheimer and Labouriau [7] give the detailed bifurcation diagrams of HH model in two-parameter space of I and VK. Bedrov reveals the possible bifurcations with changes of gNa and gK, representing the maximal conductance of sodium and potassium, respectively [8, 9]. The global structure of bifurcations in multiple-parameter space of the HH model is examined [10], and the details of the degenerate Hopf bifurcations are analyzed using the singularity theoretic approach [11]. Singularity theory offers an extremely useful approach to bifurcation problems [12]. The aim of this paper is to illustrate how constraints and parameters affect the dynamics of HH model. In the first attempt we choose I as bifurcation parameter, gK, gL as unfolding parameters, and we restrict V>0; then we use the singularity theory of bifurcations and the computing method of bifurcations with constraint to obtain the new constraint transition sets. Secondly, using the above results, we investigate the effect of constraint and parameters on the type of equilibrium point bifurcation, and we also illustrate the stability of the equilibrium points.


## 2. Hodgkin-Huxley Equations

The HH comprises the following differential equations:(1)dVdt=1CM[I-gNam3h(V-VNa)-gKn4(V-VK)-gl(V-Vl)],dmdt=αm(1-m)-βmm,dhdt=αh(1-h)-βhh,dndt=αn(1-n)-βnn.V represents the membrane potential. 0≤m≤1 and 0≤h≤1 are the gating variables representing activation and inactivation of the Na+ current, respectively. 0≤n≤1 is the gating variable representing activation of the K+ current. αm, βm, αh, βh, αn, βn are the function of V as follows:
(2)αm=2.5-0.1V[exp⁡(2.5-0.1V)-1],βm=4exp⁡(-V18.0),αh=0.07exp⁡(-V20),βh=1[exp⁡(3-0.1V)+1],αn=(0.1-0.01V)[exp⁡⁡(1-0.1V)-1],βn=0.125exp⁡(-V80).
The HH includes the following parameters: VK=-12.0 mV, VNa=115.0 mV, and Vl=10.599 mV representing the equilibrium potentials of K+, Na+, and leak currents, respectively. They are determined uniquely by the Nernst equation. gNa=120.0 mS/cm2, gK=36.0 mS/cm2, gl=0.3 mS/cm2 represent the maximum conductance of the corresponding ionic currents. Cm=1.0 μF/cm2 is the membrane capacitance. I represents the external current, in μA/cm2.


## 3. Constrained Bifurcation Theory

For the following bifurcation equation:(3)g(u,λ;α)=0,
where u, λ, α are state variable, bifurcation parameter, and auxiliary parameter (or unfolding parameter), respectively. The bifurcation equation can deal with the singularity theories developed by Golubitsky and Schaeffer [12]. However, in some case, the variation of the state variable is often subjected to restriction, here called constraint. The forms of constraints are different in different problems, of which the most popular single-sided constraint is listed here [13].The mathematical expression with single-sided constraint is(4)g(u,λ;α)=0,δ[u-U]≥0.
The following are transition sets for single-sided constraint:
(5)(B):{g(u,λ;α)=0,gu(u,λ;α)=0,gλ(u,λ;α)=0,δ[u-U]≥0;(BI):{g(U,λ;α)=0,gλ(U,λ;α)=0;(H):{g(u,λ;α)=0,gu(u,λ;α)=0,guu(u,λ;α)=0,δ[u-U]≥0;(HI):{g(U,λ;α)=0,gu(U,λ;α)=0,(DL):{g(ui,λ;α)=0,gu(ui,λ;α)=0,u1≠u2,δ[ui-U]≥0;(DLI):{g(U,λ;α)=0,g(u,λ;α)=gu(u,λ;α)=0,u≠U,δ[u-U]≥0,
where B, H, and DL are nonconstrained bifurcation point set, hysteresis point set, and double limit point set, respectively, and BI, HI, and DLI are constrained bifurcation point set, hysteresis point set, and double limit point set, respectively. Compared with nonconstrained bifurcation g(u,λ;α)=0, there exist new transiton sets which are BI, HI, and DLI. For the restriction, there come new bifurcation types, which give the system more bifurcation properties and can explain some nonlinear aspects in engineering systems and other nonlinear systems.


## 4. Singularity and Bifurcation Analysis Results

### 4.1. Constrained Transition Set

The external currentI is chosen as bifurcation parameter and gK, gL as unfolding parameters, and we restrict V>0. It is impossible to give the HH model’s analytic solutions. So, we use the singularity theory of bifurcations and the computing method of bifurcations with constraint in Section 3 to numerically construct the bifurcation diagrams; constraint transition sets are obtained in Figure 1.Figure 1
Constraint transition set.From Figure1 we can conclude that the transition sets without constraint contain only hysteresis set which divides the parameter-plane into two regions, where there are two bifurcation modes. However, the transition sets with constraint contain hysteresis set and double limit set which divide the parameter-plane into four regions, where there are four bifurcation modes. The bifurcation diagram corresponding to four different gL, gK variations taken from the above four parameters regions is obtained in Figure 2.The bifurcation diagram forgL, gK variation.
(a)
g
L
=
0.1, gK=2
(b)
g
L
=
1, gK=2
(c)
g
L
=
2, gK=4
(d)
g
L
=
2, gK=20


### 4.2. Bifurcation Analysis Results

#### 4.2.1.gL=2, gK=4

In order to show the bifurcation characteristics ofHH, it is convenient to show the bifurcation diagrams obtained by the Matcont software for the varying values of gL, gK. These are given in Figure 3(a). Using the results in Figure 3(a), the stability of equilibrium points is obtained in Figure 3(b). The solid curve denotes the equilibrium points are stable, while the dashed curve denotes the equilibrium points are unstable. Contrasting Figure 3(a) with Figure 2(c), they are identical; then it proves the validity of computing. This can be considered as a verification of the Matcont algorithms for a high order nonlinear system.The bifurcation diagram and stability of equilibrium points forgL=2,  gK=4.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsBeginning from the left side of the abscissa of Figure3(a), the first label H denotes that the equilibrium point is a Hopf bifurcation point with V=7.609434, m=0.123894, h=0.331941, n=0.437840, I=-11.231605, and first Lyapunov coefficient is positive, and there are two eigenvalues with Reλ1,2≈0, Im⁡λ1≠0; then at the Hopf bifurcation point, HH is unstable, and there is an unstable limit cycle, so it is the subcritical Hopf bifurcation (uH). Although the equilibrium points of the second and fifth are labelled as H, they are not Hopf bifurcation points. They are neutral saddle point, where the former has V=9.4260170, m=0.149260, h=0.278301, n=0.466512, I=-10.010747, and the latter has V=25.688959, m=0.518771, h=0.046889, n=0.686086, I=-6.575754. The equilibrium points labelled as LP of the third and fourth are both limit points, where the former has V=11.796299, m=0.188048, h=0.217790, n=0.503195, I=-9.438630, and the latter has V=20.373201, m=0.378788, h=0.083800, n=0.623808, I=-12.559365. The equilibrium point labelled as H of the sixth is a Hopf bifurcation point with V=31.816299, m=0.668810, h=0.025491, n=0.745429, I=20.428518, and first Lyapunov coefficient is negative, and there are two eigenvalues with Reλ1,2≈0; then at the Hopf bifurcation point, HH is stable, and there is a stable limit cycle, so it is the supercritical Hopf bifurcation (sH). Equilibrium points between the first H and the sixth H are unstable.The limit cycle emerging from sH atI=20.428518 is in Figure 4.Figure 4
The limit cycle emerging from sH atI=20.428518.The limit cycle emerging from uH atI=-11.231605 is in Figure 5.Figure 5
The limit cycle emerging from sH atI=-11.231605.In Table1, the bifurcation points found by the Matcont software are presented for gL=2, gK=4.Table 1
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
2 gK=4
V
=
7.609434 m=0.123894 h=0.331941 n=0.437840
I
=
-
11.231605
uH
V
=
9.426017 m=0.149260 h=0.278301 n=0.466512
I
=
-
10.010747
Neutral saddle
V
=
25.688959 m=0.518771 h=0.046889 n=0.686086
I
=
-
6.575754
Neutral saddle
V
=
11.796299 m=0.188048 h=0.217790 n=0.503195
I
=
-
9.438630
Limit point
V
=
20.373201 m=0.378788 h=0.083800 n=0.623808
I
=
-
12.559365
Limit point
V
=
31.816299 m=0.668810 h=0.025491 n=0.745429
I
=
20.428518
sH


#### 4.2.2.gL=0.1,gK=2

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure6. The limit cycle emerging from uH at I=-26.281425 is in Figure 7. The bifurcation points found by the Matcont software are presented in Table 2 for gL=0.1, gK=2.Table 2
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
0.1 gK=2
V
=
-
3.692856 m=0.033952 h=0.716772 n=0.262913
I
=
-
1.749367
Limit point
V
=
-
0.754488 m=0.048408 h=0.622236 n=0.306181
I
=
-
1.918161
Neutral saddle
V
=
-
1.822633 m=0.042603 h=0.658089 n=0.290154
I
=
-
1.811258
Neutral saddle
V
=
27.252348 m=0.559277 h=0.039851 n=0.702462
I
=
-
52.626190
Limit point
V
=
30.900097 m=0.648151 h=0.027784 n=0.737363
I
=
-
48.954977
Neutral saddle
V
=
37.692334 m=0.781785 h=0.015320 n=0.791063
I
=
-
26.281425
uHThe bifurcation diagram and stability of equilibrium points forgL=0.1, gK=2.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsFigure 7
The limit cycle emerging from uH atI=-26.281425.From Table2, we can find the difference in equilibrium point.


#### 4.2.3.gL=1, gK=2

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure8. The limit cycles emerging from uH at I=-13.971904 and I=-9.406580 are given in Figures 9 and 10, respectively. The bifurcation points found by the Matcont software are presented in Table 3 for gL=1, gK=2.Table 3
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
1 gK=2
V
=
4.315751 m=0.086823 h=0.442071 n=0.385365
I
=
-
9.406580
uH
V
=
5.346944 m=0.097269 h=0.406190 n=0.401801
I
=
-
9.266520
Neutral saddle
V
=
5.583327 m=0.099807 h=0.398116 n=0.405572
I
=
-
9.261244
Limit point
V
=
25.615559 m=0.516847 h=0.047253 n=0.685295
I
=
-
38.368717
Limit point
V
=
29.221394 m=0.608443 h=0.032689 n=0.721865
I
=
-
34.782328
Neutral saddle
V
=
35.263043 m=0.739314 h=0.018740 n=0.773422
I
=
-
13.971904
uHThe bifurcation diagram and stability of equilibrium forgL=1, gK=2.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsFigure 9
The limit cycle emerging from uH atI=-13.971904.Figure 10
The limit cycle emerging from uH atI=-9.406580.


#### 4.2.4.gL=2, gK=20

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure11. The limit cycle emerging from uH at I=7.131765 and sH at I=115.224276 is given in Figure 12. They both are the same. The bifurcation points found by the Matcont software are presented in Table 4 for gL=2, gK=20.Table 4
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
2 gK=20
V
=
9.688168 m=0.153228 h=0.271065 n=0.470616
I
=
7.131765
uH
V
=
24.842866 m=0.496500 h=0.051294 n=0.676858
I
=
115.224276
sHThe bifurcation diagram and stability of equilibrium point forgL=2, gK=20.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibriumFigure 12
The limit cycle emerging from uH atI=7.131765 and sH at I=115.224276.


## 4.1. Constrained Transition Set

The external currentI is chosen as bifurcation parameter and gK, gL as unfolding parameters, and we restrict V>0. It is impossible to give the HH model’s analytic solutions. So, we use the singularity theory of bifurcations and the computing method of bifurcations with constraint in Section 3 to numerically construct the bifurcation diagrams; constraint transition sets are obtained in Figure 1.Figure 1
Constraint transition set.From Figure1 we can conclude that the transition sets without constraint contain only hysteresis set which divides the parameter-plane into two regions, where there are two bifurcation modes. However, the transition sets with constraint contain hysteresis set and double limit set which divide the parameter-plane into four regions, where there are four bifurcation modes. The bifurcation diagram corresponding to four different gL, gK variations taken from the above four parameters regions is obtained in Figure 2.The bifurcation diagram forgL, gK variation.
(a)
g
L
=
0.1, gK=2
(b)
g
L
=
1, gK=2
(c)
g
L
=
2, gK=4
(d)
g
L
=
2, gK=20


## 4.2. Bifurcation Analysis Results

### 4.2.1.gL=2, gK=4

In order to show the bifurcation characteristics ofHH, it is convenient to show the bifurcation diagrams obtained by the Matcont software for the varying values of gL, gK. These are given in Figure 3(a). Using the results in Figure 3(a), the stability of equilibrium points is obtained in Figure 3(b). The solid curve denotes the equilibrium points are stable, while the dashed curve denotes the equilibrium points are unstable. Contrasting Figure 3(a) with Figure 2(c), they are identical; then it proves the validity of computing. This can be considered as a verification of the Matcont algorithms for a high order nonlinear system.The bifurcation diagram and stability of equilibrium points forgL=2,  gK=4.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsBeginning from the left side of the abscissa of Figure3(a), the first label H denotes that the equilibrium point is a Hopf bifurcation point with V=7.609434, m=0.123894, h=0.331941, n=0.437840, I=-11.231605, and first Lyapunov coefficient is positive, and there are two eigenvalues with Reλ1,2≈0, Im⁡λ1≠0; then at the Hopf bifurcation point, HH is unstable, and there is an unstable limit cycle, so it is the subcritical Hopf bifurcation (uH). Although the equilibrium points of the second and fifth are labelled as H, they are not Hopf bifurcation points. They are neutral saddle point, where the former has V=9.4260170, m=0.149260, h=0.278301, n=0.466512, I=-10.010747, and the latter has V=25.688959, m=0.518771, h=0.046889, n=0.686086, I=-6.575754. The equilibrium points labelled as LP of the third and fourth are both limit points, where the former has V=11.796299, m=0.188048, h=0.217790, n=0.503195, I=-9.438630, and the latter has V=20.373201, m=0.378788, h=0.083800, n=0.623808, I=-12.559365. The equilibrium point labelled as H of the sixth is a Hopf bifurcation point with V=31.816299, m=0.668810, h=0.025491, n=0.745429, I=20.428518, and first Lyapunov coefficient is negative, and there are two eigenvalues with Reλ1,2≈0; then at the Hopf bifurcation point, HH is stable, and there is a stable limit cycle, so it is the supercritical Hopf bifurcation (sH). Equilibrium points between the first H and the sixth H are unstable.The limit cycle emerging from sH atI=20.428518 is in Figure 4.Figure 4
The limit cycle emerging from sH atI=20.428518.The limit cycle emerging from uH atI=-11.231605 is in Figure 5.Figure 5
The limit cycle emerging from sH atI=-11.231605.In Table1, the bifurcation points found by the Matcont software are presented for gL=2, gK=4.Table 1
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
2 gK=4
V
=
7.609434 m=0.123894 h=0.331941 n=0.437840
I
=
-
11.231605
uH
V
=
9.426017 m=0.149260 h=0.278301 n=0.466512
I
=
-
10.010747
Neutral saddle
V
=
25.688959 m=0.518771 h=0.046889 n=0.686086
I
=
-
6.575754
Neutral saddle
V
=
11.796299 m=0.188048 h=0.217790 n=0.503195
I
=
-
9.438630
Limit point
V
=
20.373201 m=0.378788 h=0.083800 n=0.623808
I
=
-
12.559365
Limit point
V
=
31.816299 m=0.668810 h=0.025491 n=0.745429
I
=
20.428518
sH


### 4.2.2.gL=0.1,gK=2

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure6. The limit cycle emerging from uH at I=-26.281425 is in Figure 7. The bifurcation points found by the Matcont software are presented in Table 2 for gL=0.1, gK=2.Table 2
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
0.1 gK=2
V
=
-
3.692856 m=0.033952 h=0.716772 n=0.262913
I
=
-
1.749367
Limit point
V
=
-
0.754488 m=0.048408 h=0.622236 n=0.306181
I
=
-
1.918161
Neutral saddle
V
=
-
1.822633 m=0.042603 h=0.658089 n=0.290154
I
=
-
1.811258
Neutral saddle
V
=
27.252348 m=0.559277 h=0.039851 n=0.702462
I
=
-
52.626190
Limit point
V
=
30.900097 m=0.648151 h=0.027784 n=0.737363
I
=
-
48.954977
Neutral saddle
V
=
37.692334 m=0.781785 h=0.015320 n=0.791063
I
=
-
26.281425
uHThe bifurcation diagram and stability of equilibrium points forgL=0.1, gK=2.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsFigure 7
The limit cycle emerging from uH atI=-26.281425.From Table2, we can find the difference in equilibrium point.


### 4.2.3.gL=1, gK=2

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure8. The limit cycles emerging from uH at I=-13.971904 and I=-9.406580 are given in Figures 9 and 10, respectively. The bifurcation points found by the Matcont software are presented in Table 3 for gL=1, gK=2.Table 3
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
1 gK=2
V
=
4.315751 m=0.086823 h=0.442071 n=0.385365
I
=
-
9.406580
uH
V
=
5.346944 m=0.097269 h=0.406190 n=0.401801
I
=
-
9.266520
Neutral saddle
V
=
5.583327 m=0.099807 h=0.398116 n=0.405572
I
=
-
9.261244
Limit point
V
=
25.615559 m=0.516847 h=0.047253 n=0.685295
I
=
-
38.368717
Limit point
V
=
29.221394 m=0.608443 h=0.032689 n=0.721865
I
=
-
34.782328
Neutral saddle
V
=
35.263043 m=0.739314 h=0.018740 n=0.773422
I
=
-
13.971904
uHThe bifurcation diagram and stability of equilibrium forgL=1, gK=2.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsFigure 9
The limit cycle emerging from uH atI=-13.971904.Figure 10
The limit cycle emerging from uH atI=-9.406580.


### 4.2.4.gL=2, gK=20

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure11. The limit cycle emerging from uH at I=7.131765 and sH at I=115.224276 is given in Figure 12. They both are the same. The bifurcation points found by the Matcont software are presented in Table 4 for gL=2, gK=20.Table 4
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
2 gK=20
V
=
9.688168 m=0.153228 h=0.271065 n=0.470616
I
=
7.131765
uH
V
=
24.842866 m=0.496500 h=0.051294 n=0.676858
I
=
115.224276
sHThe bifurcation diagram and stability of equilibrium point forgL=2, gK=20.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibriumFigure 12
The limit cycle emerging from uH atI=7.131765 and sH at I=115.224276.


## 4.2.1.gL=2, gK=4

In order to show the bifurcation characteristics ofHH, it is convenient to show the bifurcation diagrams obtained by the Matcont software for the varying values of gL, gK. These are given in Figure 3(a). Using the results in Figure 3(a), the stability of equilibrium points is obtained in Figure 3(b). The solid curve denotes the equilibrium points are stable, while the dashed curve denotes the equilibrium points are unstable. Contrasting Figure 3(a) with Figure 2(c), they are identical; then it proves the validity of computing. This can be considered as a verification of the Matcont algorithms for a high order nonlinear system.The bifurcation diagram and stability of equilibrium points forgL=2,  gK=4.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsBeginning from the left side of the abscissa of Figure3(a), the first label H denotes that the equilibrium point is a Hopf bifurcation point with V=7.609434, m=0.123894, h=0.331941, n=0.437840, I=-11.231605, and first Lyapunov coefficient is positive, and there are two eigenvalues with Reλ1,2≈0, Im⁡λ1≠0; then at the Hopf bifurcation point, HH is unstable, and there is an unstable limit cycle, so it is the subcritical Hopf bifurcation (uH). Although the equilibrium points of the second and fifth are labelled as H, they are not Hopf bifurcation points. They are neutral saddle point, where the former has V=9.4260170, m=0.149260, h=0.278301, n=0.466512, I=-10.010747, and the latter has V=25.688959, m=0.518771, h=0.046889, n=0.686086, I=-6.575754. The equilibrium points labelled as LP of the third and fourth are both limit points, where the former has V=11.796299, m=0.188048, h=0.217790, n=0.503195, I=-9.438630, and the latter has V=20.373201, m=0.378788, h=0.083800, n=0.623808, I=-12.559365. The equilibrium point labelled as H of the sixth is a Hopf bifurcation point with V=31.816299, m=0.668810, h=0.025491, n=0.745429, I=20.428518, and first Lyapunov coefficient is negative, and there are two eigenvalues with Reλ1,2≈0; then at the Hopf bifurcation point, HH is stable, and there is a stable limit cycle, so it is the supercritical Hopf bifurcation (sH). Equilibrium points between the first H and the sixth H are unstable.The limit cycle emerging from sH atI=20.428518 is in Figure 4.Figure 4
The limit cycle emerging from sH atI=20.428518.The limit cycle emerging from uH atI=-11.231605 is in Figure 5.Figure 5
The limit cycle emerging from sH atI=-11.231605.In Table1, the bifurcation points found by the Matcont software are presented for gL=2, gK=4.Table 1
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
2 gK=4
V
=
7.609434 m=0.123894 h=0.331941 n=0.437840
I
=
-
11.231605
uH
V
=
9.426017 m=0.149260 h=0.278301 n=0.466512
I
=
-
10.010747
Neutral saddle
V
=
25.688959 m=0.518771 h=0.046889 n=0.686086
I
=
-
6.575754
Neutral saddle
V
=
11.796299 m=0.188048 h=0.217790 n=0.503195
I
=
-
9.438630
Limit point
V
=
20.373201 m=0.378788 h=0.083800 n=0.623808
I
=
-
12.559365
Limit point
V
=
31.816299 m=0.668810 h=0.025491 n=0.745429
I
=
20.428518
sH


## 4.2.2.gL=0.1,gK=2

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure6. The limit cycle emerging from uH at I=-26.281425 is in Figure 7. The bifurcation points found by the Matcont software are presented in Table 2 for gL=0.1, gK=2.Table 2
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
0.1 gK=2
V
=
-
3.692856 m=0.033952 h=0.716772 n=0.262913
I
=
-
1.749367
Limit point
V
=
-
0.754488 m=0.048408 h=0.622236 n=0.306181
I
=
-
1.918161
Neutral saddle
V
=
-
1.822633 m=0.042603 h=0.658089 n=0.290154
I
=
-
1.811258
Neutral saddle
V
=
27.252348 m=0.559277 h=0.039851 n=0.702462
I
=
-
52.626190
Limit point
V
=
30.900097 m=0.648151 h=0.027784 n=0.737363
I
=
-
48.954977
Neutral saddle
V
=
37.692334 m=0.781785 h=0.015320 n=0.791063
I
=
-
26.281425
uHThe bifurcation diagram and stability of equilibrium points forgL=0.1, gK=2.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsFigure 7
The limit cycle emerging from uH atI=-26.281425.From Table2, we can find the difference in equilibrium point.


## 4.2.3.gL=1, gK=2

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure8. The limit cycles emerging from uH at I=-13.971904 and I=-9.406580 are given in Figures 9 and 10, respectively. The bifurcation points found by the Matcont software are presented in Table 3 for gL=1, gK=2.Table 3
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
1 gK=2
V
=
4.315751 m=0.086823 h=0.442071 n=0.385365
I
=
-
9.406580
uH
V
=
5.346944 m=0.097269 h=0.406190 n=0.401801
I
=
-
9.266520
Neutral saddle
V
=
5.583327 m=0.099807 h=0.398116 n=0.405572
I
=
-
9.261244
Limit point
V
=
25.615559 m=0.516847 h=0.047253 n=0.685295
I
=
-
38.368717
Limit point
V
=
29.221394 m=0.608443 h=0.032689 n=0.721865
I
=
-
34.782328
Neutral saddle
V
=
35.263043 m=0.739314 h=0.018740 n=0.773422
I
=
-
13.971904
uHThe bifurcation diagram and stability of equilibrium forgL=1, gK=2.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibrium pointsFigure 9
The limit cycle emerging from uH atI=-13.971904.Figure 10
The limit cycle emerging from uH atI=-9.406580.


## 4.2.4.gL=2, gK=20

The bifurcation diagrams and the stability of equilibrium points are obtained in Figure11. The limit cycle emerging from uH at I=7.131765 and sH at I=115.224276 is given in Figure 12. They both are the same. The bifurcation points found by the Matcont software are presented in Table 4 for gL=2, gK=20.Table 4
Bifurcation analysis results derived by the Matcont software.
Parameter
Equilibrium points
I
Type of condition
g
L
=
2 gK=20
V
=
9.688168 m=0.153228 h=0.271065 n=0.470616
I
=
7.131765
uH
V
=
24.842866 m=0.496500 h=0.051294 n=0.676858
I
=
115.224276
sHThe bifurcation diagram and stability of equilibrium point forgL=2, gK=20.
(a)
Bifurcation diagram forI variation
(b)
Stability of equilibriumFigure 12
The limit cycle emerging from uH atI=7.131765 and sH at I=115.224276.


## 5. Conclusion

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 restrictV>0, then there are new transition sets, and new bifurcation type constrating to the nonconstraint case. The Matcont toolbox software environment was used for analysis of the bifurcation points in conjunction with Matlab. We give four different parameters of gL, gK. In each case, we give equilibrium point bifurcation and also illustrate the stability of the equilibrium points. This study increases our knowledge of HH model.



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*Source: 545236-2014-07-07.xml*