Equilibrium Point Bifurcation and Singularity Analysis of HH Model with Constraint

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.


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][3][4][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 and K . Bedrov reveals the possible bifurcations with changes of Na and K , 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 as bifurcation parameter, K , as unfolding parameters, and we restrict > 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.

Hodgkin-Huxley Equations
The HH comprises the following differential equations: represents the membrane potential. 0 ≤ ≤ 1 and 0 ≤ ℎ ≤ 1 are the gating variables representing activation 2 Abstract and Applied Analysis and inactivation of the Na + current, respectively. 0 ≤ ≤ 1 is the gating variable representing activation of the K + current. , , ℎ , ℎ , , are the function of as follows: The HH includes the following parameters: K = −12.0 mV, Na = 115.0 mV, and = 10.599 mV representing the equilibrium potentials of K + , Na + , and leak currents, respectively. They are determined uniquely by the Nernst equation. Na = 120.0 mS/cm 2 , K = 36.0 mS/cm 2 , = 0.3 mS/cm 2 represent the maximum conductance of the corresponding ionic currents. = 1.0 F/cm 2 is the membrane capacitance. represents the external current, in A/cm 2 .

Constrained Bifurcation Theory
For the following bifurcation equation: where , , 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 The following are transition sets for single-sided constraint: 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 ( , ; ) = 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.

Constrained Transition Set.
The external current is chosen as bifurcation parameter and K , as unfolding parameters, and we restrict > 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.
From Figure 1 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 , K variations taken from the above four parameters regions is obtained in Figure 2.

4.2.1.
= 2, = 4. In order to show the bifurcation characteristics of HH, it is convenient to show the bifurcation diagrams obtained by the Matcont software for the varying values of , K . These are given in Figure 3(a). Using the results in Figure 3(a), the stability of equilibrium points is obtained in Figure 3 coefficient is positive, and there are two eigenvalues with Re 1,2 ≈ 0, Im 1 ̸ = 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 limit cycle emerging from sH at = 20.428518 is in Figure 4.
The limit cycle emerging from uH at = −11.231605 is in Figure 5.
In Table 1, the bifurcation points found by the Matcont software are presented for = 2, K = 4.

4.2.2.
= 0.1, = 2. The bifurcation diagrams and the stability of equilibrium points are obtained in Figure 6. The limit cycle emerging from uH at = −26.281425 is in Figure 7. The bifurcation points found by the Matcont software are presented in Table 2 for = 0.1, K = 2.
From Table 2, we can find the difference in equilibrium point.

4.2.3.
= 1, = 2. The bifurcation diagrams and the stability of equilibrium points are obtained in Figure 8. The limit cycles emerging from uH at = −13.971904 and = −9.406580 are given in Figures 9 and 10, respectively. The bifurcation points found by the Matcont software are presented in Table 3 for = 1, K = 2.

4.2.4.
= 2, = 20. The bifurcation diagrams and the stability of equilibrium points are obtained in Figure 11. The limit cycle emerging from uH at = 7.131765 and sH at  Figure 12. They both are the same. The bifurcation points found by the Matcont software are presented in Table 4 for = 2, K = 20.

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 restrict > 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 , K . 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.