Chaos, Solitons and 33 (2007) 217–224 www.elsevier.com/locate/chaos

Bifurcation control of the Hodgkin–Huxley equations

Jiang Wang *, Liangquan Chen, Xianyang Fei

School of Electrical and Automation Engineering, , 300072 Tianjin, PR

Accepted 6 December 2005

Abstract

The Hodgkin–Huxley equations (HH) are parameterized by a number of parameters and show a variety of qualita- tively different behaviors. This paper finds that when the externally applied current Iext varies the bifurcation would occur in the HH equations. The HH model’s Hopf bifurcation is controlled by permanent or interval Washout filters (WF), which can transform the subcritical bifurcations into supercritical bifurcations, and can make the HH equations stable directly. Simulation results show the validity of those controllers. We choose the membrane voltage V as an input to the washout filter because V can be readily measured, and the controller can be realized easily. The controller designs described here may boost the development of electrical stimulation systems for patients suffering from different neuron- system dysfunctions. 2006 Elsevier Ltd. All rights reserved.

1. Introduction

The Hodgkin–Huxley (HH) equations represent a mathematical model which describes the electrical excitations of the squid giant axon [1–4]. The HH is parameterized by a number of parameters, and each of these parameters corre- sponds to the intrinsic property or physical environment of the membrane. When the original values are set to param- eters used by Hodgkin and Huxley, the HH behaves as an excitable membrane, and the membrane potential V shows a rapid and transient increase in response to current pulse stimulation. However, when parameter values are appropri- ately varied, the HH can show a variety of qualitatively different behaviors. Bifurcations have close relations with diseases like Parkinson disease, epilepsy or pathological heart rhythms, etc. To cure such diseases scientists have done many works [7,8,14,15]. Bifurcation control [9–13] through electrical stimulation is one of them. The study of electrical stimulation of nerve cell activity has a long history, many existing nerve stimu- lation protocols are basically simple feedforward schemes where the exact dynamics of the biophysical states of the tar- geted neurons is disregarded. To design controllers allowing dynamic time-course control of biophysical state variables, advanced control algorithms are required, since inherent nonlinearities and biological constraints limit the application of linear feedback control strategies. The nonlinearities of HH nerve cell dynamics require nonlinear control strategies to achieve efficient and precise control. Furthermore, such a controller will eventually interact with biological tissue, which is sensitive to disturbances [5]. Feedback control schemes based on biophysical states in nerve cells might lead to the development of new electrical stimulation systems as neural prostheses for patients suffering from loss of function

* Corresponding author. Tel.: +86 22 27 402293; fax: +86 22 27 401101. E-mail address: [email protected] (J. Wang).

0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.035 218 J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 217–224 or aberrant electrical signal generation in the nervous system caused by accident or disease. For example, single cell stimulation is of major interest for suppression of undesired neural oscillation as it occurs in patients with Parkinson’s disease or epilepsy [6]. This paper firstly analyzes the Hopf bifurcation in Hodgkin–Huxley equations, then, applies washout filter (WF) to such a model and successfully achieves the control of nonlinear nerve cell Hopf bifurcation by using permanent or inter- val controller, respectively. Simulation results verify the validity of those theoretic analysis and control methods.

2. Hodgkin–Huxley equations

The HH comprise the following differential equations and can be modeled by an equivalent circuit of the form shown in Fig. 1: 8 > dV 1 3 4 > ¼ ½Iext gNam hðV V NaÞgKn ðV V KÞglðV V lÞ > dt CM > > dm < ¼½a ðV Þð1 mÞb ðV ÞmU dt m m > ð1Þ > dh > ¼½ahðV Þð1 hÞbhðV ÞhU > dt > :> dn ¼½a ðV Þð1 nÞb ðV ÞnU dt n n V represents the membrane potential. 0 6 m 6 1 and 0 6 h 6 1 are the gating variables representing activation and inactivation of the Na+ current, respectively. 0 6 n 6 1 is the gating variable representing activation of the K+ current. an, bn, am, bm, ah and bh are the functions of V as follows: 8 > a ðV Þ¼0:1ð25:0 V Þ=½expðð25:0 V Þ=10:0Þ10 > m > > bmðV Þ¼4:0 expðV =18:0Þ <> ahðV Þ¼0:07 expðV =20:0Þ > ð2Þ > bhðV Þ¼1:0=½expððV þ 30:0Þ=10:0Þþ1:0 > > a V 0:01 10:0 V = exp 10:0 V =10:0 1:0 :> nð Þ¼ ð Þ ½ ðð Þ Þ bnðV Þ¼0:125 expðV =80:0Þ

The HH include the following parameters: VNa = 115.0 mV, Vk = 12.0 mV, V1 = 10.599 mV representing the equi- librium potentials of Na+,K+ and the leak currents, respectively. They are determined by the Nernsts equation. 2 2 2 gNa ¼ 120 mS=cm , gK ¼ 36:0mS=cm , gl ¼ 0:3mS=cm represent the maximum conductance of the corresponding io- 2 nic currents. They reflect the ionic-channel density distributed over the membrane. Cm = 1.0 lF/cm is the membrane capacitance. U =3(T6.3)/10 modifies the time constants of the gating variables depending on the temperature T (T = 6.3 C). Iext represents the externally applied current [7].

3. Hopf bifurcation of the HH

Bifurcation indicated a change in the number of candidate operating conditions of a when a parameter is quasistatically varied. The candidate operating condition is either an equilibrium point, a periodic solu-

Fig. 1. Electrical equivalent circuit proposed of Hodgkin and Huxley. J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 217–224 219 tion, or other invariant subset of its , without regard to its stability properties. The parameter being varied is referred to as the bifurcation parameter. A nonlinear can exhibit many different kinds of bifurcations as one or more parameters are varied. As to the HH, it has three kinds of codimension one bifurcations [8]: Hopf bifur- cation. Saddle-node bifurcation (Sn). Double cycle bifurcation (saddle-node of periodics) [9].

2 2 2 2 Fig. 2. The response of V with different Iext: (a) Iext = 2.25 lA/cm ; (b) Iext = 6.26 lA/cm ; (c) Iext =10lA/cm ; (d) Iext = 100 lA/cm ; 2 (e) Iext = 160 lA/cm ; (f) of HH. 220 J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 217–224

Here we merely analyze its Hopf bifurcation when Iext changes. Applying bifurcation theorem and using Maple as a tool we get two bifurcation points of HH when Iext changes:

I1 ¼ 9:77963800; I2 ¼ 154:52663365 ð3Þ Substitute them into the Jacobian matrix of HH [22,23], respectively; we get their eigenvalues as follows: 10 k1ðI1Þ¼6 10 þ 0:5862338120i k1ðI2Þ¼0 þ 1:062921807i k ðI Þ¼6 1010 0:5862338120i k ðI Þ¼0 1:062921807i 2 1 1 2 ð4Þ k3ðI1Þ¼0:1384747299 k3ðI2Þ¼0:3108668507

k4ðI1Þ¼4:764282435 k4ðI2Þ¼9:412063383 10 where 6 · 10 = 0 if we omit some small errors. Both ki(I1) and ki(I2)(i = 1,2,3,4) have a pair of pure imaginary eigenvalues and two negative eigenvalues. After further numerical analysis we find that the equations bifurcate between those two points. Their waveforms are as follows at different values of Iext. Fig. 2 shows the responses of membrane voltage to different Iext. (a) and (b) show the different response diagram of V 2 when Iext is lower than 6.265 lA/cm , they behave differently for different values of input current and eventually con- 2 verge to a stable point. (c) and (d) show the different bifurcation waveforms when Iext 2 (6.265,154.585) lA/cm ,in which the limit cycle amplitudes decrease and the surge frequency increases along with the increasing of Iext. (e) shows 2 that the HH returns to the resting potential when Iext are larger than 155 lA/cm . (f) shows the overall bifurcation dia- 2 gram of HH at the range of Iext 2 (0,180) lA/cm .

4. Bifurcation control with WF

Bifurcation refers to qualitative changes in the solution structure of dynamical systems occurring with slight varia- tion in system parameters. One important attribute of a bifurcation is the direction, or stability, of the bifurcation. Supercritical bifurcations permit smooth transition of system states; while subcritical, transcritical and saddle-node bifurcations normally lead to hysteresis and ‘‘jump’’ behaviors, which are undesirable. Therefore converting the bifur- cation to supercritical or eliminating it [10,11] is a better selection. Bifurcation control means to design a controller that can modify the bifurcation properties of a given nonlinear sys- tem, so as to obtain some desired dynamical behaviors [28,29]. Typical examples include delaying the onset of an inher- ent bifurcation, relocating an existing bifurcation point, modifying the shape or type of a bifurcation chain, introducing a new bifurcation at a preferable parameter value, stabilizing a bifurcated periodic trajectory, changing the multiplicity, amplitude, and/or frequency of some limit cycles emerging from bifurcation, optimizing the system performance near a bifurcation point, or a certain combination of some of these. System bifurcation can be controlled by using different methods, such as linear delayed state-feedback [15,16], or nonlinear state-feedback [17], using a washout filter [18], hybrid control for discrete time model [9,24–27], employing harmonic balance approximation [19], and applying the quadratic invariants in the normal form [20]. Lemma 4.1 [21]. Assume a stable system G(s) 2 sp·q can be represented by a rational proper transfer function matrix of s mn np nn with degree n. If G(s) is a high-pass filter, i.e., lims!0G(s) = 0, then 9K 2 R ; C 2 R ; D 2 R such that the LTI system z_ ¼ Cx Dz ð5Þ y ¼ KðCx DzÞ has the transfer function G(s), where z 2 Rn is the state vector, and x 2 Rp is the input vector, and y 2 Rq is the output vector. With the above extended WF, the controller can be written as following: 8 <> z_ ¼ Cx Dz y ¼ Cx Dz ð6Þ :> u ¼ gðyÞ C is the output measurement matrix and D determines the dynamic properties of the controller. The control function g(y) is assumed to be

gðyÞ¼Quðy; yÞþCuðy; y; yÞ ð7Þ which can represent all quadratic (Qu(y,y)) and cubic functions (Cu(y,y,y)). J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 217–224 221

With the dynamic feedback control law, the overall closed-loop system of the HH is given by

dV 1 3 4 ¼ ½Iext gNam hðV V NaÞgKn ðV V KÞglðV V lÞ þ u dt CM dm ¼½a ðV Þð1 mÞb ðV ÞmU dt m m dh ¼½ahðV Þð1 hÞbhðV ÞhU dt ð8Þ dn ¼½a ðV Þð1 nÞb ðV ÞnU dt n n z_ ¼ Cx Dz y ¼ Cx Dz u ¼ gðyÞ

Here, z is the washout filter state variable, z_ is the output function, 0 < d < 2 is the reciprocal of the filter time constant, and g(y) is the control function to be designed. When using Iext as a bifurcation parameter, bifurcation analysis of the HH model reveals a very rich bifurcation 2 structure as Iext increases (from resting level to 180 lA/cm ). Fig. 2 shows them in detail. In the following discussion, we use the electrical stimulus Iu as a control input. Following the results in [13], control laws are designed to transform the subcritical Hopf bifurcations into supercrit- ical Hopf bifurcations, or make the bifurcation points stable directly. We again employ washout filters to preserve the equilibrium structure of the system as well as to save control energy. The structure of the controller is 8 <> z_ ¼ V dz y ¼ V dz ð9Þ :> Iu ¼ gðyÞ where z and y are the state variable and output of the washout filter, respectively, while d denotes the time constant of the controller. Here, we merely employ one washout filter associated with V. It is a dynamic output feedback controller. The reason for choosing the membrane voltage V as an input to the washout filter is that V can be readily measured. It is well known that only the quadratic and cubic terms in a nonlinear system undergoing a Hopf bifurcation influ- ence the bifurcation stability coefficient [11,12]. Here we take our controller in the form of a nonlinear law with only a cubic term

3 Iu ¼ Ky ð10Þ

Fig. 3. Response of V under control with different gains: (a) d = 0.5, k = 0; (b) d = 0.5, k = 0.10. 222 J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 217–224

Fig. 4. Response of V under control with different gains: (a) d = 0.7, k = 0.2; (b) d = 0.7, k = 1.0.

Fig. 5. Response of V and phase plan of m, h and V under control: (a) time response of V; (b) trace of parameter m, h and V. J. Wang et al. / Chaos, Solitons and Fractals 33 (2007) 217–224 223

4.1. Permanent control

The control gain can be determined by a detailed calculation of the closed-loop bifurcation stability coefficient [13]. Because the structure of the HH model is very complex, it is not easy to calculate the gain analytically. The effect of varying the feedback gain K on the controller performance was explored first. Repeated simulations with different K values revealed that the interval K 2 [0,10] constitutes a reasonable choice for good bifurcation annihilation. Smaller values of K caused high steady-state errors, whereas values of K > 10 caused very high peaks in controller output Iu. Here, we select the control gain via experiments. We first fix the time constant of the washout filter, d and then use a series of control gains to test and see when the bifurcations are changed into supercritical or stable. In Fig. 4,we fix d = 0.5 and set k to be 0, 0.10, respectively. Moreover, the amplitude of limit cycle is decreased while increasing the control gain, which implies that one can control the amplitude of oscillation using this nonlinear controller. Fig. 3 shows the time response of the system under nonlinear control with different gains. While if we fix d = 0.7, and set k to be 0.2, 1.0, respectively, it can be seen that the bifurcation of HH equations turns out to be stable as Fig. 4. By transforming the Hopf bifurcations into supercritical bifurcations, the multistability near the Hopf bifurcations is eliminated and hence the occurrence of jump behavior of the HH under perturbation is prevented. Further, since the amplitude of the bifurcation becomes rather small under control, the probability of disadvantage will be reduced. And if we choose better parameters we can eliminate the whole bifurcation as we want.

4.2. Interval control

Another possibility to achieve stability with potential less energy is via an interval controller. Applying such interval controller when the bifurcation occurs we can switch on the controller to move the state of the system into a much lim- ited region, and then at a different time we can switch on another controller to annihilate the bifurcation. That means, we can control it directly or firstly transform it from subcritical to supercritical then make it stable, the latter complex one was shown in Fig. 5, which we apply the controller to transform it into supercritical and eliminate bifurcations at 200 ms and 400 ms, respectively.

5. Conclusion

In this paper, we analyze the Hopf bifurcation in HH equations when the externally applied current Iext changes, then apply a extended WF to control this kind of bifurcation, firstly transform the subcritical bifurcations into super- critical bifurcations, then adjust the parameters to make the HH equations stable. Simulation results show the validity of those controllers. In addition to our analysis and simulation studies, we can use electrical stimulation device to use the controller in practice. This may have an important clinical implications since some undesired neural oscillations are very threatening, as they occur in patients with Parkinson’s disease or epilepsy. The controller designs described here may lead in the future to smart clinical machines such as pacemakers, etc.

Acknowledgement

The authors gratefully acknowledge the support of the NSFC (No. 50177023 and 50537030).

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