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Bifurcation Control of the Hodgkin–Huxley Equations

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Bifurcation Control of the Hodgkin–Huxley Equations Chaos, Solitons and Fractals 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, Tianjin University, 300072 Tianjin, PR China 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. Ó 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 Ó 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 À gNam hðV À V NaÞgKn ðV À V KÞglðV À V lÞ > dt CM > > dm < ¼½a ðV Þð1 À mÞb ðV ÞmU dt m m > ð1Þ > dh > ¼½ahðV Þð1 À hÞbhðV ÞhU > dt > :> dn ¼½a ðV Þð1 À 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 À V Þ=½expðð25:0 À 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 À Þ ½ ðð À Þ Þ bnðV Þ¼0:125 expðV =80:0Þ The HH include the following parameters: VNa = 115.0 mV, Vk = À 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(TÀ6.3)/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 nonlinear system 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 limit set, without regard to its stability properties. The parameter being varied is referred to as the bifurcation parameter. A nonlinear dynamical system 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) bifurcation diagram 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: À10 k1ðI1Þ¼6 Â 10 þ 0:5862338120i k1ðI2Þ¼0 þ 1:062921807i k ðI Þ¼6 Â 10À10 À 0:5862338120i k ðI Þ¼0 À 1:062921807i 2 1 1 2 ð4Þ k3ðI1Þ¼0:1384747299 k3ðI2Þ¼0:3108668507 k4ðI1Þ¼4:764282435 k4ðI2Þ¼9:412063383 À10 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].
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