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1 Neurocomputing 81 (212) Contents lists available at SciVerse ScienceDirect Neurocomputing journal homepage: Bifurcations in the Hodgkin Huxley model exposed to DC electric fields Yanqiu Che a,n, Jiang Wang b, Bin Deng b, Xile Wei b, Chunxiao Han a a Tianjin Key Laboratory of Information Sensing & Intelligent Control, School of Automation and Electrical Engineering, Tianjin University of Technology and Education, Tianjin 3222, China b School of Electrical Engineering and Automation, Tianjin University, Tianjin 372, China article info Article history: Received 1 March 211 Received in revised form 23 September 211 Accepted 1 November 211 Communicated by J. Torres Available online 27 December 211 Keywords: Hodgkin Huxley model Hopf bifurcation Electric fields abstract Diverse behaviors of the original Hodgkin Huxley (HH) model, depending on the parameter values, have been studied extensively. This paper proposes modified HH equations exposed to externally applied extremely low frequency (ELF) electric fields. We investigate the effect of the DC electric fields on the dynamics of the modified HH model using bifurcation analysis. The obtained bifurcation sets partition the two dimensional parameter space, representing intensity of externally applied DC current and trans-membrane voltage induced by external DC electric fields, in terms of the qualitatively different behaviors of the HH model. Thus the neuronal information encodes the stimulus information, and vice versa. We also illustrate that the multi-stability phenomena in the HH model are associated with Hopf and double cycle bifurcations. & 211 Elsevier B.V. All rights reserved. 1. Introduction A neuron s response to external electrical stimulation varies depending on the neuron s characteristics as well as the stimulation. With a constant current, a single spike followed by resting potential (i.e. excitability) and periodic tonic spiking are two common types of responses. For the dynamical system [1], considering the steady states for these two behaviors, they correspond to a stable equilibrium and a stable limit cycle, respectively. The Hodgkin Huxley (HH) model [2] which quantitatively describes electrical excitations of squid giant axons is a prototype for excitable cell models. Under the HH formalism (the method used in deriving the HH model), a large family of mathematical models (HH-type models) for diverse neurons and excitable cells are established [3 7]. Since these HH-type models share many common dynamical characteristics with HH model, a detailed analysis of the dynamics in HH model under electric fields will also provide a basic understanding of that in HH-type models under electric fields. A bifurcation is a qualitative change in the behavior of a nonlinear dynamical system as its parameters pass through critical values [1]. The study of bifurcations in neural models is a keystone for understanding the dynamical origin of many single-neuron and circuit phenomena involved in neural information processing and the organization of behavior [8]. So far, many studies have been done on the bifurcation analysis of the original n Corresponding author. Tel./fax: þ address: yqche@tju.edu.cn (Y. Che). HH model [9 21]. With a change in the externally applied DC current I ext, Hopf bifurcation would occur at the equilibrium of HH model, generating limit cycles [9 11]. The corresponding stable and unstable periodic solutions are numerically calculated [12]. With appropriate I ext and V K, representing the equilibrium potential of potassium, two stable equilibrium potentials coexist in the HH model [13], which coincides with the experimental result [14]. Guckenheimer and Labouriau [15] give the detailed bifurcation diagrams of HH model in two-parameter space of I ext and V K. Bedrov et al. reveal the possible bifurcations with changes of g Na and g K, representing the maximal conductance of sodium and potassium, respectively [16,17]. The global structure of bifurcations in multiple-parameter space of the HH model is examined [18], and the details of the degenerate Hopf bifurcations are analyzed using the singularity theoretic approach [19]. It has also been shown that the chaotic solutions can exist in the HH equations by varying the time constant [2] or the original parameters [21]. On the other hand, the interaction between electric fields and biological tissues has been of longstanding interest. The electric fields can alter neuronal excitability in normal physiological conditions [22 27], change the neuronal ensemble activity of epileptiform [28 32]. The electric field stimulation becomes a powerful physiotherapy [33]. In spite of the importance of HH model, the detailed bifurcation analysis of the HH model exposed to external electric fields has not been carried out to the best of our knowledge. The aim of this paper is to illustrate how the DC electric fields affect the dynamics of neural activity in the context of HH equations. Based on the results we obtained in the literature /$ - see front matter & 211 Elsevier B.V. All rights reserved. doi:1.116/j.neucom

2 42 Y. Che et al. / Neurocomputing 81 (212) [34], the modified HH model exposed to extremely low frequency (ELF) external electric field is given. In the modified HH model a new parameter, induced trans-membrane potential, is introduced to denote the effect of external electric fields. We are concerned with the bifurcations of the modified HH model in two-parameter space of the externally applied DC current and the induced transmembrane voltage, while the other parameters are kept as the original values in HH. Computational tools based upon the theory of nonlinear dynamical systems are used. The researches will throw some light on the interference mechanism between biological systems and electric field exposure. 2. Neuron models 2.1. The original HH model The original HH neuron model consists of the following ordinary differential equations: dv C m dt ¼ I ext ½g Na m 3 hðv V Na Þþg K n 4 ðv V K Þþg L ðv V L ÞŠ dm dt ¼½a mðvþð1 mþ b m ðvþmšf dh dt ¼½a hðvþð1 hþ b h ðvþhšf dn dt ¼½a nðvþð1 nþ b n ðvþnšf ð1þ where V represents membrane potential in mv which is produced by the accumulation and transportation of ions such as sodium, potassium and chlorine across the membrane. The time t is measured in ms, the variables m, h and n represent the sodium activation, the sodium inactivation and the potassium activation, respectively. Parameters V Na, V K and V L are the equilibrium potentials of sodium, potassium and leak current, respectively. They are determined uniquely by Nernst s equations. g Na, g K and g L are the maximal conductance of the corresponding ionic currents. They reflect the ionic channel density distributed over the membrane. C m is the membrane capacitance, I ext is the externally applied current. The factor f ¼ 3 ðt 6:3Þ=1 alters the reaction rate constants a y ðvþ and b y ðvþðy ¼ m,h,nþ. a y ðvþ and b y ðvþðy ¼ m,h,nþ are nonlinear functions of V, given by the following equations: a m ðvþ¼:1 ð25 VÞ=½expðð25 VÞ=1Þ 1Š b m ðvþ¼4 expð V=18Þ a h ðvþ¼:7 expð V=2Þ b h ðvþ¼1:=½expðð V þ3þ=1þþ1š a n ðvþ¼:1 ð1 VÞ=½expðð1 VÞ=1Þ 1Š b n ðvþ¼:125 expð V=8Þ 2.2. The modified HH model exposed to external electric field For biological cells under external ELF electric fields, the capacitance effect of both cytoplasm and outside medium can be neglected because of their large reactance, while the capacitance effect of membrane must be taken into account due to its high resistance. Thus at cellular level, an exposure of a cell to an external electric field results in the induced membrane potential V E that superimposes to the ionic membrane potential V [34 36], ð2þ i.e. the overall membrane potential becomes V þv E. Then the original HH model is modified as follows: dðv þv E Þ C m ¼ I ext ½g dt Na m 3 hðv þv E V Na Þþg K n 4 ðv þv E V K Þ þg L ðv þv E V L ÞŠ dy dt ¼ a yðvþð1 yþ b y ðvþy ðy ¼ m,h,nþ ð3þ The induced membrane potential V E can be calculated as [34,35] V E ¼ KEðtÞR cos y½1 expð t=tþš ð4þ where E is the strength of ELF electric fields, K is a constant determined by the geometric and electrical characteristics of the cell, t is the time constant of membrane, R is the radius of the cell, and y is the angle between V E ðtþ and E(t). Since the effect of AC electric fields on the dynamics of HH model has been studied elsewhere [37], we only study the effect of DC electric fields in this paper, then in the left of Eq. (3), the term C m ðdv E =dtþ¼. The ELF electric fields directly act over membrane potential through an additive perturbation V E, which in turn influence the activity of voltage-dependent ionic channels. V E does not change the basic structure of HH model, but simultaneously shifts the three ionic reverse potentials, which is different from the situation studied in [38]. According to control theory, the introduction of V E could be regarded as a disturbance applied to the original system and its dynamic performance under disturbance should be investigated. Fixed parameter values used in this paper are listed in Table 1, with which the neuron keeps silent without externally applied electric currents and electric fields. 3. Definition and detection of bifurcations A bifurcation is a change of qualitative behavior of a dynamical system at special values of the parameters. The codimension of a bifurcation is the number of parameters which must be varied for the bifurcation to occur [15]. In this study, we consider the following bifurcations [15,18] Codimension one bifurcations Hopf bifurcation (H): A continuous fundamental path of an equilibrium point loses its stability as it intersects a secondary path of a periodic solution. The location of a Hopf bifurcation on the equilibrium point is characterized by a complex conjugate pair of linear eigenvalues of the Jacobian matrix whose real part passes through zero. When the secondary path is stable, it is the supercritical Hopf bifurcation (sh). Conversely, when the secondary path is unstable, it is the subcritical Hopf bifurcation (uh). Table 1 Fixed parameters for the HH model. Parameters Values Units Variables C m 1. mf=cm 2 Membrane capacitance V Na 115 mv Sodium equilibrium potential V K 12 mv Potassium equilibrium potential V L mv Leak equilibrium potential g Na 12 ms/cm 2 Maximal sodium conductance g K 36 ms/cm 2 Maximal potassium conductance g L.3 ms/cm 2 Maximal leak conductance T 6.3 1C Temperature

3 Y. Che et al. / Neurocomputing 81 (212) Saddle-node bifurcation (sn): Two equilibrium points coalesce and disappear. At this bifurcation point, the Jacobian matrix of the equations at the equilibrium point has a zero eigenvalue. Double cycle bifurcation (dc, saddle-node of cycles): Two periodic solutions with finite amplitude coalesce and disappear Codimension two bifurcations Cusp (c): Three equilibrium points coalesce into one. Cusp of periodic (cp): Three periodic solutions coalesce into one. On an appropriate Poincare section, this bifurcation corresponds to the cusp of the equilibrium points of the Poincare map. Takens Bogdanov bifurcation (TB): The Jacobian matrix of the equations at the equilibrium point has two zero eigenvalues. On a two-dimensional bifurcation diagram (2BD), TB locates on the sn curve and the Hopf bifurcation curve is tangent to the sn curve at this point. Degenerate Hopf bifurcation (dh): The stability of the periodic solution which bifurcates at the Hopf bifurcation point changes. A dc curve is terminated at this point on a 2BD. In this study, numerical computations for the bifurcation diagrams are conducted using MATCONT, a new MATLAB continuation package for the study of parameterized ODE systems. Matcont can detect several bifurcation points automatically and can trace both stable and unstable branches of equilibria and periodic solutions [39]. Numerical integration of the system equations for obtaining the trajectories is conducted in MATLAB using ode Bifurcation analysis In this section, we first present a global bifurcation diagram for the modified HH model (Eq. (3)) in I ext,v E two-dimensional parameter space. Then, we illustrate how the parameter V E affects the bifurcation process by analyzing a series of one parameter bifurcation diagrams for I ext with different fixed values of V E Bifurcations in I ext,v E parameter space Fig. 1 shows the two-parameter bifurcation diagram of the modified HH model. The abscissa and the ordinate are I ext and V E, 5 5 A C Fig. 1. Bifurcation diagram of HH model in (I ext, V E ) parameter space. The abscissa is constant DC externally applied current I ext and the ordinate is the electric fields parameter V E. The supercritical Hopf, subcritical Hopf, double-cycle, and saddle-node bifurcation curves are labeled as sh, uh, dc and sn, respectively. The parameter planes are divided into regions A I (regions F I are shown in Fig. 2) bythesebifurcation curves. In each region, a schematic phase portrait is shown. Solid dots and crosses represent stable and unstable equilibrium points, respectively. The curved arrows represent schematic trajectories of the equations. Stable and unstable periodic solutions are represented by solid and dashed circles, respectively. B A respectively. Fig. 2 gives detailed structure for some regions in Fig. 1. Three types of curves of codimension one bifurcations partition the whole parameter plane into several regions (A I). The solid one at the center is the Hopf bifurcation curve (uh and sh). Two dashed curves, lying in upper right, are double cycle bifurcation curves (dc). And the dash-dotted curve in the downleft corner is saddle-node bifurcation curve (sn). Hopf bifurcation changes stability at dh points and terminates at TB points. Two branches of sn join at a cusp (c) point, creating a region of coexistence of three equilibria. Two cp points are joint points of dc curves. Schematic phase portraits for each region are given. Stable equilibrium points are shown as solid dots, unstable ones are crosses; stable limit cycles are closed curves with solid lines, and unstable periodic orbits are dashed lines. Different phase portraits indicate qualitatively different behaviors of HH model. Fig. 3 shows some examples of waveforms of the membrane potentials in HH model for parameters located in the different regions of bistability (Fig. 3a e, h) or tri-stability (Fig. 3f, g). Properly added perturbations (arrows in figures) to the membrane potentials at certain time induce transitions from one stable state to another. In region A, the unique steady state is an stable equilibrium point. Parameter set ði ext,v E Þ¼ð; Þ locates in this region. The membrane is excitable, i.e. the membrane potential returns to the steady state after an excitation in respond to pulse stimulation. In region B, the periodic solution is the unique stable steady state and an unstable equilibrium point exist within it. The asymptotic dynamics of the modified HH is the periodic oscillation. In C, three equilibrium points coexist and two of them are stable, i.e. bistability of equilibrium points. Asymptotic dynamics is either one of them, depending on its initial conditions. Fig. 3(a) and (b) gives two examples of membrane potential waveforms when the parameter values are set in C ði ext,v E Þ¼ð 5, 5Þ and ði ext,v E Þ¼ð 1:755, 15:87Þ, respectively. Perturbations to the membrane potential induce transitions between the two steady states. In D, two stable attractors, an equilibrium point and a periodic solution, coexist with an unstable periodic solution in-between, i.e. bistability of an equilibrium point and a limit cycle. In E (Fig. 1), there is another bistability of an equilibrium point and a limit cycle but it is qualitatively different from that in region D. As illustrated by examples in Fig. 3(c) for region D, ði ext,v E Þ¼ð5; 25Þ and in Fig. 3(d) for region E, ði ext,v E Þ¼ð 3, 26Þ, respectively, the limit cycle encircles the equilibrium in D while it does not in E. In F (Fig. 2(a)), bistability of limit cycles is observed. In Fig. 3(e) for region F, ði ext,v E Þ¼ð115,22:9Þ, perturbations induce transitions between the small amplitude oscillation and the larger one. In G (Fig. 2(b)), four limit cycles with different periods and amplitudes coexist with one equilibrium point. Two of the limit cycles and the equilibrium point are stable (tri-stability) and the others are unstable. In Fig. 3(f) for region G, ði ext,v E Þ¼ð98:735,23:1993Þ, properly delivered perturbations transform membrane potential from resting state to with very small amplitude to oscillation with large amplitude. Note that the attractor regions of the equilibrium and the small limit cycle are very small. Region H(Fig. 2(c)) is another tri-stability region. It s boundary consists of dc 1, sn and uh. One stable and one unstable equilibrium points locate outside the stable limit cycle which encloses another stable equilibrium point and an unstable limit cycle. Fig. 3(g) for region H, ði ext,v E Þ¼ð 1:7497, 15:8776Þ, illustrates transitions between the three stable states. The narrow zone enclosed by curves of dc 6 and dc 7 between dc 1 and uh is denoted as Region I (Fig. 2(d)). In this region, between a stable equilibrium point and a stable limit cycle, there exist three even more unstable limit cycles [12,2]. Fig. 3(h) exemplifies a membrane potential waveform when the parameter values are set in I ði ext,v E Þ¼ð 1:, 5:43Þ.

4 44 Y. Che et al. / Neurocomputing 81 (212) Fig. 2. Detail bifurcation structures of regions F I shown in Fig. 1. Perturbations to the membrane potential induce transitions between the two steady states One parameter bifurcations for I ext To illustrate how V E affects the dynamics of HH model and how the multi-stabilities emerge, we investigate the changes of the one parameter bifurcation diagrams for I ext with different values of V E. The additive term V E is fixed to certain values and the bifurcation parameter I ext is continuously changed. Fig. 4(a i) shows the results for V E ¼, 2, 23, , , 23.25, 23.3, and 3 mv, respectively. In each diagram, extrema of membrane potential V extrema of the steady solutions of HH are plotted as functions of I ext. The central curve represents the equilibrium points and the upper and lower ones represent the maximum and minimum values of periodic solutions, respectively. Stable and unstable solutions are shown by solid and dotted curves, respectively. The I ext axis is divided into sub-intervals which are associated with regions in Figs. 1 and 2. Without or with very weak electric fields, e.g. V E ¼ mv (Fig. 4(a)), as I ext increases from small value, first reaches a dc bifurcation dc 1,andapairofstableandunstableperiodicsolutions emerge. Since dc does not affect the existing stable equilibrium point, the bistability of limit cycle and equilibrium point occurs. As I ext increases further, in a small interval of I ext, three branches of unstable limit cycles show azshapecurvejoinedbydc 6 and dc 7 bifurcation. When I ext reaches the left Hopf point uh, the stability of the equilibrium changes from stable to unstable, the bistability disappears. With further increase of I ext, amplitude of the stable limit cycle decreases and vanishes at the right Hopf point sh point, and the equilibrium becomes stable again. The neuron becomes to be quiescent. For a relative larger V E (e.g. V E ¼ 2 mv in Fig. 4(b)), the only different feature of the bifurcation diagram from that in Fig. 4(a) is the disappearance of the Z shape curve, leaving a single unstable periodic solution branch, due to collision and disappearance of dc 6 and dc 7.ForlargerV E, the bifurcation diagrams change dramatically in a small region, due to the coexistence of several codimension two bifurcations (see Fig. 2(b)). As shown in Fig. 4(c) for V E ¼ 23 mv, the right half of the stable limit cycle branch changes to be a Z shape, caused by the occurrence of bifurcations dc 2 and dc 3 between which two stable and one unstable limit cycles coexist with an unstable equilibrium point. When I ext is changed gradually passing this bistability region, the modified HH shows a hysteresis phenomenon, i.e. its steady state jumps from one type of periodic firing to the other at points dc 6 and dc 7. Hopf points uh and sh become closer. For V E ¼ 23:199 mv and V E ¼ 23:1993 mv (Fig. 4(d, e)), the left half of the unstable limit cycle branch creates a Z shape in a narrow region close to uh as shown in the enlargements. dc 4 and dc 5 bifurcations account for that changes. The middle limit cycle is stable rather than unstable as that occurs in Fig. 4(a), thus in the narrow parameter range, the modified HH exhibits tri-stability. The difference between (d) and (e) is the relative position of dc 4 and uh. For V E ¼ 23:25 mv (Fig. 4, dc 5 meets uh point, dc 5 disappears and uh changes into sh. The periodic solution bifurcates form this sh point is stable rather than unstable and the bifurcation direction changes. The dc 3 and dc 4 get closer and nearly collide. For V E ¼ 23:3mV (Fig. 4, afterdc 3 and dc 4 coalesce and disappear, an isolate periodic solution branch which is not associated with Hopf point (named isola [1]) exists in a wide range of I ext. The periodic solutions bifurcate from the two Hopf points join at certain value of I ext.forv E ¼ 23:455 mv (Fig. 4, the two sh points both occur before dc 2 and get much closer. For large V E (in Fig. 4, V E ¼ 3 mv), the two Hopf points coalesce and disappear after a dh bifurcation dh 2, leaving an isola and a stable equilibrium point. In a summary, as V E increases, the dc bifurcations cause the occurrence and disappearance of Z-shaped branch of the periodic

5 Y. Che et al. / Neurocomputing 81 (212) C 5 C D 1 E F 8 G H 15 I Fig. 3. Examples of membrane potential waveforms in bistability or tri-stability regions C I in Figs. 1 and 2. The abscissa and the ordinate are time (ms) and the membrane potential (mv), respectively. In each case, all the parameters of the HH are fixed, and appropriate impulsive perturbations delivered to the membrane potential V lead to transitions from one steady state to another (see text). solution, while the Hopf bifurcation points uh and sh get closer, and the left uh becomes sh. Then, the periodic solution branch is disconnected with equilibrium, leading to the appearance of the isola. As V E increases further, the left and right sh collide and disappear by dh 2. The isola remains for higher values of V E.SothedcandHopf bifurcations mainly contribute to the emergence of multi-stabilities.

6 46 Y. Che et al. / Neurocomputing 81 (212) Fig. 4. One parameter bifurcation diagrams for I ext with different values of V E. The abscissa and the ordinate are I ext and the extrema of membrane potential V extrema, respectively. In each diagram, the central curve represents the equilibrium potential. The upper and lower curves represent the maximum and minimum values, respectively, of the membrane potential of periodic solutions. Solid and dashed curves represent stable and unstable solutions, respectively. sh, uh and dc are supercritical, subcritical Hopf bifurcation, and double-cycle bifurcation, respectively.

7 Y. Che et al. / Neurocomputing 81 (212) Conclusion Many studies have been done on the diverse behaviors and bifurcations in the original HH model. This paper have investigated the effect of DC electric fields on the bifurcation structure of the HH model. We summarize the results on a two-dimensional I ext, V E bifurcation diagram. The loci of codimension one bifurcations, Hopf, saddle-node and double cycle considered in this paper, partition the parameter space into several qualitatively different regions. The stimulus parameters can be quantified by the various neuronal firing patterns. On the one hand, one can interpret the evoked spike sequences with various patterns as a representation of the parameter values. On the other hand, one can determine the parameter values to make the neuron elicit certain specific spike sequences, which is significant in neural control. By analyzing the systematic changes of bifurcation diagrams for I ext with different values of V E,weillustrate that the Hopf and double cycle bifurcations account for the bistabilities and multi-stabilities phenomena. Acknowledgement This work is supported by the Key Program of the National Natural Science Foundation of China (Grant No ), the Young Scientists Fund of the National Natural Science Foundation of China (Grants No and No ), the National Natural Science Foundation of China (Grant No ) and the National Science Foundation for Post-doctoral Scientists of China (Grant No ). References [1] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer, [2] A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane and its application to conduction and excitation in nerve, J. Physiol. 117 (1952) [3] J. Cronin, Mathematical Aspects of Hodgkin Huxley Neural Theory, Cambridge University Press, [4] T.R. Chay, J. 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Edward (Eds.), Clinical Aspects of Electroporation, 1st ed.,springer, 211, pp [36] C. Polk, E. Postow (Eds.), CRC Handbook of Biological Effects of Electromagnetic Fields, second ed., CRC Press, Boca Raton, [37] Y.Q. Che, J. Wang, W.J. Si, X.Y. Fei, Phase-locking and chaos in a silent Hodgkin Huxley neuron exposed to sinusoidal electric field, Chaos Solitons Fractals 39 (29) [38] Y.A. Bedrov, G.N. Akoev, O.E. Dick, Partition of the Hodgkin Huxley type model parameter space into the regions of qualitatively different solutions, Biol. Cybern. 66 (5) (1991) [39] A. Dhooge, W. Govaerts, Y.A. Kuznetsov, MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software 29 (23) Yanqiu Che received BS, M.Eng and the Ph.D. degrees in Control Theory and Engineering from Tianjin University, Tianjin, China in 23, 25 and 28, respectively. During28 21hewasapostdocfellowatTianjin University. Since 21, He is an associate professor at the School of Automation and Electrical Engineering, Tianjin University of Technology and Education. His research interests focus on Neurodynamics, Computational Neuroscience, Neural Control Engineering, Bioelectromagnetics and Brain Computer Interface. Jiang Wang received the Ph.D. degree in Control Theory and Engineering from Tianjin University in He is currently a professor at the School of Electrical Engineering and Automation, Tianjin University. His research interests are nonlinear dynamical systems, computational neuroscience, and information processing and detecting.

8 48 Y. Che et al. / Neurocomputing 81 (212) Bin Deng received BS, M.Eng and the Ph.D. degrees in Control Theory and Engineering from Tianjin University, Tianjin, China in 21, 24 and 27, respectively. During he was a postdoc fellow at Tianjin University. Since 29, He is an associate professor of School of Electrical Engineering and Automation, Tianjin University. His research interests are analysis and control of nonlinear system, the dynamics of neural system, and the analysis of neural signals. Chunxiao Han received the BS and Ph.D. degrees in Control Theory and Engineering from Tianjin University, Tianjin, China in 23 and 28, respectively. She is currently a lecturer at the School of Automation and Electrical Engineering, Tianjin University of Technology and Education. Her research interests focus on biosignal processing and neurodynamics. Xile Wei received BS, M.Eng and the Ph.D. degrees in Control Theory and Engineering from Tianjin University, Tianjin, China in 1999, 24 and 27, respectively. During he was a postdoc fellow at Tianjin University. Since 29, He is an associate professor of School of Electrical Engineering and Automation, Tianjin University. His current research interests are in the areas of Bioelectromagnetics, Neuromorphic Modeling, and Neural Control Engineering.