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Electric Field-induced dynamical evolution of spiral wave in the regular networks of Hodgkin–Huxley neurons
Applied Mathematics and Computation 218 (2011) 4467–4474
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Electric Field-induced dynamical evolution of spiral wave in the regular networks of Hodgkin–Huxley neurons Chun-Ni Wang a, Jun Ma a,⇑, Wu-Yin Jin b,c, Ying Wu d a
Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China School of Physics and Mechanical & Electrical Engineering, Shao Guan University, Shaoguan 512005, China c College of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050, China d State Key Laboratory for Strength and Vibration of Mechanical Structures; School of Aerospace, Xian Jiaotong University, Xian 710049, China b
a r t i c l e
i n f o
Keywords: Spiral wave Breakup Factor of synchronization Hodgkin–Huxley neurons Channel noise Regular networks
a b s t r a c t An additional gradient force is often used to simulate the polarization effect induced by the external field in the reaction–diffusion systems. The polarization effect of weak electric field on the regular networks of Hodgkin–Huxley neurons is measured by imposing an additive term VE on physiological membrane potential at the cellular level, and the dynamical evolution of spiral wave subjected to the external electric field is investigated. A statistical variable is defined to study the dynamical evolution of spiral wave due to polarization effect. In the numerical simulation, 40000 neurons placed in the 200 200 square array with nearest neighbor connection type. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical value, otherwise, spiral wave keeps alive completely. On the other hand, breakup of spiral wave occurs as the intensity of electric field exceeds the critical value in the presence of weak channel noise, otherwise, spiral wave keeps robustness to the external field completely. The critical value can be detected from the abrupt changes in the curve for factors of synchronization vs. control parameter, a smaller factor of synchronization is detected when the spiral wave keeps alive. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Spiral waves are peculiar spatiotemporal patterns far from equilibrium state and are observed in the reaction–diffusion systems, networks of neurons and oscillators [1–10]. The propagation of chemical wave in the Belousov–Zhabotinsky reaction [11] is often investigated to detect the dynamical evolution of spiral wave in an experimental way. On the other hand, numerical studies are also effective to study the dynamics of these nonlinear waves, for example, Amdjadi [3] presented an effective numerical scheme to study the dynamics and stability of spiral waves. Ramos [12] investigated the propagation of spiral waves in two-dimensional reactive–diffusive media subject to a velocity field with straining and pattern formation [13] in two-dimensional excitable media subject to Robin boundary conditions in numerical way. It does show some significance to study the spiral wave formation, selection, breakup and control in the reaction–diffusion systems or coupled arrays [14–19], because it is helpful to remove spiral wave in the cardiac tissue and thus prevent the occurrence of ventricular fibrillation [20]. Many interesting works have been carried out and most of them are confirmed to be effective to remove spiral wave and prevent the breakup of spiral wave [21–30]. It is believed that the spiral wave in the cardiac tissue is harmful and thus so many schemes are proposed to suppress the spiral wave and prevent the breakup of spiral wave. Sinha et al. [31] ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Ma). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.027
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simulated the formation of spiral wave in the excitable media with small-world connection type, and other interesting works [5–10,32] were also carried out to explain the mechanism and potential role of spiral wave in the networks of neurons since the important experimental results [33,34]. It is confirmed that spiral waves emerge in disinhabited mammalian cortex and the appearance of spiral wave could play positive role in signal communicating among neurons. For example, the author of this paper discussed the transition of spiral wave to other coherent states by changing some bifurcation parameters, noiseinduced transition of spiral wave etc. [5,6,32]. Neuronal diseases occur when the normal electric signals communication among neurons and domains are destructed abnormally. The original Hodgkin–Huxley (HH) equations are reliable theoretical model to measure the electric activities of neurons [35]. The response of neurons depend on the collective behaviors of all neurons because each domain can consist a large number of neurons, and these neurons can couple with each other in different connection types. Stable rotating spiral waves in rat neocortical slices visualized by voltage-sensitive dye imaging are observed in experiments [33,34], and it is thought that spiral waves might serve as emergent population pacemakers to generate periodical activities in a nonoscillatory network without individual cellular pacemakers. However, the theoretical study about the potential mechanism for formation and breakup of spiral wave in neocortical slices keeps open and it is challengeable to simulate the development and transition of spiral wave in the networks of neurons in theoretical way [9,10,19]. Jr and Brunnet [8] ever investigated the stability of spiral wave in the networks of Hindmarsh–Rose (HR) neurons. In fact, noise can change the dynamics of pattern formation, for example, optimized noise is helpful to support the formation of spiral wave [36] and breakup of spiral wave occurs when the intensity of noise exceeds certain threshold. Therefore, the HH neuron model could be more suitable to construct the networks of neurons and simulate the dynamics of spiral wave in the networks because channel noise [37,38] can be measured. A potential source of electrical ‘channel noise’ in neurons [37] could result from the probabilistic gating of voltage-dependent ion channels. Schmid et al. [38] confirmed that channel noise reduction in stochastic HH systems could be induced by capacitance fluctuations, and the effect of channel noise on the electrical activities of neurons is measured by autocorrelation functions of channel noise [39]. As a result, it is critical to consider the effect of channel noise in the formation and development of spiral wave in the networks. The traveling wave can be changed when the electric field is introduced into the media due to its polarization of field on the media and its potential mechanism is that the excitability is changed. A gradient force is often introduced into the reaction–diffusion equations to simulate the polarization effect of external field on the reaction–diffusion systems while few works were presented to investigate this problem in the networks of neurons. Polk and Postow ever suggested that a membrane potential perturbation at the cellular level [40] could be suitable to measure the polarization effect when the external electric field is imposed on the neuron due to the medium characteristics of cell membrane. Furthermore, Wang et al. [41] and Che et al. [42] ever proposed a modified Hodgkin–Huxley neuron model by introducing an additive term VE over physiological membrane potential to measure the polarization effect of electric field on the neurons and the simulated circuit [41] was also presented to study the bifurcation and synchronization of Hodgkin–Huxley neuron exposed to extremely low frequency electric field. In this paper, the death, robustness and breakup of spiral wave in the regular networks of Hodgkin–Huxley neurons subjected to the external electric field are investigated, and the channel noise is also considered. A statistical factor of synchronization [6] is used to detect the critical value inducing phase transition of spiral wave. A stable rotating spiral wave is selected as the initial state to be changed by the external electric field. This problem is studied in a numerical way. In the case of no channel noise being considered, death of spiral wave in the networks can be observed while breakup of spiral wave occurs in the presence of weak channel noise. Some dynamical evolution of spiral wave is discussed and these results can give some useful information to simulate the response of neuron to exposed electromagnetic field. 2. Mathematical model and discussion The regular networks of Hodgkin–Huxley neurons in the presence of channel noise [39] is described by
Cm
dV ij ¼ g~K n4ij ðV K V ij Þ þ g~Na m3ij hij ðV Na V ij Þ þ g~L ðV L V ij Þ þ Iij þ DðV iþ1j þ V i1j þ V ijþ1 þ V ij1 4V ij Þ; dt
ð1Þ
dmij ¼ am ðV ij Þð1 mij Þ bm ðV ij Þmij þ nm ðtÞ; dt
ð2Þ
dhij ¼ ah ðV ij Þð1 hij Þ bh ðV ij Þhij þ nh ðtÞ; dt
ð3Þ
dnij ¼ an ðV ij Þð1 nij Þ bn ðV ij Þnij þ nn ðtÞ; dt
ð4Þ
am ¼
0:1ðV ij þ 40Þ ; 1 expððV ij þ 40Þ=10Þ
bm ¼ 4expððV ij þ 65Þ=18Þ;
ð5Þ
C.-N. Wang et al. / Applied Mathematics and Computation 218 (2011) 4467–4474
ah ¼ 0:07expððV ij þ 65Þ=20Þ; 1 bh ¼ ; 1 þ expððV ij þ 35Þ=10Þ an ¼
0:01ðV ij þ 55Þ ; 1 expððV ij þ 55Þ=10Þ
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ð6Þ
ð7Þ
bn ¼ 0:125expððV ij þ 65Þ=80Þ; where the variable Vi,j describes the membrane potential of the neuron in the site (i, j), and all the neurons are arranged in a twodimensional square array in space. m, n and h are parameters for gate channel, the capacitance of membrane is Cm = 1 lF/cm2 and the temperature of the membrane is fixed at T = 6.3oC. D is the intensity of coupling, the maximal conductance of potassium is g~K ¼ 36 ms=cm2 , the maximal conductance of sodium is g~Na ¼ 120 ms=cm2 , the conductance of leakage current is g~L ¼ 0:3 ms=cm2 and the external injection current Iij = 0. The reversal potential VK = 77 mV, VNa = 50 mV and VL = 54.4 mV. nm(t), nh(t), nn(t), are independent Gaussian white noise and the statistical properties [39] of the channel noise are defined by
2am bm dðt t 0 Þ ¼ Dm dðt t0 Þ; NNa ðam þ bm Þ 2an bn hnn ðtÞi ¼ 0; hnn ðtÞnn ðt0 Þi ¼ dðt t 0 Þ ¼ Dn dðt t0 Þ; NK ðan þ bn Þ 2ah bh dðt t 0 Þ ¼ Dh dðt t 0 Þ; hnh ðtÞi ¼ 0; hnh ðtÞnh ðt0 Þi ¼ NNa ðah þ bh Þ
hnm ðtÞi ¼ 0; hnm ðtÞnm ðt 0 Þi ¼
ð8Þ ð9Þ ð10Þ
where Dm, Dn, and Dh describes the intensity of noise, respectively. The function d(t t0 ) = 1 at t = t0 and d(t t0 ) = 0 at t – t0 , NNa, NK is the total numbers of sodium and potassium channels presented in a given patch of the membrane, respectively. In the case of homogeneous ion channel density, qNa = 60 lm2 and qK = 18 lm2, the total channels number is decided by NNa = qNas and NK = qKs, and s describes the membrane patch. Based on the mean field theory, a statistical variable is defined to study the collective behaviors and statistical property
F¼ R¼
N X N 1 X
N2
j¼1
V ij ¼ hVis ;
ð11Þ
i¼1
hF 2 i hFi2 ; P P N N 2 2 1 j¼1 i¼1 hV ij i hV ij i N2
ð12Þ
where R is factor of synchronization, the number of neurons is N2 and the variable Vij is the membrane potential of neuron in site (i, j). As reported in the previous works, optimized channel noise and multiplicative noise are helpful to support the survival of spiral wave in the networks of neurons, which can play positive role in signal communicating in breakthrough these quiescent areas; and the curve for factor of synchronization vs bifurcation parameters measures the phase transition of spiral wave by detecting the abrupt changing points in this curve. By using the modified Hodgkin–Huxley neuron model exposed to external weak electric field [41,42], the networks of the modified Hodgkin–Huxley neuron model can be rewritten by
Cm
dV ij ¼ DðV iþ1j þ V i1j þ V ijþ1 þ V ij1 4V ij Þ þ g~K n4ij ðV K V ij V E Þ þ g~Na m3ij hij ðV Na V ij V E Þ IE þ g~L ðV L V ij V E Þ; dt ð13Þ
dyij ¼ ay ðV ij Þð1 yij Þ by ðV ij Þyij þ þny ðtÞ; ðy ¼ m; n; hÞ; dt
ð14Þ
where the parameter VE is the induced transmembrane potential from the external electric field which could be calculated in Refs. [41–45] and IE ¼ C m dVdtE is used to measure the parameter current flowing through the capacitor due to the electric field stimulus. 3. Numerical results and discussion In this section, the external electric filed-induced changes of stable rotating spiral wave in the networks of improved Hodgkin–Huxley neurons model is investigated. An appropriate initial states are used to develop a stable rotating spiral wave in the networks of neurons. In the numerical studies, the time step h = 0.001, coupling intensity D = 1.0, membrane temperature T = 6.3oC, external forcing current Iij = 0, 40000 neurons are arranged in a two-dimensional array with 200 200 sites, and no-flux boundary condition is used. In Fig. 1, the developed stable rotating spiral wave in the networks is shown in gray and this stable rotating spiral wave will be regarded as initial state to be controlled by the external electric field. The stable rotating spiral wave presented in Fig. 1 can be developed with certain transient period in the networks of Hodgkin–Huxley neuron model. Spiral wave becomes sparse at fixed bigger coupling intensity D, the selection of initial values for the four variables can refer to our previous works[32].
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Fig. 1. Stable rotating spiral wave is developed to cover the networks of neurons without channel noise for 600 time units (a), and in the presence of channel noise (membrane patch s = 4) (b), the snapshots are plotted in gray from black (about 80 mV) to white (about 40 mV) and the coupling coefficient D = 1.
The curve in Fig. 2 illustrates the correlation of synchronization factor and the intensity of external electric field, and abrupt changing point close to the peak is observed. Spiral wave is removed and the networks become homogenous completely when the intensity of electric field exceeds certain threshold, otherwise, the spiral wave keeps alive. As a result, the corresponding snapshots are plotted in Fig. 3 to illustrate the death and robustness of spiral wave at different conditions. The results in Fig. 3 confirm that spiral wave encounters death when the intensity of electric field exceeds certain threshold about VE = 2 107, and the time series of average membrane potentials of neurons in the networks decrease to certain stable values, otherwise, the spiral wave keeps alive and the time series of the average membrane potentials of neurons in the networks oscillate periodically vs.time. In fact, channel noise plays important role in changing the electric activities of neurons, therefore, it is important to study this problem in the presence of channel noise. As it is known, breakup of spiral wave occurs when the channel noise exceeds certain thresholds. Our aim in this paper is to study the electric field-induced transition of spiral wave, therefore, weak channel noise is considered, for example, the membrane patch s = 4 is used to develop a stable rotating spiral wave, then the effect of external electric field is considered. The curve in Fig. 4 shows that correlation between the factors of synchronization and the intensity of electric field in the presence of channel noise, and peak is also observed. Spiral wave keeps alive when the intensity of field is less than the critical value about VE = 7 108, otherwise, breakup of spiral wave occurs. Furthermore, snapshots of spiral wave are plotted to illustrate the state when the spiral wave is controlled by the external electric field with the intensity close to the critical value. The results in Fig. 5 confirm that spiral wave used to keep alive when the intensity of electric field is less than the critical value about VE = 7 108, and the time series of average membrane potentials of neurons in the networks oscillate regu-
Fig. 2. Factors of synchronization under different intensities of external electric field (VE) within a transient period about 600 time units are plotted. The critical point is about VE = 1 107, the red line with arrow indicates that higher factor of synchronization is associated to the death of spiral wave in this case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
C.-N. Wang et al. / Applied Mathematics and Computation 218 (2011) 4467–4474
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Fig. 3. The developed state of the stable rotating spiral wave at t = 600 time units under different intensities of external electric field, for VE = 8 108(a,e1), VE = 9 108(b,e2), VE = 1 107(c,e3) and VE = 2 107(d,e4); the time series of average membrane potentials of neurons in the networks are plotted vs. time under different intensities of external electric field close to the critical value, which can be measured from Fig. 2. The snapshots are plotted in gray from black (about 80 mV) to white (about 40 mV) and the coupling coefficient D = 1.
Fig. 4. Factors of synchronization under different intensities of external electric field (VE) within a transient period about 800 time units are plotted, and the channel noise intensity is measured by membrane patch s = 4. The critical point is about VE = 7 108, the red line with arrow indicates that higher factor of synchronization is associated to the breakup of spiral wave in this case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 5. The developed state of the stable rotating spiral wave at t = 800 time units under different intensities of external electric field, for VE = 5 108(a,e1), VE = 6 108(b,e2), VE = 7 107(c,e3) and VE = 8 107(d,e4); the time series of average membrane potentials of neurons in the networks are plotted vs. time under different intensities of external electric field close to the critical value, which can be measured from Fig. 4, the membrane patch s = 4 is used to denote the intensity of channel noise. The snapshots are plotted in gray from black (about 80 mV) to white (about 40 mV) and the coupling coefficient D = 1.
larly, otherwise, spiral wave encounters breakup and the time series of average membrane potentials of neurons oscillate irregularly when the channel noise is being considered. In summary, the death, breakup and survival of spiral wave depends on the electric field intensity greatly. In the case of no channel noise being considered, the spiral wave is removed when the intensity of electric field exceeds the critical value about VE = 2 107 whiles it encounters breakup when the intensity of electric field exceeds the critical value about VE = 7 108; otherwise, the spiral wave used to keep alive completely. On the other hand, the factor of synchronization can denote the development of spiral wave, smaller factor of synchronization often indicates that spiral wave keeps alive, and the critical value can be measured from the curve of factors of synchronization. 4. Conclusions The external electric field is introduced into the regular networks of HH neurons, the polarization effect of field is described by an additive membrane potential perturbation at the cellular level. A statistical factor of synchronization is used
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to measure the abrupt phase transition of spiral wave in the networks, the curve for the factors of synchronization vs. the intensity of electric field are calculated and used to detect the critical threshold of electric field (breakup or death of spiral wave),which is associated with an abrupt changing point. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical threshold, otherwise, the spiral wave keeps alive completely. In the presence of weak channel noise, spiral wave encounters breakup when the intensity of electric field exceed the certain critical intensity of electric field, otherwise, the spiral wave keeps alive well. The curve for factors of synchronization vs. control parameter (e.g. intensity of electric field etc) can be calculated and a stable rotating spiral wave often presents a smaller factor of synchronization. When the biological body is exposed to the electromagnetic field, the normal electric activities of neurons are often perturbed, some neuronal disease could be induced by the electromagnetic radiation. This work could give some potential explanation about the abnormality of neuronal activity because the spiral wave in the domain of neuron system is destroyed and thus normal signal communication is disturbed. Acknowledgments This work is partially supported by the National Nature Science Foundation of China under the Grant Nos. 10972179, 11072099 and the Educational tutors fund projects of Gansu Province under the Grant No. 1010ZTC088 and the Natural Science of Foundation of Lanzhou University of Technology under Grant No. Q200706. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
B.J. Chung, Ashwin Vaidya, Non-equilibrium pattern selection in particle sedimentation, Appl. Math. Comput. 218 (7) (2011) 3451–3465. J.I. Ramos, Robustness of spiral waves in two-dimensional reactive-diffusive media, Appl. Math. Comput. 148 (2004) 681–695. F. Amdjadi, A numerical method for the dynamics and stability of spiral waves, Appl. Math. Comput. 217 (2010) 3385–3391. Z.G. Zheng, X.Q. Feng, B. Ao, et al, Synchronization of groups of coupled oscillators with sparse connections, Europhys. Lett. 87 (2009) 50006. J. Ma, C.N. Wang, W.Y. Jin, et al, Transition from spiral wave to target wave and other coherent structures in the networks of Hodgkin–Huxley Neurons, Appl. Math. Comput. 217 (2010) 3844–3852. J. Ma, J. Tang, A.H. Zhang, et al, Robustness and Breakup of spiral wave in a two-dimensional lattice networks of neurons, Sci. China Ser.: Phys. Mech. Astron. 53 (2010) 72–679. S. Weber, M.T. Hütt, M. Porto, Pattern formation and efficiency of reaction–diffusion processes on complex networks, Europhys. Lett. 83 (2008) 28003. R.E. Jr, L.G. Brunnet, Multistability in networks of Hindmarsh–Rose neurons, Phys. Rev. E 78 (2008) 061917. M. Perc, Effects of small-world connectivity on noise-induced temporal and spatial order in neural media, Chaos Solitons Fract. 31 (2007) 280–291. Q.Y. Wang, M. Perc, Z.S. Duan, Delay-enhanced coherence of spiral waves in noisy Hodgkin–Huxley neuronal networks, Phys. Lett. A 372 (2008) 5681– 5687. A.N. Zaikin, A.M. Zhabotinsky, Concentration wave propagation in two-dimensional liquid-phase self-oscillating system, Nature (London) 225 (1970) 535. J.I. Ramos, Spiral wave break-up and planar front formation in two-dimensional reactive–diffusive media with straining, Appl. Math. Comput. 154 (2004) 697–711. J.I. Ramos, Spiral wave break-up and planar front formation in two-dimensional reactive–diffusive media with straining, Appl. Math. Comput. 146 (2003) 55–72. G.Z. Li, Y.Q. Chen, et al, Spiral wave dynamics in a response system subjected to a spiral wave forcing, Chin. Phys. Lett. 28 (2011) 020504. J.H. Gao, L.L. Xie, H.C. Nie, et al, Novel type of amplitude spiral wave in a two-layer system, Chaos 20 (2010) 043132. Y.Q. Bai, X. Zhu, Numerical simulations of calcium ions spiral wave in single cardiac myocyte, Chin. Phys. Lett. 27 (2010) 048201. Y.Y. Li, H.M. Zhang, C.L. Wei, et al, Stochastic signal induced multiple spatial coherence resonances and spiral waves in excitable media, Chin. Phys. Lett. 26 (2010) 30504. G.F. Gu, Y.P. Lü, G.N. Tang, Reflection and refraction of waves in oscillatory media, Chin. Phys. B 19 (2010) 050515. M. Gosak, M. Marhl, M. Perc, Pacemaker-guided noise-induced spatial periodicity in excitable media, Phys. D 238 (2009) 506–515. F.H. Samie, J. Josem, Mechanisms underlying ventricular tachycardia and its transition to ventricular fibrillation in the structurally normal heart, Cardiovasc. Res. 50 (2001) 42–250. O.U. Kheowan, S.C. Müller, Control of spiral waves in excitable media, Appl. Math. Comput. 164 (2005) 373–390. H.C. Lu, M. Zhong, J.X. Liu, et al, The inverse excitation-induced abnormal spiral wave, Phys. Lett. A 374 (2009) 241–246. C.Y. Yang, G.N. Tang, J.X. Liu, Suppression of spatiotemporal chaos under a constant electric potential signal, Phys. Lett. A 372 (2008) 2230–2236. J.X. Chen, B.B. Hu, Spiral breakup and consequent patterns induced by strong polarized advective field, Europhys. Lett. 84 (2008) 34002. J.M. Luo, M. Zhan, Destruction of spiral waves in chaotic media, Phys. Rev. E 78 (2008) 016214. M. Zhan, J.M. Luo, J.H. Gao, Chirality effect on the global structure of spiral-domain patterns in the two-dimensional complex Ginzburg–Landau equation, Phys. Rev. E 75 (2007) 016214. Z.J. Cao, P. F Li, H. Zhang, et al, Turbulence control with local pacing and its implication in cardiac defibrillation, Chaos 17 (2007) 015107. N.J. Wu, B.W. Li, H.P. Ying, Effects of periodic forcing amplitude on the spiral wave resonance drift, Chin. Phys. Lett. 23 (2006) 2030–2033. J. Garcı´a-Ojalvo, L. Schimansky-Geier, Noise-induced spiral dynamics in excitable media, Europhys. Lett. 47 (2005) 298–303. A. Belmonte, J.M. Flesselles, Q. Ouyang, Spiral instability to line sources in forced chemical pattern turbulence, Europhys. Lett. 35 (1996) 665–670. S. Sinha, J. Saraäki, K. Kaski, Emergence of self-sustained patterns in small-world excitable media, Phys. Rev. E 76 (2007) 015101. R. J. Ma, L.J. Yang, Y. Wu, et al, Spiral wave in the small-world networks of Hodgkin–Huxley neurons, Commun. Theor. Phys. 54 (2010) 583–588. X.Y. Huang, W.C. Troy, Q.C. Yang, et al, Spiral waves in disinhabited mammalian cortex, J. Neurosci. 24 (2004) 9897. S.J. Schiff, X.Y. Huang, J.Y. Wu, Dynamical evolution of spatiotemporal patterns in mammalian middle cortex, Phys. Rev. Lett. 98 (2007) 178102. A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London) 117 (1952) 00–544. W.G. Xu, D.P. Hu, A.Z. Lei, et al, Single spiral wave induced by noise, Appl. Math. Comput. 171 (2005) 703–709. J.A. White, J.T. Rubinstein, A.R. Kay, Channel noise in neurons, Trends. Neurosci. 23 (2000) 131–137. G. Schmid, I. Goychuk, P. Hänggi, Capacitance fluctuations causing channel noise reduction in stochastic Hodgkin–Huxley systems, Phys. Biol. 3 (2006) 248–254. R.F. Fox, Y.N. Lu, Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels, Phys. Rev. E 49 (1994) 3421–3431. C. Polk, E. Postow (Eds.), CRC Handbook of Biological Effects of Electromagnetic Fields, CRC Press, Boca Raton, 1996. J. Wang, K.M. Tsang, H. Zhang, Hopf bifurcation in the Hodgkin–Huxley model exposed to ELF electrical fileld, Chaos Solitons Fract. 20 (2004) 759–764.
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[42] Y.Q. Che, J. Wang, S.S. Zhou, et al, Synchronization control of Hodgkin–Huxley neurons exposed to ELF electric field, Chaos Solitons Fract. 40 (2009) 1588–1598. [43] T. Kotnik, D. Miklavcic, Time course of transmembrane voltage induced by time-varying electric fields – a method for theoretical analysis and its application, Bioelectrochem. Bioenerg. 45 (1998) 3–16. [44] T. Kotnik, D. Miklavcic, Second-order model of membrane electric field induced by alternating external electric fields, IEEE Trans. Biomed. Eng. 45 (2000) 1974–1981. [45] T. Kotnik, D. Miklavcic, Theoretical evaluation of the distributed power dissipation in biological cells exposed to electric fields, Bioelectromagnetics 21 (2000) 385–394.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Electric Field-induced dynamical evolution of spiral wave in the regular networks of Hodgkin–Huxley neurons Chun-Ni Wang a, Jun Ma a,⇑, Wu-Yin Jin b,c, Ying Wu d a
Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China School of Physics and Mechanical & Electrical Engineering, Shao Guan University, Shaoguan 512005, China c College of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050, China d State Key Laboratory for Strength and Vibration of Mechanical Structures; School of Aerospace, Xian Jiaotong University, Xian 710049, China b
a r t i c l e
i n f o
Keywords: Spiral wave Breakup Factor of synchronization Hodgkin–Huxley neurons Channel noise Regular networks
a b s t r a c t An additional gradient force is often used to simulate the polarization effect induced by the external field in the reaction–diffusion systems. The polarization effect of weak electric field on the regular networks of Hodgkin–Huxley neurons is measured by imposing an additive term VE on physiological membrane potential at the cellular level, and the dynamical evolution of spiral wave subjected to the external electric field is investigated. A statistical variable is defined to study the dynamical evolution of spiral wave due to polarization effect. In the numerical simulation, 40000 neurons placed in the 200 200 square array with nearest neighbor connection type. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical value, otherwise, spiral wave keeps alive completely. On the other hand, breakup of spiral wave occurs as the intensity of electric field exceeds the critical value in the presence of weak channel noise, otherwise, spiral wave keeps robustness to the external field completely. The critical value can be detected from the abrupt changes in the curve for factors of synchronization vs. control parameter, a smaller factor of synchronization is detected when the spiral wave keeps alive. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Spiral waves are peculiar spatiotemporal patterns far from equilibrium state and are observed in the reaction–diffusion systems, networks of neurons and oscillators [1–10]. The propagation of chemical wave in the Belousov–Zhabotinsky reaction [11] is often investigated to detect the dynamical evolution of spiral wave in an experimental way. On the other hand, numerical studies are also effective to study the dynamics of these nonlinear waves, for example, Amdjadi [3] presented an effective numerical scheme to study the dynamics and stability of spiral waves. Ramos [12] investigated the propagation of spiral waves in two-dimensional reactive–diffusive media subject to a velocity field with straining and pattern formation [13] in two-dimensional excitable media subject to Robin boundary conditions in numerical way. It does show some significance to study the spiral wave formation, selection, breakup and control in the reaction–diffusion systems or coupled arrays [14–19], because it is helpful to remove spiral wave in the cardiac tissue and thus prevent the occurrence of ventricular fibrillation [20]. Many interesting works have been carried out and most of them are confirmed to be effective to remove spiral wave and prevent the breakup of spiral wave [21–30]. It is believed that the spiral wave in the cardiac tissue is harmful and thus so many schemes are proposed to suppress the spiral wave and prevent the breakup of spiral wave. Sinha et al. [31] ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Ma). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.027
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simulated the formation of spiral wave in the excitable media with small-world connection type, and other interesting works [5–10,32] were also carried out to explain the mechanism and potential role of spiral wave in the networks of neurons since the important experimental results [33,34]. It is confirmed that spiral waves emerge in disinhabited mammalian cortex and the appearance of spiral wave could play positive role in signal communicating among neurons. For example, the author of this paper discussed the transition of spiral wave to other coherent states by changing some bifurcation parameters, noiseinduced transition of spiral wave etc. [5,6,32]. Neuronal diseases occur when the normal electric signals communication among neurons and domains are destructed abnormally. The original Hodgkin–Huxley (HH) equations are reliable theoretical model to measure the electric activities of neurons [35]. The response of neurons depend on the collective behaviors of all neurons because each domain can consist a large number of neurons, and these neurons can couple with each other in different connection types. Stable rotating spiral waves in rat neocortical slices visualized by voltage-sensitive dye imaging are observed in experiments [33,34], and it is thought that spiral waves might serve as emergent population pacemakers to generate periodical activities in a nonoscillatory network without individual cellular pacemakers. However, the theoretical study about the potential mechanism for formation and breakup of spiral wave in neocortical slices keeps open and it is challengeable to simulate the development and transition of spiral wave in the networks of neurons in theoretical way [9,10,19]. Jr and Brunnet [8] ever investigated the stability of spiral wave in the networks of Hindmarsh–Rose (HR) neurons. In fact, noise can change the dynamics of pattern formation, for example, optimized noise is helpful to support the formation of spiral wave [36] and breakup of spiral wave occurs when the intensity of noise exceeds certain threshold. Therefore, the HH neuron model could be more suitable to construct the networks of neurons and simulate the dynamics of spiral wave in the networks because channel noise [37,38] can be measured. A potential source of electrical ‘channel noise’ in neurons [37] could result from the probabilistic gating of voltage-dependent ion channels. Schmid et al. [38] confirmed that channel noise reduction in stochastic HH systems could be induced by capacitance fluctuations, and the effect of channel noise on the electrical activities of neurons is measured by autocorrelation functions of channel noise [39]. As a result, it is critical to consider the effect of channel noise in the formation and development of spiral wave in the networks. The traveling wave can be changed when the electric field is introduced into the media due to its polarization of field on the media and its potential mechanism is that the excitability is changed. A gradient force is often introduced into the reaction–diffusion equations to simulate the polarization effect of external field on the reaction–diffusion systems while few works were presented to investigate this problem in the networks of neurons. Polk and Postow ever suggested that a membrane potential perturbation at the cellular level [40] could be suitable to measure the polarization effect when the external electric field is imposed on the neuron due to the medium characteristics of cell membrane. Furthermore, Wang et al. [41] and Che et al. [42] ever proposed a modified Hodgkin–Huxley neuron model by introducing an additive term VE over physiological membrane potential to measure the polarization effect of electric field on the neurons and the simulated circuit [41] was also presented to study the bifurcation and synchronization of Hodgkin–Huxley neuron exposed to extremely low frequency electric field. In this paper, the death, robustness and breakup of spiral wave in the regular networks of Hodgkin–Huxley neurons subjected to the external electric field are investigated, and the channel noise is also considered. A statistical factor of synchronization [6] is used to detect the critical value inducing phase transition of spiral wave. A stable rotating spiral wave is selected as the initial state to be changed by the external electric field. This problem is studied in a numerical way. In the case of no channel noise being considered, death of spiral wave in the networks can be observed while breakup of spiral wave occurs in the presence of weak channel noise. Some dynamical evolution of spiral wave is discussed and these results can give some useful information to simulate the response of neuron to exposed electromagnetic field. 2. Mathematical model and discussion The regular networks of Hodgkin–Huxley neurons in the presence of channel noise [39] is described by
Cm
dV ij ¼ g~K n4ij ðV K V ij Þ þ g~Na m3ij hij ðV Na V ij Þ þ g~L ðV L V ij Þ þ Iij þ DðV iþ1j þ V i1j þ V ijþ1 þ V ij1 4V ij Þ; dt
ð1Þ
dmij ¼ am ðV ij Þð1 mij Þ bm ðV ij Þmij þ nm ðtÞ; dt
ð2Þ
dhij ¼ ah ðV ij Þð1 hij Þ bh ðV ij Þhij þ nh ðtÞ; dt
ð3Þ
dnij ¼ an ðV ij Þð1 nij Þ bn ðV ij Þnij þ nn ðtÞ; dt
ð4Þ
am ¼
0:1ðV ij þ 40Þ ; 1 expððV ij þ 40Þ=10Þ
bm ¼ 4expððV ij þ 65Þ=18Þ;
ð5Þ
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ah ¼ 0:07expððV ij þ 65Þ=20Þ; 1 bh ¼ ; 1 þ expððV ij þ 35Þ=10Þ an ¼
0:01ðV ij þ 55Þ ; 1 expððV ij þ 55Þ=10Þ
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ð6Þ
ð7Þ
bn ¼ 0:125expððV ij þ 65Þ=80Þ; where the variable Vi,j describes the membrane potential of the neuron in the site (i, j), and all the neurons are arranged in a twodimensional square array in space. m, n and h are parameters for gate channel, the capacitance of membrane is Cm = 1 lF/cm2 and the temperature of the membrane is fixed at T = 6.3oC. D is the intensity of coupling, the maximal conductance of potassium is g~K ¼ 36 ms=cm2 , the maximal conductance of sodium is g~Na ¼ 120 ms=cm2 , the conductance of leakage current is g~L ¼ 0:3 ms=cm2 and the external injection current Iij = 0. The reversal potential VK = 77 mV, VNa = 50 mV and VL = 54.4 mV. nm(t), nh(t), nn(t), are independent Gaussian white noise and the statistical properties [39] of the channel noise are defined by
2am bm dðt t 0 Þ ¼ Dm dðt t0 Þ; NNa ðam þ bm Þ 2an bn hnn ðtÞi ¼ 0; hnn ðtÞnn ðt0 Þi ¼ dðt t 0 Þ ¼ Dn dðt t0 Þ; NK ðan þ bn Þ 2ah bh dðt t 0 Þ ¼ Dh dðt t 0 Þ; hnh ðtÞi ¼ 0; hnh ðtÞnh ðt0 Þi ¼ NNa ðah þ bh Þ
hnm ðtÞi ¼ 0; hnm ðtÞnm ðt 0 Þi ¼
ð8Þ ð9Þ ð10Þ
where Dm, Dn, and Dh describes the intensity of noise, respectively. The function d(t t0 ) = 1 at t = t0 and d(t t0 ) = 0 at t – t0 , NNa, NK is the total numbers of sodium and potassium channels presented in a given patch of the membrane, respectively. In the case of homogeneous ion channel density, qNa = 60 lm2 and qK = 18 lm2, the total channels number is decided by NNa = qNas and NK = qKs, and s describes the membrane patch. Based on the mean field theory, a statistical variable is defined to study the collective behaviors and statistical property
F¼ R¼
N X N 1 X
N2
j¼1
V ij ¼ hVis ;
ð11Þ
i¼1
hF 2 i hFi2 ; P P N N 2 2 1 j¼1 i¼1 hV ij i hV ij i N2
ð12Þ
where R is factor of synchronization, the number of neurons is N2 and the variable Vij is the membrane potential of neuron in site (i, j). As reported in the previous works, optimized channel noise and multiplicative noise are helpful to support the survival of spiral wave in the networks of neurons, which can play positive role in signal communicating in breakthrough these quiescent areas; and the curve for factor of synchronization vs bifurcation parameters measures the phase transition of spiral wave by detecting the abrupt changing points in this curve. By using the modified Hodgkin–Huxley neuron model exposed to external weak electric field [41,42], the networks of the modified Hodgkin–Huxley neuron model can be rewritten by
Cm
dV ij ¼ DðV iþ1j þ V i1j þ V ijþ1 þ V ij1 4V ij Þ þ g~K n4ij ðV K V ij V E Þ þ g~Na m3ij hij ðV Na V ij V E Þ IE þ g~L ðV L V ij V E Þ; dt ð13Þ
dyij ¼ ay ðV ij Þð1 yij Þ by ðV ij Þyij þ þny ðtÞ; ðy ¼ m; n; hÞ; dt
ð14Þ
where the parameter VE is the induced transmembrane potential from the external electric field which could be calculated in Refs. [41–45] and IE ¼ C m dVdtE is used to measure the parameter current flowing through the capacitor due to the electric field stimulus. 3. Numerical results and discussion In this section, the external electric filed-induced changes of stable rotating spiral wave in the networks of improved Hodgkin–Huxley neurons model is investigated. An appropriate initial states are used to develop a stable rotating spiral wave in the networks of neurons. In the numerical studies, the time step h = 0.001, coupling intensity D = 1.0, membrane temperature T = 6.3oC, external forcing current Iij = 0, 40000 neurons are arranged in a two-dimensional array with 200 200 sites, and no-flux boundary condition is used. In Fig. 1, the developed stable rotating spiral wave in the networks is shown in gray and this stable rotating spiral wave will be regarded as initial state to be controlled by the external electric field. The stable rotating spiral wave presented in Fig. 1 can be developed with certain transient period in the networks of Hodgkin–Huxley neuron model. Spiral wave becomes sparse at fixed bigger coupling intensity D, the selection of initial values for the four variables can refer to our previous works[32].
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Fig. 1. Stable rotating spiral wave is developed to cover the networks of neurons without channel noise for 600 time units (a), and in the presence of channel noise (membrane patch s = 4) (b), the snapshots are plotted in gray from black (about 80 mV) to white (about 40 mV) and the coupling coefficient D = 1.
The curve in Fig. 2 illustrates the correlation of synchronization factor and the intensity of external electric field, and abrupt changing point close to the peak is observed. Spiral wave is removed and the networks become homogenous completely when the intensity of electric field exceeds certain threshold, otherwise, the spiral wave keeps alive. As a result, the corresponding snapshots are plotted in Fig. 3 to illustrate the death and robustness of spiral wave at different conditions. The results in Fig. 3 confirm that spiral wave encounters death when the intensity of electric field exceeds certain threshold about VE = 2 107, and the time series of average membrane potentials of neurons in the networks decrease to certain stable values, otherwise, the spiral wave keeps alive and the time series of the average membrane potentials of neurons in the networks oscillate periodically vs.time. In fact, channel noise plays important role in changing the electric activities of neurons, therefore, it is important to study this problem in the presence of channel noise. As it is known, breakup of spiral wave occurs when the channel noise exceeds certain thresholds. Our aim in this paper is to study the electric field-induced transition of spiral wave, therefore, weak channel noise is considered, for example, the membrane patch s = 4 is used to develop a stable rotating spiral wave, then the effect of external electric field is considered. The curve in Fig. 4 shows that correlation between the factors of synchronization and the intensity of electric field in the presence of channel noise, and peak is also observed. Spiral wave keeps alive when the intensity of field is less than the critical value about VE = 7 108, otherwise, breakup of spiral wave occurs. Furthermore, snapshots of spiral wave are plotted to illustrate the state when the spiral wave is controlled by the external electric field with the intensity close to the critical value. The results in Fig. 5 confirm that spiral wave used to keep alive when the intensity of electric field is less than the critical value about VE = 7 108, and the time series of average membrane potentials of neurons in the networks oscillate regu-
Fig. 2. Factors of synchronization under different intensities of external electric field (VE) within a transient period about 600 time units are plotted. The critical point is about VE = 1 107, the red line with arrow indicates that higher factor of synchronization is associated to the death of spiral wave in this case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
C.-N. Wang et al. / Applied Mathematics and Computation 218 (2011) 4467–4474
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Fig. 3. The developed state of the stable rotating spiral wave at t = 600 time units under different intensities of external electric field, for VE = 8 108(a,e1), VE = 9 108(b,e2), VE = 1 107(c,e3) and VE = 2 107(d,e4); the time series of average membrane potentials of neurons in the networks are plotted vs. time under different intensities of external electric field close to the critical value, which can be measured from Fig. 2. The snapshots are plotted in gray from black (about 80 mV) to white (about 40 mV) and the coupling coefficient D = 1.
Fig. 4. Factors of synchronization under different intensities of external electric field (VE) within a transient period about 800 time units are plotted, and the channel noise intensity is measured by membrane patch s = 4. The critical point is about VE = 7 108, the red line with arrow indicates that higher factor of synchronization is associated to the breakup of spiral wave in this case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 5. The developed state of the stable rotating spiral wave at t = 800 time units under different intensities of external electric field, for VE = 5 108(a,e1), VE = 6 108(b,e2), VE = 7 107(c,e3) and VE = 8 107(d,e4); the time series of average membrane potentials of neurons in the networks are plotted vs. time under different intensities of external electric field close to the critical value, which can be measured from Fig. 4, the membrane patch s = 4 is used to denote the intensity of channel noise. The snapshots are plotted in gray from black (about 80 mV) to white (about 40 mV) and the coupling coefficient D = 1.
larly, otherwise, spiral wave encounters breakup and the time series of average membrane potentials of neurons oscillate irregularly when the channel noise is being considered. In summary, the death, breakup and survival of spiral wave depends on the electric field intensity greatly. In the case of no channel noise being considered, the spiral wave is removed when the intensity of electric field exceeds the critical value about VE = 2 107 whiles it encounters breakup when the intensity of electric field exceeds the critical value about VE = 7 108; otherwise, the spiral wave used to keep alive completely. On the other hand, the factor of synchronization can denote the development of spiral wave, smaller factor of synchronization often indicates that spiral wave keeps alive, and the critical value can be measured from the curve of factors of synchronization. 4. Conclusions The external electric field is introduced into the regular networks of HH neurons, the polarization effect of field is described by an additive membrane potential perturbation at the cellular level. A statistical factor of synchronization is used
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to measure the abrupt phase transition of spiral wave in the networks, the curve for the factors of synchronization vs. the intensity of electric field are calculated and used to detect the critical threshold of electric field (breakup or death of spiral wave),which is associated with an abrupt changing point. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical threshold, otherwise, the spiral wave keeps alive completely. In the presence of weak channel noise, spiral wave encounters breakup when the intensity of electric field exceed the certain critical intensity of electric field, otherwise, the spiral wave keeps alive well. The curve for factors of synchronization vs. control parameter (e.g. intensity of electric field etc) can be calculated and a stable rotating spiral wave often presents a smaller factor of synchronization. When the biological body is exposed to the electromagnetic field, the normal electric activities of neurons are often perturbed, some neuronal disease could be induced by the electromagnetic radiation. This work could give some potential explanation about the abnormality of neuronal activity because the spiral wave in the domain of neuron system is destroyed and thus normal signal communication is disturbed. Acknowledgments This work is partially supported by the National Nature Science Foundation of China under the Grant Nos. 10972179, 11072099 and the Educational tutors fund projects of Gansu Province under the Grant No. 1010ZTC088 and the Natural Science of Foundation of Lanzhou University of Technology under Grant No. Q200706. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
B.J. Chung, Ashwin Vaidya, Non-equilibrium pattern selection in particle sedimentation, Appl. Math. Comput. 218 (7) (2011) 3451–3465. J.I. Ramos, Robustness of spiral waves in two-dimensional reactive-diffusive media, Appl. Math. Comput. 148 (2004) 681–695. F. Amdjadi, A numerical method for the dynamics and stability of spiral waves, Appl. Math. Comput. 217 (2010) 3385–3391. Z.G. Zheng, X.Q. Feng, B. Ao, et al, Synchronization of groups of coupled oscillators with sparse connections, Europhys. Lett. 87 (2009) 50006. J. Ma, C.N. Wang, W.Y. Jin, et al, Transition from spiral wave to target wave and other coherent structures in the networks of Hodgkin–Huxley Neurons, Appl. Math. Comput. 217 (2010) 3844–3852. J. Ma, J. Tang, A.H. Zhang, et al, Robustness and Breakup of spiral wave in a two-dimensional lattice networks of neurons, Sci. China Ser.: Phys. Mech. Astron. 53 (2010) 72–679. S. Weber, M.T. Hütt, M. Porto, Pattern formation and efficiency of reaction–diffusion processes on complex networks, Europhys. Lett. 83 (2008) 28003. R.E. Jr, L.G. Brunnet, Multistability in networks of Hindmarsh–Rose neurons, Phys. Rev. E 78 (2008) 061917. M. Perc, Effects of small-world connectivity on noise-induced temporal and spatial order in neural media, Chaos Solitons Fract. 31 (2007) 280–291. Q.Y. Wang, M. Perc, Z.S. Duan, Delay-enhanced coherence of spiral waves in noisy Hodgkin–Huxley neuronal networks, Phys. Lett. A 372 (2008) 5681– 5687. A.N. Zaikin, A.M. Zhabotinsky, Concentration wave propagation in two-dimensional liquid-phase self-oscillating system, Nature (London) 225 (1970) 535. J.I. Ramos, Spiral wave break-up and planar front formation in two-dimensional reactive–diffusive media with straining, Appl. Math. Comput. 154 (2004) 697–711. J.I. Ramos, Spiral wave break-up and planar front formation in two-dimensional reactive–diffusive media with straining, Appl. Math. Comput. 146 (2003) 55–72. G.Z. Li, Y.Q. Chen, et al, Spiral wave dynamics in a response system subjected to a spiral wave forcing, Chin. Phys. Lett. 28 (2011) 020504. J.H. Gao, L.L. Xie, H.C. Nie, et al, Novel type of amplitude spiral wave in a two-layer system, Chaos 20 (2010) 043132. Y.Q. Bai, X. Zhu, Numerical simulations of calcium ions spiral wave in single cardiac myocyte, Chin. Phys. Lett. 27 (2010) 048201. Y.Y. Li, H.M. Zhang, C.L. Wei, et al, Stochastic signal induced multiple spatial coherence resonances and spiral waves in excitable media, Chin. Phys. Lett. 26 (2010) 30504. G.F. Gu, Y.P. Lü, G.N. Tang, Reflection and refraction of waves in oscillatory media, Chin. Phys. B 19 (2010) 050515. M. Gosak, M. Marhl, M. Perc, Pacemaker-guided noise-induced spatial periodicity in excitable media, Phys. D 238 (2009) 506–515. F.H. Samie, J. Josem, Mechanisms underlying ventricular tachycardia and its transition to ventricular fibrillation in the structurally normal heart, Cardiovasc. Res. 50 (2001) 42–250. O.U. Kheowan, S.C. Müller, Control of spiral waves in excitable media, Appl. Math. Comput. 164 (2005) 373–390. H.C. Lu, M. Zhong, J.X. Liu, et al, The inverse excitation-induced abnormal spiral wave, Phys. Lett. A 374 (2009) 241–246. C.Y. Yang, G.N. Tang, J.X. Liu, Suppression of spatiotemporal chaos under a constant electric potential signal, Phys. Lett. A 372 (2008) 2230–2236. J.X. Chen, B.B. Hu, Spiral breakup and consequent patterns induced by strong polarized advective field, Europhys. Lett. 84 (2008) 34002. J.M. Luo, M. Zhan, Destruction of spiral waves in chaotic media, Phys. Rev. E 78 (2008) 016214. M. Zhan, J.M. Luo, J.H. Gao, Chirality effect on the global structure of spiral-domain patterns in the two-dimensional complex Ginzburg–Landau equation, Phys. Rev. E 75 (2007) 016214. Z.J. Cao, P. F Li, H. Zhang, et al, Turbulence control with local pacing and its implication in cardiac defibrillation, Chaos 17 (2007) 015107. N.J. Wu, B.W. Li, H.P. Ying, Effects of periodic forcing amplitude on the spiral wave resonance drift, Chin. Phys. Lett. 23 (2006) 2030–2033. J. Garcı´a-Ojalvo, L. Schimansky-Geier, Noise-induced spiral dynamics in excitable media, Europhys. Lett. 47 (2005) 298–303. A. Belmonte, J.M. Flesselles, Q. Ouyang, Spiral instability to line sources in forced chemical pattern turbulence, Europhys. Lett. 35 (1996) 665–670. S. Sinha, J. Saraäki, K. Kaski, Emergence of self-sustained patterns in small-world excitable media, Phys. Rev. E 76 (2007) 015101. R. J. Ma, L.J. Yang, Y. Wu, et al, Spiral wave in the small-world networks of Hodgkin–Huxley neurons, Commun. Theor. Phys. 54 (2010) 583–588. X.Y. Huang, W.C. Troy, Q.C. Yang, et al, Spiral waves in disinhabited mammalian cortex, J. Neurosci. 24 (2004) 9897. S.J. Schiff, X.Y. Huang, J.Y. Wu, Dynamical evolution of spatiotemporal patterns in mammalian middle cortex, Phys. Rev. Lett. 98 (2007) 178102. A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London) 117 (1952) 00–544. W.G. Xu, D.P. Hu, A.Z. Lei, et al, Single spiral wave induced by noise, Appl. Math. Comput. 171 (2005) 703–709. J.A. White, J.T. Rubinstein, A.R. Kay, Channel noise in neurons, Trends. Neurosci. 23 (2000) 131–137. G. Schmid, I. Goychuk, P. Hänggi, Capacitance fluctuations causing channel noise reduction in stochastic Hodgkin–Huxley systems, Phys. Biol. 3 (2006) 248–254. R.F. Fox, Y.N. Lu, Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels, Phys. Rev. E 49 (1994) 3421–3431. C. Polk, E. Postow (Eds.), CRC Handbook of Biological Effects of Electromagnetic Fields, CRC Press, Boca Raton, 1996. J. Wang, K.M. Tsang, H. Zhang, Hopf bifurcation in the Hodgkin–Huxley model exposed to ELF electrical fileld, Chaos Solitons Fract. 20 (2004) 759–764.
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[42] Y.Q. Che, J. Wang, S.S. Zhou, et al, Synchronization control of Hodgkin–Huxley neurons exposed to ELF electric field, Chaos Solitons Fract. 40 (2009) 1588–1598. [43] T. Kotnik, D. Miklavcic, Time course of transmembrane voltage induced by time-varying electric fields – a method for theoretical analysis and its application, Bioelectrochem. Bioenerg. 45 (1998) 3–16. [44] T. Kotnik, D. Miklavcic, Second-order model of membrane electric field induced by alternating external electric fields, IEEE Trans. Biomed. Eng. 45 (2000) 1974–1981. [45] T. Kotnik, D. Miklavcic, Theoretical evaluation of the distributed power dissipation in biological cells exposed to electric fields, Bioelectromagnetics 21 (2000) 385–394.