亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Dynamics of Spiral Waves Induced by Periodic Mechanical Deformation with Phase Di ff erence?

        2018-12-13 06:33:28PengFeiLi李鵬飛LiYanQiao喬麗顏YeHuaZhao趙葉華JiangXingChen陳江星andJunMa馬軍
        Communications in Theoretical Physics 2018年12期
        關(guān)鍵詞:馬軍李鵬

        Peng-Fei Li(李鵬飛),Li-Yan Qiao(喬麗顏), Ye-Hua Zhao(趙葉華),,? Jiang-Xing Chen(陳江星),and Jun Ma(馬軍)

        1Department of Mathematics,Hangzhou Dianzi University,Hangzhou 310018,China

        2Department of Physics,Hangzhou Dianzi University,Hangzhou 310018,China

        3Department of Physics,Lanzhou University of Technology,Lanzhou 730050,China

        AbstractThe dynamics of spiral waves under the in fluences of periodic mechanical deformation are studied.Here,the mechanical deformation propagating along the medium with phase differences are considered.It is found that weak mechanical deformation may lead to resonant drift of spiral waves.The drift direction and velocity can be changed by the wave length of the deformation.Strong mechanical deformation may result in breakup of spiral waves.The characteristics of breakup are discussed.The critical amplitudes are determined by two factors,i.e.the wave length and frequency of the periodic mechanical deformation.When the wave length of mechanical deformation is comparable to the spiral wave,simulation shows that the critical amplitude is substantially increased.As the frequency of the mechanical deformation is around 1.5 times of the spiral wave,the critical amplitudes are minimal.

        Key words:drift,breakup of spiral waves,periodic mechanical deformation,phase difference

        1 Introduction

        Spatiotemporal patterns arising from reactiondiffusion mechanism are ubiquitous in the nature.[1?3]Spiral waves represent typical two-dimensional spatiotemporal patterns in nonlinear media.Therefore,they have been observed and studied in a wide variety of physical,biological,and chemical far-from-equilibrium systems.[4?9]Wellknown examples include chemical waves in the Belousov-Zabotinsky(BZ)reaction,[10?11]CO oxidation on platinum surfaces,[12]activities of neurons in the brain neural network,[13]and electrical waves in cardiac tissue.[14?15]A major avenue of research is concerned with the formation of spiral waves and the transitions that lead to spatiotemporal chaos,which is principally from the origin of ventricular fibrillation,a lift-threatening arrhythmia.

        Due to intrinsic properties of the media,spiral waves have exhibited rich and complicated dynamics.[16?18]The propagation of spiral waves is frequently accompanied by periodic mechanical deformation(PMD)of the medium,which in turns greatly affects their dynamics.It was found that weak PMD may lead to resonant drift if its frequency is equal to that of the spiral wave.[19]In our recent studies,it was shown that the effect of PMD-induced-drift can unpin a spiral wave from an obstacle.[20]On the other hand,strong PMD may result in breakup of spiral waves[21]while a high degree of homogeneous PMD is favored to prevent the low-excitability-induced breakup of spiral waves.[22]Those effects are also reported in mechanoeletrical feedback(MEF)models developed by Pan filov et al.as well as formation of spiral waves.[23]

        In the normal physiological situation,the natural pacemaker originates electrical waves of excitation that subsequently propagate through the cardiac tissue. The contraction of a cardiac muscle cell is then caused by its excitation via the process of excitation-contraction coupling.[24]Therefore,the electric waves need time to propagate out,which results in phase differences of mechanical deformation along the cardiac tissue.Despite the rapid progress of mathematical model simulating the activity of PMD,the in fluences of PMD resulting from propagation phase differences have not been explored so far,which makes the studies meaningful.

        Here,we address the effects from the phase difference of a PMD on the dynamics of a spiral wave.The drift behavior is in fluenced by the wave length of the PMD.The critical amplitude of the PMD above which the spiral wave breaks up.Two factors are considered,i.e.the wave length and the frequency of the PMD.

        2 Model and Method

        Our numerical simulations are carried by means of the B?r model based on a two-variable reaction-diffusion equation,[25]which is generally used to describe the excitation dynamics of general excitable media

        The function g(u,v)takes the form as[u(1?u)/ε][u ?(v+b)/a]and f(u,v)is a piecewise function expressed by f(u)=0,if 0≤u<1/3;f(u)=1?6.75u(u?1)3,if 1/3≤u≤1;f(u)=1,if 1

        The periodic mechanical deformation of a medium is represented by an operation where any fixed point r of the medium is changed to r′.Here,we consider only one direction(x-direction)and adopt a simple oscillation x′=x[1+Acos(ωdt?2πy/λ+?)]with A the amplitude,ωdthe frequency,and ? the initial phase,respectively.The term 2πy/λ denotes the propagation of PMD along the y-direction.Then,Eq.(1)is modified to

        The 2D system is discretized into 256×256 grids with spacial step ?r=0.39.Equations(1)–(2)are integrated numerically using the Euler algorithm with time step?t=0.02.No- flux boundary condition is applied.A spiral wave with frequency ω0and wave length λ0is developed in the system as the initial condition.

        3 Results and Discussion

        For a given weak amplitude of the PMD,the system resonates at frequency ω/ω0=1.0,which results in a spiral drift in a straight line.The corresponding trajectories of the spiral tip are shown in Fig.1 where the drift behavior of a counterclockwise and a clockwise rotating spiral wave is presented(σ = ±1 represent the chirality of the spiral,σ=1 for counterclockwise rotating spiral,while σ= ?1 for the clockwise one).For a counterclockwise spiral in Fig.1(a),the drift direction changes clockwise when the wave length λ is decreased.The case for a clockwise spiral is opposite,as shown in Fig.1(b).The drift velocity is also in fluenced by the wave length.In the inset of Fig.1(a),we plot the dependence of velocity on the wave length.One can see that the drift velocity of spiral decreases with ω/ω0.

        Fig.1 (Color online)Dependence of the drift on the wave length(λ/λ0)of PMD at resonant frequency ω/ω0=1.0 for σ=1(a)and σ= ?1(b)with A=0.01.The inset shows the drift velocities at different wave lengths.

        There is a critical curvature Kcabove,which an excitable wave may break up in this local region with subsequent emergence of two free tips.Mikhailov et al.analytically pointed out that the dependence of Kcon the diffusion constant D is Kc~ D?1.[26]From Eq.(2),one can see that the PMD results in a time-dependent diffusion coefficient,i.e.,

        Consequently,the value of Kcis also time-dependent.When the PMD is applied on the medium,it leads to the deformation of the rotating spiral wave.In Fig.2,a process showing a rotating spiral wave under the in fluences of a PMD with big λ (=14λ0)is presented.At t=8,the spiral wave is deformed,which leads to temporal KT.In this moment,one can find that the wave is flat along the y-direction,which means temporal KTis small.If the amplitude A is big enough and other conditions are suitable,eventually,the temporal KTwill be bigger than time-dependent Kc.Consequently,the spiral wave breaks up,as illustrated at t=12 in Fig.2.A common feature is that the first region where a spiral wave begins to break up locates in the outer spiral arm tangent to the y-axis,which can be seen form the pattern at t=12.The development of breakup reaches a chaotic state at t=90.

        Then,we decrease the wave length λ to study its in fluence on spiral breakup.In Fig.3,we present the Ac?λ/λ0curves where Acis critical amplitude above which spiral breakup occurs.Three curves are presented to shows cases with different frequencies of the PMD.The changes of Acare not obvious even the values of λ are decreased greatly.However,as the value of λ/λ0approaches around 4.0,the values of Acare increased substantially.Three cases in Fig.3 show this point.

        Fig.2 (Color online)Evolution of spiral breakup under the in fluence of a PMD with A=0.53,ω =1.5ω0,and λ =14λ0.The spiral wave starts to break up after short time transient rotations.Two free ends appearing at t=12 results in subsequent breakup to a turbulent state at t=90.

        Fig.3 (Color online)Dependence of critical amplitude of Acabove spiral waves break up on λ/λ0.The triangle,square,and circle curves are plotted from the frequencies with ω/ω0=1.0,ω/ω0=2.0,and ω/ω0=1.5,respectively.

        To understand the mechanism which underpins the increases of Acas λ is decreased to value comparable to λ0,we present a process of spiral breakup by a PMD with λ =1.0λ0in Fig.4.Note that the amplitude in this case is much bigger than that in Fig.2.Di ff erent from the case in Fig.2,the small λ (1.0λ0)ruffles the spiral wave,which can be seen from the pattern at t=3,8,10.The same as that in Fig.2,the convex part with positive curvature inclines to spiral breakup.

        However,the concave part of the wave with negative curvature,may increase the velocity of wave propagation according to the relation V=c?DK where c is the velocity of a plane wave.[27]Compared to the cases with big wave length,the presented spiral wave with small wave length shows that the concave part is close to the convex part.Consequently,the convex parts are dragged by the close-by concave parts,which makes it resistant to be broken with bigger Ac.Compared to the case in Fig.2,the time sequence in Fig.4 shows the difference process illustrating the dynamics of spiral breakup.

        Fig.4 (Color online)Time sequence illustrating the dynamics of spiral breakup induced by a PMD with A=0.64,ω/ω0=1.5,and λ =1.0λ0.

        Fig.5 (Color online)Dependence of critical amplitude Acon the frequency of the PMD ω/ω0.The square and circle curves are plotted from the wave length λ/λ0=1.0 and 2.0,respectively.

        The frequency of the PMD(ω)also plays an important role in determining the value of Ac.In Fig.5,we present the Ac?ω/ω0curve with wave length comparable to that of the spiral(λ/λ0=1.0 and 2.0).One can see a tonguelike region in both cases.An interesting point is that Acis smallest when the ω is around 1.5ω0.From the evolution of spiral breakup,simulation shows that the plausible mechanism is that the deformation of spiral wave by PMD is serious at this frequency.

        4 Conclusion

        In conclusion,we have studied the dynamics of spiral waves induced by periodic mechanical deformation.The phase difference of PMD is firstly introduced.Weak PMD may induce resonant drift of spiral waves.The drift direction changes clockwise for a counterclockwise rotating spiral while the case is opposite for a counterclockwise spiral.The drift velocity decreases with the wave length λ/λ0.It is found that strong PMD results in breakup of spiral waves.The process and underlying mechanism of breakup are discussed.When the wave length of the PMD is decreased to that comparable to that of a spiral,simulation shows the critical value(Ac)of amplitude of the PMD increases substantially.The mechanism is attributed to the appearance of ruffles and consequent dynamics.The frequency of the PMD plays another important role in determining the values of Ac.It is found that the values of Acare smallest when the frequency of PMD is around 1.5 times of the spiral wave.We believe our results can be observed in chemical experiments such as BZ reaction in gels.Furthermore,since our simulation bases on the consideration of realistic conditions in cardiac tissue where the heart is stretching and contracting induced by the pacemaker all the time,we hope our results will contribute to understand the dynamics of spiral breakup in these cases.

        猜你喜歡
        馬軍李鵬
        Enhance sensitivity to illumination and synchronization in light-dependent neurons?
        Control of firing activities in thermosensitive neuron by activating excitatory autapse?
        Estimation of biophysical properties of cell exposed to electric field
        請您來給小李解疑惑
        中國儲運(2019年9期)2019-09-16 08:42:52
        人工智能的困惑
        中國儲運(2019年2期)2019-04-29 03:56:06
        Synergy and Redundancy in a Signaling Cascade with Different Feedback Mechanisms?
        Talk about music content and emotion of music movie "The Legend of 1900"
        東方教育(2017年12期)2017-08-23 05:49:54
        “賭”還是不“賭”?
        中國儲運(2017年2期)2017-02-24 08:27:41
        無人機配送,看上去很美
        中國儲運(2016年4期)2016-06-28 02:16:01
        Multi—parameter real—time monitoring scheme for powertransmission lines based on FBG sensors
        亚洲性av少妇中文字幕| 亚洲精品免费专区| 99re国产电影精品| 一区二区三区在线日本视频 | 极品美女调教喷水网站| 亚洲乱亚洲乱妇无码麻豆| 亚洲国产人在线播放首页| 亚洲欧美久久婷婷爱综合一区天堂| 国产自拍三级黄片视频| 日日麻批免费40分钟无码| 国产精品麻豆成人av电影艾秋| 久久中文字幕久久久久| 精品黑人一区二区三区久久hd| 麻豆亚洲av熟女国产一区二| 日日躁夜夜躁狠狠久久av| 在线a人片免费观看国产| 激情亚洲不卡一区二区| 国产精品欧美久久久久久日本一道 | 精品亚洲一区二区三区四| 色欲人妻综合网| 精品国偷自产在线不卡短视频| 亚洲一区在线二区三区| 东京热人妻系列无码专区 | 国产91对白在线观看| 偷窥偷拍一区二区三区| 极品老师腿张开粉嫩小泬| 亚洲人成亚洲精品| 极品av在线播放| 青青草在线免费播放视频| 国产成人亚洲精品| 久久精品一品道久久精品9 | 亚洲悠悠色综合中文字幕| 中文字幕在线观看亚洲日韩| 无码精品一区二区三区超碰 | 人妻有码中文字幕在线不卡| 内射爆草少妇精品视频| 亚洲精品久久久久中文字幕| 久久综合网天天 | 亚洲熟女av一区少妇| 久久精品亚洲精品国产色婷| 日韩欧美亚洲综合久久影院d3|