心肌细胞同步化节律的实验观察及其动力学研究
|
摘要
生命活动的正常进行依赖于心脏有节律的搏动。但是心脏的搏动节律不是均匀的,而是复杂的、非线性的。目前,人们认识到心脏节律复杂性表现在三个层次上。整体心脏中,神经、体液调节下的心脏节律的波动是内源性的,并具有混沌的性质；在心脏内兴奋的传导过程中,各级单元的失匹配会导致节律紊乱,如异位节律和折返性心率失常。在起搏节律上,窦房结中的细胞具有异质性,它们耦合而成的同步化的心脏起搏节律也具有复杂性。心脏起搏节律的复杂性是心脏节律复杂性的重要原因。因此,研究同步化的起搏节律的形成是很必要的。心肌单细胞的搏动节律极不稳定,节律变异系数大,较难从中发现可辨识的、统一的规律。同步化收缩的心肌细胞片在正常灌流液中呈现多种模式的搏动节律,周期节律(周期1、周期2、周期3)、阵发节律、整数倍节律和非周期节律等。以前,人们研究从单细胞到细胞团的同步化形成过程都是在大的网络中观察几个细胞从不同步搏动到同步化搏动的。但是这种对同步化形成过程的研究方法不能够反映同步化形成过程的细节现象。本文将选取相邻的两个或三个细胞所在的小网络,通过两个或三个细胞从不同步到同步化搏动的同步化过程的观察来说明同步化形成的具体过程。此外,通过网络的数学模型,仿真了在培养心肌细胞网络中观察到的已经达到同步化的周期1节律,用以解释心脏正常起步节律的形成机制。
本文通过活细胞工作站,采用光密度记录手段,定位记录两个或三个细胞从不同步博动到同步搏动的过程。本研究还采用钙荧光记录手段,观察到心肌细胞小网络形成的同步化节律是多样的,其中包括同步化程度高的周期一节律,但不是完全同步化的,有较小的平均相位差。为了研究心脏正常起搏节律的产生机制,我们运用非线性动力学理论与方法。研究结果如下：
1.两个或三个独立搏动的细胞随着培养时间的增加,也就是耦合强度的增加,会形成同步化的搏动节律。
2.同步初期的同步化节律是复杂多样的。
3.搏动节律从不同步到同步化的过程是复杂多样的。例如,在一些两个细胞网络中,部分搏动先达到同步,然后所有搏动达到同步。在三个细胞的网络中,两个细胞会先到达同步,然后三个细胞都达到同步。
4.随着培养时间的增加和心肌细胞网络中细胞个数的增加可以稳定网络的同步化节律为类似周期节律。
5.选用不同参数配置的Morris-Lecar(ML)模型模拟心肌细胞的异质性,在邻近耦合的异质振子构成的心肌细胞网络,振子参数远离Hopf分岔点时,可以仿真与实验相似的同步化周期一节律。
6.耦合强度越大,不同细胞的相位差就会越小但不为零；噪声作用下的细胞节律间的相位差略有增大。
本研究结果不仅揭示了同步化节律形成过程中的节律变化特征,还提供生物系统的节律同步的实验例证。仿真结果提示耦合强度是节律同步的原因,而振子的异质性和噪声是产生相位差的原因。研究结果给出了心肌细胞网络同步化周期一节律形成的动力学解释。这就从理论上解释了同步化周期节律的产生机制,有助于认识正常心脏的起搏节律。
The normal activity of life depends on rhythmic beating of the heart. However, the beating of heart is not of the uniformity but complex and nonlinear. People know that cardiac beating rhythms exhibit complex characteristics in multiple levels. In the whole heart, the fluctuation of rhythm which is adjusted by nerve and body fluid is endogenesis, and exhibits the characteristics of chaos. When units of conduction of heart's excitement mismatch, the rhythm will be turbulenced, such as the ectopic rhythm and arrhythmia. For the pace-making rhythm, the synchronized rhythms of cells in sinoatrial node are diverse, because they are heterogeneity. The diversity of pacemaker rhythm is the essential reason for the complex rhythm in the whole heart. In a word, it's necessary to study the synchronized pace-making rhythm. That the beating rhythm of single cardiac myocyte was unstable and exhibited bigger coefficient of variation was discovered in the previous experiment. It's difficult to find the recognizable and general rules. In normal solution, the synchronized beating rhythms of cultured cardiac cells exhibit periodic rhythms including period 1, period 2 and period 3, periodic rhythm with intermittency, integer multiple rhythms and non-periodic rhythms and so on. The procedure of the formation of synchronized process from single cells to network were studied in several cells whose behaviors were changed from non-synchronized rhythms to synchronized rhythms in a large scale network in previous studies. But the research method could not acquire the details in procedure to form synchronization. In this study, the network composed of two or three interfacing cells are employed to study the procedure of synchronized rhythms through the observation of the procedure from non-synchronized rhythm to synchronized rhythm in the cells. In addition, the dynamics of synchronized period 1 rhythm observed in the network composed of cultured cardiac cells are simulated in a mathematical network, and employed to interpret the periodic pace-making rhythms of heart.
The beating rhythms of two or three cultured single cardiac myocytes were recorded by light density on workstation of living cells. The synchronized rhythms of networks exhibited diversity including synchronous period 1 rhythms, using calcic fluorescence recording method. Synchronous period 1 rhythms exhibited non-complete synchronization and weak phase difference. The results are as follow:
1. With the increases of cultured time as well as coupling strength, independent rhythms of two or three myocytes became synchronous rhythms.
2. The synchronized rhythms are diversity in starting synchronization.
3. The changes of beating rhythm from independence to synchronization are complex and of diversity. For example, in network composed of two myocytes, some of beatings became synchronization while others were independent firstly, and then all beatings became synchronous. In networks composed of three myocytes, beating rhythms of two myocytes achieved synchronization firstly, and then rhythms of three cells acquired synchronization.
4. The synchronized rhythms were stabilized in period rhythm almostly with increasing of cultured time and the number of cells in networks.
5. Similar synchronized rhythms can be simulated in a network composed of heterogeneous cells, which is described by Morris-Lecar model with different parameters and with neighboring coupling manner when the parameters are far from the bifurcation Hopf points.
6. If the coupling strength is much stronger, the phase difference between different cells is much lower, but the value is not zero. Noise can slightly increase the phase difference, but the synchronous behavior is influenced little. The phase difference increases slightly when noise density is increased.
The results not only gave changing characteristics during the procedure of forming synchronous rhythm, but also provided experimental demonstration of rhythm synchronization in life system. The simulation results imply that coupling strength leads to synchronization and heterogeneity of cells and noise lead to phase difference. The results provided theoretical interpretation of the synchronized period 1 rhythm, helpful to recognize the normal pace-making rhythm of heart.
本文通过活细胞工作站,采用光密度记录手段,定位记录两个或三个细胞从不同步博动到同步搏动的过程。本研究还采用钙荧光记录手段,观察到心肌细胞小网络形成的同步化节律是多样的,其中包括同步化程度高的周期一节律,但不是完全同步化的,有较小的平均相位差。为了研究心脏正常起搏节律的产生机制,我们运用非线性动力学理论与方法。研究结果如下：
1.两个或三个独立搏动的细胞随着培养时间的增加,也就是耦合强度的增加,会形成同步化的搏动节律。
2.同步初期的同步化节律是复杂多样的。
3.搏动节律从不同步到同步化的过程是复杂多样的。例如,在一些两个细胞网络中,部分搏动先达到同步,然后所有搏动达到同步。在三个细胞的网络中,两个细胞会先到达同步,然后三个细胞都达到同步。
4.随着培养时间的增加和心肌细胞网络中细胞个数的增加可以稳定网络的同步化节律为类似周期节律。
5.选用不同参数配置的Morris-Lecar(ML)模型模拟心肌细胞的异质性,在邻近耦合的异质振子构成的心肌细胞网络,振子参数远离Hopf分岔点时,可以仿真与实验相似的同步化周期一节律。
6.耦合强度越大,不同细胞的相位差就会越小但不为零；噪声作用下的细胞节律间的相位差略有增大。
本研究结果不仅揭示了同步化节律形成过程中的节律变化特征,还提供生物系统的节律同步的实验例证。仿真结果提示耦合强度是节律同步的原因,而振子的异质性和噪声是产生相位差的原因。研究结果给出了心肌细胞网络同步化周期一节律形成的动力学解释。这就从理论上解释了同步化周期节律的产生机制,有助于认识正常心脏的起搏节律。
The normal activity of life depends on rhythmic beating of the heart. However, the beating of heart is not of the uniformity but complex and nonlinear. People know that cardiac beating rhythms exhibit complex characteristics in multiple levels. In the whole heart, the fluctuation of rhythm which is adjusted by nerve and body fluid is endogenesis, and exhibits the characteristics of chaos. When units of conduction of heart's excitement mismatch, the rhythm will be turbulenced, such as the ectopic rhythm and arrhythmia. For the pace-making rhythm, the synchronized rhythms of cells in sinoatrial node are diverse, because they are heterogeneity. The diversity of pacemaker rhythm is the essential reason for the complex rhythm in the whole heart. In a word, it's necessary to study the synchronized pace-making rhythm. That the beating rhythm of single cardiac myocyte was unstable and exhibited bigger coefficient of variation was discovered in the previous experiment. It's difficult to find the recognizable and general rules. In normal solution, the synchronized beating rhythms of cultured cardiac cells exhibit periodic rhythms including period 1, period 2 and period 3, periodic rhythm with intermittency, integer multiple rhythms and non-periodic rhythms and so on. The procedure of the formation of synchronized process from single cells to network were studied in several cells whose behaviors were changed from non-synchronized rhythms to synchronized rhythms in a large scale network in previous studies. But the research method could not acquire the details in procedure to form synchronization. In this study, the network composed of two or three interfacing cells are employed to study the procedure of synchronized rhythms through the observation of the procedure from non-synchronized rhythm to synchronized rhythm in the cells. In addition, the dynamics of synchronized period 1 rhythm observed in the network composed of cultured cardiac cells are simulated in a mathematical network, and employed to interpret the periodic pace-making rhythms of heart.
The beating rhythms of two or three cultured single cardiac myocytes were recorded by light density on workstation of living cells. The synchronized rhythms of networks exhibited diversity including synchronous period 1 rhythms, using calcic fluorescence recording method. Synchronous period 1 rhythms exhibited non-complete synchronization and weak phase difference. The results are as follow:
1. With the increases of cultured time as well as coupling strength, independent rhythms of two or three myocytes became synchronous rhythms.
2. The synchronized rhythms are diversity in starting synchronization.
3. The changes of beating rhythm from independence to synchronization are complex and of diversity. For example, in network composed of two myocytes, some of beatings became synchronization while others were independent firstly, and then all beatings became synchronous. In networks composed of three myocytes, beating rhythms of two myocytes achieved synchronization firstly, and then rhythms of three cells acquired synchronization.
4. The synchronized rhythms were stabilized in period rhythm almostly with increasing of cultured time and the number of cells in networks.
5. Similar synchronized rhythms can be simulated in a network composed of heterogeneous cells, which is described by Morris-Lecar model with different parameters and with neighboring coupling manner when the parameters are far from the bifurcation Hopf points.
6. If the coupling strength is much stronger, the phase difference between different cells is much lower, but the value is not zero. Noise can slightly increase the phase difference, but the synchronous behavior is influenced little. The phase difference increases slightly when noise density is increased.
The results not only gave changing characteristics during the procedure of forming synchronous rhythm, but also provided experimental demonstration of rhythm synchronization in life system. The simulation results imply that coupling strength leads to synchronization and heterogeneity of cells and noise lead to phase difference. The results provided theoretical interpretation of the synchronized period 1 rhythm, helpful to recognize the normal pace-making rhythm of heart.
引文
[1]C. M. Gray, P. Konig, A. K. Engel and W. Singer. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties[J]. Nature.1989, 338:334～337.
[2]P. N. Steinmeta, A. Roy, P. J. Fitzgerald, et al. Attention modulates synchronized neuronal firing in primate somatosensory cortex[J]. Nature.2000,404:187～190.
[3]S. Boccaletti, J. Kurths, G. Osipov,. D. L. Valladares, C. S. Zhou. The synchronization of chaotic systems[M]. Phys Rep.2002,366(1-2):1～101.
[4]S. Bahar, F. Moss. Stochastic resonance and synchronization in the crayfish caudal photoreceptor[J]. Math Biosci.2004,188:81～97.
[5]M. R. Boyett, H. Honjo,I. Kodama. The sinoatrial node, a heterogeneous pacemaker structure[J]. Cardiovasc Res.2000,47:658～687.
[6]李澈.窦房结[M].北京：北京医科大学出版社,2001:134～138.
[7]D. R. Rigney, J. E. Mietus, A. L. Goldberger. Is normal sinus rhythm "chaotic"? Measurement of lyapunov exponents(abstract). Circulation.1990,82:111～236.
[8]A. Babloyantz, A. Destexhe. Is the normal heart periodic oscillator[J]? Biol Cybern.1988,58: 203～211.
[9]M. R. Guevara, L. Glass, A. Shrier. Phase locking, perioddoubling bifurcation, and irregular dynamics in periodically stimulated cardiac cells[J]. Science.1981,214:1350～1353.
[10]L. Glass, M. R. Guevara, A. Shrier. Bifurcation and chaos in a periodically stimulated cardiac oscillator[J]. Physica D.1983,7:89～101.
[11]D. R. Chialvo, R. F. Gilmour, J. Jalife. Low dimensional chaos in cardiac tissue[J]. Nature.1990, 343:653～657.
[12]W. B. Cannon. Organization for physiological homeostatics[J]. Phys Rev.1929,9:399～431.
[13]A. L. Goldberger. Is the normal heartbeat chaotic or homeostatic[J]? News physical sciences. 1991,6:87～91.
[14]A. L. Goldberger. Nonlinear dynamics, fractals and chaos:applications to cardiac electrophysiplogy[J]. Ann Biomed Eng,1990,18(2):195～198.
[15]A. L. Goldberger, D. R. Rigney, J. Mietus, et al. Nonlinear dynamics in sudden cardiac death syndrome:heartrate oscillations and bifurcations[J]. Experientia.1988,44(11-12):983～987.
[16]W. L. Ditto, M. L. Spano, J. Neff, et al. Control of human atrial fibrillation[J]. Int J Bif Chaos. 2000,10(3):593～601.
[17]A. Garfinkel, M. L. Spano, W. L. Ditto, et al. Controlling cardiac chaos[J]. Science.1992,257: 1230～1235.
[18]M. A. Allessie, F.I. M. Bonke, F. J. G. Schoppman. Circus movement in rabbit atrail muscle as a mechanism of tachycardia. The "leading circle" concept[J]. Cir Res.1997,41:9～18
[19]M. J. Janse, F. J. van-Capelle, H. Morsink, et al. Flow of "injury" currents and patterns of .excitation during early ventricular arrhythmias in acute regional myocardial ischema in isolated heart[J]. Rev Letts.1980,47(2):151～165.
[20]G. Bub, A. Shrier, L. Glass. Spiral wave generation in heterogeneous excitable media[J]. Phys Rev Letts.2002,88:1～4.
[21]G. Bub, L. Glass, N. G. Publicover, et al. Bursting calcium rotors in cultured cardiac myocyte monolayers[J]. Proc Natl Acad Sci USA.1998,95:10283～10287.
[22]L. Glass, Y. Nagai, K. Hall, et al. Predicting the entrainment of reentrant cardiac waves using phase resetting curves[J]. Physical Rev E.2002,65:021908 1～10.
[23]G. P. Kremmydas, A. V. Holden. Spiral-wave meandering in reaction-diffusion models of ventricular muscle[J]. Chaos Solition & Fractals.2002,13:1659～1669.
[24]A. V. Panfilov. Spiral breakup in an array of coupled cells:The role of the intercellular conductance[J]. Phys Rev Letts.2002,88:1～4.
[25]M. R. Boyett, H. Honjo, M. Yamamoto. A downward gradient in action potential durition along the conduction path in and around the sinoatrial node[J]. Am J Physiol.1999,276(2): H686～H698.
[26]R. D. Nathan. Two electrophysiologically distinct types of cultured pacemaker cells from rabbit sinotrial node[J]. Am J Physiol.1986,250(2):H325～H329.
[27]H. Irisava, H. F. Brown, W. Giles. Cardiac pacemakeing in the sinoatrial node[J]. Physiol Rev. 1993,73(1):197～227.
[28]H. Honjo, M. R. Boyett, I. Kodama, et al. Correlation between electrical activity and the size of rabbit sinoatrial node cells[J]. J Phsiol Lond.1996,496(3):795～808.
[29]R. L. DeHaan, R. Hirakow. Synchronization of pulsation rates in isolated cardiac myocytes[J]. Exp Cll Res.1972,70:214～220.
[30]R. L. DeHaan, H. G. Sachs. Cell coupling in developing systems:the heart-cell paradigm[J]. Curr Topics in Develop Biol.1972,7:193～228.
[31]T. Sano, T. Sawanobori, H. Adaniya. Mechanism of rhythm determination among pacemaker cells of the mammalian sinus node[J]. Am J Physiol.1978,235:H379-H384.
[32]H. J. Jonsma, M. Masson-Pevet, L. Tsjernina. The development of beat-rate synchronization of rat myocyte pairs in cell culture[J]. Basic Res Cardiol.1987,82(5):454～464.
[33]Y. Yamauchi, A. Harada, K. Kavahara. Changes in fluctuation of interbeat intervals in spontaneously beating cultured cardiac myocytes:exprimental and modeling studies[J]. Biol Cybern.2002,86:147～154.
[34]C. Meunier. The electrical coupling of two simple oscillators:load and acceleration effects[J]. Biol Cybern.1992(2),67:155～164.
[35]N. Kawasaki, S. Doi, S. Sato. On the period of the synchronized oscillation of difusively-coupled BVP oscillators (in Japanese)[J]. Tech Rep IEIECE BME.1993,93-24: 121～128.
[36]刘红菊,刘志强,古华光,杨明浩,李莉,任维.心肌单细胞兴奋模式转迁规律和转迁过程中延迟后去极化、早后去极化的生成[J].生物物理学报,2006,22：441～44.
[37]M. Oyamada, H. Kimura, Y. Oyamada, A. Miyamoto, H. Ohshika, M. Mori. The expression, phosphorylation, and localization of connexin 43 and gap-junctional intercellular communication during the establishment of a synchronized contraction of cultured neonatal rat cardiac myocytes[J]. Exp Cell Res.1994,212:351～358.
[38]D. Matsuyama and K. Kawahara. Maintenance and characterization of spontaneous contraction rhythm in cultured cardiac myocytes fused with cardiac fibroblasts[J]. Biosystems.2008,92 (3):226～232.
[39]H. Kimura, Y. Oyamada, H. Ohshika, M. Mori, M. Oyamda. Reversible inhibition of gap junctional intercellular communication, synchronouscontratin, and synchronism of intracellular Ca2+ fluctuation incultured neonatal rat cardiac mycyotes by heptanol[J]. Exp Cell Res.1995, 220:348～356.
[40]N. Cohen, Y. Soen, E. Braun. Spatio-temporal dynamics of networks of heart cells in culture[J]. Physica A.1998,249:600～604.
[41]K. Matsuura, H. Wada, T. Nagai, Y. lijima, T. Minamino, M. Sano, H. Akazawa, J. D. Molkentin, H. Kasanuki and I. Komuro. Cardiomyocytes fuse with surrounding noncardiomyocytes and reenter the cell cycle[J]. J Cell Biol.2004,167:351～363.
[42]T. Hachiro, K. Kawahara, R. Sato, Y. Yamauchi, D. Matsuyama. Changes in the fluctuation of the contraction rhythm of spontaneously beating cardiac myocytes in cultures with and without cardiac fibroblasts[J]. Bio Systems.2007,90:707～715.
[43]Y. Nakayama, K. Kawahara, M. Yoneyama, T. Hachiro. Rhythmic contraction and intracellular Ca2+ oscillatory rhythm in spontaneously beating cultured cardiac myocytes[J]. Biol Rhythm Res:2005,36 (4):3157～324.
[44]刘志强,古华光,杨明浩,李莉,刘红菊,范少光,任维·心肌细胞团搏动的整数倍节律的实验观察[J].生物物理学报,2004,20(5)：355～362.
[45]刘志强,古华光,杨明浩,范少光,杨芬,任维.心肌细胞自发性搏动节律的分岔和混沌现象[J].生物物理学报,2003,19(3)：279～285.
[46]刘志强,古华光,杨明浩,杨芬,范少光,任维.心肌细胞自发性搏动的整数倍节律及其机理[J].航天医学与医学工程,2003,16(2)：142～144.
[47]T. R. Chay. Eyring rate theory in excitable membranes:application to neuronal oscillations[J]. J phys Chem.1983,87(15):2935～2940.
[48]C. Morris, H. Lecar. Voltage oscillations in the barnacle giant muscle fiber[J]. Biophys J.1981, 35:193～213.
[49]T. R. Chay, J. Keizer. Minimal model for membrane oscillations in the pancreatic beta-cell[J]. Biophys J.1983,42:181～190.
[50]C. Morris, H. Lecar. Voltage oscillations in the barnacle giant muscle fiber[J]. Biophys J.1981, 35:193～213.
[51]盛昭翰,马军海.非线性动力系统分析引论[M].北京：科学出版社,2001.
[52]高普云.非线性动力学-分叉、混沌与孤立子[M].长沙：国防科技大学出版社,2005.
[53]Y. A. Kuznetsov. Elements of Applied Bifurcation Theory[M]. New York:Springer-Verlag, 1995.
[54]陆启韶.分岔与奇异性[M].上海：上海科技教育出版社,1995.
[55]E. M. Izhikevich. Dynamical systems in Neuroscience:The geometry of excitability and bursting[M]. The MIT Press,2005.
[56]E. M. Izhikevich. Neural Excitability, Spiking and Bursting[J]. International Journal of Bifurcation & Chaos in Applied Sciences & Engineering.2000,10(6):1～96.
[57]M. R. Guevara, A. Shrier, L. Glass. Phase-locked rhythms in periodically stimulated heart cell aggregates[J]. Am J Physiol.1988,254:H1～H10.
[58]A. Shrier, J. R. Clay. Repolarization currents in chick embryonic chick atrial heart cell aggregate[J]. Biophys J.1986,50(5):861～874.
[59]J. R. Clay, C. E. Hill, D. Rodman, et al. Repolarization current in embryonic chick atrial heart cells[J]. J Physiol Lond.1988,403:525～537.
[60]R. M. Brochu, J. R. Clay, A. Shrier. Pacemaker current in single cells and in aggregates of cells dissociated from the embryonic chick heart[J]. J Physiol Lond.1992,454:503～515.
[61]刘志强,古华光,杨明浩,李莉,刘红菊,范少光,任维.心肌细胞团搏动整数倍节律的非线性动力学机制[J].生物物理学报,2004,20(5)：363～370.
[62]G. Bub, A. Shrier, L. Glass. Spiral wave generation in heterogeneous excitable media[J]. Phys Rev Letts. 2002,88(5):1～4.
[63]范雪晖,王文锋,徐振平.新生大鼠心肌细胞体外培养及其与心肌成纤维细胞形态比较[J].新乡医学院学报,2009,26(2)：129～132.
[64]H. L. Yang, Z. Q. Huang and E. J. Ding. Early crisis induced in maps with parametric noise[J]. Phys Rev Letts.1996,77:4899～4902.
[65]Y. Soen, N. Cohen, D. Lipson and E. Braun. Emergence of spontaneous rhythm disorders in self-assembled networks of heart cells[J]. Phys Rev Lett.1999,82 (17):3556～3559.
[66]刘红菊,刘志强,古华光,杨明浩,李莉,任维.心肌单细胞兴奋模式转迁的非线性动力学机理[J].生物物理学报,2006,22(6)：449～454.
[67]N. Zhang, H. M. Zhang, Z. Q. Liu, X. L. Ding, M. H. Yang, H. G. Gu, W. Ren. Stochastic Alternating Dynamics for Synchronous EAD-Like Beating Rhythms in Cultured Cardiac Myocytes[J]. Chin Phys Lett.2009,26 (11):110501.
[68]P. N. Steinmeta, A. Roy, P. J. Fitzgerald, et al. Attention modulates synchronized neural firing in primate somatosensory cortex[J]. Nature.2000,404:187～190.
[69]H. G. Gu, W. Ren, Q. S. Lu, S. G. Wu, M. H. Yang, W. J. Chen. Integer multiple spiking in neural pacemakers without external periodic stimulation[J]. Phys Lett A.2001,285:63～68
[70]Q. Y. Wang, Q. S. Lu. Phase synchronization in small world chaotic neural networks[J]. Chin Phys Lett.2005,22(6):1329～1332.
[71]李玉叶,张慧敏,魏春玲,杨明浩,古华光,任维.随机信号在神经网络中诱发的双空间相干共振[J].动力学与控制学报,2009,7(3)：230～234.
[2]P. N. Steinmeta, A. Roy, P. J. Fitzgerald, et al. Attention modulates synchronized neuronal firing in primate somatosensory cortex[J]. Nature.2000,404:187～190.
[3]S. Boccaletti, J. Kurths, G. Osipov,. D. L. Valladares, C. S. Zhou. The synchronization of chaotic systems[M]. Phys Rep.2002,366(1-2):1～101.
[4]S. Bahar, F. Moss. Stochastic resonance and synchronization in the crayfish caudal photoreceptor[J]. Math Biosci.2004,188:81～97.
[5]M. R. Boyett, H. Honjo,I. Kodama. The sinoatrial node, a heterogeneous pacemaker structure[J]. Cardiovasc Res.2000,47:658～687.
[6]李澈.窦房结[M].北京：北京医科大学出版社,2001:134～138.
[7]D. R. Rigney, J. E. Mietus, A. L. Goldberger. Is normal sinus rhythm "chaotic"? Measurement of lyapunov exponents(abstract). Circulation.1990,82:111～236.
[8]A. Babloyantz, A. Destexhe. Is the normal heart periodic oscillator[J]? Biol Cybern.1988,58: 203～211.
[9]M. R. Guevara, L. Glass, A. Shrier. Phase locking, perioddoubling bifurcation, and irregular dynamics in periodically stimulated cardiac cells[J]. Science.1981,214:1350～1353.
[10]L. Glass, M. R. Guevara, A. Shrier. Bifurcation and chaos in a periodically stimulated cardiac oscillator[J]. Physica D.1983,7:89～101.
[11]D. R. Chialvo, R. F. Gilmour, J. Jalife. Low dimensional chaos in cardiac tissue[J]. Nature.1990, 343:653～657.
[12]W. B. Cannon. Organization for physiological homeostatics[J]. Phys Rev.1929,9:399～431.
[13]A. L. Goldberger. Is the normal heartbeat chaotic or homeostatic[J]? News physical sciences. 1991,6:87～91.
[14]A. L. Goldberger. Nonlinear dynamics, fractals and chaos:applications to cardiac electrophysiplogy[J]. Ann Biomed Eng,1990,18(2):195～198.
[15]A. L. Goldberger, D. R. Rigney, J. Mietus, et al. Nonlinear dynamics in sudden cardiac death syndrome:heartrate oscillations and bifurcations[J]. Experientia.1988,44(11-12):983～987.
[16]W. L. Ditto, M. L. Spano, J. Neff, et al. Control of human atrial fibrillation[J]. Int J Bif Chaos. 2000,10(3):593～601.
[17]A. Garfinkel, M. L. Spano, W. L. Ditto, et al. Controlling cardiac chaos[J]. Science.1992,257: 1230～1235.
[18]M. A. Allessie, F.I. M. Bonke, F. J. G. Schoppman. Circus movement in rabbit atrail muscle as a mechanism of tachycardia. The "leading circle" concept[J]. Cir Res.1997,41:9～18
[19]M. J. Janse, F. J. van-Capelle, H. Morsink, et al. Flow of "injury" currents and patterns of .excitation during early ventricular arrhythmias in acute regional myocardial ischema in isolated heart[J]. Rev Letts.1980,47(2):151～165.
[20]G. Bub, A. Shrier, L. Glass. Spiral wave generation in heterogeneous excitable media[J]. Phys Rev Letts.2002,88:1～4.
[21]G. Bub, L. Glass, N. G. Publicover, et al. Bursting calcium rotors in cultured cardiac myocyte monolayers[J]. Proc Natl Acad Sci USA.1998,95:10283～10287.
[22]L. Glass, Y. Nagai, K. Hall, et al. Predicting the entrainment of reentrant cardiac waves using phase resetting curves[J]. Physical Rev E.2002,65:021908 1～10.
[23]G. P. Kremmydas, A. V. Holden. Spiral-wave meandering in reaction-diffusion models of ventricular muscle[J]. Chaos Solition & Fractals.2002,13:1659～1669.
[24]A. V. Panfilov. Spiral breakup in an array of coupled cells:The role of the intercellular conductance[J]. Phys Rev Letts.2002,88:1～4.
[25]M. R. Boyett, H. Honjo, M. Yamamoto. A downward gradient in action potential durition along the conduction path in and around the sinoatrial node[J]. Am J Physiol.1999,276(2): H686～H698.
[26]R. D. Nathan. Two electrophysiologically distinct types of cultured pacemaker cells from rabbit sinotrial node[J]. Am J Physiol.1986,250(2):H325～H329.
[27]H. Irisava, H. F. Brown, W. Giles. Cardiac pacemakeing in the sinoatrial node[J]. Physiol Rev. 1993,73(1):197～227.
[28]H. Honjo, M. R. Boyett, I. Kodama, et al. Correlation between electrical activity and the size of rabbit sinoatrial node cells[J]. J Phsiol Lond.1996,496(3):795～808.
[29]R. L. DeHaan, R. Hirakow. Synchronization of pulsation rates in isolated cardiac myocytes[J]. Exp Cll Res.1972,70:214～220.
[30]R. L. DeHaan, H. G. Sachs. Cell coupling in developing systems:the heart-cell paradigm[J]. Curr Topics in Develop Biol.1972,7:193～228.
[31]T. Sano, T. Sawanobori, H. Adaniya. Mechanism of rhythm determination among pacemaker cells of the mammalian sinus node[J]. Am J Physiol.1978,235:H379-H384.
[32]H. J. Jonsma, M. Masson-Pevet, L. Tsjernina. The development of beat-rate synchronization of rat myocyte pairs in cell culture[J]. Basic Res Cardiol.1987,82(5):454～464.
[33]Y. Yamauchi, A. Harada, K. Kavahara. Changes in fluctuation of interbeat intervals in spontaneously beating cultured cardiac myocytes:exprimental and modeling studies[J]. Biol Cybern.2002,86:147～154.
[34]C. Meunier. The electrical coupling of two simple oscillators:load and acceleration effects[J]. Biol Cybern.1992(2),67:155～164.
[35]N. Kawasaki, S. Doi, S. Sato. On the period of the synchronized oscillation of difusively-coupled BVP oscillators (in Japanese)[J]. Tech Rep IEIECE BME.1993,93-24: 121～128.
[36]刘红菊,刘志强,古华光,杨明浩,李莉,任维.心肌单细胞兴奋模式转迁规律和转迁过程中延迟后去极化、早后去极化的生成[J].生物物理学报,2006,22：441～44.
[37]M. Oyamada, H. Kimura, Y. Oyamada, A. Miyamoto, H. Ohshika, M. Mori. The expression, phosphorylation, and localization of connexin 43 and gap-junctional intercellular communication during the establishment of a synchronized contraction of cultured neonatal rat cardiac myocytes[J]. Exp Cell Res.1994,212:351～358.
[38]D. Matsuyama and K. Kawahara. Maintenance and characterization of spontaneous contraction rhythm in cultured cardiac myocytes fused with cardiac fibroblasts[J]. Biosystems.2008,92 (3):226～232.
[39]H. Kimura, Y. Oyamada, H. Ohshika, M. Mori, M. Oyamda. Reversible inhibition of gap junctional intercellular communication, synchronouscontratin, and synchronism of intracellular Ca2+ fluctuation incultured neonatal rat cardiac mycyotes by heptanol[J]. Exp Cell Res.1995, 220:348～356.
[40]N. Cohen, Y. Soen, E. Braun. Spatio-temporal dynamics of networks of heart cells in culture[J]. Physica A.1998,249:600～604.
[41]K. Matsuura, H. Wada, T. Nagai, Y. lijima, T. Minamino, M. Sano, H. Akazawa, J. D. Molkentin, H. Kasanuki and I. Komuro. Cardiomyocytes fuse with surrounding noncardiomyocytes and reenter the cell cycle[J]. J Cell Biol.2004,167:351～363.
[42]T. Hachiro, K. Kawahara, R. Sato, Y. Yamauchi, D. Matsuyama. Changes in the fluctuation of the contraction rhythm of spontaneously beating cardiac myocytes in cultures with and without cardiac fibroblasts[J]. Bio Systems.2007,90:707～715.
[43]Y. Nakayama, K. Kawahara, M. Yoneyama, T. Hachiro. Rhythmic contraction and intracellular Ca2+ oscillatory rhythm in spontaneously beating cultured cardiac myocytes[J]. Biol Rhythm Res:2005,36 (4):3157～324.
[44]刘志强,古华光,杨明浩,李莉,刘红菊,范少光,任维·心肌细胞团搏动的整数倍节律的实验观察[J].生物物理学报,2004,20(5)：355～362.
[45]刘志强,古华光,杨明浩,范少光,杨芬,任维.心肌细胞自发性搏动节律的分岔和混沌现象[J].生物物理学报,2003,19(3)：279～285.
[46]刘志强,古华光,杨明浩,杨芬,范少光,任维.心肌细胞自发性搏动的整数倍节律及其机理[J].航天医学与医学工程,2003,16(2)：142～144.
[47]T. R. Chay. Eyring rate theory in excitable membranes:application to neuronal oscillations[J]. J phys Chem.1983,87(15):2935～2940.
[48]C. Morris, H. Lecar. Voltage oscillations in the barnacle giant muscle fiber[J]. Biophys J.1981, 35:193～213.
[49]T. R. Chay, J. Keizer. Minimal model for membrane oscillations in the pancreatic beta-cell[J]. Biophys J.1983,42:181～190.
[50]C. Morris, H. Lecar. Voltage oscillations in the barnacle giant muscle fiber[J]. Biophys J.1981, 35:193～213.
[51]盛昭翰,马军海.非线性动力系统分析引论[M].北京：科学出版社,2001.
[52]高普云.非线性动力学-分叉、混沌与孤立子[M].长沙：国防科技大学出版社,2005.
[53]Y. A. Kuznetsov. Elements of Applied Bifurcation Theory[M]. New York:Springer-Verlag, 1995.
[54]陆启韶.分岔与奇异性[M].上海：上海科技教育出版社,1995.
[55]E. M. Izhikevich. Dynamical systems in Neuroscience:The geometry of excitability and bursting[M]. The MIT Press,2005.
[56]E. M. Izhikevich. Neural Excitability, Spiking and Bursting[J]. International Journal of Bifurcation & Chaos in Applied Sciences & Engineering.2000,10(6):1～96.
[57]M. R. Guevara, A. Shrier, L. Glass. Phase-locked rhythms in periodically stimulated heart cell aggregates[J]. Am J Physiol.1988,254:H1～H10.
[58]A. Shrier, J. R. Clay. Repolarization currents in chick embryonic chick atrial heart cell aggregate[J]. Biophys J.1986,50(5):861～874.
[59]J. R. Clay, C. E. Hill, D. Rodman, et al. Repolarization current in embryonic chick atrial heart cells[J]. J Physiol Lond.1988,403:525～537.
[60]R. M. Brochu, J. R. Clay, A. Shrier. Pacemaker current in single cells and in aggregates of cells dissociated from the embryonic chick heart[J]. J Physiol Lond.1992,454:503～515.
[61]刘志强,古华光,杨明浩,李莉,刘红菊,范少光,任维.心肌细胞团搏动整数倍节律的非线性动力学机制[J].生物物理学报,2004,20(5)：363～370.
[62]G. Bub, A. Shrier, L. Glass. Spiral wave generation in heterogeneous excitable media[J]. Phys Rev Letts. 2002,88(5):1～4.
[63]范雪晖,王文锋,徐振平.新生大鼠心肌细胞体外培养及其与心肌成纤维细胞形态比较[J].新乡医学院学报,2009,26(2)：129～132.
[64]H. L. Yang, Z. Q. Huang and E. J. Ding. Early crisis induced in maps with parametric noise[J]. Phys Rev Letts.1996,77:4899～4902.
[65]Y. Soen, N. Cohen, D. Lipson and E. Braun. Emergence of spontaneous rhythm disorders in self-assembled networks of heart cells[J]. Phys Rev Lett.1999,82 (17):3556～3559.
[66]刘红菊,刘志强,古华光,杨明浩,李莉,任维.心肌单细胞兴奋模式转迁的非线性动力学机理[J].生物物理学报,2006,22(6)：449～454.
[67]N. Zhang, H. M. Zhang, Z. Q. Liu, X. L. Ding, M. H. Yang, H. G. Gu, W. Ren. Stochastic Alternating Dynamics for Synchronous EAD-Like Beating Rhythms in Cultured Cardiac Myocytes[J]. Chin Phys Lett.2009,26 (11):110501.
[68]P. N. Steinmeta, A. Roy, P. J. Fitzgerald, et al. Attention modulates synchronized neural firing in primate somatosensory cortex[J]. Nature.2000,404:187～190.
[69]H. G. Gu, W. Ren, Q. S. Lu, S. G. Wu, M. H. Yang, W. J. Chen. Integer multiple spiking in neural pacemakers without external periodic stimulation[J]. Phys Lett A.2001,285:63～68
[70]Q. Y. Wang, Q. S. Lu. Phase synchronization in small world chaotic neural networks[J]. Chin Phys Lett.2005,22(6):1329～1332.
[71]李玉叶,张慧敏,魏春玲,杨明浩,古华光,任维.随机信号在神经网络中诱发的双空间相干共振[J].动力学与控制学报,2009,7(3)：230～234.