耦合振子奇异态研究
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摘要
目前网络上耦合振子模型的集体行为已经被研究了许多年,特别是对于同步化的研究工作,这在生物学、神经科学等有着很广泛的应用前景。其中一个比较著名的模型就是Kuramoto相振子模型。最近几年来,有几个研究小组发现了在全同Kuramoto相振子系统中出现了一种“奇异态(Chimera state)”的集体行为,即在系统中有一部分振子处于同步化状态而同时另一部分振子则处于去同步化状态。这种集体行为被Abrams等人称为奇异态。来自于许多生物每次睡眠时都只有半个大脑处于睡眠状态的现象[1]。在海豚和其他某些海洋哺乳动物中首次发现只有半个大脑睡眠的例子,并相继在鸟类和蜥蜴中也发现了这种现象[2]。在实验上通过记录生物大脑的脑电波发现,醒着的那一半大脑表现出了去同步电活动，对应于数以百万计的神经元不协调的振动,而休眠的那部分则处于高度同步化状态。这在生物学上被称为半脑睡眠。目前有很多专家学者都在研究这种现象[53][54][55]。Abrams等人从物理学的角度出发构造了一个最简单的Kuramoto相振子模型试图以奇异态来解释这种现象[3]。本文就是在Abrams等人的基础之上从耦合振子间的时间延迟和集团大小的非对称性来讨论系统的集体行为,并提出了一个新的模型来解释海洋生物中的交替奇异态现象。
首先,由于考虑到神经元信号传输速度的有限性,导致信号在神经元之间传递时需要一定的时间,因而神经元之间的相互耦合作用是不同时的,存在一个时间延迟,所以在Abrams等人的系统中我们引入了时间延迟,考察在引入时间延迟后对系统三种典型的奇异态行为的影响。我们发现固定的时间延迟会引起系统行为的相变,而系统的奇异态对于随机分布的时间延迟则具有鲁棒性。
其次,在Abrams等人的系统中我们考虑子集团大小的非对称性所带来的对系统行为的影响。因为在原来的Abrams等人的系统中,两个子集团是相同大小的,并目.第一个子集团始终处于同步化的状态,那么我们就改变第一个子集团的大小,来观察系统行为的变化。我们发现系统的奇异态对于集团大小的非对称性也具有鲁棒性。
第三,在某些海洋哺乳动物中,它们大脑的实际睡眠方式是一边半脑工作一边半脑睡眠,然后再轮换过来。这一边睡眠另一边处于工作状态,两个半脑总是处于这种交替睡眠的状态。因而,我们建立一个模型试图去了解这种现象的机制和解释这种奇异的生物现象,以增强我们对于这些生物现象的认识和理解。
通过以上的研究,我们发现了Abrams等人的系统对于时间延迟和集团大小的非对称具有鲁棒性。提出了我们自己的新的模型去解释海洋生物中的交替奇异态行为。发现系统的交替奇异态行为对系统的一些参数以及外界信号都具有鲁棒性。
The collective behavior in coupled oscillators has been researched for years, especially in the aspect of synchronization. It has very wide applications in the biology and neuroscience. One of the most famous models is the Kuramoto model. In recent years,some research groups have found the“Chimera state”behavior in the identical Kuramoto model, where some of the oscillators are synchronized, and other oscillators are asynchronous. This collective behavior has been named“Chimera state”. Many creatures sleep with only half of their brain at a time[1]. Such unihemispheric sleep was first reported in dolphins and some other sea mammals, and has now been seen in birds and referred in lizards [2]. When brain wave is recorded in the experiment, the awake side of the brain shows desynchronized electrical activity, corresponding to millions of neuros oscillating out of phase, whereas the sleeping side is highly synchronized. This phenomenon is called half-brain sleep in biology. Many scholars are studying this phenomenon. Abrams et al. use the simplest system of Kuramoto model to explain this phenomenon[3]. Our work is to observe the collective behavior from the aspect of time delay in coupling term and size asymmetry based on the Abrams model. After that we propose a new model to explain the alternating chimera states in the most mammals.
Fisrtly, considering that the finite velocity of transmission between the neurons will result a time delay, we introduce time delay in the Abrams model to study the influence to the three kinds of chimera state. We find that the fixed time delay can result in phase transition, but the Chimera states of system have robustness to the random time delay.
Secondly, we consider the influence of the size asymmetry in the Abrams model. Because it has two subgroups, and the size of every subgroup is equal, the first subgroup is always synchronized. So we try to change the size of the first subgroup, and observe the collective behavior. We find that the Chimera state of system has robustness to the size asymmetry.
Thirdly, the sleep mode is one half brain working and another half brain sleeping in some mammals. After some time it turn over. That is, the two half brain are always alternating sleep. So we propose a new model to understand this mechanism and explain the phenomenon. It is useful for us to recognize and understand the real biological phenomenon.
From the research of the above, we find that Abrams model has robustness to the time delay and the size asymmetry. We propose the new model to explain the alternating chimera state in many sea mammls. We find the system has robustness to the some parameters of the system and the external signal.
首先,由于考虑到神经元信号传输速度的有限性,导致信号在神经元之间传递时需要一定的时间,因而神经元之间的相互耦合作用是不同时的,存在一个时间延迟,所以在Abrams等人的系统中我们引入了时间延迟,考察在引入时间延迟后对系统三种典型的奇异态行为的影响。我们发现固定的时间延迟会引起系统行为的相变,而系统的奇异态对于随机分布的时间延迟则具有鲁棒性。
其次,在Abrams等人的系统中我们考虑子集团大小的非对称性所带来的对系统行为的影响。因为在原来的Abrams等人的系统中,两个子集团是相同大小的,并目.第一个子集团始终处于同步化的状态,那么我们就改变第一个子集团的大小,来观察系统行为的变化。我们发现系统的奇异态对于集团大小的非对称性也具有鲁棒性。
第三,在某些海洋哺乳动物中,它们大脑的实际睡眠方式是一边半脑工作一边半脑睡眠,然后再轮换过来。这一边睡眠另一边处于工作状态,两个半脑总是处于这种交替睡眠的状态。因而,我们建立一个模型试图去了解这种现象的机制和解释这种奇异的生物现象,以增强我们对于这些生物现象的认识和理解。
通过以上的研究,我们发现了Abrams等人的系统对于时间延迟和集团大小的非对称具有鲁棒性。提出了我们自己的新的模型去解释海洋生物中的交替奇异态行为。发现系统的交替奇异态行为对系统的一些参数以及外界信号都具有鲁棒性。
The collective behavior in coupled oscillators has been researched for years, especially in the aspect of synchronization. It has very wide applications in the biology and neuroscience. One of the most famous models is the Kuramoto model. In recent years,some research groups have found the“Chimera state”behavior in the identical Kuramoto model, where some of the oscillators are synchronized, and other oscillators are asynchronous. This collective behavior has been named“Chimera state”. Many creatures sleep with only half of their brain at a time[1]. Such unihemispheric sleep was first reported in dolphins and some other sea mammals, and has now been seen in birds and referred in lizards [2]. When brain wave is recorded in the experiment, the awake side of the brain shows desynchronized electrical activity, corresponding to millions of neuros oscillating out of phase, whereas the sleeping side is highly synchronized. This phenomenon is called half-brain sleep in biology. Many scholars are studying this phenomenon. Abrams et al. use the simplest system of Kuramoto model to explain this phenomenon[3]. Our work is to observe the collective behavior from the aspect of time delay in coupling term and size asymmetry based on the Abrams model. After that we propose a new model to explain the alternating chimera states in the most mammals.
Fisrtly, considering that the finite velocity of transmission between the neurons will result a time delay, we introduce time delay in the Abrams model to study the influence to the three kinds of chimera state. We find that the fixed time delay can result in phase transition, but the Chimera states of system have robustness to the random time delay.
Secondly, we consider the influence of the size asymmetry in the Abrams model. Because it has two subgroups, and the size of every subgroup is equal, the first subgroup is always synchronized. So we try to change the size of the first subgroup, and observe the collective behavior. We find that the Chimera state of system has robustness to the size asymmetry.
Thirdly, the sleep mode is one half brain working and another half brain sleeping in some mammals. After some time it turn over. That is, the two half brain are always alternating sleep. So we propose a new model to understand this mechanism and explain the phenomenon. It is useful for us to recognize and understand the real biological phenomenon.
From the research of the above, we find that Abrams model has robustness to the time delay and the size asymmetry. We propose the new model to explain the alternating chimera state in many sea mammls. We find the system has robustness to the some parameters of the system and the external signal.
引文
[1]N. C. Rattenborg, C. J. Amlaner and S. L. Lima, Neurosci. Biobehav. Rev.24,817 (2000).
[2]C. G. Mathews, J. A. Lesku, S. L. Lima and C. J. Amlaner, Ethology 112,286 (2006).
[3]D. M. Abrams, R. Mirollo, S. H. Strogatz and D. A. Wiley, Phys. Rev. Lett.101, 084103(2008).
[4]G. C. Sethia, A. Sen and F. M. Atay, Phys. Rev. Lett.100,144102 (2008).
[5]O. E. Omel'chenko, Y. L. Maistrenko and P. A. Tass, Phys. Rev. Lett.100,044105 (2008).
[6]A. Pikovsky and M. Rosenblum, Phys. Rev. Lett.101,264103 (2008).
[7]M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comput. Commun. Rev.29,251, (1999).
[8]R. Albert, H. Jeong, and A.-L. Baraba si, Nature (London) 401,130 (1999).
[9]J. Abello, P. M. Pardalos, and M. G. C. Resende, in External Memory Algorithms, edited by J. Abello and J. Vitter, DIMACS Series in Discrete Mathematics Theoretical Computer Science (American Mathematical Society), p.119 (1999).
[10]D. J. Watts, and S. H. Strogatz, Nature (London) 393,440 (1998).
[11]L. A. N. Amaral, A. Scala, M. Barthe'le'my, and H. E. Stanley, Proc. Natl. Acad. Sci. U.S.A.97,11 149(2000).
[12]王树禾,《图论》,(科学出版社,2004)
[13]P. Erdos and A. Re nyi, Publ. Math. (Debrecen) 6,290 (1959).
[14]刘宗华,《混沌动力学基础及其应用》,(高等教育出版社,2006)。
[15]A.-L. Baraba'si, and R. Albert, Science 286,509 (1999). A.-L. Baraba si, R. Albert, and H. Jeong, Physica A 281,69 (2000).
[16]R. Albert and A. L. Barabasi, Rev.Mod.Phys 74,47(2002)
[17]M. E. J. Newman, SIAM.Rev 45,167(2003)
[18]S. Boccaletti,V. latora,Y. Moreno. M. Chavez, Phys.Rep 424,175(2006)
[19]何大韧,刘宗华,汪秉宏,复杂系统和复杂网络,北京：高等教育出版社,2009
[20]B. Bolloba's, Random Graphs (Academic, London,1985). Bolloba's, B., and O. Riordan, Combinatorica, Volume 24, Number 1, January 2004, pp.5-34(30) (2001).
[21]P. Erdos,and A. Renyi,Publ.Math.Debrecen6,290(1959).
[22]赵明,汪秉宏,蒋品群,周涛,物理学进展,25(3),273-295(2005).
[23]M. E. J. Newman,M.Girvan,Phys.Rev.E 69,026113(2004).
[24]S. Guan, X. Wang, Y. Lai and C. Lai, Phys.Rev.E 77,046211(2008).
[25]D. Li, I. Leyva, J. A. Almendral, I. Sendina-Nadal, J. M. Buldu, S. Havlin and S. Boccaletti, Phys. Rev. Lett.101,168701(2008).
[26]Z. Liu and B. Hu,Europhys.Lett,72(2),315(2005)
[27]M. E. J. Newman and J.Park,Phys.Rev.E 68,036122(2003)
[28]Z.-H.Liu and P. M. Hui, Physica A 383,714(2007)
[29]A. Acebron, L. L. Bonilla, C. J. Perez Vicente, F. Ritort and R. Spigler, Rev. Mod. Phys.77,137(2005).
[30]S. H. Strogatz, Physica D 143 (2000) 1-20.
[31]A. Pikovsky,M. Rosenblum and J. Kurths,Synchronization-a universal concept in nonlinear science (Cambridge University Press,Cambridge,2001).
[32]A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Phys. Rep.469, 93(2008).
[33]J. Buck, Quart. Rev. Biol.63,265(1998).
[34]M. F. Claridge, Ann. Rev. Entomol.30,297(1985).
[35]J. Buck and E. buck, Science 159,1319(1968).
[36]R. E. Mirollo and S. H. Strogatz, SIAM J. Appl. Math.50,1645(1990).
[37]http://www.scienceblog.com/community/article4716.html.
[38]T. J. Walker, Science 166,891(1969).
[39]Z. Liu and P. M. Hui, Physica A 383,714(2007).
[40]C. Gu, J. Wang and Z. Liu, Phys. Rev. E 80,030904(2009)(R).
[41]J. A. Acebron, S. Lozano and A. Arenas, Phys. Rev. Lett.99,128701 (2007).
[42]Z. Liu and T. Munakata, Phys. Rev. E 78,046111 (2008); F. Lu and Z. Liu, Chin.
Phys. Lett.26,040503(2009); T. Kondo, Z.Liu and T. Munakata, Phys. Rev. E***.
[43]M. Perc, Phys. Rev. E 78,036105 (2008).
[44]D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett.93,174102(2004).
[45]E. Ott and T. M. Antonsen, Chaos 18,037113 (2008).
[46]J. H. Sheeba, V. K. Chandrasekar and M. Lakshmanan, Phys.Rev. E 79, 055203(R) (2009).
[47]J. H. Sheeba, V. K. Chandrasekar, A. Stefanovska and P. V. E.McClintock, Phys. Rev. E 79,046210(2009).
[48]C.R. Laing, Chaos 19,013113 (2009).
[49]J.Restrepo, E. Ott and B. Hunt, Phys. Rev. E.71,036151 (2005).
[50]D.Li, I. Leyva, J. A. Almendral, I. Sendina-Nadal, J. M. Buldu, S. Havlin and S. Boccaletti, Phys. Rev. Lett.101,168701(2008).
[51]E. Montbrio, J. Kurths and B. Blasius, Phys. Rev. E 70,056125(2004).
[52]H. Sheeba, V. K. Chandrasekar, A. Stefanovska and P. V. E. McClintock, Phys. Rev. E 79,046210(2009).
[53]N. C. Rattenborg, C. J. Amlaner and S. L. Lima, Neurosci. Biobehav. Rev.24, 817(2000).
[54]C. G. Mathews, J. A. Lesku, S. L. Lima and C. J. Amlaner, Ethology 112,286 (2006).
[55]O. I. Lyamin, P. R.Manger, S. H. Ridgway, L.M.Mukhametov and J. M. Siegel, Neurosci. Biobehav. Rev.32,1451 (2008).
[56]郭雷,许晓鸣,《复杂网络》,上海：上海科技教育出版社,2006.
[2]C. G. Mathews, J. A. Lesku, S. L. Lima and C. J. Amlaner, Ethology 112,286 (2006).
[3]D. M. Abrams, R. Mirollo, S. H. Strogatz and D. A. Wiley, Phys. Rev. Lett.101, 084103(2008).
[4]G. C. Sethia, A. Sen and F. M. Atay, Phys. Rev. Lett.100,144102 (2008).
[5]O. E. Omel'chenko, Y. L. Maistrenko and P. A. Tass, Phys. Rev. Lett.100,044105 (2008).
[6]A. Pikovsky and M. Rosenblum, Phys. Rev. Lett.101,264103 (2008).
[7]M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comput. Commun. Rev.29,251, (1999).
[8]R. Albert, H. Jeong, and A.-L. Baraba si, Nature (London) 401,130 (1999).
[9]J. Abello, P. M. Pardalos, and M. G. C. Resende, in External Memory Algorithms, edited by J. Abello and J. Vitter, DIMACS Series in Discrete Mathematics Theoretical Computer Science (American Mathematical Society), p.119 (1999).
[10]D. J. Watts, and S. H. Strogatz, Nature (London) 393,440 (1998).
[11]L. A. N. Amaral, A. Scala, M. Barthe'le'my, and H. E. Stanley, Proc. Natl. Acad. Sci. U.S.A.97,11 149(2000).
[12]王树禾,《图论》,(科学出版社,2004)
[13]P. Erdos and A. Re nyi, Publ. Math. (Debrecen) 6,290 (1959).
[14]刘宗华,《混沌动力学基础及其应用》,(高等教育出版社,2006)。
[15]A.-L. Baraba'si, and R. Albert, Science 286,509 (1999). A.-L. Baraba si, R. Albert, and H. Jeong, Physica A 281,69 (2000).
[16]R. Albert and A. L. Barabasi, Rev.Mod.Phys 74,47(2002)
[17]M. E. J. Newman, SIAM.Rev 45,167(2003)
[18]S. Boccaletti,V. latora,Y. Moreno. M. Chavez, Phys.Rep 424,175(2006)
[19]何大韧,刘宗华,汪秉宏,复杂系统和复杂网络,北京：高等教育出版社,2009
[20]B. Bolloba's, Random Graphs (Academic, London,1985). Bolloba's, B., and O. Riordan, Combinatorica, Volume 24, Number 1, January 2004, pp.5-34(30) (2001).
[21]P. Erdos,and A. Renyi,Publ.Math.Debrecen6,290(1959).
[22]赵明,汪秉宏,蒋品群,周涛,物理学进展,25(3),273-295(2005).
[23]M. E. J. Newman,M.Girvan,Phys.Rev.E 69,026113(2004).
[24]S. Guan, X. Wang, Y. Lai and C. Lai, Phys.Rev.E 77,046211(2008).
[25]D. Li, I. Leyva, J. A. Almendral, I. Sendina-Nadal, J. M. Buldu, S. Havlin and S. Boccaletti, Phys. Rev. Lett.101,168701(2008).
[26]Z. Liu and B. Hu,Europhys.Lett,72(2),315(2005)
[27]M. E. J. Newman and J.Park,Phys.Rev.E 68,036122(2003)
[28]Z.-H.Liu and P. M. Hui, Physica A 383,714(2007)
[29]A. Acebron, L. L. Bonilla, C. J. Perez Vicente, F. Ritort and R. Spigler, Rev. Mod. Phys.77,137(2005).
[30]S. H. Strogatz, Physica D 143 (2000) 1-20.
[31]A. Pikovsky,M. Rosenblum and J. Kurths,Synchronization-a universal concept in nonlinear science (Cambridge University Press,Cambridge,2001).
[32]A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Phys. Rep.469, 93(2008).
[33]J. Buck, Quart. Rev. Biol.63,265(1998).
[34]M. F. Claridge, Ann. Rev. Entomol.30,297(1985).
[35]J. Buck and E. buck, Science 159,1319(1968).
[36]R. E. Mirollo and S. H. Strogatz, SIAM J. Appl. Math.50,1645(1990).
[37]http://www.scienceblog.com/community/article4716.html.
[38]T. J. Walker, Science 166,891(1969).
[39]Z. Liu and P. M. Hui, Physica A 383,714(2007).
[40]C. Gu, J. Wang and Z. Liu, Phys. Rev. E 80,030904(2009)(R).
[41]J. A. Acebron, S. Lozano and A. Arenas, Phys. Rev. Lett.99,128701 (2007).
[42]Z. Liu and T. Munakata, Phys. Rev. E 78,046111 (2008); F. Lu and Z. Liu, Chin.
Phys. Lett.26,040503(2009); T. Kondo, Z.Liu and T. Munakata, Phys. Rev. E***.
[43]M. Perc, Phys. Rev. E 78,036105 (2008).
[44]D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett.93,174102(2004).
[45]E. Ott and T. M. Antonsen, Chaos 18,037113 (2008).
[46]J. H. Sheeba, V. K. Chandrasekar and M. Lakshmanan, Phys.Rev. E 79, 055203(R) (2009).
[47]J. H. Sheeba, V. K. Chandrasekar, A. Stefanovska and P. V. E.McClintock, Phys. Rev. E 79,046210(2009).
[48]C.R. Laing, Chaos 19,013113 (2009).
[49]J.Restrepo, E. Ott and B. Hunt, Phys. Rev. E.71,036151 (2005).
[50]D.Li, I. Leyva, J. A. Almendral, I. Sendina-Nadal, J. M. Buldu, S. Havlin and S. Boccaletti, Phys. Rev. Lett.101,168701(2008).
[51]E. Montbrio, J. Kurths and B. Blasius, Phys. Rev. E 70,056125(2004).
[52]H. Sheeba, V. K. Chandrasekar, A. Stefanovska and P. V. E. McClintock, Phys. Rev. E 79,046210(2009).
[53]N. C. Rattenborg, C. J. Amlaner and S. L. Lima, Neurosci. Biobehav. Rev.24, 817(2000).
[54]C. G. Mathews, J. A. Lesku, S. L. Lima and C. J. Amlaner, Ethology 112,286 (2006).
[55]O. I. Lyamin, P. R.Manger, S. H. Ridgway, L.M.Mukhametov and J. M. Siegel, Neurosci. Biobehav. Rev.32,1451 (2008).
[56]郭雷,许晓鸣,《复杂网络》,上海：上海科技教育出版社,2006.