振子网络上的同步及其在生物学中的应用
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摘要
在自然界和人类社会中,广泛的存在着各种各样的复杂网络,如电力网、因特网、基因调控网等。近年来,许多研究者从系统学的观点研究了网络的动力学行为与网络的拓扑结构之间的关系,使得复杂网络动力学成为复杂性科学的新兴方向,并受到国内外各领域的广泛关注,尤其是物理、数学、生物,计算机等领域.在这方面的研究中,耦合非线性振子网络是一种典型模型,其中的各个节点的动力学行为主要取决于个体的特征,同时受到来自其他节点的耦合作用的影响,从而使得整个网络表现出一定的同步行为.研究网络同步行为不仅对揭示各种群体现象的演化机理十分重要,更有助于人们利用这些机理为实际生产和生活服务.
本文结合耦合非线性振子网络的同步研究中的现有方法,提出了一些研究振子网络同步的新思路,得到了振子网络实现同步的充分条件,指出了影响振子网络同步的各种因素,在理论上揭示了振子网络实现同步的机理,并结合物理学、生物学等领域的经典模型进行了理论分析和数值验证.本文的主要内容和创新点主要概括为如下四个方面.
一.对耦合非线性振子网络的完全同步的研究,本文的工作主要表现在两个方面.a).在研究对象上,考虑到现有结果主要针对线性耦合振子网络,而非线性耦合振子网络的同步现象更为普遍,本文研究了两类非线性耦合作用下的振子网络的完全同步行为:对一类常见的耦合函数,当它在孤立节点的吸引域内是增函数时,给出了振子网络实现同步的充分条件;对一般形式的非线性耦合函数,本文提出了耦合函数的平衡假设,验证了大多数实际网络的耦合函数均满足该假设,且在该假设下给出了这类耦合振子网络实现完全同步的充分条件.b).在研究方法上,考虑到耦合振子网络中每个振子的动力学行为由振子自身动力学和耦合作用项两部分构成,而这分别是目前两种常用方法的处理困难所在,本文利用两种现有方法的优点将它们合理的结合起来,得到的结果体现了这种新的研究思路的优越性,在一定程度上解决了现有结果在实际应用时可能遇到的困难.
二.对于耦合非线性振子网络相同步的研究,现有结果基本都是借助于平均场的思想,定义一个衡量相同步程度的序参数进行数值模拟或实验观测.本文利用相约化方法将振子网络的状态方程转化为相应的相方程,然后通过对其相方程进行分析,证明了耦合极限环振子网络在很弱的平衡耦合下就可以实现相同步.
三.利用Lur'e系统的理论研究了由不同的细胞构成的多细胞系统在群体感应机制作用下的实用同步,即各细胞的最终动力学行为相似但又有一定误差,这是一种常见的同步现象.本文给出了影响其同步误差的因素,分析了群体感应机制实现同步的机理,并做了大量的数值验证.
四.研究了群体感应机制作用下的多细胞系统的频率同步问题.考虑到前人的工作普遍采用了拟稳态近似假设,而该假设只有在一定条件下才表现出其合理性.本文通过微分方程理论,借助于环境中的信号分子的精确解代替了拟稳态近似假设,给出了该模型实现同步的严格的理论分析.
最后,结合目前该领域的研究进展,对本文的工作做了全面的总结,并指出了今后该领域进一步工作的展望.
Complex networks are ubiquitous,ranging from nature to society,such as the internet,the power grid,genetic regulatory networks,etc.In recent years,many researches have been carried out to discuss the relationship between the topological structure and collective dynamical behaviors of oscillator networks.Therefore,studying network dynamics has been considered as a new and developing subject of Complexity Science.It has attracted much attention from many fields such as physics,mathetics,biology and computer science.In order to explore collective dynamical behaviors in complex networks,researchers built the model of coupled oscillator networks. The researches of collective behaviors in oscillator networks are important not only to explore the mechanisms of those natural phenomena existing widely but also to utilize these mechanisms to serve people's life.
This paper combines several new methods to study synchronization of oscillator networks and obtains many original results.These results offer sufficient conditions to relealize synchronization and point out the mechanisms of synchronization.Further investigation is carried out to combine theory with practice by simulating many classical models numerically.The main contents and innovative points of this paper are listed as follows.
(1).The study of complete synchronization in coupled oscillator networks can be generalized as the following two aspects.a).As far as the studying objects are concerned,the paper studies coupled oscillator networks with two different types of nonlinear coupling.Most of previous reseaches focused on networks coupled linearly and few researches were carried out to study oscillator networks with nonlinear coupling.Moreover,the study of oscillator networks with nonlinear coupling is interesting and important. For the case that the nonlinear coupling funchtions are increasing functions in the absorbing basin of individual oscillator,this paper gives the sufficient conditions for such type of networks to realize synchronization.For another more general type of coupling flmctions,reseaches are also carried out under the hypothesis of coupling blance,b).As we know,the dynamic behavior of each oscillator is composed of two parts:the inherent dynamic behavior of the uncoupled oscillator and the dynamic behavior influenced by coupling.It happened that two previous methods have difficulties to deal with the stability of these two parts,respectively.The paper combines the two methods by utilizing their merits and solves the respective practical difficulties of the two methods to a certain extent.
(2).The paper studies phase synchronization in oscillator networks. Most of previous researches defined an order parameter based on the idea of mean field approach and carried out numerical simulations or experiment survey.Few researches have been carried out theoretically.In this paper, the dynamics of networks is reduced to phase equations by phase reduced method.Analyzing the phase equations through the master stability function method,one can prove theoretically that the oscillators with identical frequency can be in-phase synchronized by weak balanced coupling.
(3).Utilizing the theory of Lur'e system,the paper studies the practical synchronization of the system composed by nonidentical cells coupled through quorum sensing.This type of synchronization,which implies the dynamics of each cell is similar but with small difference,is a common kind of phenomenon.The factors influencing the synchronization errors are also pointed out,which can help us understand the mechanisms of collective dynamical behaviors caused by quorum sensing.Numerical simulations are carried out to verify these theoretical results.
(4).Frequency synchronization of multicell systems coupled by quorum sensing is also discussed.Two unreasonable hypotheses are often necessary for the previous studies,which aren't adopted here.Without the hypothesis of identical cells,the multicell system is composed by similar but with small difference cells.The second hypothesis,quasi-steady-state approximation, is reasonable in the sense of biology but invalid in the rigorous sense.Exact solution of the signaling molecules in the environment is obtained by the theory of differential equations to replace the hypothesis.
At last,a compact summary of this paper is given by combining the advances of the previous researches in this fields.The prospect for future study and the possible difficulties are also given.
本文结合耦合非线性振子网络的同步研究中的现有方法,提出了一些研究振子网络同步的新思路,得到了振子网络实现同步的充分条件,指出了影响振子网络同步的各种因素,在理论上揭示了振子网络实现同步的机理,并结合物理学、生物学等领域的经典模型进行了理论分析和数值验证.本文的主要内容和创新点主要概括为如下四个方面.
一.对耦合非线性振子网络的完全同步的研究,本文的工作主要表现在两个方面.a).在研究对象上,考虑到现有结果主要针对线性耦合振子网络,而非线性耦合振子网络的同步现象更为普遍,本文研究了两类非线性耦合作用下的振子网络的完全同步行为:对一类常见的耦合函数,当它在孤立节点的吸引域内是增函数时,给出了振子网络实现同步的充分条件;对一般形式的非线性耦合函数,本文提出了耦合函数的平衡假设,验证了大多数实际网络的耦合函数均满足该假设,且在该假设下给出了这类耦合振子网络实现完全同步的充分条件.b).在研究方法上,考虑到耦合振子网络中每个振子的动力学行为由振子自身动力学和耦合作用项两部分构成,而这分别是目前两种常用方法的处理困难所在,本文利用两种现有方法的优点将它们合理的结合起来,得到的结果体现了这种新的研究思路的优越性,在一定程度上解决了现有结果在实际应用时可能遇到的困难.
二.对于耦合非线性振子网络相同步的研究,现有结果基本都是借助于平均场的思想,定义一个衡量相同步程度的序参数进行数值模拟或实验观测.本文利用相约化方法将振子网络的状态方程转化为相应的相方程,然后通过对其相方程进行分析,证明了耦合极限环振子网络在很弱的平衡耦合下就可以实现相同步.
三.利用Lur'e系统的理论研究了由不同的细胞构成的多细胞系统在群体感应机制作用下的实用同步,即各细胞的最终动力学行为相似但又有一定误差,这是一种常见的同步现象.本文给出了影响其同步误差的因素,分析了群体感应机制实现同步的机理,并做了大量的数值验证.
四.研究了群体感应机制作用下的多细胞系统的频率同步问题.考虑到前人的工作普遍采用了拟稳态近似假设,而该假设只有在一定条件下才表现出其合理性.本文通过微分方程理论,借助于环境中的信号分子的精确解代替了拟稳态近似假设,给出了该模型实现同步的严格的理论分析.
最后,结合目前该领域的研究进展,对本文的工作做了全面的总结,并指出了今后该领域进一步工作的展望.
Complex networks are ubiquitous,ranging from nature to society,such as the internet,the power grid,genetic regulatory networks,etc.In recent years,many researches have been carried out to discuss the relationship between the topological structure and collective dynamical behaviors of oscillator networks.Therefore,studying network dynamics has been considered as a new and developing subject of Complexity Science.It has attracted much attention from many fields such as physics,mathetics,biology and computer science.In order to explore collective dynamical behaviors in complex networks,researchers built the model of coupled oscillator networks. The researches of collective behaviors in oscillator networks are important not only to explore the mechanisms of those natural phenomena existing widely but also to utilize these mechanisms to serve people's life.
This paper combines several new methods to study synchronization of oscillator networks and obtains many original results.These results offer sufficient conditions to relealize synchronization and point out the mechanisms of synchronization.Further investigation is carried out to combine theory with practice by simulating many classical models numerically.The main contents and innovative points of this paper are listed as follows.
(1).The study of complete synchronization in coupled oscillator networks can be generalized as the following two aspects.a).As far as the studying objects are concerned,the paper studies coupled oscillator networks with two different types of nonlinear coupling.Most of previous reseaches focused on networks coupled linearly and few researches were carried out to study oscillator networks with nonlinear coupling.Moreover,the study of oscillator networks with nonlinear coupling is interesting and important. For the case that the nonlinear coupling funchtions are increasing functions in the absorbing basin of individual oscillator,this paper gives the sufficient conditions for such type of networks to realize synchronization.For another more general type of coupling flmctions,reseaches are also carried out under the hypothesis of coupling blance,b).As we know,the dynamic behavior of each oscillator is composed of two parts:the inherent dynamic behavior of the uncoupled oscillator and the dynamic behavior influenced by coupling.It happened that two previous methods have difficulties to deal with the stability of these two parts,respectively.The paper combines the two methods by utilizing their merits and solves the respective practical difficulties of the two methods to a certain extent.
(2).The paper studies phase synchronization in oscillator networks. Most of previous researches defined an order parameter based on the idea of mean field approach and carried out numerical simulations or experiment survey.Few researches have been carried out theoretically.In this paper, the dynamics of networks is reduced to phase equations by phase reduced method.Analyzing the phase equations through the master stability function method,one can prove theoretically that the oscillators with identical frequency can be in-phase synchronized by weak balanced coupling.
(3).Utilizing the theory of Lur'e system,the paper studies the practical synchronization of the system composed by nonidentical cells coupled through quorum sensing.This type of synchronization,which implies the dynamics of each cell is similar but with small difference,is a common kind of phenomenon.The factors influencing the synchronization errors are also pointed out,which can help us understand the mechanisms of collective dynamical behaviors caused by quorum sensing.Numerical simulations are carried out to verify these theoretical results.
(4).Frequency synchronization of multicell systems coupled by quorum sensing is also discussed.Two unreasonable hypotheses are often necessary for the previous studies,which aren't adopted here.Without the hypothesis of identical cells,the multicell system is composed by similar but with small difference cells.The second hypothesis,quasi-steady-state approximation, is reasonable in the sense of biology but invalid in the rigorous sense.Exact solution of the signaling molecules in the environment is obtained by the theory of differential equations to replace the hypothesis.
At last,a compact summary of this paper is given by combining the advances of the previous researches in this fields.The prospect for future study and the possible difficulties are also given.
引文
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