复杂网络体系重要时空动力学的理论研究
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摘要
近十年来,作为一门新兴的交叉科学,复杂网络在各个学科中有着广泛的应用。从整个人类社会到细胞内部的蛋白质,网络的概念为我们提供了一个认识现实复杂系统的新视角。
现实中的复杂系统的复杂性大体上包含两个方面,一方面是个体动力学行为的复杂性,现实的个体往往是非线性系统,它会有周期地或混沌地振荡,还会发生各种复杂的分岔行为。另一方面是个体间的相互作用的复杂性,即使是简单的个体组成的复杂系统,也会涌现出丰富的整体现象。那么,个体间的相互作用和整体的复杂现象有着怎样的关系,就是复杂网络研究的内容之一。
耦合神经元网络就是一个典型的复杂系统,一方面,神经元本身就是一个具有多时间尺度的可激发的非线性动力系统,它在不同的外界刺激下,会表现出不同动力学行为;当外界刺激改变时,还会出现丰富的分岔现象.另一方面,神经元的相互连接也具有两种复杂性,一是神经元的突触连接本身的复杂性,例如神经元的化学突触在传递信号时需要在胶质细胞的辅助下释放和接受神经递质,同时还需要ATP提供能量,这个过程中就涉及到非常复杂的生物化学反应。二是神经元网络拓扑结构的复杂性,大量的实验数据表明,大脑中神经元的结构网络和功能网络都不是一个简单的规则网络或者完全随机的网络,而是具有小世界性质和无标度性质的复杂网络。在个体和相互作用的复杂性的共同作用下,耦合神经元网络表现出丰富的时空动力学现象,簇放电同步态就是其中之一。本论文中的一个主要研究方面就是关于耦合神经元网络上的簇放电同步态。
在研究复杂体系的时空行为时,除了常/偏微分方程外,我们还可以使用耦合映象格子。与微分方程相比,耦合映象格子的计算量更小,更有利于数值模拟。本论文的另一个方面就是研究耦合logistic映射的时空斑图的行为.
总的来说,本论文主要研究了耦合神经元网络以及耦合映象格子上的时空动力学问题,包括以下几部分内容:
◆复杂网络上的簇放电同步过程
与全同振子的完全同步态不同,耦合神经元的簇放电同步态不是一个同步流形。因此,对于神经元的簇放电同步态的研究不能直接使用全同振子的完全同步态的主稳定方程分析方法。我们研究了在Newman-Watts(Nw)型小世界网络上通过电突触耦合的Hindmash-Rose(HR)神经元的簇放电同步过程。随着耦合强度的增大,系统先从时空混沌态转变到第一种属于fold-homoclinic类型的簇放电同步态,然后这种簇同步态经历一个spike-adding过程到达另一种fold-Hopf类型的簇放电同步态.在这个过程中,神经元的度起着关键性的作用,所有度k＞k_c的神经元都会从fold-homoclinic类型的簇放电态转变为fold-Hopf类型的簇放电态。这种动力学成簇的现象可以通过局域平均场近似来得到定性解释.我们还简要的讨论了通过化学突触耦合的神经元系统的簇放电同步态的转变行为.(这部分内容发表在Physical Review E上)
在此基础上,我们研究了一系列基于NW型算法生成的复杂网络上耦合HR神经元的同步过程。通过数值模拟,我们给出了在二维相图上的几种不同的斑图类型的区域以及过渡区域的范围,并且与理论的完全同步以及簇同步阈值做了对比.我们还发现当网络属于小世界类型时,系统最容易出现时空规则的簇放电同步态。
◆从神经元放电行为推测网络拓扑性质
在很多实际网络中,尤其是神经元网络,有时难以获得网络的具体结构。这时,如何通过动力学性质来推测网络的拓扑结构的“逆向工程”,就成为了一个令人关心的话题。在这里.我们就提出了两种通过神经元放电的信息来推测网络度分布性质的方法,一种是根据神经元放电的振幅,而另一种是根据神经元在一次簇放电过程中放电的次数。
我们发现通过电突触耦合的Morris-Lecar神经元网络在发生簇同步时,其放电的振幅与神经元的度有着非常好的线性关系,这种线性关系的关键之处在于在发生簇同步时各个神经元的尖峰放电的无关性。我们根据这一关系提出了用放电的振幅来推测网络的度分布的方法,最后在无标度网络和小世界网络上验证了这一方法。(这方面的部分内容已经提交给Physical Review E)
我们在研究耦合HR神经元通过簇同步的过程中发现,HR神经元在fold-homoclinic型簇放电模式和fold-Hopf型簇放电模式下在一个簇放电过程中有着明显不同簇放电数目。我们还发现神经元在达到fold-Hopf型簇放电所需的耦合强度与它的度成倒数关系。因此在神经元发生分岔类型转变的过程中,我们可以根据的神经元在一个簇放电过程中的放电数目来推测出各个神经元的相对度的大小,从而得到整个网络的度分布的性质。
◆复杂网络上耦合映象格子的同步与时空斑图
耦合映象格子是另一种重要的时空体系模型。我们在这里研究了NW型复杂网络上的耦合logistic模型。首先,结合数值模拟和主方程分析,我们给出了在以耦合强度和随机长程连接几率为参量的二维相空间内的同步区域。然后我们发现在不同的耦合强度下,系统的时空斑图会随着长程连接的增多而发生不同的变化:在强耦合条件下,系统会从时空无序斑图先变成两种斑图的共存态,然后变成同步混沌斑图;在弱耦合条件下,系统会变成三种斑图的共存态,并且最终会变成时间有序空间也较同步的时空有序斑图.在这个过程中,最终的两种斑图——时空有序和同步混沌斑图——是规则网络上不曾出现的.因此,我们认为拓扑无序会导致耦合映象格子从时空混沌态中产生新的斑图,并且在某种程度上也有“随机长程连接驯服时空混沌”的效果。(这方面的工作发表在Chinese PhysicsLctter上面)
In Recent decades,as an important interdisciplinary science,the concept of complex networks has been applied in many subjects.Ranging from the whole human society to protein in cells,the concept of complex networks provides us a new point of view in the research of complex systems.
Basically,the complexity of complex systems can be treated as two parts:One is the complexity of the components.The other one is the complexity of interactions. The complex system will emerge a lot of collective behavior even it is composed of the simplest components.Therefore,the relationship between the interaction of the components and the complex behavior of the whole system is one of the main topics in the research on complex networks.
A coupled neuron network is a typical complex system.First,a neuron is a excitable nonlinear dynamical system of multi-time scale.It shows different dynamical behaviors while the outer stimulations are different,and it also have abundant types of bifurcations during the changing of outer stimulations.Second,the connections between neurons are also complicated.On one hand,the dynamics of the neurons' synapses are complex.E.g.When a neuron transmits a signal to another neuron via chemical synapse.The axon of the first neuron releases a neurotransmitter, adjusted by the astrocytes,and then the neurotransmitter reaches the dendrite of the second neuron.During this process,the energy is provided by ATP moleculars.This process has so many biological and chemical reactions that the process itslef is complicated enough.On the other hand,A lot of experiment results indicate that either the function or the structure of a neuron system is not a regular or random network. Actually,the structure or the function network of the brain has both small world and scale free characteristics.Thus,the neurons complexity and the connections complexity make the coupled neuron networks have abundant of spatio-temporal dynamical phenomena.One of these nontrivial dynamical phenomena is burst synchronization(BS),which is one of this essay's topics.
As well as the differential equation models,the coupled map lattice model is also a useful tool to study the spatio-temporal behavior of complex systems.The coupled map lattice model is more suitable for numeral study because it has less calculated amount than differential equation models.The other topic of this essay is focused on the behavior of spatio-temporal patterns of coupled logistic maps.
Overall,The spatio-temporal dynamical behaviors of coupled neurons networks and coupled map lattice are studied in this essay,which includes the following parts:
The bursting synchronization process on complex networks
The full synchronization state of homogeneous oscillators is a synchronous manifold,but the BS state is not.Therefore the Master Stability Function(MSF) can not be used to analyse the BS behavior.Here,we studied the electrical coupled Hindmash-Rose neurons on small world networks proposed by Newman arts Watts. With the increasing coupling strength,the spatio-temporal chaos of the system can be tamed into one type of BS states with fold-homoclinic bursting,and then undergoes spiking-adding and transits into another type of BS states with fold-Hopf bursting. During the latter transition,The neuron's degree plays a key role.All the neurons with degree k＞k_c change to show fold-Hopf bursting,withεk_c nearly a constant.This is a dynamic cluster separation process,which can be explained by using local mean field aproximation.The case when the neurons are coupled via chemical synapses is also briefly discussed.
Based on this result,we studied the transiion process of coupled HR neurons on a series of complex networks by adding random shortcuts to a regular ring.Using numeral simulation,we got the phase diagram in the parameter space of coupling strength and random links probability.The numeral result is capable with theorical result about complete synchronization and BS state.More interestingly,when the network is of small world type,the system is most likely to be at BS state,which is spatio-temporal ordered.
Revealing Network's topology property by investigating the spike and burst behaviors
For many real systems,especially neuron systems,sometimes it is not easy to figure out the networks structure directly.Therefore,it is interesting to establish a 'reverse engineering' method to probe the network topology properties by studing the various dynamic behaviors of the system.Here we present two methods to reveal topology property based on the information of spike amplitude and frequency, respectively.
First we studied a Morris-Lecar neuron network coupled via electrical synapse. Under the weak coupling condition,where the system is at BS state,the spike amplitude of a given neuron have a excellent linear relationship with its degree.The keypoint of this relationship is the incoherence of the spikes inside the burst among the neurons.Based on this relationship,we proposed a possible method to estimate the degree distribution of the network by simple statistics of the spike amplitudes.We demonstrate the validity of this scheme on scale free as well as small world networks.
Second we studied the electrical coupled HR neuron network.During the transition to BS of coupled HR neurons,We found out the spike number per burst(SPB) is quite different between fold-homoclinic neurons and fold-Hopf neurons. and the critical coupling strength for a neuron to become fold-Hopf type has a reciprocal relationship with the neuron's degree.Thus,combining the information of the SPB and critical coupling strength,we can get the degree property of the network.
Synchronization and pattern dynamics of coupled map lattice on complex networks.
Coupled map lattice is another important spatio-temporal model.Here we studied the coupled logistic maps on complex networks of N-W type.First,combining numerical simulations and master function analysis,we figured out the synchronization zone in the parameter space of coupling strength and the fraction of random shortcuts.Second,we studied the pattern formation and selection process. Under strong coupling condition,the system will change from spatio-temporal order pattern to two different patterns and finally changed to the full synchonized chaotic pattern.Under weak coupling condition,the system undergoes the patterns of three types and finelly stays at the spatio-temporal ordered pattern,which is one of the former patterns.During these processes,there are two new pattems,spatio-temporal ordered and synchronized chaotic patterns,which are not existed in regular networks. Hence,is seems that the topological disorder has induced somewhat pattern formation and has an effect of'random shortcut can tame spatiotemporal chaos'.
现实中的复杂系统的复杂性大体上包含两个方面,一方面是个体动力学行为的复杂性,现实的个体往往是非线性系统,它会有周期地或混沌地振荡,还会发生各种复杂的分岔行为。另一方面是个体间的相互作用的复杂性,即使是简单的个体组成的复杂系统,也会涌现出丰富的整体现象。那么,个体间的相互作用和整体的复杂现象有着怎样的关系,就是复杂网络研究的内容之一。
耦合神经元网络就是一个典型的复杂系统,一方面,神经元本身就是一个具有多时间尺度的可激发的非线性动力系统,它在不同的外界刺激下,会表现出不同动力学行为;当外界刺激改变时,还会出现丰富的分岔现象.另一方面,神经元的相互连接也具有两种复杂性,一是神经元的突触连接本身的复杂性,例如神经元的化学突触在传递信号时需要在胶质细胞的辅助下释放和接受神经递质,同时还需要ATP提供能量,这个过程中就涉及到非常复杂的生物化学反应。二是神经元网络拓扑结构的复杂性,大量的实验数据表明,大脑中神经元的结构网络和功能网络都不是一个简单的规则网络或者完全随机的网络,而是具有小世界性质和无标度性质的复杂网络。在个体和相互作用的复杂性的共同作用下,耦合神经元网络表现出丰富的时空动力学现象,簇放电同步态就是其中之一。本论文中的一个主要研究方面就是关于耦合神经元网络上的簇放电同步态。
在研究复杂体系的时空行为时,除了常/偏微分方程外,我们还可以使用耦合映象格子。与微分方程相比,耦合映象格子的计算量更小,更有利于数值模拟。本论文的另一个方面就是研究耦合logistic映射的时空斑图的行为.
总的来说,本论文主要研究了耦合神经元网络以及耦合映象格子上的时空动力学问题,包括以下几部分内容:
◆复杂网络上的簇放电同步过程
与全同振子的完全同步态不同,耦合神经元的簇放电同步态不是一个同步流形。因此,对于神经元的簇放电同步态的研究不能直接使用全同振子的完全同步态的主稳定方程分析方法。我们研究了在Newman-Watts(Nw)型小世界网络上通过电突触耦合的Hindmash-Rose(HR)神经元的簇放电同步过程。随着耦合强度的增大,系统先从时空混沌态转变到第一种属于fold-homoclinic类型的簇放电同步态,然后这种簇同步态经历一个spike-adding过程到达另一种fold-Hopf类型的簇放电同步态.在这个过程中,神经元的度起着关键性的作用,所有度k＞k_c的神经元都会从fold-homoclinic类型的簇放电态转变为fold-Hopf类型的簇放电态。这种动力学成簇的现象可以通过局域平均场近似来得到定性解释.我们还简要的讨论了通过化学突触耦合的神经元系统的簇放电同步态的转变行为.(这部分内容发表在Physical Review E上)
在此基础上,我们研究了一系列基于NW型算法生成的复杂网络上耦合HR神经元的同步过程。通过数值模拟,我们给出了在二维相图上的几种不同的斑图类型的区域以及过渡区域的范围,并且与理论的完全同步以及簇同步阈值做了对比.我们还发现当网络属于小世界类型时,系统最容易出现时空规则的簇放电同步态。
◆从神经元放电行为推测网络拓扑性质
在很多实际网络中,尤其是神经元网络,有时难以获得网络的具体结构。这时,如何通过动力学性质来推测网络的拓扑结构的“逆向工程”,就成为了一个令人关心的话题。在这里.我们就提出了两种通过神经元放电的信息来推测网络度分布性质的方法,一种是根据神经元放电的振幅,而另一种是根据神经元在一次簇放电过程中放电的次数。
我们发现通过电突触耦合的Morris-Lecar神经元网络在发生簇同步时,其放电的振幅与神经元的度有着非常好的线性关系,这种线性关系的关键之处在于在发生簇同步时各个神经元的尖峰放电的无关性。我们根据这一关系提出了用放电的振幅来推测网络的度分布的方法,最后在无标度网络和小世界网络上验证了这一方法。(这方面的部分内容已经提交给Physical Review E)
我们在研究耦合HR神经元通过簇同步的过程中发现,HR神经元在fold-homoclinic型簇放电模式和fold-Hopf型簇放电模式下在一个簇放电过程中有着明显不同簇放电数目。我们还发现神经元在达到fold-Hopf型簇放电所需的耦合强度与它的度成倒数关系。因此在神经元发生分岔类型转变的过程中,我们可以根据的神经元在一个簇放电过程中的放电数目来推测出各个神经元的相对度的大小,从而得到整个网络的度分布的性质。
◆复杂网络上耦合映象格子的同步与时空斑图
耦合映象格子是另一种重要的时空体系模型。我们在这里研究了NW型复杂网络上的耦合logistic模型。首先,结合数值模拟和主方程分析,我们给出了在以耦合强度和随机长程连接几率为参量的二维相空间内的同步区域。然后我们发现在不同的耦合强度下,系统的时空斑图会随着长程连接的增多而发生不同的变化:在强耦合条件下,系统会从时空无序斑图先变成两种斑图的共存态,然后变成同步混沌斑图;在弱耦合条件下,系统会变成三种斑图的共存态,并且最终会变成时间有序空间也较同步的时空有序斑图.在这个过程中,最终的两种斑图——时空有序和同步混沌斑图——是规则网络上不曾出现的.因此,我们认为拓扑无序会导致耦合映象格子从时空混沌态中产生新的斑图,并且在某种程度上也有“随机长程连接驯服时空混沌”的效果。(这方面的工作发表在Chinese PhysicsLctter上面)
In Recent decades,as an important interdisciplinary science,the concept of complex networks has been applied in many subjects.Ranging from the whole human society to protein in cells,the concept of complex networks provides us a new point of view in the research of complex systems.
Basically,the complexity of complex systems can be treated as two parts:One is the complexity of the components.The other one is the complexity of interactions. The complex system will emerge a lot of collective behavior even it is composed of the simplest components.Therefore,the relationship between the interaction of the components and the complex behavior of the whole system is one of the main topics in the research on complex networks.
A coupled neuron network is a typical complex system.First,a neuron is a excitable nonlinear dynamical system of multi-time scale.It shows different dynamical behaviors while the outer stimulations are different,and it also have abundant types of bifurcations during the changing of outer stimulations.Second,the connections between neurons are also complicated.On one hand,the dynamics of the neurons' synapses are complex.E.g.When a neuron transmits a signal to another neuron via chemical synapse.The axon of the first neuron releases a neurotransmitter, adjusted by the astrocytes,and then the neurotransmitter reaches the dendrite of the second neuron.During this process,the energy is provided by ATP moleculars.This process has so many biological and chemical reactions that the process itslef is complicated enough.On the other hand,A lot of experiment results indicate that either the function or the structure of a neuron system is not a regular or random network. Actually,the structure or the function network of the brain has both small world and scale free characteristics.Thus,the neurons complexity and the connections complexity make the coupled neuron networks have abundant of spatio-temporal dynamical phenomena.One of these nontrivial dynamical phenomena is burst synchronization(BS),which is one of this essay's topics.
As well as the differential equation models,the coupled map lattice model is also a useful tool to study the spatio-temporal behavior of complex systems.The coupled map lattice model is more suitable for numeral study because it has less calculated amount than differential equation models.The other topic of this essay is focused on the behavior of spatio-temporal patterns of coupled logistic maps.
Overall,The spatio-temporal dynamical behaviors of coupled neurons networks and coupled map lattice are studied in this essay,which includes the following parts:
The bursting synchronization process on complex networks
The full synchronization state of homogeneous oscillators is a synchronous manifold,but the BS state is not.Therefore the Master Stability Function(MSF) can not be used to analyse the BS behavior.Here,we studied the electrical coupled Hindmash-Rose neurons on small world networks proposed by Newman arts Watts. With the increasing coupling strength,the spatio-temporal chaos of the system can be tamed into one type of BS states with fold-homoclinic bursting,and then undergoes spiking-adding and transits into another type of BS states with fold-Hopf bursting. During the latter transition,The neuron's degree plays a key role.All the neurons with degree k＞k_c change to show fold-Hopf bursting,withεk_c nearly a constant.This is a dynamic cluster separation process,which can be explained by using local mean field aproximation.The case when the neurons are coupled via chemical synapses is also briefly discussed.
Based on this result,we studied the transiion process of coupled HR neurons on a series of complex networks by adding random shortcuts to a regular ring.Using numeral simulation,we got the phase diagram in the parameter space of coupling strength and random links probability.The numeral result is capable with theorical result about complete synchronization and BS state.More interestingly,when the network is of small world type,the system is most likely to be at BS state,which is spatio-temporal ordered.
Revealing Network's topology property by investigating the spike and burst behaviors
For many real systems,especially neuron systems,sometimes it is not easy to figure out the networks structure directly.Therefore,it is interesting to establish a 'reverse engineering' method to probe the network topology properties by studing the various dynamic behaviors of the system.Here we present two methods to reveal topology property based on the information of spike amplitude and frequency, respectively.
First we studied a Morris-Lecar neuron network coupled via electrical synapse. Under the weak coupling condition,where the system is at BS state,the spike amplitude of a given neuron have a excellent linear relationship with its degree.The keypoint of this relationship is the incoherence of the spikes inside the burst among the neurons.Based on this relationship,we proposed a possible method to estimate the degree distribution of the network by simple statistics of the spike amplitudes.We demonstrate the validity of this scheme on scale free as well as small world networks.
Second we studied the electrical coupled HR neuron network.During the transition to BS of coupled HR neurons,We found out the spike number per burst(SPB) is quite different between fold-homoclinic neurons and fold-Hopf neurons. and the critical coupling strength for a neuron to become fold-Hopf type has a reciprocal relationship with the neuron's degree.Thus,combining the information of the SPB and critical coupling strength,we can get the degree property of the network.
Synchronization and pattern dynamics of coupled map lattice on complex networks.
Coupled map lattice is another important spatio-temporal model.Here we studied the coupled logistic maps on complex networks of N-W type.First,combining numerical simulations and master function analysis,we figured out the synchronization zone in the parameter space of coupling strength and the fraction of random shortcuts.Second,we studied the pattern formation and selection process. Under strong coupling condition,the system will change from spatio-temporal order pattern to two different patterns and finally changed to the full synchonized chaotic pattern.Under weak coupling condition,the system undergoes the patterns of three types and finelly stays at the spatio-temporal ordered pattern,which is one of the former patterns.During these processes,there are two new pattems,spatio-temporal ordered and synchronized chaotic patterns,which are not existed in regular networks. Hence,is seems that the topological disorder has induced somewhat pattern formation and has an effect of'random shortcut can tame spatiotemporal chaos'.
引文
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[19]Watts,D.J.,1999 Small Worlds:The Dynamics of Networks between Order and Randomness(Princeton University,Princeton,NJ)
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[57]J.D.Murray,1998 Mathematical Biology(2nd ed.)Springer-Verlag世界图书出版公司北京公司
[58]Ant(?)nio M.Batista et.al.,2002 Phys.Rev.E 65 056209
[59]Kaneko K 1989 Physica D 34 1
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