复杂网络上神经动力学研究
摘要
本文主要研究了复杂网络的拓扑属性对神经网络动力学的影响，神经网络中限制同步性的一种动力学机制，并且使用神经网络研究了复杂网络本身的一个普遍性问题：稀疏性特征的意义。具体内容包括下面几个部分：
一、研究了复杂网络结构对于两层神经网络之间的同步的影响。在构建由两层神经网络构成的系统时，采用了三种耦合方式：在度大的节点对之间建立耦合，在随机选定的节点对之间建立耦合和在度小的节点对之间建立耦合。通过计算机模拟，发现度大节点之间的耦合在同步中起着重要的作用。使用度大节点之间耦合的方法，在随机网络和无标度网络中都仅需要少的耦合来实现同步；而切断两个网络之间以度大节点为端点的耦合可以有效的阻止网络的同步。解析的分析表明子系统度分布的差异将导致系统同步性的不同，验证了数值结果并解释了其产生原因。此外，还发现网络存在一个同步效率最高的稀疏连接密度，在此连接下系统可用最少的耦合实现同步。
二、研究了神经元之间突触耦合的效能对于兴奋性神经网络中放电同步的影响。首先研究了单个神经元响应突触耦合电流的动力学，发现了一种新的动力学现象，即神经元的状态可以从极限环转变到固定点或一个绕着固定点运动的暂态。我们将这种现象称为放电熄灭。当突触耦合电流强度大且持续时间长时，在兴奋突触电流扰动下的神经元将表现出这种行为。这种转变的机制是，突触耦合电流的下降能够减弱放电所需要的离子通道电流正反馈，从而阻止神经元放电。进一步，我们发现这种动力学现象能够影响神经网络中的同步。在具有放电熄灭属性的神经网络中，神经元放电的同步程度更低并且对于神经元的差异不敏感。这种影响的机制是，神经元活动状态的转变打断了神经元振荡节律的调节，阻止了同步性的进一步升高。这是一种不同于以往的振子同步的新集体行为，并与相关实验相符合，从而丰富了同步动力学的研究。
三、研究了度关联无标度吸引子网络对刺激的响应。许多真实的复杂网络同时具有无标度性和度关联性，我们发现了度关联无标度网络和度无关网络一样对于随机刺激具有鲁棒性。但是正匹配无标度网络对于定向刺激更加敏感，而反匹配无标度网络对定向刺激的敏感性下降。产生这种差异的机制是，度无关的网络作为一个整体响应刺激，而无标度网络的度关联性导致吸引子网络中系统状态收敛的动力学与度无关网络中不同。在定向刺激下度关联无标度网络可以不再以一个整体做出响应，而是一个部分先做出响应。这样的动力学变化是度关联影响敏感性的机制。
四、使用吸引子网络模型作为例子研究复杂网络稀疏特征的功能意义。通过模拟研究了无标度网络上和随机网络上吸引子模型的计算性能的差异，和这个差异随着网络平均度的变化。模拟发现，随机网络和无标度网络上斑图稳定性的差异在一个稀疏的网络连接密度上有极大值。这个结果表明在一个稀疏的连接密度上网络的拓扑属性对于网络上动力学的影响最明显。使用信噪比分析，我们证明非单调的差异是由网络度分布的差异性和信号强度的竞争导致的。此工作有助于深刻理解具有网络结构的复杂系统往往是稀疏的这一普遍现象。
This thesis investigate the dynamics of neural networks. The main investigations are the influence of the topological properties of networks on the dynamics of neural networks, and the role of the sparse nature of complex networks in the functions of neural networks. Meanwhile, the dynamical origin of preventing the synchrony in neural networks is studied. Before the main investigations, from a nonlinear dynamics point of view, the features of the study on neural activities are introduced and the theory of complex networks invoked in this thesis is introduced. The studies in this thesis are as following:
First, the effects of the degree distribution on mutual synchronization of two-layer neural networks are studied. In the study three coupling strategies are carried out: large-large coupling, random coupling, and small-small coupling. By computer simulations and analytical methods, it is find that couplings between nodes with large degree play an important role in the synchronization. For large-large coupling, less couplings are needed for inducing synchronization for both random and scale-free networks. In contrast, cutting couplings between nodes with large degree is very efficient for preventing neural systems from synchronization, especially when subnetworks are scale free. The analysis treatment shows that the degree distribution of subsystems affects the synchronization between two networks.
Second, we investigate the influence of efficacy of synaptic interaction on firing synchronization in excitatory neuronal networks. We find spike death phenomena: namely, the state of neurons transits from the limit cycle to a fixed point or transient state. The phenomena occur under the perturbation of an excitatory synaptic interaction, which has a high efficacy. We show that the decrease of synaptic current results in spike death through depressing the feedback of the sodium ionic current. In the networks with the spike death property the degree of synchronization is lower and insensitive to the heterogeneity of neurons. The mechanism of the influence is that the transition of the neuron state disrupts the adjustment of the rhythm of the neurons oscillation and prevents a further increase of the firing synchronization.
Third, the response of degree-correlated scale-free attractor networks to stimuli is studied. We show that degree correlated scale-free networks are robust to random stimuli as well as the uncorrelated scale-free networks, while assortative (disassortative) scale-free networks are more (less) sensitive to directed stimuli than uncorrelated networks. We find that the degree correlation of scale-free networks makes the dynamics of attractor systems different from uncorrelated ones. The-dynamics of correlated scale-free attractor networks results in the effects of degree correlation on the response to stimuli.
Fourth, we use the Hopfield attractor networks as an example to study the role of sparse connection density in the difference of functional performance of complex networks. In simulations, we find that the stability of patterns between random and scale-free networks has a maximum difference with a specific sparse connection density. The result suggests that there exists a sparse density with which the dynamics of networks are affected most significantly by topological properties. Using the signal-to-noise-ratio analysis, we show that the non-monotonicity is induced by the competition between the distinction of degree distribution and the signal strength.
一、研究了复杂网络结构对于两层神经网络之间的同步的影响。在构建由两层神经网络构成的系统时，采用了三种耦合方式：在度大的节点对之间建立耦合，在随机选定的节点对之间建立耦合和在度小的节点对之间建立耦合。通过计算机模拟，发现度大节点之间的耦合在同步中起着重要的作用。使用度大节点之间耦合的方法，在随机网络和无标度网络中都仅需要少的耦合来实现同步；而切断两个网络之间以度大节点为端点的耦合可以有效的阻止网络的同步。解析的分析表明子系统度分布的差异将导致系统同步性的不同，验证了数值结果并解释了其产生原因。此外，还发现网络存在一个同步效率最高的稀疏连接密度，在此连接下系统可用最少的耦合实现同步。
二、研究了神经元之间突触耦合的效能对于兴奋性神经网络中放电同步的影响。首先研究了单个神经元响应突触耦合电流的动力学，发现了一种新的动力学现象，即神经元的状态可以从极限环转变到固定点或一个绕着固定点运动的暂态。我们将这种现象称为放电熄灭。当突触耦合电流强度大且持续时间长时，在兴奋突触电流扰动下的神经元将表现出这种行为。这种转变的机制是，突触耦合电流的下降能够减弱放电所需要的离子通道电流正反馈，从而阻止神经元放电。进一步，我们发现这种动力学现象能够影响神经网络中的同步。在具有放电熄灭属性的神经网络中，神经元放电的同步程度更低并且对于神经元的差异不敏感。这种影响的机制是，神经元活动状态的转变打断了神经元振荡节律的调节，阻止了同步性的进一步升高。这是一种不同于以往的振子同步的新集体行为，并与相关实验相符合，从而丰富了同步动力学的研究。
三、研究了度关联无标度吸引子网络对刺激的响应。许多真实的复杂网络同时具有无标度性和度关联性，我们发现了度关联无标度网络和度无关网络一样对于随机刺激具有鲁棒性。但是正匹配无标度网络对于定向刺激更加敏感，而反匹配无标度网络对定向刺激的敏感性下降。产生这种差异的机制是，度无关的网络作为一个整体响应刺激，而无标度网络的度关联性导致吸引子网络中系统状态收敛的动力学与度无关网络中不同。在定向刺激下度关联无标度网络可以不再以一个整体做出响应，而是一个部分先做出响应。这样的动力学变化是度关联影响敏感性的机制。
四、使用吸引子网络模型作为例子研究复杂网络稀疏特征的功能意义。通过模拟研究了无标度网络上和随机网络上吸引子模型的计算性能的差异，和这个差异随着网络平均度的变化。模拟发现，随机网络和无标度网络上斑图稳定性的差异在一个稀疏的网络连接密度上有极大值。这个结果表明在一个稀疏的连接密度上网络的拓扑属性对于网络上动力学的影响最明显。使用信噪比分析，我们证明非单调的差异是由网络度分布的差异性和信号强度的竞争导致的。此工作有助于深刻理解具有网络结构的复杂系统往往是稀疏的这一普遍现象。
This thesis investigate the dynamics of neural networks. The main investigations are the influence of the topological properties of networks on the dynamics of neural networks, and the role of the sparse nature of complex networks in the functions of neural networks. Meanwhile, the dynamical origin of preventing the synchrony in neural networks is studied. Before the main investigations, from a nonlinear dynamics point of view, the features of the study on neural activities are introduced and the theory of complex networks invoked in this thesis is introduced. The studies in this thesis are as following:
First, the effects of the degree distribution on mutual synchronization of two-layer neural networks are studied. In the study three coupling strategies are carried out: large-large coupling, random coupling, and small-small coupling. By computer simulations and analytical methods, it is find that couplings between nodes with large degree play an important role in the synchronization. For large-large coupling, less couplings are needed for inducing synchronization for both random and scale-free networks. In contrast, cutting couplings between nodes with large degree is very efficient for preventing neural systems from synchronization, especially when subnetworks are scale free. The analysis treatment shows that the degree distribution of subsystems affects the synchronization between two networks.
Second, we investigate the influence of efficacy of synaptic interaction on firing synchronization in excitatory neuronal networks. We find spike death phenomena: namely, the state of neurons transits from the limit cycle to a fixed point or transient state. The phenomena occur under the perturbation of an excitatory synaptic interaction, which has a high efficacy. We show that the decrease of synaptic current results in spike death through depressing the feedback of the sodium ionic current. In the networks with the spike death property the degree of synchronization is lower and insensitive to the heterogeneity of neurons. The mechanism of the influence is that the transition of the neuron state disrupts the adjustment of the rhythm of the neurons oscillation and prevents a further increase of the firing synchronization.
Third, the response of degree-correlated scale-free attractor networks to stimuli is studied. We show that degree correlated scale-free networks are robust to random stimuli as well as the uncorrelated scale-free networks, while assortative (disassortative) scale-free networks are more (less) sensitive to directed stimuli than uncorrelated networks. We find that the degree correlation of scale-free networks makes the dynamics of attractor systems different from uncorrelated ones. The-dynamics of correlated scale-free attractor networks results in the effects of degree correlation on the response to stimuli.
Fourth, we use the Hopfield attractor networks as an example to study the role of sparse connection density in the difference of functional performance of complex networks. In simulations, we find that the stability of patterns between random and scale-free networks has a maximum difference with a specific sparse connection density. The result suggests that there exists a sparse density with which the dynamics of networks are affected most significantly by topological properties. Using the signal-to-noise-ratio analysis, we show that the non-monotonicity is induced by the competition between the distinction of degree distribution and the signal strength.
引文
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