复杂生物神经网络的建模及其动力学特性研究
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摘要
英国生物学家Hodgkin和Huxley于1952年建立的著名长枪乌贼巨大轴突非线性动力学微分方程,即Hodgkin-Huxley(H-H)方程,成功的表述了神经放电的电化学机制。该模型可用来描述神经膜中所发生的非线性现象如自激振荡、混沌及多重稳定性等问题,为人们探索神经元的兴奋性提供了基本框架。关于成千上万的神经元之间是如何相互连接、如何协调有序放电的问题,在理论研究和实验探索中都是一个很重要的课题。近几年来,复杂网络理论的发展为探索这一问题提供了一种很好的新方法。本论文将统计方法、复杂网络理论、非线性系统理论等理论和方法应用到生物神经网络的研究中,建立了三种复杂的人工生物神经网络模型并讨论了它们的动力学行为。研究结果表明这些模型表现出了一定的特征,与真实生物神经网络的某些特性存在一定的相似性。我们的研究工作对于进一步深入认识神经网络的兴奋、发放电节律、同步等有一定的参考价值。
本文的主要内容和创新之处可概述如下:
(1)关于生物神经网络的一致共振及最优系统的研究
网络的集群行为是一个热点研究问题。本文讨论了生物神经网络中的长程连接(long-range connection)和本地连接(local connection)在网络中的相互作用。我们采用侧抑制机制,建立了一个全局耦合的侧抑制生物神经网络模型。并用数值模拟的方法研究了这种模型在侧抑制机制下的尖峰行为。研究发现,当连接强度A和抑制比μ达到最佳大小值时,系统的发放序列最有规律。但是在噪声强度较弱时,没有发现最佳的连接强度A和最佳抑制比μ。另外,我们还研究了在不同的连接强度A和抑制比μ条件下的一致共振现象和系统大小共振现象。通过研究发现,当耦合强度较小时,抑制比μ对一致共振和系统大小共振几乎没有什么影响。随着耦合强度逐渐增大,抑制比μ对一致共振和系统大小共振的影响也将越来越大。
(2)关于变权小世界生物神经网络的兴奋、优化特性及同步问题研究
目前对生物神经网络模型的研究,往往都是无权模型或者连接权值为一个固定不变的常量,而实际生物神经元之间的连接权值是随着时间不断变化的。本论文提出了一个变权的小世界生物神经网络模型,并进一步数值研究了这种模型在受外界刺激下的兴奋统计特性,得出了与真实脑神经系统受外界刺激所表现出的类似结果。最有意义的结果是:在同样的网络结构、网络参数及外部刺激信号的条件下,学习系数b存在一个最优值b*,使生物神经网络的兴奋度在b=b*时达到最大。在此基础上,我们又进一步深入研究了该模型的权值变化对同步的影响。通过数值模拟结果我们发现,网络权值的学习系数同样存在一个最优值,使得网络发放电出现尖峰同步现象。另外,我们还考虑了网络连接概率和外加刺激电流的影响,我们发现外加刺激和加边概率对网络同步的影响非常小。
(3)关于外界刺激引起的小世界生物神经网络同步现象的研究
同步在理论和实际应用中都具有很重要的价值,因此,生物神经系统中的同步现象受到越来越多的关注。本文的研究发现不同的外界刺激频率可以使生物神经网络产生同步现象,网络节点耦合强度也可以对同步产生影响,而两者所产生的同步现象的原因是不相同的。
Hodgkin and Huxley who are the biologist of Britain put forward the notability nonlinear dynamical equation of giant axon of Loligo, namely Hodgkin-Huxley(H-H)equation, and described rightly the electrochemistry mechanism of nerve discharging in 1952. This model can be used to describing nonlinear phenomenon appearing in neurilemma, such as the problem of self-oscillation, chaos, and multi-stability etc. It also provides the basic frame to study the excitement of neuron. How do thousands upon thousands neurons couple with each other to fire is an important issue both theoretically and experimentally. Recently, the development of complex networks theory provide another measure to research that problem. In this dissertation, we applied statistical method, nonlinear system theory to the complex biological neural networks, constructed three artificial neural networks models, and then studied their dynamic properties. The studies show that these neural models exhibit some characteristic, which have some extent comparability with the real biological neural networks. Our study might shed some light on studying the excitement, the discharge rhythm, synchronization of neural networks.
The main contents and originalities in this dissertation can be summarized as follows:
(1) The study on coherence resonance and optimal system size in biological neural networks. The network of collective dynamical behaviors have beening the hot topics for the study. In this dissertation, we studied the reciprocal relationship between long-range connection and local connection in the biological neural network. We construct a globally coupled biological neural network model with side-inhibit mechanism, and numerically investigate its coherence resonance. We have considered side-inhibitory mechanism in a globally coupled H-H neurons model, and then studied the spike coherence behavior. We find that the collective behavior of the system is the most regular when the connection strength A and ratioμhave an optimal value, but the optimal island does not exist if the noise strength is small. The CR and size resonance have also been investigated at the different connection strength A and different ratioμ. The ratioμhas little effect on R when the coupling strength A is small. As the coupling strength increases, the influence ofμbecome more distinguishable.
(2) The study on excitement, optimality properties and synchronization in small-world biological neural networks with time-varying weights
Recently, the biological neural network models are unweighted models or the connection weights are just a constant independent time. But, the connection weights of real-world biological neural network are updated dependent on time. In this dissertation, according to biological neural networks have small-world property and updating connections weights with time, we propose a new model of small-world biological neural networks with time-varying weights. Then we numerically study excitement statistical properties of this model under the external stimulus, and get some results which is consistent with real-world biological neural network properties. The significance results show that there is optimal learning rat value b* with the same condition of structure of networks, parameters employed and external stimulating, which make the excitement strength of biological neural networks reach the biggest. On the other hand, we study the effect of updated weights on synchronization in this model. We find there exist an optimal synchronization state when the coupling connection weights of neurons are adjusted reasonably by changing the learning rate. The larger external stimulus will make it easier to tame the synchronization and the connection-rewiring probability which possessed the small-world networks structure just has little effect on the synchronization.
(3) The study on synchronization induced by external stimulus in biological neural network
The synchronization has very importance both in theory and in practical applications. So, the synchronization of biological neural system has attracted increasing attentions. In this dissertation, the results shown that different frequency of external stimulus can induce synchronous activities in neural network. We also found that the coupling strength of network have effect on the synchronization. But the reason that induced this phenomenon is different.
本文的主要内容和创新之处可概述如下:
(1)关于生物神经网络的一致共振及最优系统的研究
网络的集群行为是一个热点研究问题。本文讨论了生物神经网络中的长程连接(long-range connection)和本地连接(local connection)在网络中的相互作用。我们采用侧抑制机制,建立了一个全局耦合的侧抑制生物神经网络模型。并用数值模拟的方法研究了这种模型在侧抑制机制下的尖峰行为。研究发现,当连接强度A和抑制比μ达到最佳大小值时,系统的发放序列最有规律。但是在噪声强度较弱时,没有发现最佳的连接强度A和最佳抑制比μ。另外,我们还研究了在不同的连接强度A和抑制比μ条件下的一致共振现象和系统大小共振现象。通过研究发现,当耦合强度较小时,抑制比μ对一致共振和系统大小共振几乎没有什么影响。随着耦合强度逐渐增大,抑制比μ对一致共振和系统大小共振的影响也将越来越大。
(2)关于变权小世界生物神经网络的兴奋、优化特性及同步问题研究
目前对生物神经网络模型的研究,往往都是无权模型或者连接权值为一个固定不变的常量,而实际生物神经元之间的连接权值是随着时间不断变化的。本论文提出了一个变权的小世界生物神经网络模型,并进一步数值研究了这种模型在受外界刺激下的兴奋统计特性,得出了与真实脑神经系统受外界刺激所表现出的类似结果。最有意义的结果是:在同样的网络结构、网络参数及外部刺激信号的条件下,学习系数b存在一个最优值b*,使生物神经网络的兴奋度在b=b*时达到最大。在此基础上,我们又进一步深入研究了该模型的权值变化对同步的影响。通过数值模拟结果我们发现,网络权值的学习系数同样存在一个最优值,使得网络发放电出现尖峰同步现象。另外,我们还考虑了网络连接概率和外加刺激电流的影响,我们发现外加刺激和加边概率对网络同步的影响非常小。
(3)关于外界刺激引起的小世界生物神经网络同步现象的研究
同步在理论和实际应用中都具有很重要的价值,因此,生物神经系统中的同步现象受到越来越多的关注。本文的研究发现不同的外界刺激频率可以使生物神经网络产生同步现象,网络节点耦合强度也可以对同步产生影响,而两者所产生的同步现象的原因是不相同的。
Hodgkin and Huxley who are the biologist of Britain put forward the notability nonlinear dynamical equation of giant axon of Loligo, namely Hodgkin-Huxley(H-H)equation, and described rightly the electrochemistry mechanism of nerve discharging in 1952. This model can be used to describing nonlinear phenomenon appearing in neurilemma, such as the problem of self-oscillation, chaos, and multi-stability etc. It also provides the basic frame to study the excitement of neuron. How do thousands upon thousands neurons couple with each other to fire is an important issue both theoretically and experimentally. Recently, the development of complex networks theory provide another measure to research that problem. In this dissertation, we applied statistical method, nonlinear system theory to the complex biological neural networks, constructed three artificial neural networks models, and then studied their dynamic properties. The studies show that these neural models exhibit some characteristic, which have some extent comparability with the real biological neural networks. Our study might shed some light on studying the excitement, the discharge rhythm, synchronization of neural networks.
The main contents and originalities in this dissertation can be summarized as follows:
(1) The study on coherence resonance and optimal system size in biological neural networks. The network of collective dynamical behaviors have beening the hot topics for the study. In this dissertation, we studied the reciprocal relationship between long-range connection and local connection in the biological neural network. We construct a globally coupled biological neural network model with side-inhibit mechanism, and numerically investigate its coherence resonance. We have considered side-inhibitory mechanism in a globally coupled H-H neurons model, and then studied the spike coherence behavior. We find that the collective behavior of the system is the most regular when the connection strength A and ratioμhave an optimal value, but the optimal island does not exist if the noise strength is small. The CR and size resonance have also been investigated at the different connection strength A and different ratioμ. The ratioμhas little effect on R when the coupling strength A is small. As the coupling strength increases, the influence ofμbecome more distinguishable.
(2) The study on excitement, optimality properties and synchronization in small-world biological neural networks with time-varying weights
Recently, the biological neural network models are unweighted models or the connection weights are just a constant independent time. But, the connection weights of real-world biological neural network are updated dependent on time. In this dissertation, according to biological neural networks have small-world property and updating connections weights with time, we propose a new model of small-world biological neural networks with time-varying weights. Then we numerically study excitement statistical properties of this model under the external stimulus, and get some results which is consistent with real-world biological neural network properties. The significance results show that there is optimal learning rat value b* with the same condition of structure of networks, parameters employed and external stimulating, which make the excitement strength of biological neural networks reach the biggest. On the other hand, we study the effect of updated weights on synchronization in this model. We find there exist an optimal synchronization state when the coupling connection weights of neurons are adjusted reasonably by changing the learning rate. The larger external stimulus will make it easier to tame the synchronization and the connection-rewiring probability which possessed the small-world networks structure just has little effect on the synchronization.
(3) The study on synchronization induced by external stimulus in biological neural network
The synchronization has very importance both in theory and in practical applications. So, the synchronization of biological neural system has attracted increasing attentions. In this dissertation, the results shown that different frequency of external stimulus can induce synchronous activities in neural network. We also found that the coupling strength of network have effect on the synchronization. But the reason that induced this phenomenon is different.
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