Rulkov和Head direction神经元及神经元网络的动力学分析

发布时间：2018-06-17 18:27

本文选题：Rulkov神经元模型 + 电耦合下的Rulkov神经元网络　；参考：《北京交通大学》2016年博士论文

【摘要】：本文首先从研究单个二维混沌的Rulkov神经元模型的参数空间入手,利用定性分析、分岔理论、中心流形定理、标准型等对单个二维混沌Rulkov神经元模型的参数空间进行划分,其目的是解释该神经元模型在二维参数空间中的动力学行为和放电机制.我们的研究结果表明：系统唯一的不动点在二维参数平面内不同区域内存在非常丰富的非线性动力学行为和复杂的生物放电机制,例如参数取值选在不同区域时,不仅影响系统不动点处的稳定性,而且可能使系统在不动点处发生鞍结点分岔、超临界/亚临界的Neimark-Sacker(NS)分岔或是出现拟周期解等.通过这些分析我们很容易理解二维单个混沌的Rulkov神经元模型所产生的放电行为,以及这些不同放电行为相互转化的机理.其次,我们考虑含有双向电耦合的由两个混沌的Rulkov神经元构成的简单神经元网络的稳定性和混沌行为.通过分析我们发现系统不动点的个数和位置并没有因为电耦合结构的产生或耦合强度的变化发生改变,也就是说在双向电耦合条件下,系统始终只存在唯一的不动点.与此同时,我们一方面给出函数中的控制参数特别是耦合强度对这个简单神经元网络不动点局部稳定性的影响；另外一方面从理论上严格证明了当参数满足相应条件时系统存在Marotto意义下混沌,从而解释了为什么该模型在数值模拟过程中当控制参数选取特定范围内会出现混沌的放电机制.本文的最后一部分工作主要是研究大尺度的头朝向(head direction)神经元网络的动态行为.与人工神经元网络模型不同的是头朝向神经元网络是依赖于头朝向神经元的生物特点建立的神经元网络模型,因此它更具有生物意义和功能.头朝向神经元网络是由头朝向神经元群体构成的,该种神经元在许多哺乳动物的大脑边缘系统都有发现.它们的特点是当动物的头朝向特定方向时异常兴奋,表现形式为放电频率高于基本线,且每个头朝向神经元只有一个偏好方向使得其放电频率达到最大值.通过分析头朝向神经元网络的动态行为,我们从理论上得到了关于头朝向神经元网络的一些性质.通过定性分析我们给出头朝向神经元网络所有平均输出有界的条件；通过傅里叶分析结合头朝向神经元网络耦合权重函数的生物特点及平均输入和输出的关系,我们得到该模型傅里叶变换后对应项系数的关系,从而首次提出该神经元网络平衡态的一般形式.与此同时,根据头朝向神经元模型动力学形式,我们准确的构造了该系统的Lyapunov函数,进而从理论上说明了整个系统流形的稳定性,即头朝向神经元网络随时间变化的趋势.通过研究所构建的Lyapunov函数的特性可知,无论系统的初值是什么样的,无论系统得到的输入信息是什么样的,只要耦合突触权重函数满足特定的条件,那么系统的所有流形都将随着时间的推移达到一个相应的平衡态,从而使该流形的能量函数达到最小值,也就是说系统过任意初始值的解均收敛于相应的平衡态.因此,从理论上解释了头朝向神经元网络模型具有稳定偏好方向的机理.更重要的是我们通过选取相应的耦合突触权重函数和S形增益函数,首次求出了该网络此时所有可能存在的平衡解,而且通过严格的数学推理证明了这些平衡解的稳定性.这些结果一方面进一步验证了头朝向神经元网络的一般性质,另外一方面比较意外的是我们不仅求出了与头朝向神经元网络生物特性相对应的具有对称结构的平衡态,而且求出了一组非对称的平衡态.这些结果对头朝向神经元网络的研究具有非常重要的影响.因为这些结果提供了理解神经生理数据的更多可能性,从而为发展研究新数据的方法、设计新的区分不同神经机制的实验提供了理论基础.
[Abstract]:In this paper, the parameter space of a single two-dimensional chaotic Rulkov neuron model is first studied, and the parameter space of a single two-dimensional chaotic Rulkov neuron model is divided by qualitative analysis, bifurcation theory, central manifold theorem and standard type. The purpose is to explain the dynamic behavior of the neural element model in the two-dimensional parameter space. Our research results show that the only fixed point of the system has a very rich nonlinear dynamic behavior and a complex biological discharge mechanism in the different regions of the two-dimensional parameter plane. For example, when the parameters are selected in different regions, the system not only affects the stability of the fixed point in the system, but also may make the system at the fixed point. There is a saddle node bifurcation, a supercritical / subcritical Neimark-Sacker (NS) bifurcation or a quasi periodic solution. Through these analyses, it is easy to understand the discharge behavior of the two dimensional single chaotic Rulkov neuron model and the mechanism of the mutual conversion of these different discharge behaviors. Secondly, we consider the two-way electrical coupling. The stability and chaotic behavior of a simple neural network consisting of two chaotic Rulkov neurons. Through analysis we find that the number and position of the fixed point of the system does not change because of the generation of the electric coupling structure or the change of the coupling strength, that is to say, the system always only exists only under the condition of two-way electric coupling. At the same time, on the one hand, we give the effect of the control parameters in the function, especially the coupling strength on the fixed point local stability of the simple neuron network. On the other hand, we theoretically prove that the system has the Marotto sense chaos when the parameter satisfies the corresponding condition, thus explaining why the model is in the case. In the process of numerical simulation, a chaotic discharge mechanism will appear when the control parameters are selected. The last part of this paper is to study the dynamic behavior of the large scale head direction neural network. Unlike the artificial neural network model, the head toward neuron network is dependent on the head oriented neuron. A neural network model established by biological characteristics, so it has a more biological meaning and function. The head toward the neuron network is made up of a head toward the neuron group, which is found in the cerebral marginal system of many mammals. Their characteristic is that when the head of the animal is in a specific direction, it is excitability and form. The discharge frequency is higher than the basic line, and each head has only one preference direction to the neuron to maximize the discharge frequency. By analyzing the dynamic behavior of the head toward the neuron network, we theoretically get some properties about the head toward the neuron network. By qualitative analysis, we turn the head toward the neuron network. By combining the biological characteristics of the head toward the neural network coupled weight function and the relationship between the average input and output, we get the relationship between the coefficients of the corresponding term after Fourier transform and the general form of the equilibrium state of the neuron network for the first time. The head toward the dynamic form of neuron model, we accurately constructed the Lyapunov function of the system, and then theoretically explained the stability of the whole system manifold, that is, the trend of the head toward the neuron network with time. By the study of the characteristics of the Lyapunov function constructed, no matter what the initial value of the system is, no matter what the system's initial value is, What is the input information obtained by the system, as long as the coupling synaptic weight function satisfies a specific condition, all the manifolds of the system will reach a corresponding equilibrium state with the passage of time, thus making the energy function of the manifold reach a minimum, that is to say, the solution of any initial value of the system converges to the corresponding equilibrium. Therefore, the mechanism that the head oriented neuron network model has a stable preference direction is explained theoretically. More importantly, we first find out all possible equilibrium solutions of the network at this time by selecting the corresponding coupling synaptic weight function and S shape gain function, and prove these equilibria by strict mathematical reasoning. On the one hand, these results further verify the general properties of the head oriented neural network. On the other hand, it is more surprising that we not only find a equilibrium state with a symmetric structure corresponding to the biological characteristics of the head toward the neuron network, but also find a group of asymmetrical equilibrium states. These results are the head orientation. The study of neural networks has a very important impact, because these results provide more possibilities for understanding neurophysiological data, thus providing a theoretical basis for the development of new data methods and the design of new experiments to distinguish different neural mechanisms.
【学位授予单位】：北京交通大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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