本文選題：神經(jīng)元 + 混合模式振蕩��； 參考：《華南理工大學(xué)》2016年博士論文

【摘要】：神經(jīng)動力學(xué)以利用動力學(xué)原理研究神經(jīng)元的電生理活動為基礎(chǔ),通過構(gòu)建神經(jīng)元模型來研究神經(jīng)系統(tǒng)的分岔,混沌,振蕩等動力學(xué)性質(zhì).本文以混合模式振蕩為主要研究對象,分析神經(jīng)元模型中此類現(xiàn)象的變化規(guī)律,并且解釋其中兩個模型中混合模式振蕩的產(chǎn)生機理.同時分析神經(jīng)元模型中平衡點的分支,峰峰間距序列的分岔以及峰的變化等問題,得到豐富的動力學(xué)結(jié)果.論文分為四部分：開篇第一章介紹神經(jīng)元中混合模式振蕩現(xiàn)象的研究進(jìn)展.從幾何奇異攝動理論入手,介紹基本概念和基本理論,并且解釋混合模式振蕩的產(chǎn)生機制,最后給出典型的具有混合模式振蕩的神經(jīng)元及其模型.接下來的兩章重點分析神經(jīng)元模型中混合模式振蕩的產(chǎn)生機理.在第二章中,基于Av-Ron-Parnas-Segel模型,研究閡下混合模式振蕩以及模型的動力學(xué)行為Av-Ron-Parnas-Segel模型是一個衡量龍蝦心臟神經(jīng)節(jié)的四維模型.首先利用幾何奇異攝動理論中的折結(jié)點原理,分析簡化后三維神經(jīng)元模型中的混合模式振蕩的產(chǎn)生機理.然后研究簡化模型以及原模型中混合模式振蕩的變化規(guī)律.得到動作電位峰峰間距序列的倍周期分岔圖與加周期分岔圖,以及發(fā)放數(shù)的階梯狀改變等結(jié)果.最后探討神經(jīng)元模型中的參數(shù)對系統(tǒng)的影響.利用峰峰間距分岔圖,發(fā)現(xiàn)離子的平衡電位與最大電導(dǎo)都能改變系統(tǒng)的發(fā)放模式.在第三章中,研究模擬胰腺β-細(xì)胞生理活動的Chay-Keizer模型的高位平臺簇放電與其周期解的轉(zhuǎn)遷.首先利用快慢動力學(xué)分析方法將系統(tǒng)分為快慢兩部分,以慢變量為參數(shù)分析簇發(fā)放以及混合模式振蕩的產(chǎn)生原因.然后進(jìn)行全系統(tǒng)的平衡點分支分析,重點給出Bogdanov-Takens分支附近的拓?fù)浣Y(jié)構(gòu).最后探討參數(shù)對模型的影響,其結(jié)果以峰峰間距序列和極值的分岔圖以及雙參數(shù)下單簇內(nèi)峰數(shù)目變化圖給出.在第四章中,研究高維神經(jīng)元模型的動力學(xué)性質(zhì).第一部分研究一類七維固有簇放電類神經(jīng)元的數(shù)學(xué)模型.固有簇放電類神經(jīng)元是在構(gòu)建丘腦-皮層回路的過程中提出的一類神經(jīng)元.對于此類模型首先給出一種降維方式將方程簡化成一個三維模型.然后研究三維模型中余維1分支與余維2分支,給出Bogdanov-Takens分支附近的分支行為.最后利用快慢動力學(xué)并借助相平面解釋系統(tǒng)簇發(fā)放的成因.第二部分主要研究受外電場影響的Chay模型與神經(jīng)回路的混合模式振蕩.文中觀察到豐富的混合模式振蕩現(xiàn)象.
[Abstract]:Neurodynamics is based on the study of the electrophysiological activities of neurons by using the principle of dynamics, and the dynamical properties of the nervous system, such as bifurcation, chaos, oscillation and so on, are studied by constructing a neuron model. In this paper, the mixed mode oscillation is taken as the main research object, the variation law of this kind of phenomenon in the neuron model is analyzed, and the mechanism of the mixed mode oscillation in the two models is explained. At the same time, the bifurcation of the equilibrium point, the bifurcation of the peak-to-peak interval sequence and the variation of the peak in the neuronal model are analyzed, and rich dynamic results are obtained. The thesis is divided into four parts: the first chapter introduces the research progress of mixed mode oscillation in neurons. Based on the theory of geometric singularity perturbation, the basic concepts and theories are introduced, and the mechanism of mixed mode oscillation is explained. Finally, the typical neurons with mixed mode oscillation and their models are given. The next two chapters focus on the mechanism of mixed mode oscillation in the neuron model. In the second chapter, based on the Av-Ron-Parnas-Segel model, the mixed mode oscillation and the dynamic behavior of the model are studied. The Av-Ron-Parnas-Segel model is a four-dimensional model to measure the lobster cardiac ganglion. Firstly, the mechanism of mixed mode oscillation in the simplified three-dimensional neuron model is analyzed by using the folded node principle in the geometric singular perturbation theory. Then the variation law of the simplified model and the mixed mode oscillation in the original model is studied. The double period bifurcation diagram and the additive period bifurcation diagram of the peak-to-peak interval sequence of the action potential are obtained, as well as the ladder changes of the number of emitters. Finally, the influence of the parameters in the neuron model on the system is discussed. It is found that both the equilibrium potential and the maximum conductance of ions can change the distribution mode of the system by using the peak-to-peak bifurcation diagram. In Chapter 3, the Chay-Keizer model, which simulates the physiological activities of pancreatic 尾 -cells, is used to study the high platform discharge and the transition of its periodic solutions. Firstly, the system is divided into two parts by using the fast and slow dynamics analysis method. The slow variable is used as the parameter to analyze the distribution of the cluster and the cause of the mixed mode oscillation. Then the equilibrium bifurcation of the whole system is analyzed and the topological structure near Bogdanov-Takens bifurcation is given. Finally, the effect of the parameters on the model is discussed. The results are given by the bifurcation diagram of the peak-to-peak interval sequence and extreme value, and the variation diagram of the number of peaks in a single cluster under two parameters. In chapter 4, the dynamic properties of high dimensional neuron model are studied. In the first part, the mathematical model of a class of seven dimensional intrinsic cluster discharge neurons is studied. Intrinsic cluster discharge neurons are proposed in the process of constructing thalamic-cortical circuits. For this kind of model, a dimensionality reduction method is presented to simplify the equation into a three-dimensional model. Then the bifurcation behavior near Bogdanov-Takens bifurcation is given. Finally, the causes of cluster distribution are explained by using fast and slow dynamics and phase plane. In the second part, the mixed mode oscillation of Chay model and neural circuit affected by external electric field is studied. A wealth of mixed mode oscillations have been observed.
【學(xué)位授予單位】：華南理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：R338

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