μ波是記錄於皮質運動感覺區上的腦電波中10赫茲左右的節律訊號，由μ波事件相關動態行為實驗結果已證實其與大腦的運動和感覺功能有關，但到今日還是不清楚產生μ波的來源。如果能夠知道μ波的產生機制，就能藉由μ波事件相關動態行為來推測大腦是如何處理運動或是感覺資訊，因此為了這個目的，本論文的目標在用大腦神經網路模型來模擬的μ波，藉此可調查產生μ波的機制。
神經質量模型是用來模擬μ波最盛行的群體模型，但因為其並不在意神經元的動態行為，因此此模型不能建立視丘內神經元動態行為和μ波產生機制這兩者間的關係，雖然已有實驗證實μ波的產生與視丘內神經元的特定動態行為有關。為了彌補這樣的不足，本論文將採用神經群體密度模型來揭露這兩者的關係，因為神經群體密度模型可在神經網路模型中保留神經元動態行為。神經群體密度模型是利用群體密度函數來描述μ波神經網路模型中單一神經群內所有神經元狀態變數的統計分佈，藉此可算出神經群的動態行為，當中群體密度函數的統御方程式為一階雙曲線偏微分方程式，必須使用數值方法來求解，所以本論文另一個目標就是探討解神經群體密度模型最佳數值方法。本論文比較四種不同的數值方法，其中以不連續Galerkin有限元素法就計算的準確度和效率這兩方面上有最佳的表現。
從μ波的模擬過程中，我們可驗證視丘內丘腦皮質神經群和網狀神經群兩種神經群在μ波出現或是消失時各自的激發模式。根據非線性動態分析，可得到下列結論：當人處在清醒狀態時，視丘皮質神經網路的動態呈現出point attractor的行為模式，當在睡眠時，則呈現limit cycle的行為模式。而在清醒狀態時，若出現μ波，此時丘腦皮質神經群處於burst激發模式，若μ波消失時，丘腦皮質神經群處於tonic激發模式，然而網狀神經群不管μ波是否出現皆處在tonic激發模式，唯有在睡眠狀態時，網狀神經群才會轉換成burst激發模式。從這樣結果可推測μ波事件相關動態行為是反映丘腦皮質神經群激發模式的改變，而這改變會由刺激輸入、任務需求、或是心理狀態的不同所引起。

The μ wave is a rhythm at around 10 Hz recorded in the cortical motosensory areas by means of scalp electroencephalogram, which, through experiments of its event-related dynamics, has proved dependent on behavioral states of motor and sensory functions in the brain. But, its origin is still not clear today. Knowing the generation mechanism of μ wave help us to understand how information about motor or sensory functions is processed in the brain if event-related dynamics of μ wave is known. In this end, therefore, the goal of this thesis is to explore the generation mechanism of μ wave through modeling studies of neuronal networks of brain.
Because of unconcern of neuronal dynamics in the neural mass model that is a popular population model of simulating μ wave, the relationship between the dynamics of neurons in the thalamocortical neuronal network and the generation of the μ wave has never constructed in μ wave simulation today, which leads us to employ the population density model, a novel population model of neuronal populations, for uncovering this relationship through the preservation of neuronal dynamics in this type of models. In our simulation of μ wave, the dynamics of each neuronal population of neurons has been described by the population density model, in which a first-order hyperbolic partial differential equations governs the time course of a population density function stating the distribution of state variables of neurons in a state space and must be solved by a numerical method. To find out the best numerical method for the population density model, four numerical methods available are utilized and compared. By comparison, the numerical method, called discontinuous Galerkin finite element method, has the best performances in terms of computing accuracy and efficiency.
We identity neuronal dynamics of thalamocotical relay neurons and thalamic reticular neurons, two types of critical neurons in the thalamocortical neuronal network, during appearance or disappearance of the μ wave. It is concluded that, according to nonlinear dynamical analysis, the dynamics of the thalamocortical neuronal network forms a point attractor during wakeful state, and a limit cycle attractor during sleep state. During wakeful state, thalamocotical relay neurons react in the burst firing mode when the μ wave appears, and, on the contrary, they react in the tonic firing mode when the μ wave disappears. Whatever the μ wave appears or not, thalamic reticular neurons react in the tonic firing mode, and they can react in the burst firing mode only during sleep state. As consequences, the event-related dynamics of the μ wave reflects alteration in the neuronal dynamics, i.e., the firing mode of thalamocotical relay neurons, which can be induced by stimulus- or task-factors and psychological states.

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