長期突觸可塑性是重要的神經生理現象，與大腦的認知系統及學習系統有關，其持續時間長度通常為數十秒到小時級距，目前研究已知當中的分子機制為鈣離子濃度與NMDA受體所主導。有學者提出鈣離子突觸可塑性模型，用以描述突觸傳遞效力隨著突觸後神經元的鈣離子濃度升降而增減，產生長期增益或長期抑制現象。上述的鈣離子可塑性模型僅針對單一神經元作模擬，然而人類各腦區的神經元數量多不可數，因此有必要建立一個神經群體網路動力學模型。群體密度模型是一種神經群體網路的建模方法，以機率密度函數代表一個神經群體，因此在神經群體動力學的模擬，具有較快的計算速度。然而隨著神經電生理模型變數之增加，群體密度模型的狀態空間維度隨之上升，因此需要將模型降維。本研究之目的為結合神經群體密度模型與長期突觸可塑性模型，發展群體密度模型合併平均場模型之神經群體模型架構。
本文使用CbAdEx模型結合突觸電導的平均場模型建立色突觸群體密度模型，此模型是基於電導式突觸的擴散過程近似法及神經適應電流的絕熱近似法。除此之外，本研究亦使用擴散係數查表法，估測EIF模型與AdEx模型之色突觸群體密度模型之擴散係數。神經群體網路動力學的標準模型是蒙地卡羅模擬法。EIF模型與AdEx模型的蒙地卡羅模擬結果顯示擴散係數查表法的建模相對誤差小於色突觸群體密度模型的建模相對誤差。蒙地卡羅法與平均場模型之結果相比，相對誤差大多在10%以下。CbAdEx模型之色突觸群體密度模型模擬結果顯示此模型能模擬長期增益與長期抑制，且能在定性上估測神經群體網路的動作電位頻率演化趨勢。
本文所提出之色突觸群體密度模型為第一個結合長期突觸可塑性之群體密度模型，計算速度高於蒙地卡羅模擬法，且能夠定性上估測三種神經元模型的蒙地卡羅模擬的動作電位頻率，以及平均場模型能夠定量估測各神經元狀態變數的總體平均值。

Long-term synaptic plasticity is an important neurophysiological phenomenon, which is related to the cognitive and learning functions of the brain. Its duration ranges from tens of seconds to several hours. The molecular mechanism is believed to be the inflow of calcium ion and binding of N-methyl-D-aspartate (NMDA) receptor. Following this idea, this thesis adopts the calcium-based adaptive exponential integrate and fire (CbAdEx) model as a single neuron model manifesting synaptic plasticity. The population density model is a method of modeling a collection of neurons. When incorporating more complex synaptic dynamics, the state-space dimension of master equation of the population density model increases, resulting in an increase of computational load. In order to maintain the computational efficiency, the dimension of the model needs to be reduced. In this thesis, the reduction of dimension is achieved by using a mean-field model of synaptic conductance. Based on both diffusion approximation of synaptic conductance and adiabatic approximation of neuronal adaptation current, a colored-synapse population density model (csPDM) is proposed. In addition, the feasibility of using a table lookup method to estimate the diffusion coefficients of csPDM is investigated in the cases of simpler single neuron models without synaptic plasticity, including the exponential integrate and fire (EIF) model and AdEx model. The simulation results show that, when compared with other methods of estimating diffusion coefficients, the table lookup method has a smaller relative error, mostly below 10%, in both EIF and AdEx models. The simulation results of the csPDM of CbAdEx model reveal that this model can simulate long-term potentiation and long-term depression, and can qualitatively estimate the firing rates of the CbAdEx population. The csPDM proposed in this thesis is the first population density model that possesses long-term synaptic plasticity. The computational load is smaller than that by Monte Carlo simulation (MCS). Moreover, it can qualitatively estimate the firing rate of neuronal populations.

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