在生物系統中，電壓門控離子通道扮演著重要的角色，負責調控細胞內外的離子流動。儘管人類在離子通道研究方面已取得一些重要進展，然而，由於生物系統的複雜性和實驗技術上的限制，多電壓門控離子通道下的動態運作機制仍然存在諸多未解之謎，而其中一個問題就是其在電壓鉗實驗中所呈現出的遲滯現象，這意味著通道的離子流大小會與過去的影響有關，當系統受過刺激後，接著再輸入相同的刺激，可能會得到不同的結果。而在本研究中，我們的目標即是建立能夠描述多電壓門控離子通道系統中的遲滯現象的物理模型，考慮了遲滯現象以及電壓門控離子通道本身的性質，我們選擇利用朗之萬動力學與電路理論的概念來建立一個理論模型，接著，我們引用了一組電壓門控鈉離子通道系統遲滯現象的數據cite{exp}，並成功透過我們的模型擬合出符合實驗數據的結果，提供了可用於描述遲滯現象，甚至於探究多電壓門控離子通道非線性動態運作的理論框架。

In biological systems, voltage-gated ion channels play a crucial role in regulating the flow of ions across cell membranes. Despite significant advances in ion channel research, due to the complexity of biological systems and limitations of experimental techniques, there are still many unsolved issues about the dynamic operating mechanism of multiple voltage gated ion channels. One of the issues is the “hysteresis” observed in voltage clamp experiments, indicating that the ionic current through these channels is influenced by their historical states. The identical stimuli can yield different results after prior stimulation. In this study, we aim to develop a physical model that describes the hysteresis in multiple voltage-gated ion channel systems. Considering the hysteresis and the properties of voltage-gated ion channels, we first employed concepts from Langevin dynamics and circuit theory to construct a theoretical model. Then, we cite a dataset of hysteresis in a multiple voltage-gated sodium ion channel system cite{exp} and successfully fitted the data using our model. This provides a theoretical framework that can be used to describe hysteresis and further investigate the nonlinear dynamic behavior of multiple voltage-gated ion channels.

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