研究表明心室顫動（VF）與單顆心臟細胞的不規律心跳之間存在聯繫。雖然生理離子模型能夠重現觀察到的不規則刺激（雙週期現象），但其複雜性源於動作電位和細胞內鈣離子循環通過L型鈣通道等敏感通道的耦合。通過合理的近似和觀察例如動作電位和細胞內鈣離子循環。該4變量動力學系統僅涉及鈣離子循環，我們提出了一個簡化的4變量模型，其成功捕捉到了離子模型中的雙週期現象以及一致/不一致交替。值得注意的是，我們的模型強調了細胞內鈣離子循環在心臟動力學中的關鍵作用。鈣離子儲存器（SR）和細胞質中的鈣濃度對鈣誘導的鈣釋放（CICR）現象產生影響，導致攝取電流與釋放電流之間的競爭，從而引發交替現象。此外，L型鈣通道中的鈣依賴性失活門是決定一致或不一致交替的主要因素。此外，4變量模型和離子模型均清楚地顯示了鈣離子循環對動作電位持續時間（APD）的影響。通過這簡化模型，我們對複雜的心臟系統獲得了有價值的看法。

Studies have indicated that ventricular fibrillation (VF), a severe heart condition, could be associated with irregular and incoherent beats of a single cardiac cell. A physiological ionic model that accounts for the intricate interplay of organelles in a single cardiac cell can reproduce the observed irregular pacing such as the period-2 cycle. Despite the success of the ionic model, we still search for a simplified model that can be used to explain the underlying mechanics. Therefore, we propose a reduced 4-variable model that is derived from the ionic model. This 4-variable model is based on numerical observations and approximations of the ionic model, and it is significantly simpler to analyze and understand. The 4-variable dynamical system successfully captures the bifurcation behavior, period-2 cycle, and concordant/discordant alternans consistent with the ionic model. Notably, our model highlights the critical role of intracellular calcium cycling in cardiac dynamics. Specifically, the calcium concentrations in the sarcoplasmic reticulum (SR) and cytosol exert the influence on the calcium-induced calcium-release (CICR) phenomenon, which gives rise to the competition between the uptake current and release current, leading to the alternans. Moreover, we show that the calcium-dependent inactivation gate in the L-type calcium channel emerges as a dominant factor determining concordant or discordant alternans. Additionally, both the 4-variable and ionic models reveal the clear influence of calcium cycling on action potential duration (APD). We gain valuable insights into the cardiac mechanism through this approach.

Contents ii
List of Tables iii
List of Figures viii
1 Introduction 1
2 Review of the Ionic Model 4
2.1 Membrane potential and ion channels . . . . . . . . . . . . . . . . . 4
2.2 Calcium cycling inside the cell . . . . . . . . . . . . . . . . . . . . . 6
2.3 Ionic model structure . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Approximation Methods for the 4-Variable Model 15
3.1 L-type calcium current ICa . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Sodium-calcium exchanger (NCX) current IN aCa . . . . . . . . . . . 20
3.3 Calcium concentration in submembrane space cs . . . . . . . . . . . 22
3.4 Cytosolic calcium buffering . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Calcium concentration in the NSR cj . . . . . . . . . . . . . . . . . 24
3.6 4-variable model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.7 Membrane Potential V and Sodium Current IN a . . . . . . . . . . . 27
4 Result 29
4.1 Period-2 cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Concordant and Discordant alternans . . . . . . . . . . . . . . . . . 32
4.2.1 γ and Concordant/Discordant alternans . . . . . . . . . . . 32
4.2.2 τf -u Phase Diagram and Concordant/Discordant alternans . 37
5 Conclusion and Future Work 45
Appendix 47
A. Details of the Ionic Model . . . . . . . . . . . . . . . . . . . . . . . . 48
B. Details of the 4-Variable Model . . . . . . . . . . . . . . . . . . . . . 52
C. Physiological Parameters of a Cardiac Cell . . . . . . . . . . . . . . . 55
Reference 58

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