This paper investigates mixed-mode oscillations (MMOs) with a three-dimensional conductance-based cardiac action potential model, which makes the heart beat in a nonrenewable way. The 3D model was entailed by utilizing voltage-dependent timescales to describe the mechanism in which MMOs are generated. As expected, motivated by geometric singular perturbation theory, our analysis explains in detail the geometric mechanisms that there is a range of parameters under which the cardiac model highlights that the presence of MMOs is induced by the intrinsical canard phenomenon. Much is currently known about the geometric mechanisms, for a folded saddle, the two singular canards perturb to maximal canards. Characteristics of the stimulus current such as frequency and duration determine which early afterdepolarizations (EADs), as a special case of MMOs bears, as well as the article compares the detailed manifold structures of original and dimensionless systems with square wave pulses by setting the pacing cycle length. An exceedingly vital technique of the analysis is the slow–fast dynamics analysis by which the system governs multiple timescale structures analytically. A more novel and successful multiple-timescale approach divides the system so that there is only one fast variable and demonstrates that the MMOs arise from canard dynamics, such as using a three-variable model in which two variables are treated as “slow” and one treated as “fast”, which the layer problem and the reduced problem are considered to explain the trajectory on the critical manifold. Meanwhile, if one variable was regarded as the single slow variable, substantial bifurcation properties are discovered for slow–fast system, as well as general one-parameter bifurcation type is discussed for the whole system similarly. By focusing on the first Lyapunov coefficient of the Hopf bifurcation, which decides whether the bifurcation is supercritical or subcritical, it was shown that an unstable limit cycle can arise via a delayed subcritical Hopf bifurcation for the original system. Meanwhile, the dynamical studies of cardiac model have major implications for further elaborating the complex dynamic behaviors, such as EADs, which can lead to tissue-level arrhythmias. Ultimately, it has turned these researches into a considerable player in the signal and information transmission for underlying nervous systems.

本文研究了基于三维电导的心脏动作电位模型的混合模式振荡（MMO），该模型使心脏搏动不可更新。通过利用依赖于电压的时标来描述3D模型，其中描述了生成MMO的机制。正如预期的那样，在几何奇异摄动理论的推动下，我们的分析详细解释了几何机制，其中存在一系列参数，在这些参数下，心脏模型突显出MMO的存在是由内在的Canard现象引起的。目前，关于几何机构，对于折叠的鞍座，两个奇异的鸭翼扰动到最大鸭翼的几何机理了解很多。刺激电流的特征（例如频率和持续时间）决定了哪些早期除极后（EAD），作为MMO熊的特例，该文章还通过设置起搏周期长度，比较了具有方波脉冲的原始系统和无量纲系统的详细流形结构。分析的一项极其重要的技术是慢速动态分析，通过该系统，系统可以分析多个时标结构。一种更新颖，更成功的多时标方法将系统划分为仅一个快速变量，并证明MMO来自卡纳德动力学，例如使用三变量模型，其中两个变量被视为“慢”变量，而一个变量被视为“快速”，层问题和简化问题被认为是解释关键流形上的轨迹。同时，如果将一个变量视为单个慢变量，对于慢速系统，我们发现了实质的分叉性质，对整个系统也同样讨论了一般的一参数分叉类型。通过关注Hopf分支的第一个Lyapunov系数，该系数决定了分支是超临界的还是亚临界的，表明对于原始系统，通过延迟的亚临界Hopf分支会产生不稳定的极限环。同时，心脏模型的动力学研究对进一步阐述复杂的动态行为（如EADs）具有重要意义，这些行为可能导致组织水平的心律不齐。最终，它使这些研究成为潜在神经系统信号和信息传输的重要参与者。对于整个系统，同样也讨论了一般的一参数分叉类型。通过关注Hopf分支的第一个Lyapunov系数，该系数决定了分支是超临界的还是亚临界的，表明对于原始系统，通过延迟的亚临界Hopf分支会产生不稳定的极限环。同时，心脏模型的动力学研究对进一步阐述复杂的动态行为（如EADs）具有重要意义，这些行为可能导致组织水平的心律不齐。最终，它使这些研究成为潜在神经系统信号和信息传输的重要参与者。对于整个系统，同样也讨论了一般的一参数分叉类型。通过关注Hopf分支的第一个Lyapunov系数，该系数决定了分支是超临界的还是亚临界的，表明对于原始系统，通过延迟的亚临界Hopf分支会产生不稳定的极限环。同时，心脏模型的动力学研究对进一步阐述复杂的动态行为（如EADs）具有重要意义，这些行为可能导致组织水平的心律不齐。最终，它使这些研究成为潜在神经系统信号和信息传输的重要参与者。结果表明，对于原始系统，通过延迟的次临界Hopf分叉会产生不稳定的极限环。同时，心脏模型的动力学研究对进一步阐述复杂的动态行为（如EADs）具有重要意义，这些行为可能导致组织水平的心律不齐。最终，它使这些研究成为潜在神经系统信号和信息传输的重要参与者。结果表明，对于原始系统，通过延迟的次临界Hopf分叉会产生不稳定的极限环。同时，心脏模型的动力学研究对进一步阐述复杂的动态行为（如EADs）具有重要意义，这些行为可能导致组织水平的心律不齐。最终，它使这些研究成为潜在神经系统信号和信息传输的重要参与者。