In our previous study, we analyzed heart rate variability and heart rate from a large variety of cardiac preparations (including humans, living animals, Langendorff-perfused isolated hearts, and single sinoatrial nodal cells) in diverse species, combining our data with those of previously published articles. The analysis revealed that regardless of conditions, heart rate variability (for the purposes of the study assessed as standard deviation of beat-to-beat intervals) vs. heart rate follows a universal exponential decay-like relationship. Numerical simulations of diastolic interval variability by adding a randomly fluctuating term (I_per) to net current revealed a similar relationship. In the present study, using a Taylor series, we found that this relationship is, in fact, inverse square, and we derive an explicit formula for the standard deviation (sd) of the cycle length (CL) as a function of heart rate (HR) with biophysically meaningful parameters: sd(CL)=sd(I_per)*(60,000/mean(HR) -APD)^2/(ΔV*C), where CL is in ms, HR in beats per minute, I_per in pA, APD in ms is an average AP duration of pacemaker cells, C in pF is cell membrane capacitance, and ΔV is the magnitude of diastolic depolarization in mV. This relationship gives direct insight into heart rate variability mechanisms at the basic level of individual pacemaker cells, i.e. their intrinsic CL variability linked to stochastic operation of ion channels (both Ca release and cell membrane channels) generating I_per. Our explicit formula may be also used for a more precise biomedical interpretation of heart rate variability after respective corrections for heart rate.

中文翻译：

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心率变异性与心率之间的普遍反平方关系

在我们之前的研究中，我们分析了来自不同物种的各种心脏制剂（包括人类、活体动物、Langendorff 灌注的离体心脏和单个窦房结细胞）的心率变异性和心率，并将我们的数据与之前的数据相结合。发表的文章。分析表明，无论条件如何，心率变异性（出于研究目的，评估为心跳间隔的标准偏差）与心率之间的关系都遵循普遍的指数衰减关系。通过添加随机波动项（I _per) 到净电流揭示了类似的关系。在本研究中，使用泰勒级数，我们发现这种关系实际上是平方反比，并且我们推导出循环长度 (CL) 的标准偏差 (sd) 作为心率函数的明确公式（ HR) 具有生物物理意义的参数：sd(CL)=sd(I _per )*(60,000/mean(HR) -APD)^2/(ΔV*C)，其中 CL 以毫秒为单位，HR 以每分钟心跳次数为单位，I_每以 pA 为单位，以 ms 为单位的 APD 是起搏器细胞的平均 AP 持续时间，以 pF 为单位的 C 是细胞膜电容，而 ΔV 是以 mV 为单位的舒张去极化幅度。这种关系可以直接洞察单个起搏器细胞基本水平的心率变异性机制，即它们的内在 CL 变异性与产生 I _per的离子通道（Ca 释放和细胞膜通道）的随机操作相关。我们的显式公式还可用于在对心率进行相应校正后对心率变异性进行更精确的生物医学解释。