Universal inverse square relationship between heart rate variability and heart rate

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 (Iper) 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(Iper)*(60,000/mean(HR) -APD)^2/(ΔV*C), where CL is in ms, HR in beats per minute, Iper 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 Iper. Our explicit formula may be also used for a more precise biomedical interpretation of heart rate variability after respective corrections for heart rate.