Respiratory sinus arrhythmia (RSA) is a phenomenon where heart-rate (HR) varies with respiration. It is widely accepted that the loss of RSA is a prognostic indicator for cardiovascular disease and that the prominent presence of RSA indicates a healthy cardiac system, yet the reasons for this are still being debated (Larsen et al. 2010). One controversy is over the main mechanism that gives rise to RSA. While most investigators agree that RSA is mainly due to direct central respiratory modulation of the parasympathetic cardiac signal, others argue that RSA is mediated by the baroreflex responding to blood pressure oscillations triggered by the abdominal thoracic pump (Eckberg 2009). Another controversy is over the physiological function of RSA. Hayano et al. (1996) hypothesized that the physiological function of RSA is to match ventilation and perfusion in the lungs and thus optimize oxygen (O2) uptake and carbon dioxide (CO2) removal. Recently, using mathematical models, we showed that RSA may serve to minimize the energy expenditure of the heart while keeping arterial CO2 levels at physiological tensions (Ben-Tal et al. 2012); our theoretical study did not support Hayano’s hypothesis. Our study was performed using different mathematical techniques and different models. First, the optimal HR was calculated using techniques from optimal control theory in a simplified model of gas exchange. We found that the calculated HR was remarkably similar to RSA and that this became more profound under slow and deep breathing. Second, the HR function was prescribed and the cardiac work, as well as the volumes of O2 and CO2 taken up or removed by the blood respectively were calculated in a more detailed model of gas exchange. We found that cardiac work was minimized for RSA-like HR functions when the blood partial pressure of CO2 was controlled, most profoundly under slow and deep breathing and that although gas exchange efficiency improved with slow and deep breathing and with increased mean heart rate, this was unrelated to RSA. Third, we tested the two hypotheses using a newly developed minimal model for the neural control of HR in which RSA appears naturally and found similar results. The newly developed minimal model for the neural control of HR assumes that the heart period is affected primarily by the parasympathetic signal, with the sympathetic signal taken as a constant. We included the baroreflex, mechanical stretch-receptor feedback from the lungs, and central modulation of the cardiac vagal tone by the respiratory drive, but we omitted the chemoreflex. Our model mimics a range of experimental observations and provides several new insights. Most notably, the model can mimic the growth in the amplitude of RSA with decreasing respiratory frequency which then decreases at frequencies below 7 breaths per minute (for humans) and predicts that the decrease in the amplitude of RSA at low breathing frequencies is due to the baroreflex (we show this both numerically and analytically with a linear baroreflex). Another new prediction of the model is that the gating of the baroreflex leads to the dependency of RSA on mean vagal tone. These findings provide new insights into potential reasons and benefits of RSA under physiological conditions.