# Emotion, Respiration, and Heart Rate Variability: A Mathematical Model and Simulation Analyses
## 4 Discussion  
We developed a mathematical model by integrating emotion, the nervous system, and the cardiovascular system. As shown in previous experiments [14], the PSD of HRVs have single spectral components exactly at the respiration frequency (Figure 2A), which is considered to represent respiratory sinus arrhythmia (RSA). Moreover, the relation between HRV total power and the respiration  
Figure 5: Time evolutions of (**A**) RR interval (RRI) and (**B**) arterial blood pressures at a short-term alerting stimulus. (**A**) The result without baroreflex. (**C**) The periodogram power spectral densities (PSDs) of the systolic and diastolic blood pressures and RRI with baroreflex. frequency in our simulation showed good accordance with the experiments [14] (Figure 2C). In addition, the removal of the baroreflex increased the amplitude of the HRV fluctuation (Figure 4), possibly because the baroreflex suppressed the RRI change for homeostatic maintenance. The peak of the HRV spectral component in the range of LF (0.05-0.15 Hz) observed in Figure 4B, which moved to a lower frequency when the baroreflex was removed, is considered to be LF. Therefore, we can say that our model reproduces experimental and clinical results [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14] quite well, though the parameters describing nerve interactions in the proposed model (Table 1) were not directly measured. Further experimental studies using human subjects are needed to investigate the validity of the parameters. The simulation analyses using this model allowed us to unveil the possible cause of the decrease in HRV total power with a higher respiration frequency. The delay, which is modeled by Equation (13) and is revealed as the gradual RRI reaction at the stimulation onset in Figure 5A, works as a low-pass filter in the process where the parasympathetic activity reflects to the heart period change, and weakens the high frequency component, reflecting the blood pressure fluctuations. The filter also works on the fluctuations, reflecting the respiration (red lines in Figure 3D), and makes the fluctuation amplitude of the heart period change smaller when the respiration frequency is higher (Figure 3C). The RRI fluctuation shows good accordance with the fluctuation of the heart period change controlled by the parasympathetic nerve (Figure 3A,C), while the fluctuation of the heart period change controlled by the sympathetic nerve is considerably small (Figure 3B). This indicates that the RRI fluctuation is mainly controlled by the parasympathetic nerve. In short, the delay in the process where the parasympathetic activity controls the heart period change removes the high frequency component and decreases the HRV total power with a higher respiration frequency. By removing the delay in the process where the parasympathetic activity controls the heart period, the decrease in HRV total power disappears (Figure 2B,C). Thus, we can say that the delay in the process where the parasympathetic activity controls the heart period may be the cause of the decrease in HRV total power with a higher respiration frequency. The baroreflex feedback control contains the delay in the sinus nerve response to the arterial pressure, adding to the delay in the heart period response to the efferent nerve activities. We assume that these delays generate the oscillations of RRI (Figure 5A) and the arterial pressure (Figure 5B) as transient responses to the simple emotional stimulation. As their frequencies are 0.1 Hz (Figure 5C) and decrease when the baroreflex is removed, the oscillations of RRI and the arterial pressure may be considered the LF and the Mayer wave, respectively. Thus, we successfully reproduced the LF component of HRV (Figure 4B) by inducing continuous emotional stimulation as a constant emotional perturbation. Although this assumption follows the baroreflex feedback theory, rhythmic vasomotor activity or centrogenic rhythm in the brain stem, which are considered to be responsible for LF generation [16; 17], are not included in our simulation.