# Emotion, Respiration, and Heart Rate Variability: A Mathematical Model and Simulation Analyses
## 1 Introduction  
Despite the increasingly recognized importance of emotions in human activity, objective measurement methodologies of emotions are yet to be established. Psychological evaluations measure emotions through self-report questionnaires such as the state-trait anxiety inventory (STAI), the self-rating depression scale (SDS), and the self-assessment manikin (SAM) [1; 2; 3]. Alternatively, physiological evaluations measure the autonomous reactions of the body such as changes in heart rate (HR) and skin conductance [1; 2; 3; 4; 5; 6; 7]. However, a large inter-individual difference in the relationship between the HR and the intensity of perceived emotions measured using SAM has been verified [2], and no correlation between the HR and the intensity of a perceived emotion was found when using either STAI or SDS [1; 3]. These observations question the reliability of the use of HR to evaluate emotion. Besides HR, the ratio of low-frequency (LF, 0.05-0.15 Hz) to high-frequency (HF, 0.15-1.0 Hz) components of HR variability (HRV) is often used as the measure of autonomous reaction corresponding to emotions [3; 8; 9; 10; 11], as this ratio (LF/HF) has been related to the activity balance between the sympathetic and parasympathetic nervous systems [12; 13]. However, this relation has also been questioned [14]. Some experiments [3; 15] have proven that LF/HF increases with a random number generation task or a mental task, but this change is less significant than the change in mean HR. Besides the objective measurement methodologies of emotions, the generation mechanism of the LF component of HRV is also controversial [16; 17]. One possible and dominant mechanism is the baroreflex feedback (loop) theory. When arterial blood pressure changes due to some perturbation,the central nervous system mediates this change by adjusting cardiac responses such as HR and vascular resistance, thereby normalizing arterial pressure. This process exhibits a delay that, according to the baroreflex feedback theory, leads to an oscillatory normalization of arterial pressure (i.e., Mayer waves) and an oscillatory normalization of HR (i.e., LF). On the other hand, some researchers have proposed the central oscillator theory, in which brain stem circuits oscillate at frequencies corresponding to LF, and the oscillation is reflected in both the HR and the arterial pressure. [PERSON] et al. [18] found that the discharges of single neurons in the brain stem of cats oscillate at these frequencies. The oscillations remained after sinoaortic and vagal resection, which supports the central oscillator theory. In this study, we aimed to examine both the reliability of LF/HF usage for emotion evaluation and the generation mechanism of the LF component of HRV. We considered a short-term alerting stimulus or long-term mental stress (e.g., anxiety) as a perturbation triggering the LF component. Mathematical models are useful in dealing with such problems because they allow us to predict the physiological behavior under conditions that are not easy to reproduce experimentally [16]. To explain the effects of emotion on HRV, we designed a novel mathematical model by expanding the model formulated by [PERSON] [19]. This model and the series of expanded models [20; 21; 22] include the activity of the sympathetic and parasympathetic nervous systems that changes the cardiac response, for example, the heart period, cardiac contractility, unstressed volume of some blood vessels, and resistance of other blood vessels, resulting in changes in the blood flow and pressure at each site. They also describe the process by which arterial pressure changes the sinus nerve activity via a baroreceptor, and the expanded model [22] successfully reproduced the LF component of HRV by inducing vasomotor noise. To avoid confusions caused by using a very complicated model, we selected the most basic model [19].