# Emotion, Respiration, and Heart Rate Variability: A Mathematical Model and Simulation Analyses
## 2 Methods  
### Mathematical Model  
Figure 1 illustrates the proposed novel mathematical model that integrates emotion, the nervous system, the respiratory system, and the cardiovascular system. To expand a part of [PERSON]'s model [19], describing the enclosed area in Figure 1, we formulated relationships in the autonomic nervous system to satisfy the following conditions based on [19; 23]:  
1. A short-term alerting stimulus or long-term mental stress (e.g., anxiety) triggers excitatory synaptic projections from the CeA to the rostral ventrolateral medulla (RVLM). 2. A short-term alerting stimulus or long-term mental stress (e.g., anxiety) triggers inhibitory synaptic projections from the CeA to the nucleus tractus solitarii (NTS). 3. The RVLM activates the sympathetic nervous system via the spinal cord. 4. The NTS activates the parasympathetic nervous system via the dorsal vagal motor nucleus (DVN) and nucleus ambiguus. 5. The NTS activates the caudal ventrolateral medulla (CVLM). 6. The CVLM inhibits the RVLM. 7. The sinus nerve activates the NTS. 8. Increased spike frequency in the sinus nerve results in exponential decrease in the spike frequency in the sympathetic fibers. 9. Increased spike frequency in the sinus nerve results in exponential increase in the spike frequency in the parasympathetic fibers with upper saturation. To formulate the problem, we denote the intensity of emotion triggering the synaptic projection of the CeA as \(S\), and the intensities of the influence of the synaptic projection on the RVLM and on the NTS as \(g_{\text{CeAR}}\) and \(g_{\text{CeAN}}\), respectively. The firing frequencies of the sinus nerve, sympathetic nerve, parasympathetic nerve, RVLM, CVLM, and NTS are denoted as \(f_{\text{cs}}\), \(f_{\text{es}}\), \(f_{\text{ep}}\), \(f_{\text{RVLM}}\), \(f_{\text{CVLM}}\), and \(f_{\text{NTS}}\), respectively. A possible pure delay introduced by processing in the central nervous system and nerve conduction is implicit in the formulation describing the influence of the sympathetic and parasympathetic nervous systems on cardiac behavior [19]. Because it is not easy to measure most of these relationships in the autonomic nervous system, we determined parameters to reproduce some experimental data from previous studies [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] by using the proposed model. Emotion: In this study, (i) a state of concentration, (ii) a state of emotional perturbation, and (iii) a short-term alerting stimulus were modeled. To model (i), we set the intensity of emotion \(S\) to 0. In the model (ii), we set the intensity of emotion \(S\) as a random waveform, a part of which is shown in the upper-right of Figure 1. We obtain this waveform by setting values every 0.1 s with a uniformly distributed random number in the interval (\(-\)5, 5), which was created using the MATLAB R2019a function 'rand' (The MathWorks, Inc. Natick, MA, USA, 2019), and by defining values every timestep with spline interpolation using the MATLAB function 'interp1'. In contrast, to model (iii), we set \(S\) as  
\[S=-10\exp(-2t)\]  
Figure 1: Schematic of proposed mathematical model. The area surrounded by the double line is modeled based on [PERSON]’s model [19]. An example of the time history of the emotion intensity given in our model is shown on the upper-right of the figure. CeA, central nucleus of the amygdala; NTS, nucleus tractus solitarii; CVLM, caudal ventrolateral medulla; RVLM, rostral ventrolateral medulla; DVN, dorsal vagal motor nucleus; NA, nucleus ambigus. where \(t\) is the elapsed time since the stimulus onset (the unit of which is seconds). Respiratory system: We adopted a part of [PERSON]'s model [22] to express the intrathoracic pressure \(P_{\text{thor}}\) (mmHg) and the lung volume \(V_{\text{L}}\) (m\({}^{3}\)) during the respiratory cycle:  
\[P_{\text{thor}}=