## 2 Heart Conduction System and Mathematical Models
### Mathematical Model
In this manner, each oscillator is weighted at 50% in the generation of the signal. This is because, in a normal rhythm, the contribution of the left and the right bundle branch is the same. The parameters \(a_{i}\), \(u_{ij}\), \(f_{i}\), \(d_{i}\), \(e_{i}\), are chosen to obtain intrinsic oscillation rates of 70 bpm, 50 bpm, 35 bpm and 35 bpm for uncoupled \(SA\), \(AV\), \(RB\) and \(LB\), respectively (Figure 3) and with behaviour close to action potentials of real pacemakers: [40; 41; 42]. In particular, we use the following experimental parameters: \(a_{1}=40\), \(a_{2}=50\), \(a_{3}=50\), \(a_{1}=40\), \(u_{11}=0.83\), \(u_{21}=0.83\), \(u_{31}=0.83\), \(u_{12}=-0.83\), \(u_{22}=-0.83\), \(u_{32}=-0.83\), \(f_{1}=25\), \(f_{2}=8.4\), \(f_{3}=1.5\), \(d_{1}=3\), \(d_{2}=3\), \(d_{3}=3\), \(e_{1}=3.5\), \(e_{2}=5\), \(e_{3}=12\). The coupling coefficients are: \(K_{SA-AV}=100\), \(K_{AV-RB}=285\), \(K_{AV-LB}=285\). We describe the depolarization and repolarization process for atrial (\(AT\)) and ventricular (\(VN\)) muscles starting from the modified FitzHugh-Nagumo model [16; 43] proposed by [PERSON] in [22]:  
\[\mathbf{P\ wave}=\left\{ \begin{array}{l}\dot{z}_{1}= k_{1}(-c_{1}z_{1}(z_{1}-w_{11})(z_{1}-w_{12})-b_{1}v_{1}-d_{1}v_{1}z_{1}+I_{AT_{ De}}),\\ \dot{v}_{1}=k_{1}h_{1}(z_{1}-g_{1}v_{1}),\end{array} \right. \tag{12}\]  
\[\mathbf{Ta\ wave}=\left\{ \begin{array}{l}\dot{z}_{2}= k_{2}(-c_{2}z_{2}(z_{2}-w_{21})(z_{2}-w_{22})-b_{2}v_{2}-d_{2}v_{2}z_{2}+I_{AT_{ Be}}),\\ \dot{v}_{2}=k_{2}h_{2}(z_{2}-g_{2}v_{2}),\end{array} \right. \tag{13}\]\[\mathbf{QRS}=\left\{ \begin{array}{ll}\dot{z}_{3}=& k_{3}(-c_{3}z_{3}(z_{3}-w_{31})(z_{3}-w_{32})-b_{3}v_{3}-d_{3}v_{3}z_{3}+I_{VN _{Dc}}),\\ \dot{v}_{3}=& k_{3}h_{3}(z_{3}-g_{3}v_{3}),\end{array} \right. \tag{14}\]  
\[\mathbf{T\ wave}=\left\{ \begin{array}{ll}\dot{z}_{4}=& k_{4}(-c_{4}z_{4}(z_{4}-w_{41})(z_{4}-w_{42})-b_{4}v_{4}-d_{4}v_{4}z_{4}+I_{ VN_{Re}}),\\ \dot{v}_{4}=& k_{4}h_{4}(z_{4}-g_{4}v_{4}),\end{array} \right. \tag{15}\]  
where \(k_{i}\) are the scaling coefficients. In our work, we used the experimental parameters: \(k_{1}=2*10^{3}\), \(k_{2}=4*10^{4}\), \(k_{3}=10^{4}\), \(k_{4}=2*10^{3}\), \(c_{1}=c_{2}=0.26\), \(c_{3}=0.3\), \(c_{4}=0.1\)\(b_{1}=b_{2}=b_{4}=0\), \(b_{3}=0.015\), \(d_{m_{1}}=d_{m_{2}}=0.4\), \(d_{m_{3}}=0.09\), \(d_{m_{4}}=0.1\), \(h_{1}=h_{2}=0.004\), \(h_{3}=h_{4}=0.008\), \(g_{1}=g_{2}=g_{3}=g_{4}=1\), \(w_{11}=0.13\), \(w_{12}=1.0\), \(w_{22}=1.0\), \(w_{21}=0.19\), \(w_{31}=0.12\), \(w_{32}=1.1\), \(w_{41}=0.22\), \(w_{42}=0.8\). The activation currents \(I_{i}\) represent the coupling between the \(SA\) and the \(AT\) muscles and between the \(RB\)-\(LB\) pacemakers and the \(VN\) muscles. With respect to the Ryzhii model [22], we adjust the activation currents of the QRS complex and T wave in order to have the \(HP\) oscillator composed of the \(RB\) and \(LB\) oscillators:  
\[I_{AT_{De}}=\left\{ \begin{array}{ll}0,&\text{for $y_{1}\leq 0$},\\ K_{AT_{De}}y_{1},&\text{for $y_{1}>0$},\end{array} \right. \tag{16}\]  
\[I_{AT_{Re}}=\left\{ \begin{array}{ll}-K_{AT_{Re}}y_{1},&\text{for $y_{1}\leq 0$},\\ 0,&\text{for $y_{1}>0$},\end{array} \right. \tag{17}\]  
\[I_{VN_{De}}=\left\{ \begin{array}{ll}0,&\text{for $y_{3 Hyp}\leq 0$},\\ K_{VN_{De}}(y_{3_{R}}^{tot}+y_{3_{LB}}^{tot}),&\text{for $y_{3 Hyp}>0$},\end{array} \right. \tag{18}\]  
Figure 3: Action potentials of uncoupled oscillators. Intrinsic oscillation rates of 70 bpm, 50 bpm, 35 bpm and 35 bpm for uncoupled \(SA\), \(AV\), \(RB\) and \(LB\), respectively. \[I_{VN_{Re}}=\left\{ \begin{array}{ll}-K_{VN_{Re}}(y_{3_{RB}}^{tot}+y_{3_{LB}}^{ tot}),&\text{for}\ y_{3_{HP}}\leq 0,\\ 0,&\text{for}\ y_{3_{HP}}>0.\end{array} \right.