## 2 Heart Conduction System and Mathematical Models
### Mathematical Model  
Ryzhii in [22] describes the three natural pacemakers \(SA\), \(AV\) and the \(HP\) bundle by a system of modified VdP equations:  
\[\mathbf{SA}=\left\{ \begin{array}{l}\dot{x}_{1}=y_{1},\\ \dot{y}_{1}=-a_{1}y_{1}(x_{1}-u_{11})(x_{1}-u_{12})-f_{1}x_{1}(x_{1}+d_{1})(x_ {1}+e_{1}),\end{array} \right. \tag{1}\]  
\[\mathbf{AV}=\left\{ \begin{array}{l}\dot{x}_{2}=y_{2},\\ \dot{y}_{2}=-a_{2}y_{2}(x_{2}-u_{21})(x_{2}-u_{22})-f_{2}x_{2}(x_{2}+d_{2})(x_ {2}+e_{2})+K_{SA-AV}(y_{1}^{\tau_{SA-AV}}-y_{2}),\end{array} \right. \tag{2}\]  
\[\mathbf{HP}=\left\{ \begin{array}{l}\dot{x}_{3}=y_{3},\\ \dot{y}_{3}=-a_{3}y_{3}(x_{3}-u_{31})(x_{3}-u_{32})-f_{3}x_{3}(x_{3}+d_{3})(x_ {3}+e_{3})+K_{AV-HP}(y_{2}^{\tau_{AV-HP}}-y_{3}),\end{array} \right. \tag{3}\]  
where \(x_{i}(t)\) and \(y_{i}(t)\) correspond to the action potential and the transmembrane currents of the heart, \(a_{i}(x_{i}-u_{ij})(x_{i}-u_{ik})\) are the damping factors, \(f_{i}x_{i}(x_{i}-d_{ij})(x_{i}-e_{ik})\) are the harmonic force terms, \(a_{i}>0\), \(u_{ij}\) represent the nonlinear damping force parameters, \(f_{i}\) are the parameters related to the intrinsic frequency of the oscillator, the coupling coefficients \(K_{SA-AV}\) and \(K_{AV-HP}\) represent the unidirectional coupling between the SA, AV and HP pacemakers, \(y_{i}^{\tau_{n}}\equiv y_{i}(t-\tau_{n})\) are the velocity coupling components of the time-delay signal, and \(\tau_{n}\) are the time delays [18; 20]. The synchronism between the three oscillators depends on the coupling coefficients \(K\)[35; 36]. A lot of coupling methods can be found in the literature [37; 38; 39]. However, it is possible to avoid the use of delays \(\tau_{n}\) in the coupling terms by choosing the appropriate coupling coefficients \(K_{SA-AV}\) and \(K_{AV-HP}\). In conclusion, the VdP Equations (1)-(3) become  
\[\mathbf{SA}=\left\{ \begin{array}{l}\dot{x}_{1}=y_{1},\\ \dot{y}_{1}=-a_{1}y_{1}(x_{1}-u_{11})(x_{1}-u_{12})-f_{1}x_{1}(x_{1}+d_{1})(x_ {1}+e_{1}),\end{array} \right. \tag{4}\]  
\[\mathbf{AV}=\left\{ \begin{array}{l}\dot{x}_{2}=y_{2},\\ \dot{y}_{2}=-a_{2}y_{2}(x_{2}-u_{21})(x_{2}-u_{22})-f_{2}x_{2}(x_{2}+d_{2})(x_ {2}+e_{2})+K_{SA-AV}(x_{1}-x_{2}),\end{array} \right. \tag{5}\]\[\mathbf{HP}=\left\{ \begin{array}{l}\dot{x}_{3}=y_{3},\\ \dot{y}_{3}=-a_{3}y_{3}(x_{3}-u_{31})(x_{3}-u_{32})-f_{3}x_{3}(x_{3}+d_{3})(x_{3 }+e_{3})+K_{AV-HP}(x_{2}-x_{3}).\end{array} \right. \tag{6}\]  
Starting from this model, we propose improvements to describe the phenomenological behaviour of the right and left bundle branch. Leaving the \(SA\) and \(AV\) node equations (Equations (4) and (5)) unaltered, we describe the right bundle branch (\(RB\)) and the left bundle branch (\(LB\)) as  
\[\mathbf{SA}=\left\{ \begin{array}{l}\dot{x}_{1}=y_{1},\\ \dot{y}_{1}=-a_{1}y_{1}(x_{1}-u_{11})(x_{1}-u_{12})-f_{1}x_{1}(x_{1}+d_{1})(x _{1}+e_{1}),\end{array} \right. \tag{7}\]  
\[\mathbf{AV}=\left\{ \begin{array}{l}\dot{x}_{2}=y_{2},\\ \dot{y}_{2}=-a_{2}y_{2}(x_{2}-u_{21})(x_{2}-u_{22})-f_{2}x_{2}(x_{2}+d_{2})(x_{ 2}+e_{2})+K_{SA-AV}(x_{1}-x_{2}),\end{array} \right. \tag{8}\]  
\[\mathbf{RB}=\left\{ \begin{array}{l}\dot{x}_{3_{RB}}=y_{3_{RB}},\\ \dot{y}_{3_{RB}}=-a_{3}y_{3_{RB}}(x_{3_{ES}}-u_{31})(x_{3_{RB}}-u_{32})-f_{3}x_{ 3_{RB}}(x_{3_{RB}}+d_{3})(x_{3_{RB}}+e_{3})+K_{AV-RB}(x_{2}-x_{3_{RB}}),\end{array} \right. \tag{9}\]  
\[\mathbf{LB}=\left\{ \begin{array}{l}\dot{x}_{3_{LB}}=y_{3_{LB}},\\ \dot{y}_{3_{LB}}=-a_{3}y_{3_{LB}}(x_{3_{LB}}-u_{31})(x_{3_{LB}}-u_{32})-f_{3}x_ {3_{LB}}(x_{3_{LB}}+d_{3})(x_{3_{LB}}+e_{3})+K_{AV-LB}(x_{2}-x_{3_{LB}}).\end{array} \right. \tag{10}\]  
In the physiological condition, the right and left bundle branches are synchronized and the sum of their behaviour has to be the same as the \(HP\) oscillator:  
\[y_{3_{HP}}=\alpha_{1}y_{3_{RB}}+\alpha_{2}y_{3_{LB}}, \tag{11}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are coefficients used to adjust the oscillation amplitudes. In order to obtain the physiological signal, we set \(\alpha_{1}=0.5\) and \(\alpha_{2}=0.5\).