## 2 Heart Conduction System and Mathematical Models
### Mathematical Model
\tag{19}\]  
In particular,  
\[y_{3_{RB}}^{tot}=\alpha_{1}y_{3_{RB}}+\alpha_{3}y_{3_{LB}}^{\tau_{ LB}}, \tag{20}\]  
\[y_{3_{LB}}^{tot}=\alpha_{2}y_{3_{LB}}+\alpha_{3}y_{3_{RB}}^{\tau_{ RB}},\]  
where \(y_{3_{HP}}=(y_{3_{RB}}+y_{3_{LB}})\), \(K_{AT_{De}}=9*10^{-5}\), \(K_{AT_{Re}}=1.5*10^{-5}\), \(K_{VN_{De}}=9*10^{-5}\), \(K_{VN_{Re}}=9*10^{-5}\), \(\alpha_{1}=0.5\), \(\alpha_{2}=0.5\) and \(\alpha_{3}=0.001\). In Equation (20), the terms \(y_{3_{RB}}\) and \(y_{3_{LB}}\) represent the impulses traveling through the right and left ventricles and through the bundle branches. The terms \(y_{3_{RB}}^{\tau_{RB}}\) and \(y_{3_{LB}}^{\tau_{LB}}\) represent the impulses traveling through the right and left ventricles through the myocardium cells. These secondary pathways are slower than the main path propagated through the bundle branches. In the physiological case, the contribution of these impulses is negligible with respect to the contribution of the main pathways [12]. On the other hand, in the pathological case, where one branch is damaged, the secondary pathways guarantee the contraction of the ventricle not served by the main pathway. To provide a complete model, we need to take into account the contribution of the secondary pathways. It can be described as an impulse that travels through the bundle branches delayed and attenuated by a multiplicative constant \(\alpha_{3}\). We note that, in the actual case, these secondary impulses travel through the myocardium cells and the propagation of electrical activity should be described with a specific mathematical model. However, the scope of our work is to provide a phenomenological model. For this reason, we assumed that the impulses that travel through the myocardium cells can be modeled as an impulse attenuated and delayed in time. In the model of the left bundle branch block (LBBB), the activation currents \(I_{VN_{De}}\) and \(I_{VN_{Re}}\) represent respectively the coupling between the \(RB\) and \(LB\) pacemakers with the \(VN\) muscles; consequently, Equations (18) and (19) become  
\[I_{VN_{De}}^{LBBB}=\left\{ \begin{array}{ll}0,&\text{for}\ y_{3_{RB}}\leq 0,\\ K_{VN_{De}}(\alpha_{1}y_{3_{RB}}+\alpha_{3}y_{3_{RB}}^{\tau_{RB}}),&\text{for} \ y_{3_{RB}}>0,\end{array} \right. \tag{21}\]  
\[I_{VN_{Re}}^{LBBB}=\left\{ \begin{array}{ll}-K_{VN_{Re}}\alpha_{1}y_{3_{RB}},&\text{for}\ y_{3_{RB}}\leq 0,\\ 0,&\text{for}\ y_{3_{RB}}>0.\end{array} \right. \tag{22}\]  
For LBBB simulation, we set the experimental parameters: \(K_{SA-AV}=100\), \(K_{AV-BR}=285\), \(K_{AV-LB}=0\), \(K_{AT_{De}}=9*10^{-5}\), \(K_{AT_{Re}}=3*10^{-5}\), \(K_{VN_{De}}=9*10^{-5}\), \(K_{VN_{Re}}=-8*10^{-5}\), \(\tau_{RB}=0.78\), \(\alpha_{1}=1\), \(\alpha_{3}=1\) and \(\alpha_{2}=0\). In the model of right bundle branch block (RBBB) the activation currents, \(I_{VN_{De}}\) and \(I_{VN_{Re}}\) are  
\[I_{VN_{De}}^{RBBB}=\left\{ \begin{array}{ll}K_{VN_{De}}(\alpha_{2}y_{3_{ LB}}+\alpha_{3}y_{3_{LB}}^{\tau_{LB}}),&\text{for}\ y_{3_{LB}}<0,\\ K_{VN_{De}}\alpha_{2}y_{3_{LB}},&\text{for}\ y_{3_{LB}}>0,\end{array} \right. \tag{23}\]  
\[I_{VN_{Re}}^{RBBB}=\left\{ \begin{array}{ll}-K_{VN_{Re}}\alpha_{2}y_{3_{ LB}},&\text{for}\ y_{3_{LB}}<0,\\ 0,&\text{for}\ y_{3_{LB}}>0.\end{array} \right. \tag{24}\]For RBBB simulation, we set the experimental parameters: \(K_{SA-AV}=100\), \(K_{AV-BR}=0\), \(K_{AV-LB}=285\), \(K_{AT_{Da}}=9*10^{-5}\), \(K_{AT_{Rx}}=2*10^{-5}\), \(K_{VN_{De}}=5.5*10^{-5}\), \(K_{VN_{Re}}=7*10^{-5}\), \(\tau_{RB}=0.05\), \(\alpha_{1}=0\), \(\alpha_{2}=1\) and \(\alpha_{3}=1\). Finally, we obtain the synthetic ECG signal as a composition of the \(AT\) and \(VN\) waveforms  
\[ECG=z_{0}+z_{1}-z_{2}+k_{R}z_{3}+z_{4}, \tag{25}\]  
where \(z_{0}=0.2\) provides the adjustment of the baseline and \(k_{R}\) is a multiplicative coefficient to modulate the amplitude of the R peak.