## 3 Results  
In order to validate our model, we performed different Simulink simulations reproducing either normal and pathological signals. Simulations are performed using a fixed step Runge-Kutta solver and a fixed step size of 0.0001 seconds. Figure 4 shows the normal condition where all pacemakers are dominated by the \(SA\) node. As a consequence, the heart rate follows the SA rate (70 bpm). In this case, both the right and left bundle branches are working, and they give the same contribution. The experimental parameters are the same used in the non-pathological as described in Section 2. Then, we performed a simulation of a pathological condition of the His bundle. We obtain this condition by changing the physiological parameters with the pathological parameters in order to obtain both complete right bundle branch block (RBBB) and complete left bundle branch block (LBBB). The pathological parameters used are shown in Section 2.  
are activated sequentially (first right then left) rather than simultaneously, this produces a broad or notched ('M'-shaped) R-wave in the lateral leads (I, aVL, V5-6). Figure 6a shows simulation results for the LBBB case. 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 this case, the term \(y_{3_{LB}}\) in Equation (21) is not present because the contribution of the left bundle branch is zero. On the other hand, the impulse generated by the right bundle branch \(y_{3_{RB}}\) is able to both normally depolarize the right ventricle and to travel towards the left ventricle through the myocardium cells. As this second conduction way is slower than the main bundle of His-Purkinje fibres, the term \(y_{3_{RB}}\) in Equation (21) (that represents the left ventricle depolarization) is delayed through the coefficient \(\tau_{RB}\). This produces a broad or notched ('M'-shaped) R-wave in the ECG signal (Figure 6b). It is possible to compare simulation results with a real patient's ECG signal affected by LBBB shown in Figure 6c. Figure 5: Left Bundle Branch Block (LBBB) and the four coupled oscillators model. Figure 6: Simulation results for Left Bundle Branch Block (LBBB). (**a**) action potentials, muscles response and ECG signal in the LBBB case; (**b**) extra deflection of the QRS complex due to the rapid depolarization of the right ventricle followed by the slower depolarization of the left ventricle; (**c**) real ECG signal affected by LBBB [44]. Right Bundle Branch BlockIn the Right Bundle Branch Block (RBBB), the activation of the right ventricle is delayed because the depolarization has to spread from the left ventricle (yellow arrows in Figure 7). The left ventricle is activated normally and the early part of the QRS complex is unchanged. The delayed right ventricular activation produces a secondary R-wave (R') in the right precordial leads (V1-3) and a wide, slurred S-wave in the lateral leads [32; 33] (Figure 7). In this case, the term \(y_{3_{RB}}\) in Equation (24) is not present because the contribution of the right bundle branch is zero. In this case, the impulse is generated by the left bundle brunch \(y_{3_{LB}}\) that is able to both normally depolarize the left ventricle and to travel towards the right one through the myocardium cells. Figure 8a shows the simulation results for the RBBB case. For this simulation, we set the following experimental parameters: \(K_{SA-AV}=100\), \(K_{AV-BR}=0\), \(K_{AV-LB}=285\), \(K_{AT_{De}}=9*10^{-5}\), \(K_{AT_{Re}}=2*10^{-5}\), \(K_{VNb_{De}}=5.5*10^{-5}\), \(K_{VNb_{Re}}=7*10^{-5}\), \(\tau_{LB}=0.05\), \(\alpha_{1}=0\), \(\alpha_{2}=1\) and \(\alpha_{3}=1\). In addition, in this case, it is possible to compare simulation results with a real patient's ECG signal affected by RBBB shown in Figure 8c. Figure 8: Simulation results for Right Bundle Branch block (RBBB).