## 4 Dynamic Behaviors and Time Domain Analysis
### Time Domain  
To validate the proposed neuron model in case of timing accuracy and spiking patterns, the time domain must be considered. In this approach, based on different stimulus currents, the spiking patterns between original and proposed DSSN models are compared. As can be seen in Figure 4, spiking patterns of the proposed neuron model share a high degree of similarity with the original model. In addition, the error calculations are in a low state, which are computed in the next step.  
Figure 3: Dynamical behaviors for the original and proposed models. (**A1,A2**): Dynamics for the Class I and Class II patterns in the original model. (**B1,B2**): Dynamics for the Class I and Class II patterns in the proposed model. In these tates, \(n\) is the slow variable and \(V\) is the membrane potential. The equilibrium points of two original and proposed models are the same.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Point (Orig.)** & **Value (Orig.)** & **Point (Prop.)** & **Value (Prop.)** \\ \hline Saddle Point & \((-0.2,-0.7)\) & Saddle Point & \((-0.22,-0.68)\) \\ Spiral Source & \((0.09,0.92)\) & Spiral Source & \((0.1,0.9)\) \\ Saddle Point & \((-0.28,-1)\) & Saddle Point & \((-0.27,-0.98)\) \\ Spiral Sink & \((0.24,1.27)\) & Spiral Sink & \((0.22,1.25)\) \\ \hline \end{tabular}
\end{table}
Table 3: Equilibrium Points For the Original and Proposed Models.  
As can be seen in Figure 4, in some region of the spiking patterns, there are differences (errors) between the original and proposed DSSN models. Indeed, it is attempted to reduce this error to a near-zero value. A different method is available to calculate the error values between the original and proposed models. Some of these methods are focused on the absolute differences between two signals. In addition, some methods are emphasized on the square of root mean values. In this paper, these two basic error methods are applied for validating that the proposed model is in the low-error calculation compared with the original main model. These two methods are formulated as follows:  
\[RMSE(V_{Prop.},V_{Orig.})=\sqrt{\frac{\sum_{i=1}^{n}(V_{Prop.}-V_{Orig.})2}{n}} \tag{17}\]  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}|V_{Prop.}-V_{Orig.}| \tag{18}\]  
Moreover, as can be seen in Figure 5, the error level is based on the differences between the original and proposed functions. Indeed, when this difference is accrued, the spiking patterns between the original and proposed models may have differences. These differences in our proposed models have been optimized, so the modified model regenerates the same behaviors of spike-based behavior of the original model to a high degree of similarity and low-error computations. As it is shown in Table 4, the error levels are calculated.  
\begin{table}
\begin{tabular}{c c c c c} \hline \(I_{stimulus}\) & **RMSE (Class I)** & **MAE (Class I)** & **RMSE (Class II)** & **MAE (Class II)** \\ \hline
0.1 & 0.05 & 0.02 & 0.067 & 0.022 \\
0.2 & 0.08 & 0.01 & 0.1 & 0.018 \\
0.5 & 0.09 & 0.034 & 0.095 & 0.04 \\
1 & 0.1 & 0.028 & 0.15 & 0.039 \\
2 & 0.075 & 0.014 & 0.035 & 0.012 \\
3 & 0.11 & 0.08 & 0.16 & 0.06 \\ \hline \end{tabular}
\end{table}
Table 4: Error Calculations of the Proposed Model.  
Figure 4: Comparison between spiking patterns of two models (original and proposed). (**a**–**f**): Spiking patterns for considering the stimulus currents as: \(I_{stimulus=0.1}\), \(I_{stimulus=0.2}\), \(I_{stimulus=0.5}\), \(I_{stimulus=1}\), \(I_{stimulus=2}\), and \(I_{stimulus=3}\).