## 2 Background  
The Digital Spiking Silicon Neuron (DSSN) model is a simple practical model in terms of qualitative states. This model is capable of reproducing different classes of spiking such as Class I and Class II patterns. The DSSN model can be formulated by two coupled differential equations for voltage and recovery variables. The mathematical equations of the model are given by following statements:  
\[\frac{dV}{dt}=\frac{\phi}{\tau}(f(V)-n+I_{0}+I_{Stimulus}) \tag{1}\]  
\[\frac{dn}{dt}=\frac{1}{\tau}(g(V)-n) \tag{2}\]  
where  
\[f(V)=\left\{\begin{array}{lcl}a_{n}(V+b_{n})^{2}-c_{n}&;&V<0\\ -a_{p}(V-b_{p})^{2}+c_{p}&;&V>0\end{array}\right. \tag{3}\]  
\[g(V)=\left\{\begin{array}{lcl}k_{n}(V-p_{n})^{2}+q_{n}&;&V<r\\ k_{p}(V-p_{p})^{2}+q_{p}&;&V>r\end{array}\right. \tag{4}\]  
In these equations, \(V\) and \(n\) represents the voltage and slow variable, respectively. On the other hand, \(V\) is the membrane potential, and \(n\) is the recovery variable for producing the voltage variable. The parameter \(I_{0}\) is a bias fixed parameter, and \(I_{Stimulus}\) is the applied current for neurons. The other parameters for generating the Class I and Class II patterns are presented in Tables 1 and 2, respectively. These parameters are explained as follows:  
* \(I_{0}\): Bias constant;* \(I_{Stimulus}\): Applied current for neurons;
* \(\phi\) and \(\tau\): Time constant;
* \(r,a_{x},b_{x},c_{x},k_{x},p_{x}\), and \(q_{x}\) (\(x=n\) and \(x=p\)): Constants that control the nullclines of the variables (Dynamics of the model).  
It is emphasized that the DSSN model is a spike-based neuron model, and all the variables and constants are abstracted and do not have a physical unit. In addition, by selecting appropriate values for these parameters, both Class I and Class II neurons can be realized with parameter settings. Finally, different spiking patterns based on these two basic parameters can be simulated, as can be seen in Figure 1.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \hline \(a_{n}\) & \(8\) & \(a_{p}\) & \(8\) \\ \(b_{n}\) & \(0.25\) & \(b_{p}\) & \(0.25\) \\ \(c_{n}\) & \(0.5\) & \(c_{p}\) & \(0.5\) \\ \(k_{n}\) & \(2\) & \(k_{p}\) & \(16\) \\ \(p_{n}\) & \(-0.3125\) & \(p_{p}\) & \(-0.2187\) \\ \(q_{n}\) & \(-0.7058\) & \(q_{p}\) & \(-0.6875\) \\ \(\phi\) & \(1\) & \(\tau\) & \(0.003\) \\ \(r\) & \(-0.2053\) & \(I_{0}\) & \(-0.205\) \\ \hline \end{tabular}
\end{table}
Table 1: Different Parameters for Reproducing the Class I Pattern.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(a_{n}\) & \(8\) & \(a_{p}\) & \(8\) \\ \(b_{n}\) & \(0.25\) & \(b_{p}\) & \(0.25\) \\ \(c_{n}\) & \(0.5\) & \(c_{p}\) & \(0.5\) \\ \(k_{n}\) & \(4\) & \(k_{p}\) & \(16\) \\ \(p_{n}\) & \(-0.5625\) & \(p_{p}\) & \(-0.2187\) \\ \(q_{n}\) & \(-1.3177\) & \(q_{p}\) & \(-0.6875\) \\ \(\phi\) & \(0.5\) & \(\tau\) & \(0.003\) \\ \(r\) & \(-0.1041\) & \(I_{0}\) & \(-0.23\) \\ \hline \end{tabular}
\end{table}
Table 2: Different Parameters for Reproducing the Class II Pattern.  
Figure 1: Spike-based signals of the DSSN model. (**A1–A5**): Spiking patterns of Class I mode for different stimulation current as: \(I_{Stimulus}=0.1\), \(I_{Stimulus}=0.2\), \(I_{Stimulus}=0.5\), \(I_{Stimulus}=1\), and \(I_{Stimulus}=2\), respectively. (**B1–B5**): Spiking patterns of Class II mode for different stimulation current as: \(I_{Stimulus}=0.1\), \(I_{Stimulus}=0.2\), \(I_{Stimulus}=0.5\), \(I_{Stimulus}=1\), and \(I_{Stimulus}=2\), respectively.