# Parameter Estimation of a Class of Neural Systems with Limit Cycles
## 5 Simulations  
For purpose of illustration, we consider the FHN model [25; 26] described in the dimensionless form (2). In simulation, the parameters are taken as \(a=0.1,b=1,c_{1}=1,c_{2}=0.5,\mu=100\) and \(J=0.5\). The FHN model is firstly discretized based on the explicit forward Euler method with step size \(T=10\)ms. Then, consider the initial value \(v(0)=-0.3,w(0)=0.6\), and perform the simulations based on the identification algorithms in Section 4. Taking the data length \(L_{d}=20,000\), the forgotten factor \(\lambda\) is chosen as \(0.99\) for RLS and MIRLS algorithms. For both SG and MISG algorithms, we take the forgotten factor \(\alpha=0.8\). To compare the estimation performance of the RLS, SG, MIRLS and MISG algorithms, apply these four algorithms to estimate the parameter vector \(\theta\) of the neuron system, respectively. Note that, when \(p=1\), the MIRLS algorithm reduces to the RLS algorithm and the MISG algorithm reduces to the SG algorithm. For the MIRLS and MISG algorithms, two cases with different innovation lengths \(p=3\) and \(p=5\) are considered, respectively. To quantify the estimation accuracy and clearly compare the performance of the RLS, SG, MIRLS and MISG algorithms, we consider two different noise levels, i.e., \(\sigma^{2}=0.2^{2}\) and \(\sigma^{2}=0.5^{2}\). The parameter estimates and their errors are shown in Tables 1-4 with different noise variances. The estimation errors \(\delta\) versus \(k\) are shown in Figures 2 and 3, where \(\delta:=\|\hat{\theta}-\theta\|/\|\theta\|\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\sigma^{2}\) & \(k\) & \(\mu\) & \((a+b)\mu\) & \(ab\mu\) & \(J\) & \(c_{1}\) & \(c_{2}\) & \(\delta\) (\%) \\ \hline \multirow{6}{*}{\(0.2^{2}\)} & 10 & 3.7906 & \(-0.7233\) & 0.6077 & 2.9444 & 2.6491 & 0.3230 & 98.2022 \\  & 20 & 1.6180 & 4.6890 & \(-6.8895\) & 2.7357 & 1.8070 & 0.5639 & 97.0998 \\  & 50 & 94.7731 & 102.8236 & 8.3295 & 47.5574 & 0.9996 & 0.6066 & 5.9548 \\  & 100 & 99.1037 & 108.8017 & 9.7432 & 49.5408 & 1.0207 & 0.5576 & 1.0100 \\  & 150 & 99.2001 & 108.8906 & 9.7232 & 49.5928 & 1.0401 & 0.5390 & 0.9256 \\  & 200 & 99.5227 & 109.3771 & 9.8946 & 49.7620 & 1.0404 & 0.5346 & 0.5272 \\ \hline \multirow{6}{*}{\(0.5^{2}\)} & 10 & 19.0174 & \(-2.3066\) & 7.1256 & 10.3718 & 4.2456 & 0.4154 & 91.6760 \\  & 20 & 11.4943 & 28.0891 & \(-11.5647\) & 8.2026 & 2.0516 & 0.9504 & 82.3618 \\ \cline{1-1}  & 50 & 93.5511 & 100.3815 & 7.4373 & 47.0843 & 1.2176 & 0.9968 & 7.7788 \\ \cline{1-1}  & 100 & 98.8246 & 108.3058 & 9.5814 & 49.3799 & 1.0948 & 0.6696 & 1.4012 \\ \cline{1-1}  & 150 & 98.9660 & 108.4155 & 9.5199 & 49.4549 & 1.1355 & 0.6290 & 1.2950 \\ \cline{1-1}  & 200 & 99.7218 & 109.5163 & 9.8858 & 49.8592 & 1.1210 & 0.5971 & 0.3861 \\ \hline True values & 100.0000 & 110.0000 & 10.0000 & 50.0000 & 1.0000 & 0.5000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The RLS estimates and errors. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\sigma^{2}\) & \(k\) & \(\mu\) & \((a+b)\mu\) & \(ab\mu\) & \(J\) & \(c_{1}\) & \(c_{2}\) & \(\delta\) (\%) \\ \hline \multirow{6}{*}{\(0.2^{2}\)} & 10 & 7.9850 & \(-1.6092\) & 2.5997 & 4.8664 & 2.4115 & 0.3541 & 96.5310 \\  & 20 & 3.4710 & 8.9666 & \(-7.0001\) & 3.6340 & 1.7240 & 0.5638 & 94.2987 \\  & 50 & 98.1459 & 107.4963 & 9.4579 & 49.1372 & 0.9894