## 2 Materials and Methods
### Methodology
#### 2.1.6 Nonmaximum Suppression  
The nonmaximum suppression (NMS) is an algorithm which is generally utilized in the field of computer vision for object detection tasks. Generally, the NMS algorithm is implemented to suppress the bounding boxes and eliminate the redundant boxes. However, within this study, the circle-based NMS is implemented instead using of bounding boxes. As the datapoint or number of epochs increases, the number of GC circles also increases, which potentially results in misclassification and therefore decreases the clustering accuracy. Therefore, to reduce the occurrence of overlapping circles, NMS is applied to each GC. Figure 4a,b visualizes the before and after effect of NMS on the implementation on GCs with intersection over union (IoU) of 0.1 for two different classes with a total of 4000 epochs. Figure 3: Comparative analysis of class separation pre- and post-vertical matrix transformation. (**a**) displays Gershgorin circles pre-transformation, centered at \(\text{Var}\left(e_{i}\right)\) with radii \(\text{Cov}\left(e_{i},e_{j}\right)\). (**b**) shows post-transformation Gershgorin circles, now centered at \(\text{Cov}\left(e_{i},e_{j}\right)\) and radius \(\text{Var}\left(e_{j}\right)\). Subplots (**c,d**) depict visual representations of random neural spikes for Class 1 and Class 2, respectively. In the NMS procedure, each GC is assigned a score corresponding to the \(\text{Var}\big{(}\text{e}_{\text{j}}\big{)}\) of matrix \(\text{T}\). An IoU value \(\alpha\), ranging between 0 and 1, is selected to determine the extent of overlap or interaction between two GCs. Assume that there is GC \(-\text{A}\) with center \(\text{O}_{1}(\text{x}_{1},\text{y}_{1})\) and radius \(\text{r}_{1}\) and GC \(-\text{B}\) with center \(\text{O}_{2}(\text{x}_{2},\text{y}_{2})\) and radius \(\text{r}_{2}\), as shown in Figure 4c. The algorithm initiates with the identification of the GC with the highest score; assume that GC \(-\text{A}\) is the one with the highest score. Then, the subsequent calculations determine the distance d between the \(\text{O}_{1}\) and \(\text{O}_{2}\) as follows:  
\[\text{d}=\sqrt{\left(\text{x}_{2}-\text{x}_{1}\right)^{2}+\left(\text{y}_{2}- \text{y}_{1}\right)^{2}}\]  
If the \(\text{d}\geq\text{r}_{1}+\text{r}_{2}\), then the circles do not overlap and the IoU is zero. In cases of overlap, as depicted in Figure 4c, the IoU between GCs A and B is computed using Equation (5). If \(\text{IoU}(\text{A},\text{B})\) exceeds \(\alpha\), GC B is suppressed, under the presumption that GC A possesses a higher score. This iterative process prioritizes GCs based on their scores, suppressing those that significantly overlap with a higher-scored GC. Note that the relevant GC centers are preserved in the vector \(\text{V}_{\text{NMS}}=\text{NMS}\big{(}\big{\{}\text{c}_{\text{i}}\big{(} \text{T}_{\text{i}_{11}}\big{)  
estimating the probability density function (PDF) of a random variable. The KDE is computed for \(\mathrm{V_{NMS}}\) using Equation (6) [25]. \[\mathrm{f}\left(\mathrm{V_{x}^{NMS}}\right)=\frac{1}{\mathrm{qh}}\sum_{i=1}^{ \mathrm{q}}\mathrm{Ker}\left(\frac{\mathrm{V_{x}^{NMS}-V_{x_{i}}^{NMS}}}{\mathrm{h }}\right) \tag{6}\]  
where:  
\(\mathrm{f}\left(\mathrm{V_{x}^{NMS}}\right)=\mathrm{The~{}estimated~{}density~{} at~{}point~{}x}\);  
\(\mathrm{q}=\mathrm{The~{}number~{}of~{}datapoints}\);  
\(\mathrm{V_{x_{i}}^{NMS}}=\mathrm{The~{}datapoints}\);  
\(\mathrm{Ker}=\mathrm{The~{}Gaussian~{}kernel~{}functions}\);  
\(\mathrm{h}=\mathrm{The~{}bandwidth~{}kernel~{}function}\). The KDE of \(\mathrm{V_{NMS}}\) based on the Gaussian kernel with appropriate bandwidth is shown in Figure 1. The bandwidth parameter \(\mathrm{h}\) controls the smoothness of the density estimation in such a way that a low bandwidth may yield a noisy estimation, whereas a high bandwidth could overly smooth the data, potentially leading to underfitting. The bandwidth is typically chosen within the range of 0.0 to 1.0, based on the signal-to-noise ratio (SNR) of the spikes or signal.