## 4 Conclusions  
In conclusion, this paper introduces the innovative G-NMS clustering method, which uniquely employs the Gershgorin circle (GC) theorem in conjunction with the nonmaximum suppression (NMS) method, an approach that is not implemented in spike clustering applications. The empirical evaluations conducted with both datasets highlight the robust and excellent performance of the G-NMS method, characterized by minimal standard deviations, indicative of its reliability and consistency. While the G-NMS method consistently outperformed across most of the experimental setups, it is noteworthy that in the remaining two experiments--D1 and D2 from Dataset 2--the method experienced a modest decline in average accuracy. Specifically, there was a 14.28% and 4.12% decrease in accuracy for D1 and D2, respectively, when compared to the top-performing clustering algorithms in these categories. Despite this, G-NMS maintained lower standard deviations in these instances as well, underscoring its consistent performance across varying levels of complexity and noise. The observed dip in performance for D1 is attributed to reduced covariance between the two spike classes (Spk) involved in these particular datasets. This factor highlights the sensitivity of the G-NMS method to the inherent data characteristics, an aspect that may guide future optimizations and refinements of the algorithm. The introduction of the G-NMS method represents a significant advancement in the field of clustering. It integrates the mathematical eigenvalue inclusion theorem (GC) and a \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Experiments** & **IoU (\(\alpha\))** &
\begin{tabular}{c} **Gaussian Kernel** \\ **Bandwidth (h)** \\ \end{tabular} & **ACC** \\ \hline SNR0.5\_Spk-1,2 & 0.1 & 0.4 & 99.930 \\ SNR1.25\_Spk-2,3 & 0.1 & 0.2 & 99.938 \\ SNR2.0\_Spk-3,1 & 0.1 & 0.1 & 99.942 \\ E1\_SNR0.05\_Spk-1,2 & 0.5 & 0.4 & 99.784 \\ E1\_SNR0.2\_Spk-1,2 & 0.1 & 0.2 & 99.570 \\ E2\_SNR0.05\_Spk-2,3 & 0.3 & 0.1 & 99.517 \\ E2\_SNR0.2\_Spk-2,3 & 0.1 & 0.2 & 99.017 \\ D1\_SNR0.05\_Spk-3,1 & 0.5 & 0.1 & 79.823 \\ D1\_SNR0.2\_Spk-3,1 & 0.4 & 0.06 & 68.639 \\ D2\_SNR0.05\_Spk-1,2 & 0.4 & 0.1 & 94.855 \\ D2\_SNR0.2\_Spk-1,2 & 0.4 & 0.1 & 95.907 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Selection of IoU and kernel bandwidth parameters across SNR variants and their impact on clustering accuracy. post-processing object detection algorithm from the field of deep learning (NMS), opening new paths for exploration and application. Looking to the future, the performance of the G-NMS could be further enhanced by adopting a more robust covariance matrix instead of the sample covariance matrix currently in use. Additionally, exploring other eigenvalue inclusion theorems might offer a more precise estimation and localization of eigenvalues, thereby improving the algorithm's accuracy. Moreover, the algorithm's effectiveness could be improved by incorporating a variant of the NMS algorithm, such as a soft-NMS approach, which might provide less rigid thresholding compared to the current hard-threshold, greedy NMS strategy. These potential modifications and enhancements to the G-NMS method hold promise for even more effective clustering solutions in future applications. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/signals5020020/s1](https://www.mdpi.com/article/10.3390/signals5020020/s1)]([https://www.mdpi.com/article/10.3390/signals5020020/s1](https://www.mdpi.com/article/10.3390/signals5020020/s1)). All code related to this work is available in [30]. [PERSON] developed the approach for this research and drafted the manuscript. [PERSON] helped to revise the drafted manuscript and provided valuable advice for this research. All authors have read and agreed to the published version of the manuscript. This research received no external funding. All syntenic signal datasets supporting this study's findings come from previously reported studies, which are cited accordingly within the manuscript.