## 4 Discussion  
In this communication, we present graph-based smoothness harmonics as a powerful tool for capturing the slowly varying spatiotemporal patterns of cortical dynamics in fMRI data. The temporal smoothness index and spatial smoothness brain map derived from the smoothness harmonics provide valuable insights into the propagation of slowly varying cortical dynamics across the entire cortex over time. By applying this approach to age-development fMRI data, we demonstrated the versatility of smoothness harmonics in discerning differences in cortical dynamics between adults and children, thereby reflecting the age-related variations in the development of the theory of mind. These findings highlight the potential of smoothness harmonics as a valuable tool for investigating and understanding cortical dynamics in various contexts.  
Undoubtedly, graph analysis has been widely employed in the study of cortical dynamics, ranging from early attempts to model brain connectomes as small-world networks [6; 32] to more recent advancements in deep graph neural networks for learning nodal representations of graph signals [12; 33] and exploring complex brain topology to investigate neuropsychiatric disorders [34]. However, our proposed smoothness harmonic approach stands out from these existing methods in two key aspects. Firstly, our method is specifically designed for the analysis of fMRI data. It captures the predominant slow-frequency oscillations that are characteristic of fMRI signals through the extraction of smoothness harmonics. This targeted approach enables a more-focused examination of cortical dynamics in the context of fMRI studies. Secondly, unlike the complex learning algorithms used in other approaches, our smoothness harmonic approach relies solely on the well-established eigendecomposition framework. This ensures a straightforward and easily implementable methodology. This simplicity and convenience make our approach accessible to researchers and facilitates its application in diverse fMRI studies.  
The introduction of smoothness harmonics opens up exciting future opportunities to explore and compare various types of neuropsychiatric disorders based on their extracted smoothness harmonics. Additionally, as a potential avenue for future research, the concept of smoothness harmonics can be applied to dynamic graph analysis to address the growing demand for dynamic functional connectivity analysis [35]. However, when applying this concept to other neuroimaging modalities such as EEG and MEG data, it is important to consider incorporating additional non-slow harmonics to accommodate the specific characteristics of those modalities. Overall, the graph-based smoothness harmonic has the potential to provide valuable insights into the spatiotemporal dynamics of brain activity and contribute to advancements in the field of neuroscience.  
**Funding:** This work was funded by the Japan Agency for Medical Research and Development (AMED) under grant number JP23 dm0307008.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The utilized development fMRI data is public available and can be sourced at [[https://nilearm.github.io/stable/modules/generated/nilearm.datasets.fetch_development_fmri.html#nilearm.datasets.fetch_development_fmri](https://nilearm.github.io/stable/modules/generated/nilearm.datasets.fetch_development_fmri.html#nilearm.datasets.fetch_development_fmri)]([https://nilearm.github.io/stable/modules/generated/nilearm.datasets.fetch_development_fmri.html#nilearm.datasets.fetch_development_fmri](https://nilearm.github.io/stable/modules/generated/nilearm.datasets.fetch_development_fmri.html#nilearm.datasets.fetch_development_fmri)) (accessed on 1 May 2022).  
**Conflicts of Interest:** The author declares no conflict of interest.