_Applied_  
_sciences_  
**Smoothness Harmonic: A Graph-Based Approach to Reveal Spatiotemporal Patterns of Cortical Dynamics in fMRI Data**  
**[PERSON]**  
Department of Computational Brain Imaging, Neural Information Analysis Laboratories, Advanced  
Telecommunication Research Institute International (ATR), Kyoto 619-0288, Japan; [EMAIL_ADDRESS]  
**Featured Application: Benefiting from its convenient derivation without resorting to complex learning algorithms, our smoothness harmonic approach can be deployed in numerous neuroscientific discoveries. This includes differentiating various types of neuropsychiatric disorders by utilizing the extracted smoothness harmonics and exploring the distinctive harmonic features between sleep and awake states of in vivo neuronal recordings. **  
**Abstract: Despite fMRI data being interpreted as time-varying graphs in graph analysis, there has been more emphasis on learning sophisticated node embeddings and complex graph structures rather than providing a macroscopic description of cortical dynamics. In this paper, I introduce the notion of smoothness harmonics to capture the slowly varying cortical dynamics in graph-based fMRI data in the form of spatiotemporal smoothness patterns. These smoothness harmonics are rooted in the eigendecomposition of graph Laplacians, which reveal how low-frequency-dominated fMRI signals propagate across the cortex and through time. We showcase their usage in a real fMRI dataset to differentiate the cortical dynamics of children and adults while also demonstrating their empirical merit over the static functional connectomes in inter-subject and between-group classification analyses. **  
**Keywords: computational neuroscience; fMRI data; graph analysis; connectome Laplacian analysis; cortical dynamics**  
**1. Introduction**  
Functional magnetic resonance imaging (fMRI) methods, with their superior spatial resolution [1], have become valuable tools in both clinical and research settings [2] for studying brain activity in healthy and neuropsychiatric conditions. By leveraging the blood-oxygen-level-dependent contrast [3] or the arterial spin-labeling technique [4], these methods enable the investigation of intricate neural processes. However, the high-dimensional nature of fMRI data, coupled with the presence of dominant ultra-slow oscillations (0.01-0.25 Hz) [5], poses a significant challenge in extracting meaningful spatiotemporal patterns of cortical dynamics from fMRI data. One common approach to revealing spatiotemporal patterns of cortical dynamics in high-dimensional fMRI data is through graph analysis. The human cortical network can be interpreted as a graph [6, 7], where nodes and edges represent brain regions and the connectome between regions, respectively. Graph analysis can be a powerful tool for examining spatiotemporal patterns of neural activity associated with nodes and edges. However, the use of graph analysis in fMRI data often emphasizes identifying the (time-varying or static) network topology of the cortex [8, 9] and learning low-dimensional nodal representations [10, 11] or graph embedding [12] to reflect high-dimensional fMRI signals. A macroscopic graph-analysis-based description of cortical dynamics based on fMRI data remains largely unexplored. In this communication, we introduce the concept of smoothness harmonics: the graph-based features that capture the slowly varying oscillation patterns in fMRI signals across the cortex and over time. These smoothness harmonics are derived through Laplacian eigendecomposition analysis [13] and provide low-frequency-dominated representations of cortical dynamics in ultra-slow fMRI signals. The resulting spatial and temporal smoothness harmonics serve as direct macroscopic descriptions of slow cortical dynamics without the need for complex learning algorithms. We investigate the applicability of the proposed smoothness harmonic approach in real fMRI data, specifically using the age-development fMRI dataset [14].