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Graph Frequency Analysis of Brain Signals
Weiyu Huang, Leah Goldsberry, Nicholas F. Wymbs, Scott T. Grafton, Danielle S. Bassett, Alejandro Ribeiro
TL;DR
The paper asks how graph-based frequency analysis can characterize brain signals and functional networks during motor learning. It develops graph Fourier transforms and filters to decompose signals by spatial smoothness, then applies them to learning experiments, finding frequency-specific adaptability and associations between graph spectra and task familiarity.
Problem
Brain-signal analysis needs tools that incorporate functional connectivity and distinguish multiple spatial-variation patterns beyond PCA’s typical dimensionality-reduction use.
Method
The paper applies graph Fourier transforms and low-, medium-, and high-frequency graph filters to functional brain networks and signals collected during motor-learning tasks.
Results
Brain activities at different graph frequencies show different adaptability during learning, while graph spectral properties are strongly associated with task familiarity and frequency signatures vary in contribution across familiarity stages.
Takeaways & Limitations
Graph frequency decomposition identifies smooth and rapidly varying brain signals that are informative about learning and task familiarity.
Abstract
from arXiv · showhide
This paper presents methods to analyze functional brain networks and signals from graph spectral perspectives. The notion of frequency and filters traditionally defined for signals supported on regular domains such as discrete time and image grids has been recently generalized to irregular graph domains, and defines brain graph frequencies associated with different levels of spatial smoothness across the brain regions. Brain network frequency also enables the decomposition of brain signals into pieces corresponding to smooth or rapid variations. We relate graph frequency with principal component analysis when the networks of interest denote functional connectivity. The methods are utilized to analyze brain networks and signals as subjects master a simple motor skill. We observe that brain signals corresponding to different graph frequencies exhibit different levels of adaptability throughout learning. Further, we notice a strong association between graph spectral properties of brain networks and the level of exposure to tasks performed, and recognize the most contributing and important frequency signatures at different task familiarity.
I. INTRODUCTION
The paper treats brain activity as a graph signal supported on functional connectivity networks and applies graph signal processing to identify spatially varying patterns during visual-motor learning. It introduces graph frequencies, filters, and their relationship to PCA as tools for analyzing signals and networks across learning.
- Motivation: Brain activity is modeled as a signal on a graph whose nodes represent cortical regions and whose edges encode structural or functional connectivity.The graph provides network information for interpreting patterns in regional activity.
- Core concepts: Graph Fourier concepts generalize conventional frequency analysis to irregular brain networks, separating slowly varying from rapidly varying signal components.Low graph frequencies represent slow changes relative to connectivity, whereas high frequencies represent rapid changes.
- Relation to PCA: The paper distinguishes graph signal processing from PCA because graph frequency decomposition can separate low, medium, and high components rather than primarily reducing dimensionality.The distinction is framed for functional connectivity networks and brain-signal analysis.
- Contributions: The study introduces GSP tools, evaluates functional-network graph spectra, analyzes frequency-specific signal variation during visual-motor learning, and examines frequency contributions across learning stages.These four aims organize the paper’s methodological and empirical contributions.
- Method: The paper uses graph filters, including low-pass, band-pass, and high-pass filters, to isolate components with different levels of spatial variability.These filters support later analyses of brain signals during learning.
A. Graph Fourier Transform and Graph Frequencies
The graph Fourier transform decomposes a brain signal into orthogonal components defined by the eigenvectors of the graph Laplacian. Ordered Laplacian eigenvalues provide a graph-frequency scale in which low-frequency components are smooth across connected regions and high-frequency components vary rapidly.
- Transform: The graph Fourier transform and inverse transform represent a brain signal through orthogonal Laplacian eigenvectors and their spectral coefficients.The transform maps the vertex-domain signal to graph-frequency coefficients and the inverse reconstructs the original signal.
- Frequency interpretation: Graph frequency is determined by Laplacian eigenvalue order: eigenvectors associated with small eigenvalues vary slowly across the graph, whereas those associated with large eigenvalues vary rapidly.This ordering transfers the Fourier notion of frequency to network-supported signals.
- Total variation: Total variation measures how much a signal changes relative to network connectivity through weighted squared differences between connected regions.Large edge weights amplify differences between regions expected to have similar activity.
- Smoothness: A signal with small total variation is smooth over the graph, while a signal with large total variation changes rapidly across connected regions.The Laplacian eigenvectors inherit this variability ordering from their eigenvalues.
B. Graph Filtering and Frequency Decompositions
Graph filters partition a brain signal into mutually exclusive low-, medium-, and high-frequency components. The low-pass component can also be interpreted as localized averaging over neighboring graph regions.
- Low-pass filtering: The low-pass filter retains the lowest KL graph frequencies and sets the remaining spectral coefficients to zero.Filtering is first performed in the graph-frequency domain and then mapped back to the vertex domain.
- Vertex-domain form: The vertex-domain low-pass signal is xL = HLx, where HL is obtained by transforming the spectral filter through the graph eigenvector basis.This operation isolates low graph-frequency content directly from the original signal.
- Interpretation: Because the low-pass filter is dominated by small Laplacian powers, it acts like localized averaging across nodes connected within successive graph hops.L^0 preserves the signal, while higher powers incorporate interactions through neighboring and intermediate nodes.
- Frequency bands: Low-pass, band-pass, and high-pass filters select the lowest KL, middle KM, and highest n − KL − KM frequencies, respectively.The three frequency ranges are mutually exclusive and collectively exhaustive.
- Signal decomposition: The filtered signals satisfy x = xL + xM + xH, decomposing the original signal into slow, medium, and high variability relative to brain connectivity.The components correspond to low-, medium-, and high-frequency graph content.
III. BRAIN SIGNALS DURING LEARNING
The paper studies graph-based brain signals during two simple motor-learning experiments: a 6-week sequence-production task and a 3-day task. Participants performed visually cued finger sequences during repeated MRI scanning sessions, with regional BOLD activity used to construct functional connectivity signals and networks.
- Experimental frameworks: Two experiments examined motor learning across repeated scanning sessions: one lasted 6 weeks with 20 participants, and the other spanned 3 days with 18 participants.The first experiment included four scanning sessions, while the second included three sessions.
- Tasks: In the first experiment, participants responded with their dominant hand to sequentially presented visual stimuli using a custom response box.The task involved discrete sequence production across repeated scanning sessions.
- Tasks: In the second experiment, participants responded to visually cued sequences using the four fingers of their nondominant hand.The experiment used musical-note-like visual cues and no between-session home training.
- Task controls: Both experiments required prompt and accurate responses to sequences without repetitions or regularities such as trills and runs.Sequence order and trial number were identical across participants.
- Signal construction: The imaging pipeline parcellated the brain into 112 cortical and subcortical regions and estimated regional BOLD time series for functional-connectivity analysis.Functional connectivity matrices were constructed from magnitude-squared spectral coherence between every pair of regions, with nonsignificant 3-day connections set to zero.
IV. BRAIN NETWORK FREQUENCIES
The paper evaluates brain-network frequencies through Laplacian eigenvector variation and weighted zero crossings. Higher-index eigenvectors generally fluctuate more, although weighted zero crossings decline beyond k=100 because of network connectivity and degree structure.
- Laplacian eigenvectors with larger indexes fluctuate more across brain networks in both the 6 week and 3 day experiments.Total variation increases relatively linearly with the eigenvector index.
- Weighted zero crossings increase proportionally with graph-frequency index k for 0 ≤ k ≤ 100.Weighted zero crossings measure the weighted sum of edges linking positive and negative signal values.
- For k greater than 100, higher-frequency eigenvectors exhibit fewer weighted zero crossings.The functional brain networks’ high connectivity and nearly homogeneous degree distribution concentrate large magnitudes at high-degree vertices, reducing global zero crossings.
- The analysis groups eigenvectors into three frequency sets to examine their anatomical magnitude patterns across cortical and subcortical regions.The paper sets KL=40 and KM=32, producing roughly equally sized components.
A. Artificial Functional Brain Networks
The paper introduces an artificial-network framework for mimicking functional brain networks with few parameter inputs. The construction models within- and between-cluster edge weights while allowing variability and links independent of connecting regions.
- The proposed model generates symmetric networks with edge weights between 0 and 1 and no self-loops.It is related to weighted block stochastic models but incorporates individual variance and links independent of the connected regions.
- The construction requires expected average edge weights for within-cluster and intercluster connections.For two clusters, the inputs are μ1, μo, and μ1o.
- Edge weights are generated from uniform distributions around cluster-specific means, with contamination and intercluster variation controlled probabilistically.The same construction is generalized to networks with more regions of interest by specifying region groups and expected connection values.
- The framework ranges from node-specific expected weights to nearly fully random networks, depending on the number of node sets.Useful constructions require prior knowledge about community structure.
B. Spectral Properties of Brain Networks
The paper examines how graph-eigenvector variation differs across visual, motor, other, and cross-module connections. These spectral properties change with task exposure, including lower variation in visual and motor modules as participants become more familiar with the task.
- B. Spectral Properties of Brain Networks: The analysis measures eigenvector variation separately within visual, motor, and other modules, as well as across visual–motor and other cross-module connections.Visual and motor modules are examined separately because of their known associations with motor learning.
- B. Spectral Properties of Brain Networks: The study summarizes regional variation using low-, medium-, and high-frequency eigenvector sets.The visual-module measures are defined over the selected frequency groups, alongside analogous motor and cross-module measures.
- B. Spectral Properties of Brain Networks: At training onset, low-frequency eigenvectors show higher variation on other-module connections than high-frequency eigenvectors.The difference is statistically significant with p < 0.0001.
- B. Spectral Properties of Brain Networks: As participants become more familiar with the assignment, their brain networks display lower variation in visual and motor modules and higher variation in other modules.
C. Discussion
The discussion links graph spectral patterns to brain-network organization, task exposure, and module-specific connectivity. Artificial networks reproduce key spectral properties and help explain how visual and motor connectivity changes during training.
- Graph-frequency structure: Very high graph frequencies showed fewer weighted zero crossings because highly connected, nearly homogeneous networks concentrate magnitude at one high-degree vertex.This network structure produces similar values at other nodes and reduces global zero crossings.
- Graph-frequency structure: High-frequency eigenvectors were similar across datasets, with over 60% overlap in high-magnitude regions concentrated in visual and sensorimotor cortices.Their cross-dataset magnitude correlation was 0.6616, while middle-frequency eigenvectors showed little association.
- Training-related evolution: Spectral properties were associated with exposure level, with an average correlation coefficient of 0.8164 in the six-week experiment.Artificial networks closely reproduced real-network spectral properties using a small parameter set; correlations with training intensity were 0.6436, 0.7187, and 0.8457 for low, medium, and high frequencies.
- Connectivity mechanisms: Module variation depends on within-module, between-module, and other-module edge weights, producing lower low-frequency variation in highly connected modules.Visual and motor modules are highly connected, and visual regions are strongly linked with motor regions.
- Training-related evolution: As task exposure increased, visual and motor connections weakened, while high-frequency variation declined faster for visual than motor regions.Artificial-network analysis attributed this difference to faster weakening within visual regions, stronger motor-to-other connectivity, and relatively stable other-module associations.
- Signal adaptation: Signals with smooth and rapid spatial variations had higher temporal adaptations than moderately varying signals across the experiments.Smooth variations passed the session-level test in 95% of sessions, while rapid variations did so in 65%.
V. FREQUENCY DECOMPOSITION OF BRAIN SIGNALS
The paper decomposes brain signals into low-, middle-, and high-frequency spatial-variation components and visualizes their regional distributions across training. These distributions are compared between the first and last scans in the six-week experiment and across subjects in the three-day experiment.
- Six-week experiment: The six-week experiment maps absolute regional magnitudes for smooth xL, intermediate xM, and rapidly varying xH components at the first and last scan sessions.Only regions exceeding a fixed absolute-magnitude threshold are colored.
- Three-day experiment: The three-day experiment maps xL, xM, and xH averaged across sample points, subjects, and participants.Regions below an absolute-value threshold are left uncolored.
A. Temporal Variation of Graph Frequency Components
Temporal variation is assessed for graph-frequency components at multiple timescales in two motor-learning experiments. The analysis uses averaged component activities and their variance across complete experiments and individual training sessions.
- Measurement approach: Temporal variation is measured over days and minutes by evaluating variance in decomposed graph signals across multiple temporal scales.The method is described for xL, with analogous computations for xM and xH.
- Exposure and experiments: For the six-week dataset, selected session-sequence combinations are ordered by participant exposure to illustrate component variance at two temporal scales.Three combinations are shown from twelve total, and the remaining combinations exhibit similar properties.
B. Discussion
Graph-frequency components show distinct relationships with learning across training. Low- and high-frequency signals are most adaptable, with opposing exposure-dependent correlation trends.
- 0.5841, 0.2852, and 0.6469 are the cross-dataset correlation coefficients for low, medium, and high graph frequencies, respectively.
- Low-frequency components show rapid temporal variation despite an expectation of the smallest variation, supporting their stronger contribution during learning.The result is consistent across examined temporal scales and datasets.
- Learning-rate associations are computed by correlating participants’ learning rates with the norms of low-, medium-, and high-frequency decomposed signals across scanning sessions.Exposure is represented by task day in the 3-day experiment or completed trials in the 6-week experiment.
- ≈0.25 at initial training, low-frequency xL correlation with learning decreases to ≈−0.25 after heavy training.
- ≈−0.2 initially, high-frequency xH correlation with learning increases to ≈0.25 as participants become highly familiar with the sequence.The xH trend is opposite to the xL trend.
- Middle-frequency xM correlation generally increases with training intensity, but its trend is less pronounced than those of xL and xH.
A. Discussion
Graph-frequency analysis links smooth and rapid spatial signal variations to learning adaptability and task familiarity. The framework is presented as broadly applicable to signals defined on networks, with future neuroscience applications beyond the evaluated setting.
- The strongest learning associations come from signals varying smoothly or rapidly over the brain network, rather than primarily from middle frequencies.
- At unfamiliar stages, smooth, spread, and cooperative signals are positively associated with learning, while high-frequency signals are negatively associated.
- As task familiarity increases, smooth cooperative signal distributions become less important and can become detrimental at a later exposure level.The reported transition differs with task difficulty between the 3-day and 6-week experiments.
- Future work could apply graph-frequency analysis to other neuroscience signals and networks, including fMRI measurements on structural networks.
- The paper establishes connections between graph frequency and principal component analysis for functional-connectivity networks while identifying frequency-dependent adaptability during motor learning.