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REVIEW 5 major objections 5 minor 27 references

Topological Time Frequency Analysis of Functional Brain Signals

T0 review · 5 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A short-time Fourier transform of two scalar topological summaries extracted from sliding-window brain networks yields a topological spectrogram in which 0D topology is stable and 1D topology is non-stationary, and this spectrogram…

desk verdict A new but unvalidated pipeline—STFT of scalar birth/death means—with internal inconsistencies and a simulation that tests the wrong thing. read the letter →

arxiv 2502.05814 v2 pith:ULRDIQKS submitted 2025-02-09 q-bio.NC physics.data-an

Moo K. Chung , Aaron F. Struck This is my paper · ORCID
classification q-bio.NCphysics.data-an MSC 55N31
keywords persistenthomologytime-frequencyanalysisgraphfiltrationfunctionalbrainnetworkstopologicalspectrogramresting-statefMRIshort-timeFouriertransformbirth-deathdecomposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to turn the topology of a moving brain network into a time-frequency signal. For each sliding-window correlation network, it decomposes edge weights into 0D births and 1D deaths, averages them into two scalar time series $b(t)$ and $d(t)$, and runs a short-time Fourier transform to make a topological spectrogram. On 400 resting-state fMRI subjects, the 0D spectrogram is stable and low-frequency while the 1D spectrogram fluctuates more, and both show weak but statistically significant correlations with fluid and crystallized intelligence. The motivation is that this bypasses the need to align frames across subjects, since resting-state dynamics are asynchronous. Simulation results argue that topological clustering with the Wasserstein distance produces far fewer false positives than Euclidean k-means or hierarchical clustering on topologically equivalent shapes.

What carries the argument

The load-bearing object is the graph-filtration birth-death decomposition $W = B \cup D$ with $B \cap D = \emptyset$: for a weighted graph, sorting and deleting edge weights yields nested subgraphs, and each edge is either the birth value $b_{(i)}$ of a 0D connected component (together forming the maximum spanning tree) or the death value $d_{(i)}$ of a 1D loop. Taking per-time-point averages $b(t)$ and $d(t)$ embeds each dynamic network as a point in a 2D topological plane, and the short-time Fourier transform of those scalar series is the spectral engine that produces the topological spectrogram and the intelligence correlations.

What would settle it

Randomly permute the edges within each time point's maximum spanning tree (and separately within the non-tree edges) so that the average birth and death values are preserved while the temporal identity of each edge is destroyed, then recompute the spectrogram and its correlations with intelligence; if the correlations survive, the scalar summaries carry only average edge weight rather than topology.

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Extended reading notes

Core claim

The paper claims that every weighted graph carries a unique topological decomposition of its edge weights: the sorted edge-deletion filtration splits the edges into a birth set $B$ (the edges of the maximum spanning tree, which create 0D connected components) and a death set $D$ (the remaining edges, which close and kill 1D loops). Averaging these values at each time point produces two scalar time series, $b(t)$ and $d(t)$, and the authors' central move is to pass these through the short-time Fourier transform to obtain a topological spectrogram. On resting-state fMRI data, the 0D spectrogram shows stable power concentrated below 0.03 Hz, interpreted as a persistent tree-like backbone, while the 1D spectrogram includes more variable, higher-frequency content, interpreted as cycles forming and dissolving. The spectrogram's power correlates weakly but statistically significantly with fluid and crystallized intelligence, with 1D topology showing correlations over a broader frequency range than 0D; simulation results further claim that Wasserstein-based topological clustering produces fewer false positives than Euclidean k-means or hierarchical clustering on topologically equivalent patterns.

Load-bearing premise

The load-bearing premise is that averaging all 0D birth values and all 1D death values into two scalar time series preserves the topologically informative content of the brain network; if that averaging washes out the structure that matters, the spectrogram and its reported correlations are not really about topology.

Editorial extensions

If this is right

  • If the central claim is correct, dynamic resting-state networks can be studied without frame-wise alignment across subjects, because the topological summaries depend on within-subject evolution rather than simultaneous instants.
  • The stable low-frequency 0D band would imply that a maximum-spanning-tree backbone persists over time and across individuals, serving as a common scaffold for faster cyclic dynamics.
  • The 1D spectrogram's broader frequency range would imply that loops form and dissolve on fast timescales, and the correlation pattern would connect recurrent network activity to fluid intelligence.
  • In the simulation results, the topological Wasserstein distance would be the better tool for clustering topologically defined shapes, cutting false positives by roughly half relative to Euclidean k-means and hierarchical clustering.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable consequence the paper leaves implicit: if $b(t)$ and $d(t)$ are truly sufficient, then replacing them with quantiles or full persistence barcodes should preserve or sharpen the intelligence correlations rather than destroy them.
  • A permutation control that shuffles maximum-spanning-tree edge weights across time while keeping their mean fixed would reveal whether the spectrogram reflects topology or simply time-varying average connectivity; the paper does not report such a control.
  • The same pipeline could be moved to EEG or MEG, where temporal alignment across subjects is even harder, to see whether the stable-0D/unstable-1D split reproduces; the authors name epilepsy and Alzheimer's disease as future clinical targets.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper proposes a framework that combines graph-filtration persistent homology with the short-time Fourier transform (STFT) to analyze dynamic functional brain connectivity. For each sliding-window correlation matrix, the authors compute the average 0D birth value b(t) and the average 1D death value d(t), then compute STFT spectrograms of these two scalar time series. The method is applied to resting-state fMRI data from 400 HCP participants, with reported findings that 0D topology is stable across time while 1D topology is non-stationary, and that the topological spectrogram shows weak correlations with fluid and crystallized intelligence. A simulation on circular point clouds is used to argue that topological Wasserstein-distance clustering produces fewer false positives than k-means or hierarchical clustering.

Significance. If the central claims were fully supported, the proposal would be a useful addition to dynamic functional connectivity analysis: it would convert graph filtrations into a time-frequency representation and suggest that 0D/1D topological summaries carry cognitive information. The paper builds on previously published graph-filtration results [1,17,19], and the genuinely new element is the STFT of scalar birth/death summaries plus the HCP application. The authors are also correct that frame-wise comparisons of resting-state dynamics are statistically problematic. However, as written, the validation does not exercise the proposed pipeline, and several internal inconsistencies need to be resolved before the claims can be accepted. No code or data availability statement is provided, which further limits reproducibility.

major comments (5)
  1. The central validation does not test the proposed pipeline. Equations (3)-(4) reduce the persistence barcode to two scalar means b(t) and d(t); with p=379, d(t) is the mean of 71,253 death values per time point, and this averaging removes which edges form the MST and which non-MST edges create which cycles. The simulation in Section II.B.3, however, uses the 2-Wasserstein distance on the full sorted birth and death lists of static circular point clouds, preserving exactly the distributional information that b(t) and d(t) discard. No experiment demonstrates that the scalar summaries, or their STFT spectrograms, separate topologically different dynamic networks or track topological changes over time. The reported 0D-stable/1D-nonstationary pattern and the intelligence correlations in Figures 5-6 could therefore be reproduced by any slowly varying measure of mean edge weight; the paper needs a control analysis (e.g., comparing against mean correlation dynamics or random-edge-shuffle nulls) and a simulation that applies Equations (3)-(4) to time-varying networks with known topology.
  2. The data dimension is ambiguous. Section II.A states that AAL parcellation yields 116 time series, but Section III reports 1173 time-varying correlation matrices of size 379 x 379 for each subject. Since q0 and q1 in Equation (3) depend on p, the effective number of nodes must be stated consistently. Please correct the parcellation description or the matrix size, and report the actual p used in the analysis.
  3. The simulation result is internally inconsistent: the text first reports that topological clustering showed 37% fewer false positives than k-means/hierarchical clustering, then, a few lines later, claims a 47-50% reduction for the same comparison. With only five networks per pattern, the accuracy values carry large uncertainty; please reconcile the numbers, report per-condition results, and provide confidence intervals or resampling-based intervals.
  4. The significance threshold is mis-stated and the multiple-comparison problem is not addressed. For n=374, |r|=0.2 gives t≈3.94 on 372 degrees of freedom, whose one-sided p-value is far below 0.0018. Conversely, p=0.0018 corresponds to |r|≈0.15. Moreover, correlations are evaluated over many frequency bins and two homology dimensions with no correction for multiple testing; the phrase 'statistically significant but weak' in the captions of Figures 5 and 6 needs a defensible multiple-testing procedure (e.g., FDR or permutation over subject labels).
  5. The claim that 0D topology is stable while 1D topology is non-stationary is supported only by a single-subject spectrogram and an unsupported sentence that the pattern appears in all 400 subjects. A claim about stationarity should be quantified across subjects, e.g., with a test for time-variation of the PSD in each frequency band, or at least group-level summary statistics of b(t) and d(t). Without such evidence, the abstract's statement that the method identifies 0D and 1D features that are robust to noise and temporal misalignments is not established.
minor comments (5)
  1. The PSD equations PB(τ,ω)=20 log10 B(τ,ω) and PD(τ,ω)=20 log10 D(τ,ω) use the complex STFT directly; they should use |B| and |D| (or squared magnitudes) before taking logarithms.
  2. The abstract says features are identified 'in the signal's time-frequency domain'; the actual pipeline computes STFT of topological summaries, not persistent homology on the time-frequency representation. Please rephrase to avoid ambiguity.
  3. The STFT parameters are underspecified: 'window size equivalent to 10% of total signal duration' needs the number of samples and the overlap in samples; 'FFT length set to the next power of two greater than the window size' should be stated numerically.
  4. The number of subjects for the intelligence correlations differs between Figure 5 (400 subjects) and Figure 6 (374 subjects), but the text says 'in 374 subjects' for both analyses; please clarify the exact sample size for each correlation.
  5. There are several typographical errors: 'weighted weighted network' in the Fig. 1 caption, 'brain netework' in Section III, and 'topological data dnalysis' in reference [14].

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the spectrogram is defined from the birth/death summaries, and the intelligence correlations are descriptive rather than fitted predictions.

full rationale

The paper's derivation chain is not circular in the sense that matters here. The topological summary b(t) = (1/q0)Σb_t(i) and d(t) = (1/q1)Σd_t(i) is defined directly from the graph-filtration birth and death sets, and the topological spectrogram is then defined as the short-time Fourier transform of these two scalar time series. The stationarity/non-stationarity statements and the intelligence correlations are empirical descriptions of those computed summaries, not predictions derived from inputs that already contain the same outputs. No parameter is fitted to the cognitive measures, and no quantity is defined in terms of the claimed conclusion. The many self-citations ([1], [15]–[17], [19]) supply the graph-filtration birth/death decomposition and the topological embedding coordinates; these are published, parameter-free mathematical constructions whose assumptions do not include the paper's empirical findings, so they count as independent support rather than circular import. The main weaknesses are non-circular: the simulation validates the 2-Wasserstein distance on full sorted birth/death lists rather than the scalar-summary STFT pipeline, and the manuscript contains an unexplained discrepancy between the stated 116 AAL parcels and the reported 379×379 correlation matrices. These affect evidential weight and reproducibility, but they do not make the central claim equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The 'topological spectrogram' is a new representation built from existing graph filtration and STFT components, not an invented entity. The main load-bearing assumptions are the sufficiency of the averaged birth and death summaries and the validity of sliding-window correlation networks.

free parameters (5)
  • sliding window size = 28 TRs (20.16 s)
    Used to compute dynamic correlation matrices; changing this window alters b(t), d(t), and all downstream spectrograms and correlations.
  • STFT window size = 10% of total signal duration (~117 samples)
    Controls time-frequency resolution; no sensitivity analysis is provided for this choice.
  • FFT length = 128 (next power of two greater than window size)
    Determines frequency binning in the spectrogram; exact choice is not justified.
  • window overlap = window size minus one sample
    Affects smoothness and redundancy of the STFT; chosen without sensitivity analysis.
  • significance threshold = |r| = 0.2
    Used to declare correlations with intelligence significant; this is an arbitrary threshold and is not corrected for multiple comparisons across frequency bins.
assumptions (3)
  • standard math Graph filtration edge-weight decomposition: W = B cup D with |B| = p-1 and |D| = (p-1)(p-2)/2.
    Stated in Section II.B.1 and cited to [1],[13],[17]; this is a known property of graph filtrations.
  • domain assumption Average cumulative birth and death values b(t) and d(t) are sufficient topological summaries of the time-varying network.
    Defined in Section II.B.2; the paper provides no proof that averaging over all maximum spanning tree edges and all non-tree edges preserves discriminative topological information.
  • domain assumption Sliding-window Pearson correlation captures meaningful dynamic functional connectivity.
    Used throughout Section III; the paper cites [22] on spurious fluctuations in dynamic functional connectivity but does not address those concerns in its own analysis.

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Cite this review

Pith. "Pith review of Topological Time Frequency Analysis of Functional Brain Signals." pith.science (2026). https://pith.science/paper/ULRDIQKS

@misc{pith2026250205814,
  author       = {Pith},
  title        = {Pith review of: Topological Time Frequency Analysis of Functional Brain Signals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ULRDIQKS}},
  note         = {Machine review of arXiv:2502.05814}
}
read the original abstract

We present a novel topological framework for analyzing functional brain signals using time-frequency analysis. By integrating persistent homology with time-frequency representations, we capture multi-scale topological features that characterize the dynamic behavior of brain activity. This approach identifies 0D (connected components) and 1D (loops) topological structures in the signal's time-frequency domain, enabling robust extraction of features invariant to noise and temporal misalignments. The proposed method is demonstrated on resting-state functional magnetic resonance imaging (fMRI) data, showcasing its ability to discern critical topological patterns and provide insights into functional connectivity. This topological approach opens new avenues for analyzing complex brain signals, offering potential applications in neuroscience and clinical diagnostics.

Figures

Figures reproduced from arXiv: 2502.05814 by the authors.

Figure 1
Figure 1. The overall schematic of topological decomposition followed by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Average cumulative birth value b(t) (red) and death value d(t) (blue) for a representative subject. 0D homology, which is determined by the maximum spanning tree, remains very stable. In contrast, 1D homology fluctuates rapidly relative to 0D homology, reflecting cyclical changes in the rs-fMRI network over time. Their spectrograms will reveal that 0D homology remains stationary, whereas 1D homology is non-stationar… view at source ↗
Figure 3
Figure 3. The topological spectrogram for a representative subject. The [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Top: simulation study on topological equivalence. The correct [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]
Figure 5
Figure 5. Figure 5: Correlation between the spectrogram of 0D and 1D topologies [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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