{"id":"d50a8615-14e3-43e1-9290-2e1f49c80e8f","arxiv_id":"2502.00249","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A Hodge-FAST framework applies Hodge decomposition to filtered dynamic EEG connectivity, revealing higher-order interaction differences in MCI patients.","lead":"A new framework combines Hodge decomposition with a global correlation filter to track higher-order brain network interactions in EEG over time. Applied to two Alzheimer's-related memory-impaired groups, it finds group differences in loop and triangle flow components that pairwise analyses miss.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hodge components of non-negative connectivity weights do not support the claimed higher-order flow interpretation; curl can be large for symmetric positive weights, so significant group differences may reflect an artifact of the absolute-value transform.","rationale":"I read the paper's central claim as twofold: (1) the Hodge-FAST framework is the first to capture higher-order dynamic interactions in noisy EEG, and (2) it finds reproducible patient/control differences aligned with prior EEG studies. The reader's weakest assumption was the selection-bias risk from computing the mask M on all participants and then testing the same participants. I agree that this is a legitimate concern, but I find it less decisive than the paper's core interpretational issue: the edge flow being decomposed is non-negative by construction, which undermines the meaning of the Hodge components. Under the null hypothesis of no group difference, the mask derived from pooled data is symmetric in group labels, so permutation-based p-values remain valid; the larger issue is that Cohen's d and the subgroup-specific interpretations may be optimistically biased. More importantly, the Hodge decomposition of a non-negative weight matrix does not produce 'flows' in the directed sense. A uniform positive triangle is assigned entirely to the curl component, demonstrating that the curl component does not indicate rotational behavior. Since the paper uses absolute values or squared differences, all edge weights are non-negative, and the resulting curl/harmonic components can be large even when the underlying process is purely pairwise and sign-symmetric. The reported group differences in these components could therefore be artifacts of the absolute-value pre-processing rather than evidence of higher-order neural interactions. This directly questions the framework's novelty and the interpretation of the experimental results. A signed-correlation rerun or a phase-randomized surrogate test would settle whether the findings are genuine. I therefore keep the reader's CONDITIONAL verdict unchanged: the paper needs this additional validation before the strong claims can be accepted.","tokens_in":9212,"tokens_out":18125,"duration_ms":189403,"concrete_test":"Recompute the analysis using signed instantaneous correlations, e.g., ΘFAST_ij(t) = C_FAST_ij · (x_i(t)-\\bar{x}_i)(x_j(t)-\\bar{x}_j) without the absolute value, keeping the same mask, sparsity, windows, and FDR procedure. If the significant curl/harmonic group differences in Tables I and II disappear or reverse in sign, the reported higher-order findings are an artifact of the absolute-value encoding rather than genuine higher-order interactions. As a complementary check, run the original pipeline on phase-randomized surrogates that preserve power spectra and pairwise correlations but destroy higher-order phase structure; the curl/harmonic differences should vanish if the framework truly detects higher-order interactions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the Hodge-FAST framework reveals higher-order (triangle/loop) dynamic interactions in EEG. The load-bearing condition is that the Hodge components carry the claimed geometric meaning. However, ΘFAST(t) is non-negative by construction: Eq. (2) uses (x_i(t)-x_j(t))^2 and Eq. (3) uses |(x_i-\\bar{x}_i)(x_j-\\bar{x}_j)|, both weighted by C_FAST ≥ 0. The Hodge decomposition is a linear orthogonal projection onto subspaces associated with node gradients (im δ0), triangle boundaries (im δ1^T), and harmonic cycles. For a single triangle with equal positive edge weights, the entire weight vector is orthogonal to im δ0 and therefore is assigned to the curl component, even though there is no directed circulation—only a symmetric co-activation pattern. Thus the 'curl' component measures any edge-weight pattern orthogonal to node potentials, not rotational flow, and the absolute-value transform injects positive weights that can produce large curl/harmonic components solely as an artifact of discarding the sign of correlations. The paper interprets significant curl/harmonic group differences as evidence of higher-order interactions (e.g., 'greater dysregulation in interactions among triangles'), but this interpretation is not supported by the mathematics of the decomposition on non-negative weights. This directly threatens the claimed novelty of capturing higher-order dynamics.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a dynamic functional connectivity (DFC) analysis framework that combines Filtered Average Short-Term (FAST) functional connectivity with Hodge decomposition. It sparsifies the instantaneous connectivity tensor using a mask derived from long-term averaged correlations, then decomposes the resulting edge flow into gradient, curl, and harmonic components. The framework is applied to two independent EEG datasets from MCI patients and controls performing a visual short-term memory binding task, and significant group differences are reported in specific time windows linked to the LPP and P300 ERPs. The paper claims this is the first framework to capture higher-dimensional interaction dynamics at high temporal resolution in noisy EEG.","tokens_in":9508,"tokens_out":5625,"duration_ms":58227,"significance":"If the methodological assumptions are valid, this framework would offer a computationally efficient way to extend DFC analysis to higher-order (triangle- and loop-level) structures, with potential applications in biomarker discovery for prodromal Alzheimer's disease. The use of two independent datasets, non-parametric tests, and effect sizes is a strength, as is the integration of existing tools (FAST FC and Hodge decomposition) into a dynamic setting. However, the geometric interpretation of the Hodge components on non-negative connectivity weights, the data-derived mask selection, the lack of a pairwise baseline, and considerable cross-dataset parameter differences currently prevent the central claims from being accepted at face value.","major_comments":[{"comment":"The Hodge components are interpreted as evidence of directed higher-order flow (e.g., \"rotational behaviors\", \"dysregulation in interactions among triangles\"), but the connectivity tensor is non-negative by construction in both node functions: Eq. (2) uses squared differences and Eq. (3) uses absolute values. For a single triangle with equal positive edge weights, the entire edge vector is orthogonal to im(delta_0) and therefore lies in the curl subspace, despite being a symmetric co-activation pattern with no directed circulation. Consequently, the curl component measures any edge pattern orthogonal to node potentials, not necessarily rotational flow. The authors must either supply an oriented, signed connectivity tensor (e.g., raw correlations with an explicit edge orientation) or reinterpret the components as algebraic projections rather than directed flows; otherwise the claimed higher-order interpretation is unsupported.","section":"Section II-B, Eqs. (2)-(3); Section III"},{"comment":"The sparsification mask M is computed from CFAST, the long-term connectivity averaged over all participants, and the same participants are then used for the group-level Wilcoxon tests on the masked tensors. This is a double-dipping/selection-bias problem: the filter and the statistical test share the same data, which can inflate group differences. The paper reports no split-half, leave-one-out, or permutation-based evaluation of the mask's independence. To support the reproducibility claim, the authors should validate the results using a mask estimated from a training subset (or a null distribution) and then tested on held-out participants.","section":"Section II-B (mask M); Section III"},{"comment":"The abstract and conclusion claim that the framework reveals \"significant temporal differences related to higher-order interactions that a pairwise analysis on its own does not implicate.\" However, no pairwise-only baseline is shown anywhere in the manuscript. Without a comparison against standard pairwise DFC on the same data (e.g., the FAST connectivity tensor without Hodge decomposition, or edge-level statistics), the claimed added value over pairwise methods is unsubstantiated.","section":"Abstract; Section V"},{"comment":"The two datasets are processed with different node functions (local Dirichlet energy for MCI-FAM vs. instantaneous correlation for MCI-SPO), different sparsity thresholds (top 5% vs. top 1%), and different numbers of windows (10 vs. 15). The manuscript argues that the findings are reproducible across independent cohorts, but the differences in results could be driven by these processing choices rather than by a shared neurophysiological effect. The authors should justify the parameter selection or run the identical pipeline on both datasets before claiming reproducibility.","section":"Tables I and II; Section III"},{"comment":"The construction of the simplicial complex is underspecified. The boundary operator B_p is defined generically, but the paper does not state how triangles and higher-order simplices are obtained from the sparsified graph (e.g., clique complex, flag complex, or some other rule), nor the edge orientation convention used in the coboundary maps. These details are needed for reproducibility and for interpreting the geometric meaning of the decomposition.","section":"Section II-B"}],"minor_comments":[{"comment":"The claim of being \"the first framework capable of detecting higher-order dynamic interactions\" should be tempered, given that static Hodge decompositions of brain networks already exist [24,25]. A formulation such as \"to our knowledge, the first application to dynamic EEG at high temporal resolution\" would be more defensible.","section":"Abstract; Section I"},{"comment":"The text describes CFAST as the \"modulus of the Pearson correlation coefficient,\" but the formula in Eq. (1) does not contain an absolute value. This inconsistency should be resolved, as it affects both the value and the sign of the filtering weights.","section":"Eq. (1)"},{"comment":"The window size W is never reported, although the tables list the number of windows. Additionally, Figure 2's caption says \"top 10% of the strongest global connections\" while Table 1 states \"Sparsity: Top 5%\" for MCI-FAM; the two numbers should be reconciled.","section":"Tables I and II; Figure 2"},{"comment":"The definition of CFAST_K as the value \"above which only the top K-th percentile of connections are retained\" is ambiguous. It should be clarified whether K denotes the percentage of retained edges (e.g., K=5 means keeping the top 5%) or a percentile rank.","section":"Section II-B"},{"comment":"In the definition of the boundary operator, the notation uses both k and p inconsistently (\"Bpij = 1 if the k-simplex is part of a (k+1)-simplex\"). Using a consistent index (e.g., p for the dimension of the simplex) would improve readability.","section":"Section II-B"},{"comment":"The description of FDR correction is vague: it is unclear whether it is applied separately for each component, jointly across all windows per component, or across all tests combined. Please specify the exact correction procedure, including the total number of tests.","section":"Section III"}],"recommendation":"major_revision","confidential_remarks":"The core methodological idea is interesting and the application to two datasets is a useful step, but the current form has three problems that are likely to be raised by other reviewers: (i) the geometric interpretation of Hodge components on non-negative, unoriented weights is dubious and needs either a signed formulation or a reinterpretation; (ii) the mask-double-dipping issue undermines the statistical claims; (iii) the absence of a pairwise baseline makes the novelty claim hard to evaluate. These are fixable within the scope of a major revision, but the authors should be prepared to provide substantial reanalysis rather than cosmetic changes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the packaging: Hodge decomposition on top of FAST-filtered dynamic connectivity, applied to two independent MCI EEG datasets with FDR-corrected tests and effect sizes. That combination is not in the cited literature, and the authors deserve credit for a real attempt to move static Hodge brain-network analysis into the time-resolved EEG setting. The two-dataset design and the fact that some effects land in known ERP windows (LPP, P300) make the empirical part worth a look.\n\nThe soft spots, in ascending order of seriousness. First, the missing implementation details: window size W, sparsity threshold selection, edge orientation, triangle construction, and no code. Independent replication is not possible as written. Second, the mask double-dipping: M is built from CFAST averaged over all participants, and the same participants are then used for group comparisons. That is a selection-bias risk with no split-half or cross-validation. A simple held-out mask would fix it. Third, there is no pairwise baseline shown, so the claim that higher-order components reveal what pairwise analysis does not is not actually demonstrated.\n\nNow the one that bothers me most, and it is conceptual. The Hodge components are interpreted as gradient, curl, and harmonic flows, with curl specifically said to capture rotational behavior among triangles. But the edge weights entering the decomposition are non-negative by construction (absolute correlations or squared differences). For a single triangle with equal positive weights, the vector is orthogonal to the gradient space and therefore projects almost entirely onto the curl component, even though there is no directed circulation at all. The decomposition is a linear projection; with non-negative weights, the 'curl' component simply absorbs any pattern that cannot be written as node potentials. So the group differences in curl and harmonic components may be driven by changes in overall co-activation magnitude, not by any meaningful higher-order topological structure. The paper's central interpretive claim—that it detects 'higher-order dynamic interactions'—is not supported by the math as presented. You can make it supportable, but you need to argue why the Hodge projection on positive weights retains geometric meaning, or use signed edge flows where the decomposition has its standard interpretation.\n\nWho is this for? Anyone working on TDA for EEG or dynamic brain networks will want to know about it, mainly as a cautionary example and as a proof that the FAST filter can be scaled to tensors. It deserves serious peer review, because the combination is novel and the flaws are addressable. I would send it to a referee, but with a clear expectation of major revision: fix the interpretation, add split-half validation, show a pairwise baseline, and release the code.\n\nRecommendation: engage with it, but do not let the current interpretation pass as established.","headline":"Nice engineering, but the Hodge 'higher-order flow' reading is mathematically shaky on positive weights, and the group tests are not independent of the mask.","tokens_in":9968,"tokens_out":2176,"would_cite":false,"duration_ms":27075,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces Hodge-FAST, which breaks moment-by-moment EEG connectivity into gradient, curl, and harmonic flows and reports reproducible group differences between Alzheimer's-related MCI patients and controls that pairwise edge…","keywords":["dynamic functional connectivity","Hodge decomposition","higher-order interactions","EEG signal processing","FAST functional connectivity","mild cognitive impairment","topological data analysis","visual short-term memory binding"],"falsifier":"Recompute the mask $M$ using only control participants, or via leave-one-participant-out, and repeat the entire Hodge-FAST pipeline; if the FDR-significant curl and harmonic differences in the 0.8–1.0 s window disappear or change sign, the reported group effects depend on in-sample sparsification. A permutation test that shuffles patient and control labels and reruns the full pipeline, counting how often equally extreme component differences appear, would provide the same check in a single number.","tokens_in":9025,"feed_emoji":"🧠","tokens_out":6492,"duration_ms":59053,"temperature":0.7,"pith_summary":"The paper introduces a framework, Hodge-FAST, that uses Hodge decomposition to split instantaneous EEG connectivity tensors into three orthogonal components—gradient, curl, and harmonic—so that interactions among triangles and loops can be studied at the millisecond scale rather than only in static networks. The authors claim this is the first method to capture such higher-dimensional interactions at high temporal resolution in noisy EEG recordings, and they apply it to two independent cohorts of patients with Alzheimer's-related mild cognitive impairment performing a visual short-term memory binding task. They report significant, FDR-corrected differences between patients and controls concentrated in time windows previously tied to the LPP and P300 event-related potentials, with different components implicated in the familial and sporadic cohorts. If the framework works as claimed, higher-order connectivity dynamics could serve as reproducible biomarkers for early Alzheimer's disease and open EEG recordings to topological data analysis that pairwise graph methods cannot provide.","feed_headline":"Framework finds higher-order EEG interaction differences in MCI","feed_subtitle":"Applied to two independent cohorts, it finds reproducible curl and harmonic flow differences pairwise analysis misses.","key_machinery":"The load-bearing object is the temporal Hodge decomposition of the sparsified FAST connectivity tensor. Boundary operators $B_p$ build the Hodge Laplacian $\\Delta_p = B_p^\\top B_p + B_{p+1}B_{p+1}^\\top$, and the edge flow $\\bar{\\Theta}_{\\mathrm{FAST}}(t)$ is projected by least squares onto the gradient plane and the curl plane, with the harmonic component as the residual. The FAST matrix $C_{\\mathrm{FAST}}$, the average of absolute Pearson correlations over all participants, supplies the mask $M$ that keeps only the top $K$th percentile of global connections, and a sliding window of size $W$ averages the filtered instantaneous tensors before decomposition.","core_discovery":"On the paper's own terms, the central discovery is that after filtering instantaneous connectivity by the long-term global FAST correlation structure and sparsifying to the top percentile of connections, the resulting edge-flow tensor admits a Hodge decomposition whose temporal traces show reproducible patient-control differences in both a familial and a sporadic MCI cohort. The familial cohort shows FDR-significant curl-component increases in the 0.8–1.0 s LPP window in delta and theta bands, while the sporadic cohort shows harmonic-component changes in the same window, and both cohorts show component-specific effects in the P300 range. These differences are invisible to pairwise edge-level analysis alone, which is the paper's core claim: higher-order interactions are not merely present in transient EEG connectivity but carry task-relevant, disease-relevant signal.","pith_inferences":["A key open question the paper leaves implicit is whether the sparsifying mask $M$, computed from the same participants later used in the group tests, biases the comparisons; a split-half or leave-one-out re-estimation of $M$ would settle whether the reported effects survive an in-sample selection procedure.","The same pipeline could be applied to source-reconstructed EEG or MEG to test whether triangle- and loop-level flows reflect cortical generator geometry rather than scalp volume-conduction artefacts.","Because the framework yields per-window gradient, curl, and harmonic traces, these traces could be used as features for single-subject classification or continuous cognitive-state tracking, which the current group-level statistical design does not address.","The dissociation between cohorts (curl in MCI-FAM versus harmonic and gradient in MCI-SPO in the LPP window) suggests a testable hypothesis about distinct network pathologies, but the paper does not yet assess whether this dissociation is statistically robust beyond the reported windows."],"forward_implications":["EEG dynamic functional connectivity can be studied at the level of triangles and loops at millisecond resolution, not just through static whole-network summaries.","The three orthogonal components provide separable readouts—gradient flow for node-level integration, curl flow for triangle-bound rotational dynamics, and harmonic flow for loop-like structures—so a disease effect can be localized to a specific geometric flow.","Reproducible patient-control differences in LPP and P300 windows across two independent cohorts and two recording systems suggest that component-specific higher-order measures are candidates for early Alzheimer's biomarkers.","Because the FAST filter avoids eigen-decomposition, the framework stays computationally light enough for high-resolution temporal analysis of high-density EEG.","The method generalizes beyond EEG to any time-resolved network signal where edge flows over simplicial complexes are meaningful."],"supporting_citations":[{"why":"Supplies the FAST functional connectivity filter and the two MCI datasets used in the analysis.","marker":"[1]"},{"why":"Introduces the Hodge Laplacian for brain networks, the mathematical basis for decomposing edge flows.","marker":"[24]"},{"why":"Introduces the Hodge decomposition of brain networks into gradient, curl, and harmonic components that this framework applies over time.","marker":"[25]"},{"why":"Provides the graph-variate signal analysis framework from which FAST connectivity is derived.","marker":"[26]"},{"why":"Documents abnormal functional hierarchies in the same familial and sporadic MCI cohorts during the binding task, the comparison the results extend.","marker":"[3]"},{"why":"Established that time-resolved dynamics of the binding task carry task-relevant EEG connectivity, the temporal resolution target this method improves upon.","marker":"[9]"}],"fun_headline_variants":["Hodge decomposition exposes MCI-specific EEG couplings pairwise misses","Higher-order EEG links distinguish MCI in two cohorts","New EEG framework finds MCI signals invisible to pairwise analysis","Reproducible higher-order EEG differences mark MCI","Beyond pairwise: Hodge filtering reveals MCI brain dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sparsifying mask $M$ is built from the long-term connectivity averaged over all participants, and the same participants are then used for the patient-control significance tests, so the connection filter and the statistical comparison share the same data with no cross-validation.","fun_headline_variants_meta":{"raw":{"variants":["Hodge decomposition exposes MCI-specific EEG couplings pairwise misses","Higher-order EEG links distinguish MCI in two cohorts","New EEG framework finds MCI signals invisible to pairwise analysis","Reproducible higher-order EEG differences mark MCI","Beyond pairwise: Hodge filtering reveals MCI brain dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1220,"prompt_tokens":848,"completion_tokens":372,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":292}},"tokens_in":464,"tokens_out":372,"duration_ms":4381,"temperature":1.0,"reasoning_tokens":292,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T19:38:50.223191+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the mask $M$ using only control participants, or via leave-one-participant-out, and repeat the entire Hodge-FAST pipeline; if the FDR-significant curl and harmonic differences in the 0.8–1.0 s window disappear or change sign, the reported group effects depend on in-sample sparsification. A permutation test that shuffles patient and control labels and reruns the full pipeline, counting how often equally extreme component differences appear, would provide the same check in a single number.","supporting_citations":[{"cited_title":"Parra, Keith M","cited_arxiv_id":null,"evidence_quote":"Supplies the FAST functional connectivity filter and the two MCI datasets used in the analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Hodge Laplacian for brain networks, the mathematical basis for decomposing edge flows."},{"cited_title":"Hodge-Decomposition of Brain Networks","cited_arxiv_id":null,"evidence_quote":"Introduces the Hodge decomposition of brain networks into gradient, curl, and harmonic components that this framework applies over time."},{"cited_title":"Smith, L","cited_arxiv_id":null,"evidence_quote":"Provides the graph-variate signal analysis framework from which FAST connectivity is derived."},{"cited_title":"M., Starr, J","cited_arxiv_id":null,"evidence_quote":"Documents abnormal functional hierarchies in the same familial and sporadic MCI cohorts during the binding task, the comparison the results extend."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Established that time-resolved dynamics of the binding task carry task-relevant EEG connectivity, the temporal resolution target this method improves upon."}],"review_version":1}