Interpreting Supervised Pairwise Graph Representations: Local Evidence, Confounding, and Provable Post-hoc Correction

Interpreting Supervised Pairwise Graph Representations: Local Evidence, Confounding, and Provable Post-hoc Correction Ilya Brekin1 , Ariela Moldavskaya2,* , Elina Sidorova3 , and Ural Valeev4

arXiv:2610.32611v1 [cs.LG] 23 Oct 2026

1

HSE University, Moscow, Russia; brekinilya@gmail.com Technion – Israel Institute of Technology, Technion City, Haifa 3200003, Israel 3 Lomonosov Moscow State University, Moscow, Russia; elina_sidorova@internet.ru 4 Ufa University of Science and Technology, Ufa, Russia; valeev.ural@mail.ru * Corresponding author. E-mail address: arielam@campus.technion.ac.il 2

Abstract Supervised pairwise graph representations convert pairwise predictive relations into samplespecific graphs, but graph-level accuracy does not by itself show that higher-order topology carries the signal. In a controlled Transformer error-detection task, supervised pairwise graphs outperform attention-as-adjacency (ROC-AUC 0.839 versus 0.559; the comparable no-confidence pair is 0.687 versus 0.559), but their apparent advantage over model confidence disappears after controlling for sequence length, which is almost perfectly recoverable from the pooled hidden state (R2 = 0.9995). On published HCP Emotion ensemble graphs, signed node aggregation already acts as an ensemble of supervised pairwise predictions. Edge magnitude alone is near chance; topology-aware augmentations change ROC-AUC by at most 0.004, with only sign-separated degree showing small positive changes across all tested graph sizes, while a validation-tuned three-layer GCN provides no measurable improvement over the signed node-summary baseline. A Fisher post-hoc corrector satisfies its distribution-agnostic guarantees across the evaluated feature representations and remains stable under reduced labelled budgets; in a near-error-free fMRI calibration regime the bound correctly becomes statistically uninformative. A separate cross-attention analysis identifies a redundant reversal circuit and downstream EOS-timing and copy failures. Higher-order topology should therefore be credited only after local supervised evidence and recoverable nuisance variables have been excluded.

Keywords: supervised pairwise graphs; graph neural networks; attribution; nuisance variables; fMRI; Transformer; post-hoc correction MSC 2020: Primary 68T07; Secondary 68T05, 62H30, 68R10.

1

Introduction

Supervised pairwise graph representations turn high-dimensional observations into sample-specific graphs whose edges are produced by pairwise predictors. This creates an attribution problem: a successful graph model may exploit supervised information already present in individual edges, local node summaries, or nuisance variables rather than higher-order topology. We study this problem in two deliberately different regimes. Setup 1 is a controlled sequencereversal Transformer in which nodes are anonymous latent coordinates and the target is prediction 1

correctness. Setup 2 uses published HCP Emotion ensemble graphs, where nodes are named brain regions and signed edges are held-out pairwise evidence for cognitive state. We adopt an edge → node → higher-order information hierarchy as a standard of evidence and add nuisance-variable controls before accepting a topological interpretation. The team’s work extends beyond applying this criterion. We construct the controlled Transformer benchmark and its supervised pairwise graph representation, introduce a directioninvariant decisiveness edge score for anonymous latent coordinates, implement an independent subject-safe raw-HCP reconstruction path, perform topology and dimensional GCN stress tests on published HCP ensemble graphs, evaluate Fisher post-hoc correction under reduced labelled budgets, and conduct a separate mechanistic analysis of Transformer cross-attention. The resulting picture is asymmetric but coherent. In the Transformer regime, the graph representation is more informative than raw attention, yet its apparent edge over confidence is explained by sequence length. In HCP Emotion, supervised edge sign already carries nearly all measurable discriminative information, leaving little room for the evaluated higher-order descriptors or GCN. The corrector and cross-attention analyses provide two additional positive results independent of the topology claim.

2

Related Work

Synolitic Graph Neural Networks (SGNNs) were introduced as a general framework for constructing sample-specific supervised pairwise graphs by Zaikin et al. [14]. Related synolitic constructions were demonstrated on clinical and physiological data by Krivonosov et al. [6]. Vlasenko et al. specialize the approach to HCP fMRI ensemble graphs [12]. Their published accuracy range for logistic regression on mean incident edge weights is 97.07–99.74%, while the separately reported GNN range on ensemble graphs is 88.00–99.42%. These ranges are not paired task-by-task comparisons. For the Emotion paradigm studied here, however, the published values are directly comparable: 98.45% accuracy for logistic regression on mean incident edge weights versus 88.00% for the GNN, a 10.45 percentage-point gap. Tyukin et al. develop weakly supervised post-hoc correctors with distribution-agnostic, nonasymptotic guarantees [8, 9]. We use this framework only as a downstream decision layer: validity of the correction bound is not treated as evidence that a graph representation is necessary. Attention weights are often inspected as model structure, but Jain and Wallace [4] and Wiegreffe and Pinter [13] caution against treating attention as explanation without an explicit test. Accordingly, attention-as-adjacency is only a direct structural baseline here; the separate mechanistic analysis tests a concrete mirror-position hypothesis in cross-attention. Our attribution logic is related to control-task/selectivity ideas in representation probing (Hewitt and Liang [3]) and nuisance control in neuroimaging decoding (Snoek et al. [7]). We do not claim these diagnostics as new. The contribution is their systematic use in supervised pairwise graphs, where target information can enter through edge construction itself.

3

Model Description

For samples x(n) ∈ Rd with binary labels y (n) , a probabilistic classifier is associated with each unordered coordinate pair (i, j). Its sample-specific output defines a weighted graph G(n) = (V, E, W (n) ). Pairwise classifiers are trained within Setup 1 and inherited from the published Vlasenko construction in Setup 2. For anonymous Transformer coordinates we remove arbitrary class polarity and use the direction-invariant decisiveness score introduced here as an experimental instantiation, not as a proposed replacement for signed pairwise constructions: wij = 2 qij − 2

1 . 2

For named HCP brain regions we preserve the published signed evidence wij = P (c2 | i, j) − P (c1 | i, j). The difference between these edge definitions follows the semantics of the nodes rather than representing two competing graph constructions. In Setup 1, latent coordinates are anonymous: exchanging the arbitrary class orientation of a pairwise classifier should not change the relation represented by an edge, which motivates the direction-invariant decisiveness score. In Setup 2, nodes are named brain regions and the published edge sign explicitly encodes evidence for one cognitive state versus the other; discarding the sign would therefore remove supervised information rather than eliminate an arbitrary orientation. We distinguish individual edge evidence, local node summaries, and higher-order graph processing. The principal signed HCP summary is w̄i =

1 X wij . d − 1 j̸=i

When edges are supervised predictions of the target, this local aggregation already behaves as an ensemble of weak predictors. We operationalize attribution sequentially: first test whether predictive performance is explained by individual edge information or recoverable nuisance variables; next compare against local node summaries; only then test whether topology-aware descriptors or message passing add measurable predictive information. Detailed evaluation procedures are given in Section 5. Node identity is fixed in both settings, so the principal graph learners use identity-preserving flatten readout. Setup 1 uses GATv2 [1]; Setup 2 uses a three-layer GCN backbone [5] for comparability with the published ensemble-graph study. Both internally constructed Transformer graphs and externally supplied HCP ensemble graphs are converted to the same validated downstream graph interface. This separates graph construction from graph learning and allows the same attribution logic to be applied without implying that the two edge generators have the same provenance or semantics.

4

Datasets and Preprocessing

4.1

Controlled Transformer benchmark

Setup 1 contains 10,000 sequence-reversal examples from a Transformer [11] with two encoder and two decoder layers, four attention heads, dmodel = 64, and approximately 236K parameters. Training is stopped at 72.4% exact-match accuracy so that held-out data retain a substantial mixture of correct and erroneous predictions. Mean-pooled encoder hidden states provide the 64-dimensional representation. The fixed split is 60/20/20 for training, calibration, and test; 10% of training is reserved for GNN selection.

4.2

Published HCP ensemble graphs

Setup 2 uses published HCP Emotion ensemble graphs [12] for fear-versus-neutral classification, based on Human Connectome Project data [2, 10]: 581 subjects, right-to-left phase encoding, Glasser parcellation, and up to d = 379 named brain-region nodes. Both states from a subject remain in the same partition: 349/116/116 subjects (698/232/232 graphs) for training/calibration/test. Each published graph is held out with respect to its own subject. Because our partition is imposed after the original held-out folds were generated, a graph assigned to our test split may have been produced by an ensemble that observed subjects we place in training. This can inflate absolute AUC values, and we therefore do not interpret them as independent estimates of decoding accuracy. Our comparisons are paired: every model consumes identical 3

fixed graphs under an identical partition. They are nevertheless conditional on this precomputed bundle and do not provide an independent assessment of the full graph-generation and decoding pipeline.

4.3

Independent raw-HCP reconstruction

In parallel, we implemented an independent raw-HCP path with fixed-length preprocessing, subject-safe splitting, out-of-fold pairwise graph generation, signed edge preservation, and a common PyTorch Geometric interface. It is used for preprocessing audits rather than the principal comparative HCP results; the recording-length audit is reported in Section 6.3.1, and evaluation procedures are described in Section 5.

5

Experimental Protocol

5.1

Metrics and baseline comparisons

Held-out ROC-AUC is the primary discrimination metric; exact-match accuracy is used only to characterize the upstream Transformer and the L = 20 error regime. Paired comparisons use the same held-out samples or fixed graph realizations. Setup 1 compares confidence, raw hidden states, flattened graph features, raw-attention graphs, and synolitic GNNs. Setup 2 compares signed node-summary logistic regression, topology-aware augmentations, and a validation-tuned GCN.

5.2

Data separation and calibration

In Setup 1, pairwise classifiers are fitted on training data only; 10% of training is reserved for GNN early stopping and model selection, and the calibration split is reserved for the Fisher corrector. In Setup 2, GCN hyperparameters are selected separately for each d and compared with LogReg on the same fixed graph realizations. The distinction between the published graph bundle and the independent raw-HCP reconstruction is maintained throughout.

5.3

Transformer nuisance controls

Exact-length strata are used for the nuisance-controlled GNN-versus-confidence comparison; uncertainty is obtained by bootstrap resampling preserving the strata. The bootstrap resample count for this ROC-AUC analysis was not preserved in the available experimental record. It is distinct from the mirror-fidelity bootstrap described below.

5.4

HCP region sampling and GCN selection

HCP topology probes are evaluated at d = 34, 50, 100 over three random region subsets. The GCN ladder uses d = 34, 50, 100, 200 over five random subsets and d = 379 once on the complete graph; the two robustness experiments are not numerically compared as if they shared identical draws. Signed and magnitude-only edge information are evaluated separately. For each graph dimensionality, the three-layer GCN was tuned separately on the validation split. Table 1 reports the searched and validation-selected hidden dimensions. Configuration selection uses a single region draw while evaluation uses five, so selection is not strictly nested. This may favour the tuned GCN over the untuned logistic baseline and limits interpretation of the reported differences.

4

Table 1: Validation-selected GCN dimensions. d

h searched

h′ searched

34 50 100 200

8,12,16 12,16,24,32 16,24,32,48 24,32,48

8,12 8,16 8,16,24 8,16,24

5.5

Corrector and mechanistic evaluation

h chosen

h′ chosen

8 12 48 32

8 8 16 16

The Fisher corrector is evaluated across feature representations and labelled correction budgets, including 25 examples. Cross-attention mirror fidelity is compared within length strata using a stratified bootstrap with 2,000 resamples. Error analysis separately tests intact mirror alignment and, at L = 20, EOS timing versus conditional content-copy failure.

6

Results

6.1

Pairwise construction versus attention-as-adjacency

The raw-attention graph is a weak error detector (ROC-AUC 0.559), whereas the strongest synolitic configuration reaches 0.839 (Table 2). Because the latter also receives model confidence, the more comparable no-confidence synolitic model reaches 0.687 versus 0.559 for raw attention. However, 0.687 is essentially identical to logistic regression on the original hidden representation (0.686), and logistic regression on flattened graph features reaches 0.699. Thus supervised pairwise construction is more informative than direct attention-as-adjacency, but these comparisons do not establish a gain over the underlying hidden representation or attribute the full 0.839–0.559 gap solely to topology. Table 2: Setup 1 principal test ROC-AUC results (Ntest = 2000; base exact-match accuracy 72.4%). Method Softmax confidence LogReg on raw hidden states LogReg on flat graph features LogReg on graph features + confidence GNN on raw-attention graphs No-confidence synolitic GNN Synolitic GNN ensemble

6.1.1

Confidence

ROC-AUC

— No No Yes No No Yes

0.799 0.686 0.699 0.841 0.559 0.687 0.839

Readout and architecture ablations

The evaluated configurations do not form a complete factorial design: a mean-pool GNN without confidence was not evaluated. Table 3 therefore lists only configurations actually run.

5

Table 3: Readout and architecture ablations in Setup 1. Model / representation

Readout

LogReg on flattened graph features LogReg on mean/max-pooled graph features No-confidence synolitic GNN Synolitic GNN ensemble Mean-pool GNN LogReg on graph features + confidence

6.1.2

Confidence

ROC-AUC

Flatten Mean/max pooling

No No

0.699 0.634

Flatten Flatten Mean pooling Flatten

No Yes Yes Yes

0.687 0.839 0.834 0.841

Exploratory graph-statistic diagnostics

We compare several graph-derived summary statistics between correct and erroneous Transformer predictions. The observed differences motivated the stronger attribution controls used in the main analysis; they are not interpreted as evidence that higher-order topology causes the effect. Table 4: Correct versus erroneous Transformer predictions. Statistic

Correct

Error

Difference

p-value

Mean strength Max strength Mean closeness Mean betweenness Mean edge weight Number of decisive edges

4.376 4.131 30.14 25.97 0.242 0.199 0.0103 0.0084 0.778 0.762 168.8 105.3

+0.246 +4.17 +0.043 +0.0019 +0.016 +63.4

6.7 × 10−15 3.0 × 10−18 7.3 × 10−20 3.6 × 10−21 6.7 × 10−15 1.2 × 10−10

6.1.3

Node-feature ablation

To test whether the Setup 1 signal can be attributed to any single graph-derived node statistic, we fit logistic regression to each node feature separately and compare it with the original hidden representation. No individual graph-derived statistic (degree, strength, closeness, or betweenness) exceeds the raw-hidden-state baseline. The value coordinate is not a graph-derived statistic and is reported separately for completeness. The ablation therefore does not identify a single graphderived node statistic that explains the Setup 1 result; it supports the stronger nuisance-controlled and representation-level comparisons in Section 6.2. Table 5: Setup 1 node-feature ablation. Feature

Test ROC-AUC

value (raw hidden-state coordinate) degree strength closeness betweenness all five features (flat) raw hidden states (baseline)

0.714 0.670 0.674 0.649 0.629 0.696 0.686

6

6.2

Sequence length as a Transformer nuisance variable

In Setup 1, adding sequence length to confidence raises ROC-AUC to 0.840, essentially matching the pooled synolitic GNN result of 0.839 (Table 6). Length is almost perfectly recoverable from the pooled hidden state (ridge R2 = 0.9995). Within exact-length strata, the ranking reverses: GNN 0.806 versus confidence 0.817, with bootstrap 95% CI [−0.018, −0.003]. Thus the apparent pooled advantage over confidence does not survive nuisance control. Together with the no-confidence comparisons in Section 6.1, these results provide no clear evidence of a topologyspecific advantage over simpler representations; the no-confidence GNN may nevertheless be suboptimal, so this is not an impossibility claim. Table 6: Pooled Transformer comparisons. Predictor

Test ROC-AUC

Length only, LogReg Softmax confidence Confidence + length, LogReg Synolitic GNN ensemble

6.3 6.3.1

0.628 0.799 0.840 0.839

Local evidence and topology in HCP Raw-series record-length audit Table 7: ROC-AUC of trivial structural predictors.

Paradigm

Duration

No. blocks

After cropping

Emotion Social Motor Language

0.941 0.774 0.500 0.848

0.500 0.528 0.500 1.000

0.500 0.500 0.500 1.000 (excluded)

Emotion and Social lose duration-related predictability after deterministic, class-independent fixed-length cropping; Motor shows no measurable confound; Language remains perfectly predictable from block count and is excluded. This audit applies only to the reconstructed raw-HCP path. 6.3.2

Signed node summaries and topology-aware augmentations

The HCP result has a different mechanism. Each edge is signed supervised evidence, wij = P (c2 | i, j) − P (c1 | i, j), so signed node aggregation already ensembles pairwise predictions. Edge magnitude alone remains near chance (ROC-AUC ≈ 0.47–0.54). Across topology probes (Table 8), changes are concentrated near zero: the largest individual positive deviation is +0.004 at d = 50, while sign-separated degree is the only augmentation positive at all tested sizes (+0.003, +0.003, +0.001 for d = 34, 50, 100). 6.3.3

Dimensional scaling

Figure 1 compares the signed node-summary baseline with the validation-tuned GCN across the sampled graph sizes described in Section 5.4. For this construction and the tested GCN family, a validation-tuned three-layer GCN provides no measurable improvement over the signed node-summary baseline. LogReg/GCN ROC-AUC pairs are 0.924/0.911, 0.964/0.949, 0.982/0.978, 0.993/0.994, and 0.999/0.996 for 7

Table 8: ∆ROC-AUC relative to signed mean-incident-weight LogReg; mean of three random region draws. Added features

d = 34

d = 50

d = 100

Magnitude centralities, dense Magnitude centralities, top 20% Magnitude centralities, top 10% Sign-aware centralities Sign-separated degree

−0.000 −0.002 −0.001 −0.002 +0.000 −0.002 −0.002 +0.004 +0.003 +0.003

−0.000 −0.000 +0.001 −0.000 +0.001

d = 34, 50, 100, 200, 379. At small d, where headroom remains, the local baseline is slightly stronger; at d = 200 and 379 both models are near ceiling. Configuration selection and evaluation share region draws, so the reported GCN values may carry a small optimistic bias. The non-nested selection described in Section 5.4 may favour the tuned GCN over the untuned logistic baseline; these comparisons are therefore interpreted conditional on the evaluated configurations. The local-aggregation mechanism helps explain the specific 10.45 percentage-point Emotion gap reported by Vlasenko et al. [12]: the local statistic already aggregates supervised target evidence, leaving little measurable additional information for the tested higher-order graph processing.

Figure 1: Setup 2 dimensionality ladder. For d ∈ {34, 50, 100, 200}, points summarize repeated random region subsets; the complete-graph result at d = 379 is reported in the text and is not shown in this figure.

6.4

Provable post-hoc correction

All twenty empirical-versus-theoretical comparisons for the Transformer corrector satisfy their distribution-agnostic lower bounds. With the synolitic GNN score as Φ, empirical error rejection is 0.821 against a lower bound of 0.722. Confidence reaches 0.826 and the graph-summary Fisher representation 0.832, showing that the guarantee is not specific to the graph score. Performance remains stable with as few as 25 labelled correction examples; increasing calibration size mainly tightens the certificate (Figure 2). The near-ceiling fMRI decoder illustrates the opposite regime: test accuracy is 0.9784, but the calibration split contains only one error among 232 graphs (accuracy 0.9957). With M− = 1 and ∆ = 0.8, the theoretical rejection bound is ρ ≈ 0.025. The guarantee becomes nearly vacuous because the calibration sample contains too little error evidence, correctly refusing to assert strong selective reliability.

8

Table 9: Setup 1 Fisher corrector at ∆ = 0.8. Feature Φ

Accept bound

Accept empirical

0.650 0.512 0.350 0.279 0.151

0.704 0.559 0.436 0.332 0.269

GNN score Confidence Raw X (Fisher) Graph topology (Fisher) Graph summary (Fisher)

6.4.1

Reject bound Reject empirical 0.722 0.722 0.692 0.692 0.692

0.821 0.826 0.792 0.792 0.832

Dependence on labelled correction budget

We evaluate how empirical correction performance and its theoretical certificate change as the labelled calibration budget is reduced.

Figure 2: Setup 1 corrector versus labelled correction budget. Empirical performance is stable from 25 examples; theoretical bounds tighten with larger calibration samples.

6.4.2

Accept/reject trade-off

We additionally vary the confidence parameter ∆ to characterize the empirical accept/reject trade-off and the corresponding distribution-agnostic lower bounds.

Figure 3: Setup 1 corrector trade-off across confidence parameter ∆. Empirical accept/reject rates consistently satisfy the corresponding lower bounds.

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6.5

Cross-attention mechanism

An initial analysis correlated synolitic node centrality with attention-head specialization and found strong Spearman correlations (|r| up to 0.935). These associations do not survive stratification by sequence length: attention entropy is bounded by log L and rises with length, while node centrality derived from X falls with length, so the two can covary mechanically. This motivated the direct cross-attention analysis reported below. Direct cross-attention analysis yields a separate positive result. All four layer-2 heads preferentially attend to the mirror source position (hit rates 0.79–0.86 versus chance near 0.07), while layer-1 heads show no comparable specialization. Within fixed length, correct predictions have greater mirror fidelity than errors: sample-weighted gap +0.091, stratified-bootstrap 95% CI [+0.079, +0.104]. Yet 140 of 554 errors occur despite intact mirror alignment, showing that the reversal circuit can remain intact while decoding fails downstream. EOS-timing errors account for approximately 50–90% of errors depending on sequence length. At L = 20, per-token content accuracy drops from 0.987 to 0.941; combining the measured termination failure with conditional copy success reproduces the observed exact-match accuracy 0.377. This decomposition separates the redundant reversal circuit from downstream EOS-timing and content-copy failures.

7

Discussion

The two regimes reveal different limits of graph-based interpretation. In the Transformer, graphderived scores inherit a recoverable nuisance variable; in HCP, supervised edge sign already exposes the decisive target information before higher-order processing begins. Thus graph construction can be useful without implying that topology is the source of the predictive gain. The HCP mechanism is particularly informative. Magnitude-only information is near chance, so centralities built mainly from magnitude operate on a weak channel. Signed node aggregation, by contrast, ensembles supervised edge predictions. The small, consistently positive signseparated-degree effect suggests at most a weak local sign-structure contribution, not a broad higher-order topology gain. These conclusions are deliberately bounded. We evaluate one Transformer task, one HCP contrast, one published pairwise edge construction, and one tuned GCN family; other messagepassing architectures or graph-generation schemes may behave differently. The raw-HCP audit and published Vlasenko bundle are distinct data paths, so no preprocessing guarantee is transferred between them. The practical implication is an attribution discipline: preserve strong local baselines and plausible nuisance variables before assigning credit to graph topology. The Fisher corrector and mechanistic attention analysis are separate positive results, not evidence for the topology claim.

8

Conclusion

Supervised pairwise graphs can organize useful predictive information, but graph-level accuracy alone is insufficient evidence for higher-order topology. In the Transformer setting, pairwise construction outperforms attention-as-adjacency while its apparent advantage over confidence is explained by sequence length. In HCP Emotion, signed local aggregation already captures nearly all measurable signal available to the tested topology-aware summaries and GCN. The same study also shows that the representation can support provable post-hoc correction and that direct cross-attention analysis reveals a robust reversal circuit with downstream failure modes. The central methodological lesson is to credit topology only after simpler local evidence and recoverable nuisance variables have been excluded.

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Author Contributions All authors contributed to the common experimental framework and interpretation. Ilya Brekin designed and trained the controlled sequence-reversal Transformer and developed shared reproducibility and data-interface infrastructure. He led the sequence-length nuisance analysis and the mechanistic cross-attention/error decomposition, and contributed to the HCP recording-length and topology diagnostics. Ariela Moldavskaya developed the supervised pairwise graph construction for the Transformer, including the direction-invariant decisiveness representation, and implemented graphgeneration and contract-validation infrastructure. She also implemented the raw-attention graph baseline and developed the independent subject-safe raw-HCP path from time series through out-of-fold pairwise graph generation to topology-aware PyTorch Geometric objects. Elina Sidorova developed and evaluated the graph-neural-network models in both settings, including GATv2 for Transformer graphs and the identity-preserving three-layer GCN for HCP ensemble graphs. She performed validation-based architecture selection and the repeatedregion dimensional-scaling experiments and contributed to comparison with simpler node-level representations. Ural Valeev developed and implemented the Fisher post-hoc correction component and its calibration procedure. He carried out the theoretical and empirical evaluation of the distributionagnostic guarantees across feature representations, operating thresholds, and reduced labelled correction budgets. The authors jointly designed the sequence of controls separating edge-level, node-level, nuisance-derived, and higher-order information, interpreted the combined results, and contributed to preparation and revision of the manuscript.

Acknowledgments We thank Professor Alexey Zaikin (University College London) for formulating the research problem. We also thank the organizers and mentors of SMILES Summer School 2026 for providing the project framework and research environment. Data were provided in part by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Uğurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.

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