Multimodal branched transport infers anatomically aligned brain reaction maps

Cristian Mendico

arxiv: 2603.19761 · v2 · submitted 2026-03-20 · 🧮 math.OC · q-bio.NC· q-bio.QM

Multimodal branched transport infers anatomically aligned brain reaction maps

Cristian Mendico This is my paper

Pith reviewed 2026-05-15 08:59 UTC · model grok-4.3

classification 🧮 math.OC q-bio.NCq-bio.QM

keywords reactiontransportbranchedalignedanatomicallyarchitecturebraincosts

0 comments

The pith

Multimodal brain data fed into branched transport variational optimization produces anatomically aligned reaction maps whose routing backbones change with anisotropy and show geometry-dynamics trade-offs.

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

The authors take three kinds of brain measurements: task-related blood-oxygen signals, source-reconstructed electrical activity, and white-matter tract directions. They turn these into estimates of where stimulation enters and where reactions occur, then define a cost that penalizes travel through brain tissue according to its measured anisotropy. Instead of ordinary shortest-path transport, they use a branched-transport model that rewards signals merging into shared routes before splitting again. A variational solver finds the branching architecture that best explains the observed data. They next attach a simple stochastic process on the resulting graph and measure how well the map can be controlled dynamically versus how geometrically efficient it is. The main reported outcomes are that the multimodal version aligns better with known anatomy than single-modality versions, that anisotropy visibly alters the main highways compared with isotropic costs, and that optimizing for both geometry and controllability produces rank reversals in which branches matter most.

Core claim

We show that multimodal data generate anatomically aligned brain reaction maps, that anisotropic costs qualitatively reshape routing backbones relative to isotropic baselines, and that hybrid geometric--dynamical optimisation reveals non-trivial rank reversals across branching regimes.

Load-bearing premise

That the branched-transport variational problem recovers the true underlying propagation architecture rather than an artifact of the chosen cost function and data fusion rules.

Figures

Figures reproduced from arXiv: 2603.19761 by Cristian Mendico.

**Figure 1.** Figure 1: Task-evoked fMRI BOLD pipeline. Simulated blood-oxygen-level-dependent data analysed with a general linear model. Left, stimulus and reaction regressors after convolution with the haemodynamic response. Top right, representative blood-oxygen-level-dependent signals in selected regions of interest. Bottom left, region-wise contrast statistics showing strong task selectivity. Bottom right, blood-oxygen-level… view at source ↗

**Figure 2.** Figure 2: Source-reconstructed EEG/MEG pipeline. Top left, trial-averaged source-space ERP waveforms showing early stimulus-locked and later reaction-locked components. Top right, source map at ∼ 100 ms. Bottom left, source map at ∼ 350 ms. Bottom right, region-wise electrophysiological activity scores for the stimulation and reaction windows. This figure provides the temporally resolved component of the source and … view at source ↗

**Figure 3.** Figure 3: Multimodal fusion and probability measures. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Diffusion-informed anisotropic transport geometry. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Data-driven branched optimal transport infers the brain reaction map. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Hybrid stochastic extension on the anisotropic graph. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: Supplementary Fig. S1. α-dependence of anisotropic BOT observables. Left: anisotropic geometric cost Eα versus branching exponent α. Middle: dynamic cost Jdyn versus α. Right: support size |e ∗ | versus α. The vertical dashed line marks the main-text value α = 0.65. The main stable regime is interrupted by a narrow near-degenerate transition around α ≈ 0.4, where the support collapses and the observables s… view at source ↗

**Figure 8.** Figure 8: Supplementary Fig. S2. Anisotropic brain reaction maps across branching regimes. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: Supplementary Fig. S3. Comparison with non-branched baselines. [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: Supplementary Fig. S5. Relay-region statistics for the anisotropic solution at [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗

**Figure 11.** Figure 11: Supplementary Fig. S4. Scalability of the anisotropic BOT solver. [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

read the original abstract

How external stimulation is transformed into distributed reaction patterns remains unresolved at the level of propagation architecture. Existing large-scale control models quantify transition costs on prescribed networks but do not infer the routing map itself from source and target activity. Here we combine task-related blood-oxygen-level-dependent responses, source-reconstructed electrophysiology and tractography-derived anisotropy to estimate stimulation and reaction measures, define an anatomical transport cost, and infer a branched propagation architecture by variational optimisation. Unlike standard transport formulations, branched transport favours aggregation of signal into shared neural highways before redistribution. We further attach a stochastic graph-induced dynamics to the inferred map and quantify the trade-off between geometric efficiency and dynamical controllability. We show that multimodal data generate anatomically aligned brain reaction maps, that anisotropic costs qualitatively reshape routing backbones relative to isotropic baselines, and that hybrid geometric--dynamical optimisation reveals non-trivial rank reversals across branching regimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Branched transport on multimodal brain data is a fresh formulation but the abstract supplies only qualitative claims with no recovery benchmarks or error metrics.

read the letter

The paper's core move is to treat brain reaction maps as the output of a branched-transport variational problem whose cost comes from tractography anisotropy and whose sources and sinks are estimated from BOLD and source-reconstructed electrophysiology. That combination is not in the cited literature, and the authors show that switching from isotropic to anisotropic costs visibly alters the inferred routing backbones while the hybrid geometric-dynamical objective produces rank reversals across branching penalties. Those observations are the concrete contribution so far. The formulation itself is cleanly stated: a transport cost that favors aggregation into shared paths, plus a stochastic dynamics layer that lets them trade geometric length against controllability. If the full manuscript contains the actual optimization details and the precise fusion rules, that part is worth reading. The obvious weakness is the complete absence of quantitative checks. There are no synthetic ground-truth recovery experiments, no cross-validation numbers, no comparison against simpler models with reported error, and no demonstration that the recovered maps predict held-out activity better than the input tractography alone. Without those controls it is impossible to separate genuine inference from the cost function simply reproducing the anisotropy it was given. The circularity concern is therefore live: if the branching weight or the geometry-dynamics trade-off was tuned on the same reaction patterns used to judge alignment, the claimed anatomical fidelity could be an artifact. This work is aimed at people building large-scale control models who are already comfortable with variational network inference. A reader looking for a new way to relax fixed-graph assumptions might borrow the branched-transport setup, but only after seeing the missing validation numbers. I would send it to referees rather than desk-reject, because the modeling idea is coherent and the qualitative observations are at least suggestive, but the current version needs the quantitative backbone before it can be taken as evidence that the method recovers true propagation architecture.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a variational branched transport model that fuses task-related BOLD fMRI, source-reconstructed electrophysiology, and diffusion tractography to infer stimulation-to-reaction propagation maps in the brain. It claims that multimodal integration produces anatomically aligned branched architectures, that anisotropy from tractography qualitatively reshapes routing backbones relative to isotropic baselines, and that a hybrid geometric-dynamical objective produces non-trivial rank reversals across branching regimes.

Significance. If the inferred maps can be shown to recover known architectures on controlled data, the method would offer a principled way to extract routing backbones from combined structural and functional recordings, extending optimal transport to neural propagation with explicit branching and a geometry-dynamics trade-off. The approach addresses a gap in large-scale control models by inferring the routing map itself rather than assuming a fixed network.

major comments (2)

[Results] Results: The abstract and description claim anatomically aligned maps and non-trivial rank reversals, yet no quantitative recovery metrics (e.g., branch recovery error, Dice overlap with known tracts, or controllability error on synthetic data with planted branches) are supplied. This leaves open whether alignments arise from the variational problem or from the tractography-derived cost construction itself.
[Methods] Methods: The transport cost is defined from tractography anisotropy and multimodal fusion, but it is unclear whether the branching penalty weight or geometry-dynamics trade-off parameter is tuned on the same reaction patterns later used to evaluate alignment and rank reversals. If so, the evaluation risks circularity.

minor comments (1)

[Abstract] The abstract would benefit from an explicit statement of the variational objective functional and the number of subjects or sessions in the dataset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and indicate revisions to the manuscript.

read point-by-point responses

Referee: [Results] Results: The abstract and description claim anatomically aligned maps and non-trivial rank reversals, yet no quantitative recovery metrics (e.g., branch recovery error, Dice overlap with known tracts, or controllability error on synthetic data with planted branches) are supplied. This leaves open whether alignments arise from the variational problem or from the tractography-derived cost construction itself.

Authors: We agree that quantitative recovery metrics on synthetic data would strengthen the claims. In the revised manuscript we add a controlled synthetic experiment with planted branches, reporting branch recovery error, Dice overlap with known tracts, and controllability error. These results show that the variational problem recovers the planted architecture beyond what is encoded in the tractography cost alone. revision: yes
Referee: [Methods] Methods: The transport cost is defined from tractography anisotropy and multimodal fusion, but it is unclear whether the branching penalty weight or geometry-dynamics trade-off parameter is tuned on the same reaction patterns later used to evaluate alignment and rank reversals. If so, the evaluation risks circularity.

Authors: The branching penalty and geometry-dynamics trade-off parameters were selected via cross-validation on a held-out subset of reaction patterns that is disjoint from the data used for final evaluation of alignment and rank reversals. We have added an explicit statement of this separation in the revised Methods section. revision: yes

Circularity Check

1 steps flagged

Anatomical alignment of inferred maps enforced by tractography-derived cost construction

specific steps

self definitional [Abstract]
"combine task-related blood-oxygen-level-dependent responses, source-reconstructed electrophysiology and tractography-derived anisotropy to estimate stimulation and reaction measures, define an anatomical transport cost, and infer a branched propagation architecture by variational optimisation. [...] We show that multimodal data generate anatomically aligned brain reaction maps"

The transport cost is constructed from tractography anisotropy; the optimisation then selects the map that minimises this anatomical cost subject to the activity measures. Alignment with anatomy is therefore true by definition of the objective, reducing the central claim to a restatement of the chosen cost rather than an independent prediction.

full rationale

The derivation defines an anatomical transport cost from tractography anisotropy, then infers the branched propagation map via variational optimisation that minimises this cost while matching activity-derived measures. The reported 'anatomically aligned' property therefore follows directly from the objective rather than emerging as a non-trivial inference from functional data alone. No ground-truth recovery benchmarks are described that would separate the cost construction from the result.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that neural propagation can be usefully modeled as branched transport with an anisotropy-derived cost and that the variational optimum recovers biologically meaningful routes. No explicit free parameters are named in the abstract, but the transport cost and the weighting between geometric and dynamical terms are necessarily tuned.

free parameters (2)

branching penalty weight
Controls how strongly the model favors signal aggregation into shared highways; must be chosen or fitted to produce the reported maps.
geometry-dynamics trade-off parameter
Balances the two objectives in the hybrid optimisation; its value determines the observed rank reversals.

axioms (2)

domain assumption Brain signal propagation obeys a branched-transport cost derived from tractography anisotropy
Invoked when the variational optimisation is defined.
domain assumption The stochastic graph dynamics attached to the inferred map correctly capture controllability
Required for the hybrid optimisation step.

pith-pipeline@v0.9.0 · 5448 in / 1471 out tokens · 31429 ms · 2026-05-15T08:59:32.330835+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

min w≥0 ∑_e β_e w_e^α subject to A w = b, 0<α<1 … anisotropic edge cost β_aniso_ij = ℓ_ij √(τ_ij^⊤ (D_mid_ij + ε I)^−1 τ_ij)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

hybrid functional F_λ(G,w) = E_α(G,w) + λ J_dyn(G,w) … Pareto frontier across branching exponents α

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Mendico, Branched Optimal Transport for Stim- ulus to Reaction Brain Mapping, manuscript, 2026

C. Mendico, Branched Optimal Transport for Stim- ulus to Reaction Brain Mapping, manuscript, 2026

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Kawakita et al., Quantifying brain state transi- tion cost via Schrödinger bridge,Network Neuro- science, 6 (2022), 118–134

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T. R. Knösche and J. Haueisen,EEG/MEG Source Reconstruction, Springer, 2022

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Patow, I

G. Patow, I. Martin, Y. Sanz Perl, M. L. Kringel- bach, and G. Deco, Whole-brain modelling: an essential tool for understanding brain dynamics, Nature Reviews Methods Primers, 4 (2024), Article 53

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A. P. Pathak, D. R. Roy, and A. Banerjee, Whole- brain network models: from physics to bedside, Frontiers in Computational Neuroscience, 16 (2022), 866517

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Xia, Optimal paths related to transport prob- lems,Communications in Contemporary Mathemat- ics, 5 (2003), 251–279

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Yeh et al., Mapping structural connectivity using diffusion MRI: challenges and opportunities, Journal of Magnetic Resonance Imaging, 53 (2021), 1666–1682

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Zhang et al., Quantitative mapping of the brain’s structural connectivity using diffusion MRI tractog- raphy: a review,NeuroImage, 249 (2022), 118870

F. Zhang et al., Quantitative mapping of the brain’s structural connectivity using diffusion MRI tractog- raphy: a review,NeuroImage, 249 (2022), 118870. 10 Supplementary overview This Supplementary Information is designed to reinforce the central claims of the main manuscript along four dimensions that are especially relevant for evaluation by a broad mu...

work page 2022

[1] [1]

Mendico, Branched Optimal Transport for Stim- ulus to Reaction Brain Mapping, manuscript, 2026

C. Mendico, Branched Optimal Transport for Stim- ulus to Reaction Brain Mapping, manuscript, 2026

work page 2026

[2] [2]

Bazinet, J

V. Bazinet, J. Y. Hansen, and B. Misic, Towards a biologically annotated brain connectome,Nature Reviews Neuroscience, 24 (2023), 747–760

work page 2023

[3] [3]

Breakspear, Dynamic models of large-scale brain activity,Nature Neuroscience, 20 (2017), 340–352

M. Breakspear, Dynamic models of large-scale brain activity,Nature Neuroscience, 20 (2017), 340–352

work page 2017

[4] [4]

Deco and M

G. Deco and M. L. Kringelbach, Turbulent-like dynamics in the human brain,Cell Reports, 33 (2020), 108471

work page 2020

[5] [5]

Esteban et al., Analysis of task-based functional MRI data preprocessed with fMRIPrep,Nature Protocols, 15 (2020), 2186–2202

O. Esteban et al., Analysis of task-based functional MRI data preprocessed with fMRIPrep,Nature Protocols, 15 (2020), 2186–2202

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Fotiadis et al., Structure–function coupling in macroscale human brain networks,Nature Reviews Neuroscience, 2024

P. Fotiadis et al., Structure–function coupling in macroscale human brain networks,Nature Reviews Neuroscience, 2024

work page 2024

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K. J. Friston, Modalities, modes, and models in functional neuroimaging,Science, 326 (2009), 399– 403

work page 2009

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Gilson et al., Model-based whole-brain effective connectivity to study distributed cognition in health and disease,Network Neuroscience, 4 (2020), 338– 373

M. Gilson et al., Model-based whole-brain effective connectivity to study distributed cognition in health and disease,Network Neuroscience, 4 (2020), 338– 373

work page 2020

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Gu et al., Controllability of structural brain networks,Nature Communications, 6 (2015), 8414

S. Gu et al., Controllability of structural brain networks,Nature Communications, 6 (2015), 8414

work page 2015

[10] [10]

Hagmann et al., Mapping human whole-brain structural networks with diffusion MRI,PLoS ONE, 2 (2007), e597

P. Hagmann et al., Mapping human whole-brain structural networks with diffusion MRI,PLoS ONE, 2 (2007), e597

work page 2007

[11] [11]

S.Kamiyaetal., Optimalcontrolcostsofbrainstate transitions in linear stochastic systems,Journal of Neuroscience, 43 (2023), 270–281

work page 2023

[12] [12]

Kawakita et al., Quantifying brain state transi- tion cost via Schrödinger bridge,Network Neuro- science, 6 (2022), 118–134

G. Kawakita et al., Quantifying brain state transi- tion cost via Schrödinger bridge,Network Neuro- science, 6 (2022), 118–134

work page 2022

[13] [13]

T. R. Knösche and J. Haueisen,EEG/MEG Source Reconstruction, Springer, 2022

work page 2022

[14] [14]

Patow, I

G. Patow, I. Martin, Y. Sanz Perl, M. L. Kringel- bach, and G. Deco, Whole-brain modelling: an essential tool for understanding brain dynamics, Nature Reviews Methods Primers, 4 (2024), Article 53

work page 2024

[15] [15]

A. P. Pathak, D. R. Roy, and A. Banerjee, Whole- brain network models: from physics to bedside, Frontiers in Computational Neuroscience, 16 (2022), 866517

work page 2022

[16] [16]

Xia, Optimal paths related to transport prob- lems,Communications in Contemporary Mathemat- ics, 5 (2003), 251–279

Q. Xia, Optimal paths related to transport prob- lems,Communications in Contemporary Mathemat- ics, 5 (2003), 251–279

work page 2003

[17] [17]

Yeh et al., Mapping structural connectivity using diffusion MRI: challenges and opportunities, Journal of Magnetic Resonance Imaging, 53 (2021), 1666–1682

C.-H. Yeh et al., Mapping structural connectivity using diffusion MRI: challenges and opportunities, Journal of Magnetic Resonance Imaging, 53 (2021), 1666–1682

work page 2021

[18] [18]

Zhang et al., Quantitative mapping of the brain’s structural connectivity using diffusion MRI tractog- raphy: a review,NeuroImage, 249 (2022), 118870

F. Zhang et al., Quantitative mapping of the brain’s structural connectivity using diffusion MRI tractog- raphy: a review,NeuroImage, 249 (2022), 118870. 10 Supplementary overview This Supplementary Information is designed to reinforce the central claims of the main manuscript along four dimensions that are especially relevant for evaluation by a broad mu...

work page 2022