branched-optimal-transport-brain-mapping

name	branched-optimal-transport-brain-mapping
description	Branched optimal transport framework for inferring stimulus-to-reaction propagation architectures in brain networks. Uses anisotropic branched OT where concavity of flux cost promotes aggregation and branching. Activation: branched optimal transport, stimulus reaction brain mapping, ramified transport brain, brain propagation architecture, optimal transport neuroscience.

Branched Optimal Transport for Brain Mapping

Variational framework for inferring stimulus-to-reaction routing architectures in the brain using anisotropic branched optimal transport, treating the transport network itself as unknown rather than fixed.

Metadata

• Source: arXiv:2603.19751
• Authors: Cristian Mendico
• Published: 2026-03-20
• Categories: math.OC, q-bio.NC, q-bio.QM

Core Methodology

Key Innovation

Traditional brain state transition models control trajectories on a fixed network substrate. This work inverts the problem: the transport network itself is the inferred object, modeled as a graph/current connecting a stimulation source measure to a reaction target measure.

Technical Framework

1. Anisotropic Branched OT Formulation

   • Model as variational problem: find optimal current (graph) connecting source measure μ (stimulation) to target measure ν (reaction)
   • Flux cost function is concave → promotes aggregation and branching
   • Support of optimal current defines stimulus-to-reaction routing architecture

2. Existence Theory

   • Proved existence of minimizers in both discrete and continuous formulations
   • Discrete: finite graph with edge currents
   • Continuous: measure-theoretic formulation on manifold

3. Hybrid Stochastic Extension

   • Combines ramified transport with path-space KL control cost
   • Induced graph dynamics incorporate stochasticity
   • Provides probabilistic interpretation of routing uncertainty

Mathematical Structure

minimize: ∫ c(θ) d|J|(θ) + KL(Path || Reference)
subject to: div(J) = ν - μ (continuity equation)
where: J = optimal current, c = concave flux cost

Applications

• Inferring brain reaction maps from stimulation experiments
• Mapping neural propagation pathways without pre-defined connectivity
• TMS/tDCS stimulation response prediction
• Understanding brain network plasticity through optimal routing

Implementation Guide

Prerequisites

• Optimal transport library (POT, geomloss)
• Graph optimization tools
• Numerical PDE solvers for continuous formulation

Step-by-Step

1. Define source measure μ (stimulation region) and target measure ν (reaction region)
2. Choose concave flux cost function c(θ) (e.g., θ^α, α < 1)
3. Solve discrete branched OT on candidate graph
4. Validate with continuous formulation via level-set methods
5. Extend to stochastic version with KL control cost for uncertainty quantification

Pitfalls

• Concave optimization is non-convex → multiple local minima possible
• Discrete approximation may miss fine-scale branching structure
• Stochastic extension adds computational complexity
• Requires careful regularization for numerical stability

Related Skills

• optimal-transport-brain
• brain-network-controllability
• brain-stimulation-dynamics-state
• adaptive-flow-routing-brain-networks