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Branched Optimal Transport for Stimulus to Reaction Brain Mapping

Cristian Mendico

Abstract

A central problem in systems neuroscience is to determine how an external stimulation is propagated through the brain so as to produce a reaction. Current deterministic and stochastic control models quantify transition costs between brain states on a prescribed network, but do not treat the transport network itself as an unknown. Here we propose a variational framework in which the inferred object is a graph/current connecting a stimulation source measure to a reaction target measure. The model is posed as an anisotropic branched optimal transport problem, where concavity of the flux cost promotes aggregation and branching. The support of an optimal current defines a stimulus-to-reaction routing architecture, interpreted as a brain reaction map. We prove existence of minimizers in discrete and continuous formulations and introduce a hybrid stochastic extension combining ramified transport with a path-space Kullback--Leibler control cost on the induced graph dynamics. This approach provides a mathematical mechanism for inferring propagation architectures rather than controlling trajectories on fixed substrates.

Branched Optimal Transport for Stimulus to Reaction Brain Mapping

Abstract

A central problem in systems neuroscience is to determine how an external stimulation is propagated through the brain so as to produce a reaction. Current deterministic and stochastic control models quantify transition costs between brain states on a prescribed network, but do not treat the transport network itself as an unknown. Here we propose a variational framework in which the inferred object is a graph/current connecting a stimulation source measure to a reaction target measure. The model is posed as an anisotropic branched optimal transport problem, where concavity of the flux cost promotes aggregation and branching. The support of an optimal current defines a stimulus-to-reaction routing architecture, interpreted as a brain reaction map. We prove existence of minimizers in discrete and continuous formulations and introduce a hybrid stochastic extension combining ramified transport with a path-space Kullback--Leibler control cost on the induced graph dynamics. This approach provides a mathematical mechanism for inferring propagation architectures rather than controlling trajectories on fixed substrates.
Paper Structure (19 sections, 9 theorems, 116 equations)

This paper contains 19 sections, 9 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Notations
  4. Estimation of $\mu_{\mathrm{stim}}^{+}$, $\mu_{\mathrm{react}}^{-}$, and $c(x,\tau)$ from data
  5. General principle
  6. Estimation from fMRI and EEG/MEG data
  7. Estimation of the anatomical cost density
  8. Discrete graph model
  9. Weighted directed graphs
  10. Finite-library existence theorem
  11. Continuous current formulation
  12. Rectifiable transport currents
  13. Continuous branched transport functional
  14. Hybrid stochastic extension
  15. Graph-induced linear stochastic dynamics
  16. ...and 4 more sections

Key Result

Theorem 4.1

Under the finite-library assumption, the optimization problem eq:discrete-problem admits at least one minimizer.

Theorems & Definitions (41)

  • Definition 1: Stimulus and reaction measures
  • Definition 2: Anatomical cost density
  • Remark 1
  • Remark 2
  • Definition 3: Embedded weighted directed graph
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 3
  • Theorem 4.1
  • ...and 31 more