Branched Optimal Transport for Stimulus to Reaction Brain Mapping

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.