Doubly stochastic inter-assembly coupling via entropic optimal transport in echo-state networks for chaotic flows

We propose the Optimal-Transport Gated Echo-State Network (OT-ESN), a two-timescale reservoir that replaces ad hoc inter-module couplings with a principled, mass-conserving transport mechanism on a cortical-sheet geometry. At each step, a slow, exogenous controller computes an entropically regularized optimal-transport plan Π between the previous distribution of column activity (source) and an input-derived "intent" over columns (target), using a geometric cost that encodes anatomical or functional proximity. The resulting plan-doubly stochastic up to prescribed marginals-acts as a bounded, geometry-aware mixer that gates inter-column blocks of the reservoir at the next fast update. This one-step delay ensures that Π is absent from the time-t Jacobian, so with a 1-Lipschitz nonlinearity and fixed leak, the echo-state property collapses to a single spectral-norm inequality on pre-scaled intra- and inter-column operators, yielding a uniform contraction certificate. OT-ESN, thus, achieves interpretable, neuromodulation-like routing of assembly activity while preserving the simplicity of readout-only training. Computationally, Sinkhorn iterations on a J×J kernel provide efficient, smooth control, with the regularization parameter spanning diffuse (diffusion-like) to sharp (path-like) transports without jeopardizing stability. Ergo, via optimal transport, OT-ESN enables long, structured memory and geometry-respecting information flow in a provably stable recurrent substrate.