Title: Stable Spatiotemporal Memory in Echo-State Networks via Gliotransmitter Feedback
Primary area: learning on time series and dynamical systems
Keywords: Echo State Networks, Reservoir Computing, Neuro–glial coupling, Astrocyte lattice, Reaction–diffusion dynamics, Gliotransmitter bias, Spatiotemporal memory, Joint echo-state property, Provable stability

Abstract: Reservoir computing promises simple, stable training for sequence modeling, yet vanilla reservoirs struggle to sustain \emph{long, structured memory} without operating near fragile regimes. We seek a reservoir that retains echo–state guarantees and one–shot readout training while endowing the state with \emph{slow, spatially coherent} memory relevant to forecasting and control. The difficulty is twofold: (i) injecting slow feedback often breaks contractivity and thus the echo–state property (ESP), and (ii) encoding spatial structure (e.g., diffusion on a lattice) alongside fast neuronal recurrence typically entangles gains in ways that are hard to certify. We introduce an architecture which augments a fast ESN core with a bidirectionally coupled \emph{astrocytic reaction–diffusion lattice}. Neuronal activity produces rectified release proxies pooled onto astrocytes; astrocytic Ca$^{2+}$ evolves by a \emph{diffusive, saturating} update; a bounded gliotransmitter then feeds back to neurons as an \emph{additive bias}. A one–step delay in each cross–coupling and bounded, Lipschitz kinetics yield explicit, checkable \emph{row–wise operator–norm inequalities} that make the joint  map a uniform contraction in a block–$\ell_\infty$ norm, providing a clean \emph{ESP certificate}. The design is enforced by a norm–aware initialization; the astrocyte lattice is computed by a 5–point stencil, keeping per–step cost dominated by the neuronal multiply. We verify the approach empirically by benchmarking on canonical long–horizon tasks (chaotic forecasting and real-world time series) against ESN baselines, showing longer valid prediction horizons and stable closed–loop rollouts with modest overhead, and via ablations isolating the effect of diffusion and glial feedback. Theoretically, beyond the ESP certificate, we establish (i) input–Lipschitz and bounded–input bounded–state estimates, and (ii) a small–gain condition that ensures stability of the \emph{autoregressive} closed loop used at inference.

TL;DR: A fast ESN is bidirectionally coupled to a diffusive astrocyte lattice whose bounded reaction–diffusion waves drive gliotransmitter additive bias, yielding a provable joint ESP and an interpretable, spatially structured slow memory.