Memory and chaos in a physics-constrained relaxation substrate: phase diagram, multi-timescale forgetting, and disturbance robustness

Abstract

Reservoir computing exploits the fading-memory dynamics of a physical substrate, yet the memory–chaos trade-off is usually studied on abstract recurrent networks with hand-tuned leak rates. We characterize a physics-constrained relaxation substrate inspired by Si3N4 shallow-trap charge storage: N units with a log-normal time-constant spectrum (median τ0 ≈ 174 μs, CV = 0.20) coupled through a per-pulse, topology-dependent modulation of the injection coefficient. Sweeping the coupling strength κ across five topology families and measuring the finite-time Lyapunov exponent λ, the held-out Jaeger memory capacity, and input separation, we find a sharp order–chaos transition at κ* ∈ (25, 30): the held-out memory capacity peaks 24–53% above the uncoupled baseline just before the transition, and deep chaos destroys memory. The decay of linear memory follows an analytically derived forgetting kernel M(t) = ∫ p(τ) e^(−t/τ) dτ over the log-normal trap spectrum (Pearson r = 0.97 against the measured memory-capacity curve). The 1/e horizon stays near τ0/⟨Δt⟩ ≈ 16 pulses as the spectrum widens (numerically 12–16 pulses for CV ∈ [0.02, 1.0]), while the width CV controls the tail weight. Finally, a homeostatic regulator that estimates λ online and adjusts κ to a near-critical target improves post-disturbance held-out memory by 8–18% under temperature drift, edge damage, and readout noise.