Loops as States

State-space models (S4, S4D, Mamba) are a bank of linear dynamical systems. A superconducting optoelectronic (SOEN) loop — a synapse, dendrite, or soma — obeys one first-order leaky-integrator equation, τ · ds/dt = −s + γτ·g(φ,s). Linearize it and the two are the same object: one loop is one diagonal SSM mode. The three panels below let you feel that correspondence, and the experiment behind the note’s headline result.

1  One loop is one mode

A leaky loop is a real pole — its impulse response (the SSM convolution kernel) is an exponential. A resonant dendrite (second-order LRC) is a complex-conjugate pole pair — a damped oscillation. Move the leak time; toggle resonance.

2  A bank is an S4D layer

Stack loops with different timescales and frequencies, mix them with read-out weights C, and you build an arbitrary multi-exponential / oscillatory kernel — exactly the S4D basis. Drag the four read-out weights.

3  The binding constraint: frequency precision

Realize a target S4D layer on imperfect loops. Fabrication jitter perturbs each mode’s decay (σ_α) and frequency (σ_ω); thermal/flux noise adds state noise (σ_s). Watch the reconstruction error: leak precision is forgiving, frequency precision is not.