Dynamic Inference Limits in Plastic Recurrent Networks Under Invasive Measurement

We derive a lower bound on the error of parameter inference from invasive measurement in recurrent adaptive networks with online plasticity. The bound decomposes into a variance contribution and a bias contribution, arising from physically distinct components of measurement back-action: stochastic fluctuations injected into the state dynamics and the systematic shift of the state operating point under repeated observation. Under generic measurement families, the bias term and the measurement-dependent component of the variance channel increase with measurement precision through distinct mechanisms and cannot generally be traded off against each other. Near marginal stability, both contributions are amplified by resolvent operators that diverge as their corresponding spectral radii approach unity, so that the bound becomes more restrictive in the regime in which cortical circuits are believed to operate. Evaluated at biologically realistic scale, the bound implies that the per-neuron disturbance required for noninvasiveness falls orders of magnitude below ambient biological fluctuations, making the constraint plausibly operative for standard invasive measurement modalities. Beyond the stationary regime, a finite-time analysis converts the bound into an experimentally relevant timescale, distinguishing the ballistic accumulation of measurement-induced bias from intrinsic representational drift and yielding testable temporal and geometric signatures for optogenetic and chronic-recording protocols. The result identifies a structural limit on parameter inference in adaptive recurrent systems, induced by the dynamical coupling between measurement, recurrent amplification, and online plasticity.