Identifiability of Neural Learning Rules Under Sparse and Noisy Observations: A Virtual-Experiment Study

Backpropagation cannot run in the brain as literally implemented: it requires weight transport, non-local error signals, and strict alternating forward and backward phases. Local alternatives have been proposed for a decade but rarely tested against the data experiments produce: few neurons, noisy calcium imaging, sparse time points. We implement six candidate rules behind one shared interface, verify each against known theoretical equivalences, and train all six under one matched online regime across two tasks and three architectures. A virtual experiment then subsamples units, adds calcium-imaging-like noise calibrated against real Allen Institute recordings, and sparsifies timepoints. Classifiers identifying which rule produced a run's observables top out near 54% six-way (chance 16.7%), with a wide bootstrap 95% interval of [36.4%, 60.6%]. Five of the six rules split into two families, {backpropagation, predictive coding} against {feedback alignment, direct feedback alignment, burst-dependent plasticity}, and that split is far easier to call: 98.5% in distribution, 95.6% on an architecture never seen in training, and 81.1% on an unseen task family, reproduced across three separately trained sweeps. Behavioral-timescale synaptic plasticity (BTSP) separates from the other five at 100% precision and recall, and 122 of 128 conditions survive a grid-wide multiple-comparisons correction. The discriminability spectrum shows why the coarse question is the tractable one. It is dominated by one direction in all 256 conditions across two independently trained sweeps, and that direction is the family contrast itself (minimum alignment 0.9993). At three timepoints and 3% of units, family membership remains recoverable at 89-91% while individual rule identity falls to 15-17%, at or below chance. The signature survives controls for learning speed and pooled learning rates, and a log-normal weight initialisation matched to real MICrONS synapse statistics, and is carried mainly by population activity variance, not early weight change. We turn this into testable predictions for real recordings.