Independent noise makes delay direction invisible to power spectra in two-cell feedback models

We proved that two single-cell power spectra and coherence magnitude cannot identify how a known total delay is split between directions in a stable linear excitatory-inhibitory loop with independent stationary input noise. Characteristic roots are also unchanged. Complex cross-spectra retain the split as a phase factor. The smallest possible worst-case absolute direction-delay error from these exact magnitude summaries is half the total delay. We tested the result in models of 91 selected Allen mouse visual-cortex pairs and three human temporal-cortex pairs. Swapping measured onset delays gave a median absolute cross-correlation shift of 0.326 ms in mouse models, with no power change beyond numerical error. The corresponding median phase change at 40 Hz was 4.70 degrees. A noise correlation of 0.2 broke the symmetry and gave a median largest power change of 2.69 percent over 1 to 100 Hz. The time-shift principle is established theory; this report adds a direct spectral proof, an error certificate, and a public-data benchmark. Effective filters, assumed gains and noise, selected slice records, and no measured activity spectra limit biological inference. Code, aggregate evidence, source hashes, and external regeneration instructions support reproduction.

Allen Institute · delay inference · power spectrum · cross-spectrum · identifiability · feedback · independent noise