§2 · The cochlear amplifier: cube-root compression

The cochlear amplifier: cube-root compression

The outer-hair-cell active amplifier (prestin / SLC26A5) emerges as the inherited R19 cubic at its critical point: the response compresses as the parameter-free cube root r ∝ F^(1/3), and the 1/3 is forced by the cubic, not fitted. Detection and amplification are one cubic in two regimes.

At criticality g=0 the inherited cubic g·s−s³+h has its zero at s=F^(1/3); the integrator’s fitted exponent is 0.333333 and the gain falls as F^(−0.666667) — the compression behind the ear’s ~120 dB dynamic range (an observed human range, cited in the text below; not a model output) [observation]. The cube root is the cubic’s critical fixed point, no constant chosen.

One cubic, two regimes

The §1 detection switch and the §2 amplifier are the same equation. With g=γ_TMC1 the response is bistable (the §1 switch); with g=0 the same drives give the continuous cube root s = h^(1/3) [F].

Bistable detection runs in parallel with critical amplification — one substrate, two operating regimes. The amplifier is not a new mechanism; it is the cubic taken to its critical point.

Compression is the cube root — forced, not fitted

The 1/3 exponent is the inherited cubic’s critical fixed point. Evaluating the inherited sdot at s=F^(1/3), g=0 gives max|residual| <1×10⁻⁹ over seven decades of drive [F].

The integrator’s read-off exponent is 0.333333 and the gain exponent −0.666667 [code output] — the inherited integrator reproducing the normal form’s analytic value 1/3; this is an internal check, not a test against data. With gain ∝ F^(−2/3), faint drives are amplified ~100× more than loud over three decades of drive. The ~120 dB span of human hearing is an observation [observation] (e.g. Moore 2012, An Introduction to the Psychology of Hearing, 6th ed.), not a model output, consistent with N1 [O].

Observation [observation]: measured basilar-membrane compression at CF is roughly 0.2–0.33 dB/dB (Ruggero et al. 1997, J. Acoust. Soc. Am. 101, 2151; reviewed in Robles & Ruggero 2001, Physiol. Rev. 81, 1305), so 1/3 sits at the upper edge of the observed range. The cube root at a Hopf bifurcation is prior work: Eguíluz et al. 2000 (Phys. Rev. Lett. 84, 5232) and Camalet et al. 2000 (PNAS 97, 3183). This volume takes it over through the seam to sensory_organ §7 (Hopf cube root) [interpretation].

Uniform across the bank; magnitude is the obstacle

Every place carries the same critical cubic, so the 1/3 exponent is CF-independent — compression is uniform across frequency [interpretation] (a consequence of the model assumption that every place carries the same cubic). Observation [observation]: measured compression is weaker at the apex (low CF) than at the base (reviewed in Robles & Ruggero 2001), so this interpretation does not match the apex. SLC26A5 (γ 1.4023, spinodal 0.6392) takes its place in the emergence lineage.

The absolute gain, the dB of amplification, and the sharpness Q are [O] — a number would require tuning a constant. Only the exponent (1/3) and the direction (compression) are forced.

Honest negatives — what is not claimed

  1. N1. The absolute gain / dB / dynamic-range-in-dB / sharpness Q are [O] — a value would require tuning. Only the exponent and the direction are forced.
  2. N2. The cube root matches the FORM of measured cochlear compression; the study-dependent I/O slope (~0.2–0.5 dB/dB) and the absolute curve are [O].
  3. N3. Otoacoustic-emission frequencies and amplitudes are [O] — they need the per-place gain, Q, and an operating point just above the bifurcation.
  4. N4. Prestin’s electromotile force is NOT read from γ (firewall) — only the gene’s spinodal-order place and the compression exponent are forced.
  5. N5. The inherited integrator under-converges to the fixed point at very small drive; the rigorous [F] is the analytic fixed point, the [code output] is on the converged range.