Interrelations among distortion-product phase-gradient delays: their connection to scaling symmetry and its breaking

Distortion-product-otoacoustic-emission (DPOAE) phase-versus-frequency functions and corresponding phase-gradient delays have received considerable attention because of their potential for providing information about mechanisms of emission generation, cochlear wave latencies, and characteristics of cochlear tuning. The three measurement paradigms in common use (fixed-f1, fixed-f2, and fixed-f2/f1) yield significantly different delays, suggesting that they depend on qualitatively different aspects of cochlear mechanics. In this paper, theory and experiment are combined to demonstrate that simple phenomenological arguments, which make no detailed mechanistic assumptions concerning the underlying cochlear mechanics, predict relationships among the delays that are in good quantitative agreement with experimental data obtained in guinea pigs. To understand deviations between the simple theory and experiment, a general equation is found that relates the three delays for any deterministic model of DPOAE generation. Both model-independent and exact, the general relation provides a powerful consistency check on the measurements and a useful tool for organizing and understanding the structure in DPOAE phase data (e.g., for interpreting the relative magnitudes and intensity-dependencies of the three delays). Analysis of the general relation demonstrates that the success of the simple, phenomenological approach can be understood as a consequence of the mechanisms of emission generation and the approximate local scaling symmetry of cochlear mechanics. The general relation is used to quantify deviations from scaling manifest in the measured phase-gradient delays; the results indicate that deviations from scaling are typically small and that both linear and nonlinear mechanisms contribute significantly to these deviations. Intensity-dependent mechanisms contributing to deviations from scaling include cochlear-reflection and wave-interference effects associated with the mixing of distortion- and reflection-source emissions (as in DPOAE fine structure). Finally, the ratio of the fixed-f1 and fixed-f2 phase-gradient delays is shown to follow from the choice of experimental paradigm and, in the scaling limit, contains no information about cochlear physiology whatsoever. These results cast considerable doubt on the theoretical basis of recent attempts to use relative DPOAE phase-gradient delays to estimate the bandwidths of peripheral auditory filters.