Recruitment Curves

How can I be so sure that our hearing exhibits the recruitment behavior shown in all my graphs? Easy, that steep drop toward no hearing is merely a consequence of elevated thresholds, and the way the data are presented. The actual details of our hearing determine only second order characteristics of the curvature in these curves, but not their basic nature.

We have known for almost 200 years that our hearing is nearly linear at threshold levels, and becomes approximately cube-root compressive at louder levels. At question is merely at what level and how rapidly the transition between these two characters occurs.

My EarSpring equation shows that for human hearing, that transition occurs pretty sharply at sound levels around 20 dBHL, and by the time we reach normal everyday sound levels, 40 dBHL and higher, we are quite close to cube root compressive.

Showing the solution of the EarSpring equation for normal hearing. Note that this is not a recruitment hearing curve. Rather, it shows how our perception of sound changes from linear sensation at very faint sound levels, and into a cube-root compressive sensation for louder sounds.

The only thing needed to see recruitment hearing behavior is elevated thresholds – thresholds of hearing that are above those of persons with normal hearing. Any sounds at levels below the elevated threshold can’t be heard by impaired hearing.

Given these conditions, when we plot the apparent loudness of sounds as perceived by people with elevated thresholds, the steep recruitment near the elevated threshold is solely a consequence of the fact that the logarithm function becomes very steep around zero, and undefined for negative values.

The apparent loudness in Sones is the loudness of the sound, as perceived by a person with normal hearing, minus the elevated threshold: S_{app}(p) = S(p) - S(p_{el}), and S(p) = 10^{\frac{p-40}{30}} for sound level, p, measured in dBH.

If we plot this apparent sound level in linear Sones space, it looks like this:

Apparent Sones heard by a person with normal hearing (green), and by a person with a threshold elevation of 60 dB (red).

In other words, the impaired hearing doesn’t look that much different from normal hearing, except that it is displaced due to the threshold elevation. But the impaired curve has the same slope as the unimpaired curve at all presentation levels.

But when showing the recruitment curves in dB loudness space in this series of blog notes, I have always shown what the sound would seem like, relative to a person’s sensation who has normal hearing. That means that I use logarithmic perceptual loudness units on the vertical axis, and the vertical units are relative to what normal hearing would hear at any given presentation level.

Stated differently, whatever someone hears at any given presentation level, that labels their perceived sensation for that presentation level. For people with normal hearing, the presentation level is their perceived sound level. People with impaired hearing depart from that.

This way of presenting information does two things: First the curvature in the normal sensation curve becomes removed. A person with normal hearing hears what they hear, no matter the presentation level, and so their response curve becomes a straight line in log space. It is relative to that straight line that I present the apparent sensation for persons with elevated thresholds.

Secondly, because the impaired hearing curve dips below zero Sones, the log-space presentation gains a steep approach to zero sensation near the elevated threshold. The logarithm is undefined for negative Sones values.

Illustration of recruitment hearing due to 60 dB threshold elevation. The graph presents this information in perceptual loudness units, using logarithmic loudness measure dBHL. A person with normal hearing hears whatever they hear, but if the sound is presented at, say, 40 dBHL, then we label their sensation as 40 dBHL, which gives rise to the Ideal Hearing diagonal line.

So really, there can be nothing disputable about my recruitment curves. It only requires that impaired hearing cannot perceive sounds presented at levels below their elevated threshold – which is the definition of an elevated threshold.

But, you say… why should the slope of the Sones curves be the same between normal hearing and impaired hearing? Well, I argue that if they were not, then the sound field presented to the hair cells along the basilar membrane would be different in the two cases. Conduction hearing loss might be one such mechanism.

But we aren’t discussing conduction losses. Those simply act like a plugged entryway into the cochlea, and need only increased amplification to overcome. Conduction losses don’t affect the spectral response.

So, assuming the same sound field present in both situations, the mechanical response of hair cells to the sound field should be the same. Hence the same increment in Sones (absolute) loudness for any increment in presentation level, i.e., the same slope.

No, what I argue is that for impaired hearing, there is a neural mechanism that sets an elevated threshold – perhaps fewer responding hair cells. And while we could argue about the detailed shape of the diminished response, that can only change second order characteristics of the curvature seen in recruitment curves, but not their basic nature. All it takes for a recruitment curve to arise is an elevated threshold.

These second order differences in models dictate only the details of the hearing corrections needed, not their gross values. And we have seen that even crude approximations, such as assuming only cube-root compression, produce acceptable corrections in practice.

The principal character of recruitment hearing arises solely from an elevated threshold.