Biophysics of Hearing

Derivation: The Acoustic Wave Equation

Consider a fluid element of cross-sectional area $A$ and thickness $dx$at equilibrium position $x$. Let $\xi(x,t)$ be the displacement from equilibrium. The pressure at any point is $P = P_0 + p(x,t)$ where $p$is the acoustic pressure perturbation.

Step 1 — Conservation of mass (continuity): The fractional change in volume of the element is:

$$\frac{\Delta V}{V} = \frac{\partial \xi}{\partial x}$$

For an adiabatic process in an ideal gas, $PV^\gamma = \text{const}$, giving the bulk modulus $B = -V \, dP/dV = \gamma P_0$. The acoustic pressure relates to compression as:

$$p = -B \frac{\partial \xi}{\partial x} = -\gamma P_0 \frac{\partial \xi}{\partial x}$$

Step 2 — Newton's second law: The net force on the element is $[p(x) - p(x+dx)]A = -(\partial p / \partial x) A \, dx$. The mass is $\rho_0 A \, dx$. Therefore:

$$\rho_0 \frac{\partial^2 \xi}{\partial t^2} = -\frac{\partial p}{\partial x}$$

Step 3 — Combine: Substituting the expression for$p$ into Newton's law:

$$\rho_0 \frac{\partial^2 \xi}{\partial t^2} = B \frac{\partial^2 \xi}{\partial x^2}$$

Taking $\partial/\partial x$ of the pressure equation and combining, we get the wave equation directly for the acoustic pressure:

$$\boxed{\frac{\partial^2 p}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2}}$$

where the speed of sound is $c = \sqrt{B/\rho_0} = \sqrt{\gamma P_0 / \rho_0}$. For air at 20°C: $c \approx 343$ m/s.

Sound Intensity and the Decibel Scale

For a plane harmonic wave $p(x,t) = p_0 \sin(kx - \omega t)$, the instantaneous intensity (power per unit area) is $I_{\text{inst}} = p \cdot v$ where$v = p/(\rho_0 c)$ is the particle velocity. Time-averaging:

$$\boxed{I = \frac{p_{\text{rms}}^2}{\rho_0 c} = \frac{p_0^2}{2\rho_0 c}}$$

The product $\rho_0 c$ is the acoustic impedance of the medium. For air,$\rho_0 c \approx 413$ Pa·s/m.

The ear responds to an enormous range of intensities. The decibel scale compresses this logarithmically:

$$\boxed{L = 10 \log_{10}\!\left(\frac{I}{I_0}\right) \quad \text{dB SPL}}$$

where $I_0 = 10^{-12}$ W/m$^2$ is the reference intensity, corresponding to $p_0 = 20 \; \mu$Pa, the threshold of hearing at 1 kHz. The threshold of pain is at approximately $I = 1$ W/m$^2$(120 dB SPL), corresponding to $p_0 \approx 20$ Pa.

Derivation: The Tonotopic Map

The BM varies systematically along its 35 mm length: it is narrow and stiff at the base (width ~0.1 mm) and wide and floppy at the apex (width ~0.5 mm). Model the BM as a tapered elastic beam. The local resonant frequency depends on stiffness $k(x)$ and mass per unit length $m(x)$:

$$f(x) = \frac{1}{2\pi}\sqrt{\frac{k(x)}{m(x)}}$$

Experimentally, the stiffness decreases approximately exponentially from base to apex:$k(x) = k_0 \, e^{-2x/d}$ where $d \approx 7$ mm is the length constant. With mass varying more slowly, this gives the exponential tonotopic map:

$$\boxed{f(x) = f_{\max} \cdot e^{-x/d}}$$

where $f_{\max} \approx 20{,}000$ Hz at the base ($x = 0$) and$f_{\min} \approx 20$ Hz at the apex ($x = 35$ mm). Inverting:

$$x(f) = d \cdot \ln\!\left(\frac{f_{\max}}{f}\right)$$

This is Greenwood's equation (refined form). The logarithmic spacing means each octave occupies the same length of BM (~3.5 mm), consistent with our perception of pitch as logarithmic in frequency.

Derivation: The Traveling Wave (von Békésy)

Georg von Békésy (Nobel Prize 1961) showed that sound does not simply excite a resonant point on the BM. Instead, a traveling wave propagates from base to apex. Model the cochlea as a transmission line: two fluid chambers coupled through the elastic BM partition.

The pressure difference $\Delta p(x,t)$ across the BM drives its displacement$\eta(x,t)$. The fluid equations give:

$$\frac{\partial^2 \Delta p}{\partial x^2} = \frac{2\rho}{A} \frac{\partial^2 \eta}{\partial t^2}$$

where $A$ is the cross-sectional area of each chamber and $\rho$ is fluid density. The BM responds locally:

$$m(x)\frac{\partial^2 \eta}{\partial t^2} + r(x)\frac{\partial \eta}{\partial t} + k(x)\eta = \Delta p$$

For harmonic excitation at frequency $\omega$, the local wavenumber is:

$$\kappa(x) = \sqrt{\frac{2\rho \omega^2}{A[k(x) - m(x)\omega^2 + i r(x)\omega]}}$$

The wave propagates with increasing amplitude and decreasing wavelength until it reaches the characteristic place where $k(x) = m(x)\omega^2$(resonance). Beyond this point, the wave is evanescent and decays rapidly. The quality factor of the passive BM is $Q \approx 1$–$3$, meaning the frequency selectivity of the passive membrane is rather poor — the active process sharpens it dramatically.

Derivation: The Gating Spring Model

When the stereocilia bundle is deflected by a displacement $X$, the tip link exerts a force on the transduction channel. Model the tip link as a gating spring of stiffness $\kappa_g$. The channel has two states: closed (C) and open (O), with the open state having the gate swung through a distance $z \approx 4$ nm (the single-channel gating swing).

The energy difference between open and closed states depends on the applied force:

$$\Delta G = \Delta G_0 - \kappa_g z \left(X - X_0\right)$$

where $\Delta G_0$ is the intrinsic energy difference and $X_0$ is the set point. Using Boltzmann statistics, the open probability is:

$$\boxed{P_{\text{open}} = \frac{1}{1 + \exp\!\left(-\frac{\kappa_g z (X - X_0)}{k_B T}\right)}}$$

This is a Boltzmann sigmoid with sensitivity set by the ratio $\kappa_g z / k_B T$. With $\kappa_g \approx 1$ mN/m and $z \approx 4$ nm, the characteristic displacement is:

$$X_c = \frac{k_B T}{\kappa_g z} \approx \frac{4.1 \times 10^{-21}}{10^{-3} \times 4 \times 10^{-9}} \approx 1 \; \text{nm}$$

The operating range of the hair cell is thus only a few nanometres — astoundingly small!

Transduction Current and Adaptation

Each stereocilium tip has ~2 transduction channels with single-channel conductance$\gamma \approx 100$ pS. With a driving potential of ~80 mV (from the endolymph potential +80 mV combined with the resting potential of -60 mV), the single-channel current is:

$$i = \gamma \times V_{\text{drive}} = 100 \times 10^{-12} \times 0.14 \approx 14 \; \text{pA}$$

The total transduction current for $N \approx 100$ channels per bundle:

$$\boxed{I_{\text{trans}} = N \cdot i \cdot P_{\text{open}}(X)}$$

Adaptation is mediated by myosin-1c motors that climb along the actin core of the stereocilium, adjusting the resting tension on the tip link. This shifts $X_0$ to keep $P_{\text{open}}$ near ~5% at rest, maximising sensitivity to both positive and negative deflections. The adaptation time constant is $\tau_{\text{adapt}} \approx 1$–$10$ ms for fast adaptation and $\sim 100$ ms for slow adaptation.

Derivation: Outer Hair Cell Electromotility

Outer hair cells (OHCs) change length when their membrane potential changes. This electromotility is driven by the protein prestin (SLC26A5) densely packed in the OHC lateral wall (~8,000 motors/$\mu$m$^2$).

Prestin acts as a piezoelectric element. The fractional length change is proportional to the membrane potential change:

$$\boxed{\frac{\Delta L}{L} = \alpha \cdot \Delta V}$$

where $\alpha \approx 20$ nm/mV for an OHC of length $L \approx 30 \; \mu$m, giving a fractional change of ~$10^{-3}$/mV. The associated nonlinear capacitance follows a Boltzmann function:

$$C_{\text{NL}}(V) = C_{\max} \cdot \frac{\exp\!\left(\frac{ze(V - V_{1/2})}{k_B T}\right)}{\left[1 + \exp\!\left(\frac{ze(V - V_{1/2})}{k_B T}\right)\right]^2}$$

where $V_{1/2} \approx -40$ mV is the voltage of peak capacitance and$ze \approx 0.8e$ is the effective charge movement.

Derivation: Amplification Gain

The OHC operates in a positive feedback loop: BM motion $\to$ stereocilia deflection$\to$ transduction current $\to$ membrane potential change$\to$ OHC length change $\to$ enhanced BM motion. The loop gain is:

$$G_{\text{loop}} = \frac{\partial P_{\text{open}}}{\partial X} \cdot i \cdot N \cdot \frac{1}{C_m \omega} \cdot \alpha \cdot L \cdot \frac{k_{\text{BM}}}{\Delta p_{\text{BM}}}$$

Each factor represents a stage: mechanotransduction sensitivity, current, RC filtering, electromotility, and mechanical coupling. The net gain at the characteristic frequency can be as high as:

$$\boxed{G \approx 40 \text{--} 60 \; \text{dB} \quad (\times 100 \text{--} 1000 \; \text{in amplitude})}$$

This active amplification is what allows the ear to detect basilar membrane vibrations of less than 1 picometre — smaller than the diameter of an atom! The gain is compressive: large at low sound levels (soft sounds are amplified most) and decreasing at high levels, giving a roughly $\frac{1}{3}$-power compressive nonlinearity:

$$\eta_{\text{BM}} \propto p_{\text{input}}^{1/3}$$

This compression maps the enormous 120 dB input range onto a manageable ~40 dB range of BM vibration, enabling both sensitivity and resistance to overload.

Derivation: Distortion Product OAEs

When two tones at frequencies $f_1$ and $f_2$ ($f_2 > f_1$,$f_2/f_1 \approx 1.2$) are presented to the ear, the nonlinear mechanics of the cochlea generates distortion products at combination frequencies. The compressive nonlinearity can be expanded as a power series:

$$\eta = a_1 p + a_2 p^2 + a_3 p^3 + \cdots$$

With two-tone input $p = A_1\cos(\omega_1 t) + A_2\cos(\omega_2 t)$, the cubic term generates:

$$a_3 p^3 \supset \frac{3a_3}{4} A_1^2 A_2 \cos(2\omega_1 - \omega_2)t + \cdots$$

The dominant distortion product is at:

$$\boxed{f_{\text{dp}} = 2f_1 - f_2}$$

This distortion is generated at the overlap region of the two traveling waves, then propagates back through the cochlea, through the middle ear, and is emitted as an acoustic signal that can be recorded with a sensitive microphone in the ear canal. The DPOAE level is typically 40–60 dB below the stimulus level in a normal ear and is absent or reduced in ears with OHC damage — making it a powerful clinical tool.

Spontaneous Otoacoustic Emissions

Remarkably, about 70% of ears with normal hearing produce spontaneous OAEs (SOAEs) — tonal sounds emitted continuously without any external stimulus. These arise from the active cochlea's operating point being very close to a Hopf bifurcation (the boundary between stable and self-oscillating behaviour). The dynamics near the bifurcation are described by the normal form:

$$\frac{dz}{dt} = (\mu + i\omega_0)z - |z|^2 z + F(t)$$

where $z$ is the complex amplitude of BM oscillation, $\mu$ is the bifurcation parameter (negative = stable, positive = self-oscillating), $\omega_0$is the natural frequency, and $F(t)$ is the external forcing. When$\mu < 0$ but close to zero, the system responds to external stimuli with the characteristic $\frac{1}{3}$-power compressive nonlinearity:

$$|z| \propto |F|^{1/3} \quad \text{for } |F| \gg |\mu|^{3/2}$$

This Hopf bifurcation model elegantly unifies the cochlear amplifier's key features: frequency selectivity, compressive nonlinearity, and spontaneous emissions all arise from a single dynamical mechanism.

Clinical Significance of OAEs

DPOAEs and transient-evoked OAEs (TEOAEs) are now standard clinical tools for:

• • Newborn hearing screening: Universal screening programmes use OAEs as the first-tier test. Present OAEs indicate functional OHCs and rule out moderate-to-severe sensorineural hearing loss.
• • Monitoring ototoxicity: Aminoglycoside antibiotics and cisplatin chemotherapy damage OHCs. Serial DPOAE monitoring detects damage before it becomes symptomatic, allowing dose adjustment.
• • Differentiating cochlear vs. neural hearing loss: Present OAEs with absent auditory brainstem response suggest auditory neuropathy — a disorder of the IHC-to-nerve synapse or the auditory nerve itself.

The DPOAE "gram" plots emission amplitude as a function of $f_2$ frequency, providing a frequency-specific map of OHC function that correlates well with behavioural audiometric thresholds.

Middle Ear Mechanics: Impedance Matching

Before reaching the cochlea, sound must be transferred from air to the cochlear fluid. The acoustic impedance mismatch between air ($Z_{\text{air}} = \rho_{\text{air}} c_{\text{air}} \approx 413$ Pa·s/m) and water ($Z_{\text{water}} = \rho_{\text{water}} c_{\text{water}} \approx 1.5 \times 10^6$ Pa·s/m) would reflect most of the sound energy. The reflection coefficient is:

$$R = \left(\frac{Z_{\text{water}} - Z_{\text{air}}}{Z_{\text{water}} + Z_{\text{air}}}\right)^2 \approx 0.999$$

This means 99.9% of incident energy would be reflected — a 30 dB loss! The middle ear overcomes this through two mechanisms:

• • Area ratio: The tympanic membrane area ($A_{\text{TM}} \approx 55$ mm$^2$) is much larger than the stapes footplate ($A_{\text{stapes}} \approx 3.2$ mm$^2$). The pressure gain is $A_{\text{TM}}/A_{\text{stapes}} \approx 17$.
• • Lever ratio: The ossicular chain (malleus, incus, stapes) acts as a lever with advantage ~1.3.

Total pressure gain: $17 \times 1.3 \approx 22$, or about 27 dB — recovering most of the impedance mismatch loss. The transformer ratio is:

$$G_{\text{ME}} = \frac{A_{\text{TM}}}{A_{\text{stapes}}} \times \frac{l_{\text{malleus}}}{l_{\text{incus}}} \approx 22$$

Frequency Selectivity: Quality Factor

The sharpness of frequency tuning is characterised by the quality factor $Q$, defined as the centre frequency divided by the bandwidth at -3 dB (or -10 dB in auditory research, denoted $Q_{10\text{dB}}$):

$$Q_{10\text{dB}} = \frac{f_{\text{CF}}}{\Delta f_{10\text{dB}}}$$

In the passive cochlea (post-mortem or at high intensities),$Q_{10\text{dB}} \approx 1$–$3$. In the active cochlea at low sound levels, $Q_{10\text{dB}} \approx 5$–$15$at low frequencies, increasing to $\sim 20$–$50$ at high frequencies. This means at 10 kHz, the tuning bandwidth is only ~200–500 Hz — remarkably sharp for a biological system.

The equivalent rectangular bandwidth (ERB) of an auditory filter at centre frequency$f$ (in Hz) is well approximated by:

$$\text{ERB}(f) = 24.7 \cdot (4.37 f / 1000 + 1) \; \text{Hz}$$

At 1 kHz: ERB $\approx 133$ Hz. At 4 kHz: ERB $\approx 456$ Hz. The ERB scale provides a perceptually uniform frequency axis.

Cochlear Implants

Cochlear implants bypass damaged hair cells by electrically stimulating the auditory nerve directly. An electrode array is inserted into the scala tympani with electrodes spaced along the tonotopic axis. The electrode-nerve interface is governed by:

$$I_{\text{threshold}} = \frac{V_{\text{th}}}{\rho / (4\pi r)} = 4\pi r \sigma V_{\text{th}}$$

where $r$ is the electrode-neuron distance, $\sigma$ is tissue conductivity, and $V_{\text{th}} \approx 10$ mV is the neural threshold. Modern implants achieve 12–22 spectral channels, sufficient for speech comprehension.

The channel interaction problem limits cochlear implant performance. Current spreads through the conductive perilymph, so each electrode stimulates a broad region. The spatial spread of the electric field from a monopolar electrode decays as:

$$V(r) = \frac{I}{4\pi\sigma r} \exp(-r/\lambda)$$

where $\lambda \approx 3$–$5$ mm is the space constant in the cochlea. This broad current spread means effective spectral resolution is limited to 4–8 independent channels, well below the 22 physical electrodes. Bipolar and tripolar electrode configurations can narrow the current spread but require higher stimulation levels.

Hearing Aids: Amplification Strategy

Modern digital hearing aids implement frequency-dependent amplification that mimics the lost cochlear amplifier gain. The prescription formula maps audiometric hearing loss$HL(f)$ (in dB) to the required insertion gain:

$$G_{\text{target}}(f) \approx \frac{HL(f)}{2} \quad \text{(half-gain rule, simplified)}$$

The half-gain rule approximates the fact that OHC damage eliminates the cochlear amplifier (40–60 dB gain) but the neural threshold remains near normal. Wide dynamic range compression (WDRC) is used to restore the compressive nonlinearity of the healthy cochlea:

$$L_{\text{out}} = L_{\text{TK}} + \frac{L_{\text{in}} - L_{\text{TK}}}{CR}$$

where $L_{\text{TK}}$ is the compression threshold (knee-point, typically 40–50 dB SPL) and $CR$ is the compression ratio (typically 2:1 to 3:1). This provides more gain for soft sounds and less gain for loud sounds, restoring something like the healthy cochlea's compressive input-output function.

Noise-Induced Hearing Loss

Damage to OHCs from excessive noise exposure causes a loss of the cochlear amplifier gain. The damage threshold follows an equal-energy hypothesis: the total acoustic energy dose determines damage:

$$E = I \times t = \frac{p_{\text{rms}}^2}{\rho_0 c} \times t$$

OSHA regulations specify a permissible exposure limit of 85 dB SPL for 8 hours, with a 3 dB exchange rate: each 3 dB increase halves the permissible duration (since intensity doubles). At 100 dB, the safe exposure time is only 15 minutes.