Conceived and designed the experiments: RDR BF. Performed the experiments: RDR KDB BF. Analyzed the data: RDR SC KDB BF WEB. Contributed reagents/materials/analysis tools: RDR SC. Wrote the paper: RDR.
The authors have declared that no competing interests exist.
Cochlear outer hair cells (OHCs) are fast biological motors that serve to enhance the vibration of the organ of Corti and increase the sensitivity of the inner ear to sound. Exactly how OHCs produce useful mechanical power at auditory frequencies, given their intrinsic biophysical properties, has been a subject of considerable debate. To address this we formulated a mathematical model of the OHC based on first principles and analyzed the power conversion efficiency in the frequency domain. The model includes a mixture-composite constitutive model of the active lateral wall and spatially distributed electro-mechanical fields. The analysis predicts that: 1) the peak power efficiency is likely to be tuned to a specific frequency, dependent upon OHC length, and this tuning may contribute to the place principle and frequency selectivity in the cochlea; 2) the OHC power output can be detuned and attenuated by increasing the basal conductance of the cell, a parameter likely controlled by the brain
The sense of hearing is exquisitely sensitive to quiet sounds due to active mechanical amplification of sound-induced vibrations by hair cells within the inner ear. In mammals, the amplification is due to the motor action of “outer hair cells” that feed mechanical power into the cochlea. How outer hair cells are able to amplify vibrations at auditory frequencies has been somewhat of a paradox given their relatively large size and leaky electrical properties. In the present work, we examined the power conversion efficiency of outer hair cells based on first principles of physics. Results show that the motor is highly efficient over a broad range of auditory frequencies. Results also show that the motor is likely controlled by the brain in a way that allows the listener to focus attention on specific frequencies, thus improving the ability to distinguish sounds of interest in a noisy environment.
Outer hair cells (OHC) in the mammalian cochlea are essential to the remarkable sensitivity of hearing. These highly specialized cells actively feed mechanical power into the organ of Corti and amplify its mechanical vibrations in response to sound
OHC somatic electromotility is driven by the MET current entering the cell and likely draws thermodynamic power from the electo-chemical potential between fluid compartments in the cochlea. The apical surfaces of OHCs are bathed in high-potassium endolymph, biased to approximately +50 to +80 mV, and their basal poles bathed in high-sodium perilymph at 0 mV reference. This endocochlear potential is maintained by the stria vacularis and associated cells
A) The intracellular space was modeled as an axial conductor with resistance
New results include the force vs. velocity, and power vs. velocity curves for OHCs (c.f. skeletal muscle cells
Experimental procedures and animal care were designed to advance animal welfare and were approved by the Baylor College of Medicine animal care and use committee.
Our primary objective was to estimate what fraction of the electrical power entering the soma is converted into useful mechanical power output, and to estimate how this conversion efficiency would vary with frequency and biophysical parameters. It has not yet been technically possible to directly measure the electrical to mechanical power conversion efficiency of the OHC. The primary challenge is that one must measure the MET current, membrane potential, mechanical force generated and mechanical strain and velocity, all simultaneously and under physiologically relevant mechanical loading conditions. Therefore, we applied first principles of physics to formulate a relatively simple mathematical model of the OHC that reproduces all key published experimental data using a single set of physical parameters. The same model was then applied to compute the power conversion efficiency.
The OHC lateral wall, where the motor elements are located
We further simplified the model by treating the OHC lateral wall as a thin shell undergoing axisymmetric deformations (
This composite material is distinct from ideal piezoelectric materials due to the presence of a membrane conductance, and because the motor itself only occupies a fraction of the membrane surface area while the remainder is occupied by passive elastic material
In piezoelectric materials occurring in nature, the coefficient
Conservation of momentum in the axial direction can be written
Electrically, the OHC was modeled as a cylinder filled with conducting cytoplasm and immersed in a conducting fluid media (
Four types of stimuli and three loading conditions were considered. For stimuli, we considered voltage clamp (VC) of the basal region of the cell (
In a subset of simulations we estimated the velocity and force for physiological hair bundle movements. The OHC transduction current appears to adapt very rapidly to step hair bundle displacements (time constant on the order of 100 micro-seconds), and the adaptation may be nearly 100% complete in some cells
Equations 8–10 define the model and were solved in the frequency domain using an eigenvector expansion. The model equations were solved by first considering the a solution in the form:
The coefficients
The power efficiency was defined as the mechanical power output divided by the electrical power input. The mechanical power output is computed in the frequency domain using
The present model has some features similar to previous piezoelectric-like models of the OHC
Dissipative drag from the cytoplasm and the extracellular space are unavoidable. As a first approximation we modeled the axial component of the drag acting on the plasma membrane using a version of the Navier-Stokes equations. Assuming small displacements from the resting configuration, and ignoring the convective nonlinearity, the Navier-Stokes equations reduce to
Model parameters were estimated from known dimensions and physical constants combined with voltage clamp and mechanical data shown in
A) Model predictions for the nonlinear capacitance based on the Boltzmann piezoelectric distribution compared to data from Kakehata & Santos-Sacchi
A) Input capacitance and B) resistance of two 50 µm long OHCs measured with patch pipettes attached at the base begin to roll-off at high frequencies. Error bars denote one standard deviation of the capacitance at each frequency tested. Solid curves show model results. The capacitance begins to roll off above ∼1 kHz. The roll off is captured by the model due to a loss of space clamp that occurs at higher frequencies. These data were used to estimate the intracellular axial electrical resistance of the cell.
Sinusoidal control of the extracellular voltage around the base of OHCs (microchamber configuration) evokes movment proportional to the voltage and dependent upon cell length. Symbols show microchamber data from Frank et al.
A) The zero-load velocity gain and B) phase are shown as functions of frequency for an 80 µm long OHC. Symbols replot data from by Frank et al.
Maximum force is predicted to occur under isometric conditions (zero velocity), and maximum velocity is predicted to occur under zero load – both extremes require MET electrical power input but result in zero mechanical power output. The mechanical power output is shown as a function of velocity (solid parabolic curve) along with the electrical power input
A) The taxonomy of electrical to mechanical power conversion efficiency delineating regions where input electrical power is lost to series-elastic piezoelectric coupling, OHC stiffness, fluid viscosity, entrained mass, and OHC intracellular axial electrical resistance for a 28 µm long cell. Results are shown under control conditions when the base of the OHC has a high impedance (solid red, cross-hatch, high
OHCs vary their length systematically with the place-principle of best frequency sensitivity in the cochlea. A) Anatomical lengths of hair cells in the cochlea (red symbols connected by lines,
Experimental procedures and animal care were designed to advance animal welfare and were approved by the Baylor College of Medicine animal care and use committee. All physical parameters were deduced from the published literature, with the exception of the intracellular electrical resistance,
When the lateral wall deforms there is a compensatory electrical charge movement due to deformation in the motor portion of the membrane – behavior that is fundamentally piezoelectric in nature
The input capacitance of OHCs measured at the base is nearly constant below 1 kHz, but begins to roll-off as the interrogation frequency is increased (
The predictive capability of the model is further illustrated in
Most experimental data addressing OHC electromotility are collected under conditions of isometric length (
Classical piezoelectricity is thermodynamically conservative
Activation of the medial olivocochlear efferent bundle reduces mechanical amplification by outer hair cells
The present model also addresses how the brain likely controls mechanical power output of OHCs through efferent mediated ionic conductances at the base of the cell. Electrical current entering the MET channels is divided into two parts. The first part drives charge displacement in the lateral wall and is responsible for somatic electromotility through the piezoelectric effect (Eq. 2). The second part of the current exits the base of the cell through conductive ionic channels. If the ion channels are closed (high
Shorter cells exhibited their best efficiency at high frequencies while longer cells exhibited their best efficiency at low frequencies.
There are four major observations that can be drawn from the present work. The first addresses how OHCs operate at high frequencies given their electrical capacitance
The second observation is that OHCs may be tuned to maximize their power output at a best frequency, albeit broadly tuned. Although OHC displacement and force are quite flat over a broad range of frequencies when driven by voltage (e.g.
The third observation addresses how the MET channels would be expected to further tune output of the somatic motor. MET adaptation generates high-pass filtered MET currents
The fourth observation is that OHC somatic power output may be controlled by the brain
Finally, it is important to note that the OHC somatic motor is not present in non-mammals, yet these animals also exhibit many of the properties of the mammalian cochlear amplifier