The elongated cochlea is tonotopically organized such that high sound frequencies are detected towards the basal end and low sound frequencies towards the apical end. Hereafter, the “apex” and the “base” denote the locations in the gerbil cochlea with best frequencies (BF) of near 0.6 kHz and 19 kHz respectively. The gerbil’s basilar membrane is 12 mm long and its audible frequency range extends from about 0.25 to 45 kHz 
. The basal location, although having a BF at least an octave below the upper frequency limit for the gerbil, was chosen because it is the most basal location for which electrical and mechanical data exist. The x-axis and y-axis in the model correspond to the radial (neural-abneural) and transverse (normal to the basilar membrane plane) directions respectively ().
Amplification by OHC–Impulse Response of the Cochlear Partition
Amplification was primarily achieved by OHC somatic motility. In order to compare how two types of OHC active force, hair bundle force originating from mechano-transduction apparatus 
and somatic force attributable to the membrane protein prestin 
, affect the cochlear partition vibration, four different cases were simulated (). The four cases are: control (with both somatic and hair bundle motors), without hair bundle force (only somatic force), without somatic force (only hair bundle force) and passive (without either force). The mechanotransduction channel kinetics and the transduction current were common to all four cases. A finite element model of the cochlear partition from the basal turn of gerbil cochlear coil, a 600 µm piece centered at ~19 kHz BF, was simulated. The length of coil simulated is large compared to the space constants for longitudinal coupling measured in the gerbil cochlea 
. When a 0.5 nN-ms impulse stimulus was applied to the basilar membrane, the cochlear partition vibrated at its BF, governed by the stiffness of the basilar membrane and the mass carried. In the control, it took 41 oscillating cycles in 2.1 ms before the oscillation dissipated below 5 percent of the peak. Without the hair bundle force, it took 32 cycles in 1.6 ms. When the OHC somatic force was not fed back to the OC mechanics, the oscillations dissipated within 4–6 cycles. Thus the somatic motor made by far the largest contribution to tuning and the hair bundle force amplification only when present together with the OHC somatic force. Similar results were obtained at the apex (data not shown) where, following an impulse stimulus, the apical section oscillated at 0.6 kHz. It took 9, 8, 5 and 5 oscillations before settling down below 5 percent of the peak for control, without hair bundle force, without OHC somatic force and without any force feedback from OHCs. Therefore, the amplification is dominated by OHC somatic motility at both apex and base.
Impulse response of the cochlear partition.
In the basal section, about 40 percent of the OHC transducer channels were open at rest 
, which created a 5.1 nA resting or ‘silent’ transducer current. The balance between the resting transducer current and outwardly rectifying membrane potassium current resulted in a −54 mV resting membrane potential. After 1 ms of impulse application, one nanometer of basilar membrane displacement resulted in 0.65 mV OHC receptor potential ( bottom row). A single OHC in the middle of the simulated partition generated 6 pN force out of the mechano-transduction apparatus, and 60 pN out of the somatic motility per one nm basilar membrane displacement. The hair bundle force led the basilar membrane displacement by 61 degrees while the somatic force lagged by 7 degrees.
The inclusion of somatic force of the OHCs resulted in higher oscillation frequencies (19.5 kHz, ) than the other cases (16–19 kHz, ). All parts of the cochlear partition at a radial section (in the x-y plane) vibrated with the same frequency when the OHC somatic force was dominant (or when small stimulation was applied at the BF of the site). However, when no OHC somatic force was fed back to the OC (), two different frequencies were excited immediately after the impulse and settled down to the lower frequency. In the two to three oscillations after the impulse, the tectorial membrane vibrated faster in the transverse direction (yTM oscillates at ~20 kHz) than in the radial direction (xTM oscillates at ~17 kHz).
There was a notable phase difference between responses with and without OHC somatic motility (bottom row of ). The transverse displacement of the tectorial membrane (yTM, orange line) led the basilar membrane displacement (yBM, black line) by ~90 degrees only when the OHC somatic force was incorporated. Note that the phase of the tectorial membrane radial displacement (xTM, green line) was little affected by OHC motility. In order to explain how these phase relations are determined, we investigated the response of an individual OHC ().
Response of a single OHC to hair bundle stimuli.
Response of an Isolated OHC
An individual OHC was simulated in order to observe the relationship between hair bundle displacement, transducer current and receptor potential (). To impose the proper mechanical impedance, we obtained the OC stiffness felt by the OHC from the finite element model, which was comparable to the OHC stiffness itself. Recently measured membrane electrical properties were used throughout this study (, 
Outer hair cell electrical properties.
Three cases were tested. Firstly, the OHC hair bundle was stimulated with sinusoidal force at different frequencies and the amplitude of normalized transduction current was observed (). The mechano-transduction apparatus produces a broad band-pass filter peaking at about 1 kHz and 10 kHz (apex and base respectively, ). Three factors shape this filter: activation and adaptation kinetics of the mechanotransducer channels and viscous damping on the hair bundle (Table S1 in Supporting Information S1
). Secondly, a single OHC was electrically stimulated by modulating the mechanotransducer current to measure the membrane potential (, lines with •).
A 5 percent modulation of the resting transduction current resulted in 3 mV and 2.5 mV OHC receptor potentials at the apex and base respectively. Because the membrane behaves as a first-order low pass filter, the phase lag develops from zero to 90 degrees as the stimulation frequency increases. The 3 dB cut-off frequency was 0.4 kHz (apex) and 7.3 kHz (base). Note that the cut-off frequency of the basal OHC is more than an order of magnitude higher than was previously believed. When the full transducer current was used, the maximum receptor potential was 50 mV (apex) and 35 mV (base). When stimulated at its BF (broken lines, ), the OHC receptor potential lagged the transduction current by 64 (apex) and 69 (base) degrees. Our model parameters result in different filtering frequencies for the OHC hair bundle and for the cell body. Thirdly, the hair bundle was mechanically stimulated and the receptor potential was measured in order to observe the combined effect of hair bundle transduction and membrane current (, solid lines without marker). The hair bundle was oscillated with amplitudes of 3.3 and 1 nm (at apex and base respectively) at different frequencies and the receptor potential was observed. The hair bundle displacements were chosen to yield ~5 percent amplitude change of transducer current that is comparable to the first two simulations. The receptor potential lagged the hair bundle displacement by 49 degrees (apex) and 96 degrees (base). Because the kinetic step between the membrane voltage change and ensuing somatic motility is very fast (>50 kHz, 
), these phase relations can be considered to be the relations between the hair bundle mechanical stimulation and the OHC somatic reaction. To summarize, a single OHC responded like a low pass filter with a half-power frequency that was lower than the BF of the location. The sharpness of tuning contributed by the transduction apparatus is meager but, nevertheless, this affects the phase difference between the hair bundle displacement and the receptor potential of the OHC.
Response of the Cochlear Partition to Pure Tone Stimuli
The apical and basal cochlear partitions were stimulated with sinusoidal force applied to the basilar membrane (). With a small stimulating force, the partition movement peaked at frequencies of 0.60 kHz (apex) and 19.5 kHz (base). The amplification is close to 30 dB at the base and 10 dB at the apex. As the stimulus level was increased, the importance of amplification declined and the responses approached the passive condition. At the apex, the BF shifted to 0.55 kHz as the simulating level increased. At the base, the basilar membrane response of the passive OC had two peaks, one at ~17 kHz and the other at ~20 kHz, which correspond to the resonant frequencies of the impulse response (), with the lower frequency peak being dominant. The response of the reticular lamina for the passive basal cochlear partition peaked at ~ 20 kHz. As a result, in the intermediate stimulus level, the basilar membrane response peaked at a slightly lower frequency than the reticular lamina, agreeing with recent observations 
Response of cochlear partition to pure tone stimuli.
The basilar membrane vibrated in phase with the stimulus at low frequencies below the BF and lagged the stimulus by 180 degrees at frequencies above BF (). In the passive case, the reticular lamina was in phase with (apex) or lagged (base) the basilar membrane. When the OHC mechanical feedback was turned on, the reticular lamina displacement led that of the basilar membrane (). The extent of the phase lead decreased as the stimulus frequency and level increased. At the apical cochlear partition, the phase difference (at 0.6 kHz) decreased from 76 to 10 degrees as the stimulus level increased 1000 -fold. At the base, the reticular lamina-basilar membrane phase difference (at 19.5 kHz) decreased from 83 to 43 degrees as the stimulus level increased by 500 times. These results indicate the phase relations between the reticular lamina and basilar membrane during basilar membrane vibration depend crucially on the action of the somatic motor. For low stimulus levels, when the contribution of the OHC somatic motility is significant, the reticular laminar leads the basilar membrane motion by about 90 degrees.
Inappropriate OHC Membrane Properties Disable the Amplification by the OHCs
To test the importance of the OHC membrane electrical properties for amplification, these membrane properties were exchanged between the apical and basal cochlear partition models (). In other words, the OHC membrane capacitance and conductance measured at the base were assigned to the apex and vice versa. As a result, the OHC membrane RC filtering frequency was higher (apex, 7.3 kHz) or lower (base, 0.4 kHz) than the resonant frequencies of the cochlear partitions. In both cases, the responses became close to those of the passive system. However, the reason for disabling the amplification is different at the two locations. When the membrane electrical time constant was too high (apical OHC with basal properties), despite a comparable receptor potential to the control case (-apex), the incorrect phase difference between the RL and the BM motion (32 degrees, -apex) prevented amplification. The incorrect phase relation is ascribed to a reduced phase lag between the OHC transduction current and the receptor potential (from 64 degrees to nearly zero degrees). When the membrane electrical time constant was too low (basal OHC with apical properties), the OHC receptor potential was greatly reduced due to low pass filtering (-base). As a result, too small OHC a somatic force was recruited to amplify the vibration. This result implies that there exists an optimal range of OHC membrane electrical characteristics and the OHC membrane conductance should increase with the BF in order for the OHCs to fulfill their role as an amplifier, which agrees with the experimental data 
Effect of inappropriate electrical properties of the outer hair cell membrane.
The importance of the electrical properties was further explored by asking whether there exists an optimal range for the OHC basolateral membrane conductance (GM
) which is conferred by voltage-dependent K+
channels. The apical and basal cochlear partitions were stimulated with sinusoidal force applied to the basilar membrane (0.6 and 19.5 kHz respectively), and siumulaneously the conductance of the OHC basolateral membrane was increased with time (). For the apical model, the maximum conductance GM,max
was increased from 10 to 80 nS during the 1000 ms simulation and for the basal model, it was increased from 18 to 800 nS in 106 ms. The membrane potential changed from −7 to −50 mV at the apex and from −3 to −60 mV at the base. Due to its voltage dependence, GM
increased from 11 to 67 nS at the apex and from 30 to 480 nS at the base (). As the conductance was increased, there was a steady deflection of the basilar membrane towards the scala tympani and the vibration amplitude varied non-monotonically (). The basilar membrane vibration was greatest when the GM
was 30 nS and 160 nS for the apex and the base respectively (). The 3 dB band of the GM
was 21–45 nS for the apex and 97–292 nS for the base (double arrow). Our default GM
value is within this range (indicated with
). Variation in the OHC receptor potential with change in GM
showed a similar trend to the basilar membrane displacement (). This result indicates that there is an optimal range of OHC basolateral membrane conductances to achieve cochlear amplification, and that the optimal conductance value is higher at the high BF location which agrees with experimental observation 
Optimal OHC membrane conductance to amplify basilar membrane vibrations.
Because the mechanical and electrical data on the gerbil cochlea are available only up to about 19 kHz, it was not possible to perform experiment-based simulations at higher frequencies. However, if existing parameters were extrapolated to the upper frequency limit of the gerbil cochlea, then significant (20 dB) amplification, reliant on the OHC somatic motor, could still be achieved provided the basolateral conductance GM
was also increased. Sharp tuning at a BF of 41 kHz was generated with GM
500 nS but was reduced with larger or smaller conductance values. Interestingly, the value of GM
needed for amplification is close to that obtained by extrapolating the electrical measurements 
. This suggests that the same principle of optimizing the electrical properties will apply even at the most basal locations in the rodent cochlea.
Kinematic Gain of the OC
Because of its detailed structural and electrical representation, our fully-deformable finite element model provides more information than other lumped parameter models obtained through kinematic analysis. OC mechanical and electrical responses per 1 nm basilar membrane displacement are summarized in . Interestingly, active feedback by the OHCs did not improve the kinematic gain of the OC structures such as the displacement of hair bundle, reticular lamina and tectorial membrane per unit basilar membrane displacement. This small kinematic gain is ascribed to a compliant tectorial membrane (comparable to the lower bound of reported values such as 
). However, we previously found 
that the kinematic gain of the OC depended on the tectorial membrane stiffness. shows that the dynamic responses of the OC are different depending on whether a stiff or compliant tectorial membrane is assumed. With the former, the transverse motion predominated, whereas the radial and transverse displacement amplitudes were comparable with a compliant tectorial membrane. In this study, the compliant tectorial membrane was used for two reasons. Firstly, despite a smaller kinematic gain, the OC with a compliant tectorial membrane results in a more sharply tuned motion with active OHC feedback. Secondly, the shift in BF with the action of the OHCs corresponds better to experimental observations when the tectorial membrane is compliant, the OC response peaking at a higher frequency when the OHC somatic motor is functional.
OC responses per 1 nm basilar membrane displacement.
Effect of tectorial membrane stiffness for active and passive responses at the base.