**11. Parametric‐piezoelectric model of the cochlear amplifier**

In general, parametric amplification can be achieved in a resonant circuit, when one of its reactive elements (capacitance *C*(*t* ) or inductance *L*(*t* ), see **Figure 6**) changes in time. The variations of *C*(*t* ) and *L*(*t* ) will supply energy to the resonant circuit.

**Figure 6** shows the layout of a serial (electrical) resonance circuit whose capacitance *C*(*t* ), formed by two parallel plates, varies sinusoidally in time. The capacitance of this planar capacitor is modulated by moving up and down the upper plate of the capacitor. Here, the lower plate of the capacitor is fixed. Setting the upper plate of the capacitor into motion requires an additional external source of energy, which is called the pumping source (pump).

In the circuit of the electronic parametric amplifier (where the variable in time capacitance *C*(*t* ) is the varactor), this external energy is delivered from an electrical (AC) pumping signal

**Figure 6.** Illustration of the idea of a parametric amplifier that uses a time‐variable capacitance *C*(*t* ). The inductor *Ls* = const.

source of appropriate frequency, in relation to the frequency of the input signal *Eg* . Nonlinear (variable in time) capacitance (varactor) transfers energy from the pump circuit into the input signal circuit (*Rs* , *Ls* , *Cs* ). In this way the energy (power) of the input signal (represented by *Eg* ) is amplified and dissipated on the load resistance *RL* . At the same time, sharpening of the frequency characteristics (resonance curve) of the parametric amplifier occurs. These are char‐ acteristic features of the parametric amplifier.

#### **11.1. Proposed parametric: piezoelectric model of the OHC**

duce sustained "undamped" vibrations. The active elements in the cochlear amplifier, such as the proposed electromechanical transistors and parametric amplifiers based on the nonlinear capacitance of the OHC, can be responsible for the generation of periodic self‐sustaining vibra‐ tions in the inner ear. Under certain conditions any amplifying element can become a generator. Generation of the OAE signals in the OHCs occurs on the electric side. Subsequently, these signals through the inverse piezoelectric effect are transformed on the mechanical side. These mechanical (acoustic) signals leave the inner ear and can be received in the outer ear. The occurrence of the OAE phenomenon is an evidence that active processes in the cochlea do exist. The measurement of the OAE is now routinely employed for the detection of hearing

The phenomenon of the OAE can be treated as a side effect of operation of the cochlear

Between the inner and outer cell membrane of the OHC, there is a linear (static) capacitance and nonlinear capacitance. The linear capacitance has a constant value (∼30 pF), which does not depend on the applied voltage *u*. The nonlinear capacitance *C*(*u* ) depends on the voltage applied between the inner and outer wall of the OHC. The shape of this function resembles a bell curve, with a maximum value of approximately 25 pF. It is assumed that the nonlinear capacitance *C*(*u* ) is produced by the movement of confined charges in walls of the OHC.

This nonlinear capacitance of the OHC *C*(*u* ) will be used as an active element in the pro‐ posed parametric cochlear amplifier based on a single OHC. In classical electronics, nonlin‐ ear voltage‐dependent capacitance is called "varactor." By driving a nonlinear capacitance *C*(*u* ) with a the time‐dependent voltage *u*(*t* ), we obtain the capacitance *C*(*t* ) which is a func‐

In general, parametric amplification can be achieved in a resonant circuit, when one of its reactive elements (capacitance *C*(*t* ) or inductance *L*(*t* ), see **Figure 6**) changes in time. The

**Figure 6** shows the layout of a serial (electrical) resonance circuit whose capacitance *C*(*t* ), formed by two parallel plates, varies sinusoidally in time. The capacitance of this planar capacitor is modulated by moving up and down the upper plate of the capacitor. Here, the lower plate of the capacitor is fixed. Setting the upper plate of the capacitor into motion requires an additional external source of energy, which is called the pumping source (pump).

In the circuit of the electronic parametric amplifier (where the variable in time capacitance *C*(*t* ) is the varactor), this external energy is delivered from an electrical (AC) pumping signal

**11. Parametric‐piezoelectric model of the cochlear amplifier**

variations of *C*(*t* ) and *L*(*t* ) will supply energy to the resonant circuit.

impairment in newborns (in newborn hearing screening).

**10. Nonlinear capacitance of the OHC**

amplifier.

80 Advances in Clinical Audiology

tion of time.

A new physical (parametric‐piezoelectric) model of the OHC cochlear amplifier was pro‐ posed by the author in 2013 [43]. This model explains the mechanisms of power amplification and frequency selectivity that occur in the cochlear amplifier. The proposed model is a direct consequence of the idea of Gold. In the following a physical interpretation of the active ele‐ ment is given. The active element is related to specific physical (physiological) components of the cochlea. The model proposed by the author removes most of the deficiencies of the exist‐ ing models, presented in Sections 6–8.

Below, the proposed parametric amplifier model of the cochlea will be briefly described.

One OHC element is represented by a piezoelectric tube, see **Figure 7**. The left side of the OHC shown in **Figure 7** is the region of the OHC adjacent to the basilar membrane (BM). The right side of the OHC in **Figure 7** displays the apical part of the OHC in the vicinity of tecto‐ rial membrane (TM). Nonlinear capacitance *C*(*u* ) between the inner and the outer wall of the OHC is a key component of the proposed cochlear parametric amplifier.

#### **11.2. Operation of the proposed OHC parametric‐piezoelectric amplifier**

Force source *F*(*t* ), on the left side of **Figure 7**, represents an input acoustic signal, which acts on the OHC from the BM side. Through the direct piezoelectric effect this force source *F*(*t* ) is transformed into the electric side as an alternating current (AC) voltage or current source. The cylinder, which represents the structure of the OHC exhibits piezoelectric properties. On the right side of **Figure 7**, one can see OHC stereocilia located at the apical part of the

**Figure 7.** Simplified electromechanical diagram of a single OHC operating as a parametric amplifier. *F*(*t* ) represents the input acoustic signal, *ZL* is the electrical load impedance, *C*(*u* ) is the nonlinear capacitance of the OHC, and *u* is the voltage between the inner and outer walls of the OHC.

OHC. Movement of stereocilia causes the flow of ionic currents (through the ion channels) into the bulk of the OHC. Taking place here, the phenomenon of forward mechano‐electric transduction produces a transistor effect. As shown in **Figure 5**, the operation of the elec‐ tromechanical transistor generates an alternating current source (AC). This variable current source *I*(*t* ), which is also visible at the top right in **Figure 7**, acts as a pumping signal that pumps power to the nonlinear capacitance of the OHC *C*(*u* ), which is visible on the right side of **Figure 7**.

This nonlinear capacitance *C*(*u* ) operates in a parametric amplifier circuit. Transformed, on the electric side, the input acoustic signal is amplified in this parametric amplifier circuit. Apart from the power amplification, the phenomenon of the sound sharpening (sharp tuning) occurs here. This enhanced (on the electrical side) acoustic input signal is subsequently trans‐ formed into the mechanical side (inverse piezoelectric effect), where it performs useful work on the TM and BM. The work carried out on the mechanical side represents, on the electrical side, the power that dissipates in the output circuit on the load resistance *R <sup>L</sup>* . The electric power which is dissipated on the resistance *R <sup>L</sup>* is not a lost power, i.e., it is not transformed to heat. On the contrary, this is the useful power that represents the amplified (on the electrical side) power of the input acoustic signal.

Sequence of the physical phenomena that occur in an individual OHC is as follows:


(EMT transistor). In fact, in the EMT transistor, the changes in the mechanical deflection of stereocilia are transformed to changes in the channel conductance and consequently to changes in the ion channel current (K+ ions) flow, see **Figures 3a**, **b** and **4**. Power to the nonlinear capacitance *C*(*u*) is supplied by an AC electric pump signal represented by the variable current source *I*(*t* ). The mechanism of power gain in the EMT transistor is similar to that occurring in the electronic field effect transistor, with a modulated channel conduc‐ tance [40].


OHC. Movement of stereocilia causes the flow of ionic currents (through the ion channels) into the bulk of the OHC. Taking place here, the phenomenon of forward mechano‐electric transduction produces a transistor effect. As shown in **Figure 5**, the operation of the elec‐ tromechanical transistor generates an alternating current source (AC). This variable current source *I*(*t* ), which is also visible at the top right in **Figure 7**, acts as a pumping signal that pumps power to the nonlinear capacitance of the OHC *C*(*u* ), which is visible on the right

**Figure 7.** Simplified electromechanical diagram of a single OHC operating as a parametric amplifier. *F*(*t* ) represents

is the electrical load impedance, *C*(*u* ) is the nonlinear capacitance of the OHC, and *u* is the

This nonlinear capacitance *C*(*u* ) operates in a parametric amplifier circuit. Transformed, on the electric side, the input acoustic signal is amplified in this parametric amplifier circuit. Apart from the power amplification, the phenomenon of the sound sharpening (sharp tuning) occurs here. This enhanced (on the electrical side) acoustic input signal is subsequently trans‐ formed into the mechanical side (inverse piezoelectric effect), where it performs useful work on the TM and BM. The work carried out on the mechanical side represents, on the electrical

heat. On the contrary, this is the useful power that represents the amplified (on the electrical

**2.** The movement of BM acts on a corresponding OHC and causes its motion with respect to TM. As a consequence, deflection of the OHC stereocilia occurs, which triggers operation of the proposed electromechanical transistor. In this stage, the transformation of mechani‐

**3.** Subsequently, nonlinear capacitance of the OHC is charged (pumped) by an AC current source *I*(*t* ) ∼ cos*ωt*, generated at the output of the proposed electromechanical amplifier

. The electric

is not a lost power, i.e., it is not transformed to

side, the power that dissipates in the output circuit on the load resistance *R <sup>L</sup>*

Sequence of the physical phenomena that occur in an individual OHC is as follows:

**1.** The incoming acoustic signal sets in motion the BM and TM membranes.

power which is dissipated on the resistance *R <sup>L</sup>*

side) power of the input acoustic signal.

cal energy into electric energy occurs.

side of **Figure 7**.

the input acoustic signal, *ZL*

82 Advances in Clinical Audiology

voltage between the inner and outer walls of the OHC.

**6.** The output useful signal from the OHC amplifier is therefore a mechanical signal (force, velocity, or displacement) acting on the basilar membrane (BM) and tectorial membrane (TM) and as a consequence on stereocilia of the inner hair cells (IHCs). Finally, IHCs sen‐ sors transform mechanical signals into electrical signals (electric pulse trains) in afferent nerves connected to the central nervous system.

The power of the output mechanical signal in the OHC amplifier can surpass many times the power of an input acoustic signal. Since the parametric amplifier is a highly selective system, it can get a very narrow frequency characteristic (sharp tuning). More details con‐ cerning the operation of the proposed cochlear parametric amplifier can be found in the author's paper [43].

**Figure 8.** Nonlinear electrical equivalent (Norton) circuit of the proposed parametric cochlear amplifier built around a single OHC. *I g* is the input signal current source, *G g* is the source conductance, *GL* is the load conductance, *GS* is the loss conductance, *Ls* is the inductance of the OHC's resonator, *C*(*u* ) is the nonlinear capacitance, *I p* is the pumping current source, and *G p* is the pumping source conductance.

#### **11.3. Nonlinear Norton equivalent circuit of the proposed parametric cochlear amplifier**

The operation of a nonlinear oscillator resulting from the proposed model of the parametric cochlear amplifier can be described using the concept of a parallel (Norton) electrical equiva‐ lent circuit, see **Figure 8**.

The input electric signal, represented by the current source *I g* with an admittance *Gg* (see left side of **Figure 8**), corresponds to the input acoustic signal transformed into electrical side by the direct piezoelectric effect. The input electrical signal *I g* is subsequently amplified in the proposed parametric amplifier (formed with the nonlinear capacitance *C*(*u* )). The electric power is supplied to the circuit by the pumping current source *I p* , which is generated by the forward mechano‐electric transduction effect (see right side of **Figure 8**). After amplification, the electric signal is dissipated at the output load conductance *GL* . The dissipated power is a useful output power of the proposed parametric amplifier.

It is noteworthy that mathematical description of the operation of the electrical circuit presented in **Figure 8** is a nonlinear ordinary differential equation of the second order. This equation results from Kirchhoff's laws applied to the circuit in **Figure 8**. The solution of this nonlinear equation describes nonlinear properties of the OHC amplifier for an arbitrary level of signals (small and large). For low‐level signals, the solution of this equation should display an enhanced value of the quality factor and therefore higher value of amplification of the input signal.

#### **11.4. Negative conductance**

To explain the power amplification phenomenon in the parametric amplifier the concept of negative conductance was introduced. Negative conductance −*G <sup>a</sup>* occurs in parallel to the non‐ linear capacitance *C*(*u* ) that represents an active element in the parametric amplifier. Positive conductance (resistance) dissipates electric power. By contrast, negative conductance supplies energy to the circuit. The negative conductance represents energy transfer from an external source to the circuit of the resonator, thereby the reduction of attenuation (undamping) of the resonant circuit occurs. In this way, the negative conductance characterizes an active element in the parametric amplifier circuit (e.g., the nonlinear capacitance of the OHC). The resultant conductance of the parallel OHC resonant circuit decreases (*G* = *G <sup>s</sup>* − *G <sup>a</sup>* ) which increases the quality factor of the resonant circuit, see right side of **Figure 9**. As a result, a sharpening of the resonance curve of the resonant circuit occurs. This leads to higher sensitivity and selectivity

**Figure 9.** Operation of the active element, based on the nonlinear (time‐varying) capacitance of an OHC, produces a negative conductance −*Ga* , which reduces the loss of the OHC resonant circuit and increases its quality factor.

(i.e., the ability to distinguish two tones with nearly the same frequencies, e.g., 1000 and 1005 Hz) of the proposed cochlear amplifier, as well as its power gain.

It is known from the classical circuit theory that parametric effect introduces an effective neg‐ ative conductance (resistance) to the resonant parametric circuit, see **Figure 9**.

In summary, electromechanical parametric amplifier built with a single OHC performs the following functions:

**1.** amplifies the power of the input acoustical (electrical) signal,

**11.3. Nonlinear Norton equivalent circuit of the proposed parametric cochlear amplifier**

The input electric signal, represented by the current source *I*

by the direct piezoelectric effect. The input electrical signal *I*

power is supplied to the circuit by the pumping current source *I*

the electric signal is dissipated at the output load conductance *GL*

negative conductance was introduced. Negative conductance −*G <sup>a</sup>*

conductance of the parallel OHC resonant circuit decreases (*G* = *G <sup>s</sup>* − *G <sup>a</sup>*

useful output power of the proposed parametric amplifier.

lent circuit, see **Figure 8**.

84 Advances in Clinical Audiology

**11.4. Negative conductance**

negative conductance −*Ga*

The operation of a nonlinear oscillator resulting from the proposed model of the parametric cochlear amplifier can be described using the concept of a parallel (Norton) electrical equiva‐

side of **Figure 8**), corresponds to the input acoustic signal transformed into electrical side

the proposed parametric amplifier (formed with the nonlinear capacitance *C*(*u* )). The electric

forward mechano‐electric transduction effect (see right side of **Figure 8**). After amplification,

It is noteworthy that mathematical description of the operation of the electrical circuit presented in **Figure 8** is a nonlinear ordinary differential equation of the second order. This equation results from Kirchhoff's laws applied to the circuit in **Figure 8**. The solution of this nonlinear equation describes nonlinear properties of the OHC amplifier for an arbitrary level of signals (small and large). For low‐level signals, the solution of this equation should display an enhanced value of

To explain the power amplification phenomenon in the parametric amplifier the concept of

linear capacitance *C*(*u* ) that represents an active element in the parametric amplifier. Positive conductance (resistance) dissipates electric power. By contrast, negative conductance supplies energy to the circuit. The negative conductance represents energy transfer from an external source to the circuit of the resonator, thereby the reduction of attenuation (undamping) of the resonant circuit occurs. In this way, the negative conductance characterizes an active element in the parametric amplifier circuit (e.g., the nonlinear capacitance of the OHC). The resultant

quality factor of the resonant circuit, see right side of **Figure 9**. As a result, a sharpening of the resonance curve of the resonant circuit occurs. This leads to higher sensitivity and selectivity

**Figure 9.** Operation of the active element, based on the nonlinear (time‐varying) capacitance of an OHC, produces a

, which reduces the loss of the OHC resonant circuit and increases its quality factor.

the quality factor and therefore higher value of amplification of the input signal.

*g*

*g*

*p*

with an admittance *Gg*

is subsequently amplified in

, which is generated by the

. The dissipated power is a

occurs in parallel to the non‐

) which increases the

(see left

