**9. Power flow in the OHC**

#### **9.1. Introduction of the concept of an electromechanical transistor**

A single OHC element represents one of 20,000 active elements in the cochlear amplifier. The most important feature of the active element is its ability to amplify the power of an input acoustical signal. It is amazing that by analyzing power flow in the OHC, the author arrived to the new concept of an electromechanical transistor.

The analysis performed in this section is based on the phenomenon of forward mechano‐elec‐ tric transduction that occurs in the apical part of the OHC.

#### *9.1.1. Power amplifier*

Here, we will present the conditions which must be fulfilled by the system (device) to be a power amplifier. Such a system must include the following components:


The principle of operation of the amplifier is based on the control of a higher output power by a lower input power. In our case, see **Figure 3a** and **b**, the low power *P*in in the mechanical

**Figure 3.** (a) Structure of the proposed electromechanical biological transistor, built around a single OHC. Arrows indicate the flow of K+ ionic electric current in the electromechanical transistor (amplifier), *R*(*t* ) is a time‐varying channel resistance, which represents the electromechanical control element (EMCE), and *RL* is a load resistor. BM is the basilar membrane, TM is the tectorial membrane, and SV denotes the stria vascularis, (b) electrical equivalent circuit of the OHC's electromechanical transistor, electromotive force *E*<sup>2</sup> = 150 mV represents the difference of an endocochlear potential and intracellular potential inside the OHC.

input circuit (stereocilia of the OHC) controls the high power flow from the stria vascularis battery to the output electrical circuit, *P*out. As it will be seen later, all of the above conditions are satisfied by the system composed of the following elements:


#### *9.1.2. Electromechanical amplifier*

The active electromechanical device, presented in **Figure 3a**, is composed of the initial part of the OHC and the entire amplifying tract of the cochlear amplifier that begins with BM deflec‐ tion and terminates in IHC, where the amplified mechanical signal is transformed into an electrical signal stimulating the afferent nerves.

**Figure 3a** presents schematic view of the proposed electromechanical (biological) transis‐ tor, built around a single OHC. To illustrate the principle of operation of the new amplify‐ ing device, dimensions of channels at the upper (apical) part and in the lower (basolateral) part of the OHC in **Figure 3a** are greatly exaggerated. Operation of the channels at the apical part and at the basolateral part of the OHC is modeled by one resultant (effective) channel, i.e., one effective channel for the apical part and one effective channel for the basolateral part. The flow of potassium ions K+ is of crucial importance for electrical phenomena occur‐ ring in the OHC. The current of potassium ions K+ , see **Figure 3a**, flows in a closed circuit starting from the stria vascularis (DC voltage source) to the body of the OHC and then back to the stria vascularis. The top apical channels play a role of a controlled (time‐varying) resistance *R*(*t* ).

#### *9.1.3. Electromechanical control element (EMCE)*

The time‐variable resistance *R*(*t* ) in **Figure 3a** and **b** is controlled by a time‐varying input acoustic signal, i.e., particle velocity *v*(*t* ) and/or mechanical force *F*(*t* ). As a consequence, the current flowing through the load resistance *RL* varies in time unison with the changes of the input acoustic signal. This time‐variable resistance *R*(*t* ) can be identified as an electrome‐ chanical controlled element (EMCE), see Ref. [40].

#### *9.1.4. Load resistance*

input circuit (stereocilia of the OHC) controls the high power flow from the stria vascularis battery to the output electrical circuit, *P*out. As it will be seen later, all of the above conditions

**Figure 3.** (a) Structure of the proposed electromechanical biological transistor, built around a single OHC. Arrows

membrane, TM is the tectorial membrane, and SV denotes the stria vascularis, (b) electrical equivalent circuit of the OHC's electromechanical transistor, electromotive force *E*<sup>2</sup> = 150 mV represents the difference of an endocochlear

ionic electric current in the electromechanical transistor (amplifier), *R*(*t* ) is a time‐varying channel

The active electromechanical device, presented in **Figure 3a**, is composed of the initial part of the OHC and the entire amplifying tract of the cochlear amplifier that begins with BM deflec‐ tion and terminates in IHC, where the amplified mechanical signal is transformed into an

**Figure 3a** presents schematic view of the proposed electromechanical (biological) transis‐ tor, built around a single OHC. To illustrate the principle of operation of the new amplify‐ ing device, dimensions of channels at the upper (apical) part and in the lower (basolateral) part of the OHC in **Figure 3a** are greatly exaggerated. Operation of the channels at the apical part and at the basolateral part of the OHC is modeled by one resultant (effective) channel, i.e., one effective channel for the apical part and one effective channel for the basolateral

starting from the stria vascularis (DC voltage source) to the body of the OHC and then back to the stria vascularis. The top apical channels play a role of a controlled (time‐varying)

is of crucial importance for electrical phenomena occur‐

, see **Figure 3a**, flows in a closed circuit

is a load resistor. BM is the basilar

are satisfied by the system composed of the following elements:

resistance, which represents the electromechanical control element (EMCE), and *RL*

**2.** Stereocilia + ionic channels located in the apical part of the OHC

**3.** Ionic channels located in the basolateral part of the OHC

electrical signal stimulating the afferent nerves.

ring in the OHC. The current of potassium ions K+

**1.** Stria vascularis

indicate the flow of K+

74 Advances in Clinical Audiology

*9.1.2. Electromechanical amplifier*

potential and intracellular potential inside the OHC.

part. The flow of potassium ions K+

resistance *R*(*t* ).

The basolateral channels in **Figure 3a** are modeled by a load resistance *RL* , which represents the output power receiver. The power dissipated on the load resistance *RL* is not a lost power, but in contrary constitutes a useful power, which pumps energy into the parametric electro‐ mechanical amplifier, based on the nonlinear capacitance of the OHC. This will be shown in more details in Sections 11–13 of this chapter.

**Figure 3b** shows an equivalent circuit model of the amplifying electromechanical device pre‐ sented in **Figure 3a**. In this circuit model we can identify a source of potential energy (DC battery), an active control element (electromechanical transistor) represented by a controlled (time‐varying) resistance *R*(*t* ), and finally a load resistance *RL* .

#### *9.1.5. Power flow in the electromechanical amplifier*

Power flow in the electromechanical amplifier, based on the phenomenon of the forward mechano‐electrical transduction, which is triggered by the movement of the OHC stereocilia, is as follows:


The physical foundation of operation of the proposed electromechanical transistor is the phe‐ nomenon of forward mechano‐electrical transduction, taking place in the apical part of the OHCs (stereocilia + ion channels).

#### *9.1.6. Input and output circuits*

The input circuit of the proposed electromechanical transistor is a mechanical circuit consist‐ ing of the OHC's stereocilia. Input control signal is the particle velocity and/or the mechanical force exerted on the stereocilia. The output circuit is an electrical circuit, see **Figure 3a** and **b**. Electrical output controlled signal is the current and/or the voltage across the load resistance *RL* . The electric circuit in **Figure 3a** and **b** closes through the structures lying outside the OHC (back to the stria vascularis).

#### *9.1.7. Electromechanical transistor (EMT)*

The electromechanical amplifying system in **Figure 3a** and **b** satisfies four necessary condi‐ tions for power gain to occur, namely:


Thus, this electromechanical controlled element *EMCE* = *R*(*t* ) represents an electromechani‐ cal transistor. This transistor is a close analog of the electronic unipolar field effect transistor (FET), see **Figure 4a** and **b**.

#### **9.2. Analogy between the proposed EMT type transistor and the FET type transistor**

The proposed electromechanical transistor is analogous to the classical field effect transistor (FET). Certainly, the proposed electromechanical transistor resembles also a vacuum tube or a bipolar transistor, but according to the author, similarity in this case is less direct (explicit). This is due to the following reasons:


**d.** In case of the electromechanical transistor (EMT), the role of the channel is played by ion channels existing in the apical part of the OHC. Their resistance *R*(*t* ) varies depending on the degree of opening or closing of the channels (deviation of stereocilia). This resistance exists physically and could be measured using an ohmmeter. The resistance *R*(*t*) is modu‐ lated by an input acoustic (mechanical) signal reaching OHC.

force exerted on the stereocilia. The output circuit is an electrical circuit, see **Figure 3a** and **b**. Electrical output controlled signal is the current and/or the voltage across the load resistance

. The electric circuit in **Figure 3a** and **b** closes through the structures lying outside the OHC

The electromechanical amplifying system in **Figure 3a** and **b** satisfies four necessary condi‐

**2.** There is a device (electromechanical controlled element – electromechanical transis‐

**4.** There is an electric controlled output circuit (ion channels in the basolateral part of the OHC). Thus, this electromechanical controlled element *EMCE* = *R*(*t* ) represents an electromechani‐ cal transistor. This transistor is a close analog of the electronic unipolar field effect transistor

The proposed electromechanical transistor is analogous to the classical field effect transistor (FET). Certainly, the proposed electromechanical transistor resembles also a vacuum tube or a bipolar transistor, but according to the author, similarity in this case is less direct (explicit).

a. In the proposed electromechanical transistor, similarly as in the field effect transistor (FET), the process of electric current conduction involves only monopolar carriers of the same sign. Therefore, like in case of the classical FET electronic transistor, the electromechanical transistor is a "unipolar" transistor, since the process of current conduction in the electro‐

**b.** In the proposed electromechanical transistor, similarly as in the FET transistor, channel resistance is modulated. These channels exist physically in the structure of the OHC, i.e., they have definite dimensions, spatial positions, and play the role of the controlled

**c.** In case of the FET transistor, the channel is formed in the semiconductor material, which is sandwiched between two electrodes (source and drain). Resistance of this channel is modulated by changing voltage applied to the gate. This results in a change of channel's

**9.2. Analogy between the proposed EMT type transistor and the FET type transistor**

i.e., *EMCE* = *R*(*t* ). The value of the resistance *R*(*t* ) is controlled by the input mechanical (acoustical) signal (velocity and/or force). Ion channels in the apical part of the OHC play

).

to the load resistance *RL*

.

,

*RL*

(back to the stria vascularis).

76 Advances in Clinical Audiology

(FET), see **Figure 4a** and **b**.

resistor *R*(*t* ).

This is due to the following reasons:

cross‐section and its conductivity.

*9.1.7. Electromechanical transistor (EMT)*

tions for power gain to occur, namely:

the role of the controlled resistance *R*(*t* ).

**1.** There is a source of potential energy (voltage source *E*<sup>2</sup>

tor) to control the flow of energy from the voltage source *E*<sup>2</sup>

**3.** There is a mechanical control input circuit (stereocilia of the OHC).

mechanical transistor employs only positive potassium cations K+

**e.** By contrast, the resistance which is modulated in the vacuum tube or in the bipolar tran‐ sistor is rather an effective resistance (a phenomenological concept). This resistance is not located in a definite site. For example, in the vacuum tube (triode), the resistance which is modulated by the grid voltage is an apparent resistance of the region distributed between the cathode and anode.

Therefore, we can state that the proposed electromechanical transistor is a close analog of the classical electronic field effect transistor (FET), see **Table 1**.

The principle of operation of the proposed EMT transistor and the classical FET transistor is presented below in **Figure 4**.


**Table 1.** Comparison of the features of the electromechanical transistor (EMT) and the field effect transistor (FET).

**Figure 4.** Physical models of (a) proposed electromechanical transistor EMT, and (b) classical field effect transistor FET.

It is not surprising that the structures of both transistors presented in **Figure 4a** and **b** are almost identical. In case of the proposed EMT transistor, movement of the stereocilia modu‐ lates the conductivity of ion channels. On the other hand, in the classical FET transistor, the voltage applied to the gate modulates the conductivity of the semiconductor channel. Thus, the proposed EMT transistor and classical FET transistors are controlled, correspondingly, by mechanical and electrical signals.

The channels in the EMT and FET transistors exist physically and can be modeled by a time‐ varying resistance *R*(*t* ). It is worth noticing that the transistor itself does not generate any energy. In fact, its essential function is to control (according to changes in the input modulat‐ ing signal) the flow of energy from an external "high" energy source (such as a DC voltage battery) into the output circuit (load resistance).

#### **9.3. Small signal linearized electric equivalent circuit of the EMT**

Operation of the proposed EMT transistor can be presented in the form of an equivalent elec‐ trical circuit.

#### *9.3.1. Small‐signal electrical equivalent circuit*

The input circuit of the proposed electromechanical transistor EMT is represented by two mechanical quantities, i.e., the velocity *v*<sup>1</sup> and mechanical force *F*<sup>1</sup> on the cilia. The output cir‐ cuit of the proposed electromechanical transistor EMT is represented by two electrical quan‐ tities, i.e., output voltage *U*<sup>2</sup> and current *I* 2 . In general, the relationships between (*F*<sup>1</sup> , *v*<sup>1</sup> ) and (*U*<sup>2</sup> , *I* 2 ) are described by complex nonlinear functions. However, for small signal amplitudes, the link between (*F*<sup>1</sup> , *v*<sup>1</sup> ) and (*U*<sup>2</sup> , *I* 2 ) can be linearized, i.e., described by linear functions framed in a matrix form.

Classical circuit theory allows modeling of the transistor in the form of an equivalent circuit composed of passive admittances and active current sources. In this way, instead of inves‐ tigating a large number of complex 3D physical phenomena occurring in the actual spatial structure of the transistor, operation of the transistor can be satisfactorily described using a combinations of lumped circuit elements, such as admittances and controlled sources. The resulting circuit constitutes a small‐signal equivalent circuit of the transistor. Influence of these elements on the operation of the transistor can be calculated by applying the laws of the current flow, known from the classical circuit theory [12].

**Figure 5** shows small signal equivalent circuit of the proposed electromechanical transistor for small values of the output AC electric signals (voltage and current) and input mechanical signals (velocity and force on stereocilia).

The matrix equation linking together the input mechanical and output electrical signals in **Figure 5** can be presented using the concept of a hybrid matrix [ *h*], as follows:

$$
\begin{bmatrix} F\_1 \\ I\_2 \end{bmatrix} = \begin{bmatrix} h\_{11} & h\_{12} \\ h\_{21} & h\_{22} \end{bmatrix} \cdot \begin{bmatrix} \upsilon\_1 \\ \mathsf{U}\_2 \end{bmatrix} = \begin{bmatrix} Z\_{1u} & 0 \\ g\_{uu} & g\_{22} \end{bmatrix} \cdot \begin{bmatrix} \upsilon\_1 \\ \mathsf{U}\_2 \end{bmatrix} \tag{1}
$$

Power Amplification and Frequency Selectivity in the Inner Ear: A New Physical Model http://dx.doi.org/10.5772/66542 79

**Figure 5.** Linearized electromechanical equivalent circuit of the electromechanical transistor that uses a phenomenon of forward mechano‐electric transduction occurring in the apical part of an OHC (hair bundles + ionic channels).

The elements of the matrix [*h*] have the following physical interpretation:

*h*<sup>11</sup> = *Zin* is a mechanical input impedance,

It is not surprising that the structures of both transistors presented in **Figure 4a** and **b** are almost identical. In case of the proposed EMT transistor, movement of the stereocilia modu‐ lates the conductivity of ion channels. On the other hand, in the classical FET transistor, the voltage applied to the gate modulates the conductivity of the semiconductor channel. Thus, the proposed EMT transistor and classical FET transistors are controlled, correspondingly, by

The channels in the EMT and FET transistors exist physically and can be modeled by a time‐ varying resistance *R*(*t* ). It is worth noticing that the transistor itself does not generate any energy. In fact, its essential function is to control (according to changes in the input modulat‐ ing signal) the flow of energy from an external "high" energy source (such as a DC voltage

Operation of the proposed EMT transistor can be presented in the form of an equivalent elec‐

The input circuit of the proposed electromechanical transistor EMT is represented by two

cuit of the proposed electromechanical transistor EMT is represented by two electrical quan‐

Classical circuit theory allows modeling of the transistor in the form of an equivalent circuit composed of passive admittances and active current sources. In this way, instead of inves‐ tigating a large number of complex 3D physical phenomena occurring in the actual spatial structure of the transistor, operation of the transistor can be satisfactorily described using a combinations of lumped circuit elements, such as admittances and controlled sources. The resulting circuit constitutes a small‐signal equivalent circuit of the transistor. Influence of these elements on the operation of the transistor can be calculated by applying the laws of the

**Figure 5** shows small signal equivalent circuit of the proposed electromechanical transistor for small values of the output AC electric signals (voltage and current) and input mechanical

The matrix equation linking together the input mechanical and output electrical signals in

*Zin* 0 *gem <sup>g</sup>*22]<sup>∙</sup> [ *v*1 *U*2

**Figure 5** can be presented using the concept of a hybrid matrix [ *h*], as follows:

*h*<sup>11</sup> *h*<sup>12</sup> *h*<sup>21</sup> *h*22] ∙ [ *v*1 *U*2 ] <sup>=</sup> [

) are described by complex nonlinear functions. However, for small signal amplitudes,

2

and mechanical force *F*<sup>1</sup>

. In general, the relationships between (*F*<sup>1</sup>

) can be linearized, i.e., described by linear functions

on the cilia. The output cir‐

] (1)

, *v*<sup>1</sup>

) and

mechanical and electrical signals.

78 Advances in Clinical Audiology

trical circuit.

(*U*<sup>2</sup> , *I* 2

battery) into the output circuit (load resistance).

*9.3.1. Small‐signal electrical equivalent circuit*

mechanical quantities, i.e., the velocity *v*<sup>1</sup>

, *v*<sup>1</sup>

signals (velocity and force on stereocilia).

[

tities, i.e., output voltage *U*<sup>2</sup>

the link between (*F*<sup>1</sup>

framed in a matrix form.

**9.3. Small signal linearized electric equivalent circuit of the EMT**

and current *I*

, *I* 2

) and (*U*<sup>2</sup>

current flow, known from the classical circuit theory [12].

*F*1 *I* <sup>2</sup>] <sup>=</sup> [


#### *9.3.2. Occurrence of the alternating currents (AC) in the cochlea*

In Sections 3 and 4 of this chapter, we found that (DC) voltages and currents are present in the cochlear structure. In the following, the problem of occurrence in the structure of the cochlea (AC) voltages and currents will be analyzed in more detail.

From the analysis presented formerly by the author it follows that the transistor effect (based on the phenomenon of forward mechano‐electric transduction) generates alternating electric current (AC) in the cochlea. This current is represented in **Figure 5** by an active current source.

This analysis can serve as a theoretical description of the (AC) voltages and currents in the structure of the cochlea. These (AC) voltage and current sources are produced in the circuit: DC voltage source + time variable resistance *R*(*t* ), see **Figure 3b**.

The output circuit has the properties of the controlled current source. Output current *I* 2 is gen‐ erated by the controlled current source *I*(*t*) = *h*<sup>21</sup> *v*<sup>1</sup> (*t* ), which depends linearly on the input velocity *v*<sup>1</sup> (*t* ). This is a characteristic feature of active elements (in this case electromechanical transistors).

As it will be presented later, the controlled (AC) current source, shown in **Figure 5**, will act as the electrical signal that pumps energy into the parametric amplifier established on the basis of the nonlinear capacitance of the OHC, see **Figure 7**. More details concerning the operation of this electromechanical transistor can be found in the work of the author [40].

#### *9.3.3. Otoacoustic emission*

Discovered in 1978, phenomenon of otoacoustic emission (OAE) relies on the generation of sound waves in the inner ear. Acoustic waves generated in this way travel into the middle and outer ear [41, 42]. To explain the phenomenon of the OAE, we can examine the properties of the active elements constituting the cochlear amplifier, which in certain conditions can pro‐

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 impairment in newborns (in newborn hearing screening).

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