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Paper text:
remarkable detection capabilities appearing in animal hearing are essentially governed by both a passive mechanical and an active biophysical procedures in the cochlea  @xcite .
the active amplification of the living cochlea was conjectured by gold in 1948  @xcite and is now qualitatively studied widely through the hopf equation in mathematical models  @xcite . in order to give a better understanding to the experimental observations ,
a more accurate model is required .
furthermore the physical origin of the active oscillation remains still unclear . in mammals , the outer hair cells ( ohcs )
are known to be force generators for auditory sensitivity and frequency selectivity . proposed mechanisms of the force generation
are contraction of the ohc itself  @xcite and an active motion of the hair bundle  @xcite .
energy depot model can describe the active phenomena  @xcite , because the energy supplied by the depot can induce a negative dissipation . a similar mechanism for a negative stiffness in the bullfrog s hair bundle
has been reported  @xcite .
it is reasonable to consider the ohc as an energy depot since the ohcs play an important role in the active amplification of the cochlea even though we do not know the exact mechanism of the energy supply . in nature
all living things have some mechanisms of protection for survival .
we expect that there might be a similar mechanism to protect cochlea from damage by any external factors such as loud stimulation .
hence , we propose a braking mechanism in the cochlea to prevent the basilar membrane ( bm ) from an excessive oscillation which could damage the cochlea .
for this purpose , a critical velocity of the ohc oscillation will be introduced in our model .    in the following sections , we describe our model and reproduce the essential phenomena of the active cochlea observed in recent experiments such as the compressive nonlinearity , the dynamics of the bm response , nonlinear distortion and the spontaneous bm oscillation ( sbmo ) .
it will also be pointed ut that our model naturally includes the hopf bifurcation model  @xcite in the case of weak stimulation . however , in the regime of strong stimulation , the present model is essentially different from the hopf bifurcation model .
it shows that the response of the bm is non - linear and compressive at the cf .
( note that the slope of the solid line is less than one . ) otherwise , the response of the bm is a linear function of the stimulation yielding that the slopes are @xmath0 .
these results agree with the experimental observations  @xcite . ]
it is well known that a change of transmembrane potential in the ohc induces a somatic vibration .
this implies that a part of the electric energy is converted into the mechanical energy . in this sense , we believe that the ohc functions as an energy depot for the bm .
the ohc as the energy depot stores energy supplied and converts it into kinetic energy overcoming dissipation .
this energetics can be described by the following energy balance equation  @xcite .
@xmath1 where @xmath2 is the energy density of the energy depot , @xmath3 the rate of the energy pumping into the energy depot per unit area , @xmath4 the rate of the internal energy dissipation , @xmath5 the rate of the energy converting into the kinetic energy of the bm , and @xmath6 the velocity of the bm . considering that constant contribution of @xmath5 can be incooperated into @xmath4 and that conversion rate of the energy into the kinetic energy of the bm will not be sensitive to the sign of @xmath6 , we assume that @xmath5 is an even function of @xmath6 without a constant term . the energy depot model in which @xmath7 has been extensively discussed  @xcite . in this paper , we include the next higher contribution to @xmath5 and introduce a critical velocity of the bm to describe a braking mechanism  @xcite .
hence the conversion rate @xmath5 can be written as @xmath8 where @xmath9 is the critical velocity .
for a finite @xmath9 the oscillation of the bm can be braked by the negative conversion rate into the kinetic motion of the bm .
note that @xmath10 corresponds to a system without the braking mechanism .
the critical velocity will be roughly estimated from the contraction of the ohc later for our numerical calculation .
khz and the stimulation is @xmath11db .
the onset time is much shorter than the offset time which is not shown in full scale .
this result is qualitatively consistent with the experimental observations  @xcite .
@xmath12kgm@xmath13s@xmath14 , @xmath15m@xmath13s@xmath16 , and @xmath17 .
the stimulation is unloaded at @xmath18ms .
the lower panel shows the response represents the case without the braking mechanism , showing a continuous increase as long as the stimulation is on . ]
kgm@xmath13s@xmath14 , @xmath15m@xmath13s@xmath16 , @xmath19 m/s ( solid thin line ) ; @xmath20kgm@xmath13s@xmath14 , @xmath21m@xmath13s@xmath16 , @xmath22 m/s ( solid thick line ) ; @xmath23kgm@xmath13s@xmath14 , @xmath15m@xmath13s@xmath16 , @xmath24 ( dashed thin line ) , and @xmath25kgm@xmath13s@xmath14 , @xmath26m@xmath13s@xmath16 , @xmath27 ( dashed thick line ) .
the velocity of the bm is ( not ) saturated when the stimulation is sufficiently large with ( without ) the braking mechanism . ]
now it is worthwhile to compare our newly proposed model with the one previously introduced in ref .  @xcite . whereas we treat the energy density ( @xmath2 ) of the energy depot(ohc ) , in ref .
@xcite the energy density ( @xmath28 ) of the propagating hydrodynamic wave is considered .
the model in ref .
@xcite describes the coupling of the active elements to the propagating wave , while the present model describes the coupling between the energy depot(ohc ) and the bm oscillation .
the emphasis in this paper is on showing that the active and compressive response arises naturally from the generalized energy depot concept .
an adiabatic approximation in which the adaptation of the energy depot is very fast  @xcite yields that @xmath29 it should bear in mind that @xmath2 is not constant .
the oscillation of the bm is then governed by @xmath30 where @xmath31 is the density , @xmath32 the physical damping coefficient , @xmath33 the stiffness , and @xmath34 the displacement of the bm .
@xmath35 includes the contributions of a sound wave ( external stimulation ) and a noise .
the noise is assumed to be a gaussian white noise .
@xmath36 is the active force density acting on the bm by the ohc , @xmath37 note that the langevin equation , eq .
( [ full ] ) , describes the passive and the active response in a unified and natural way .
it should be noted that the active force , eq .
( [ active ] ) , can be either positive or negative depending on the value of @xmath6 . when @xmath38 is smaller than @xmath39 ( weak stimulation ) , a positive active force
is provided on the bm ( active mechanism ) . on the other hand , when @xmath38 is larger than @xmath39 ( strong stimulation ) , a negative active force , hence an extra drag force , is exerted on the bm ( braking mechanism ) .
the mass density of the bm is known to be about @xmath40kg / m@xmath16  @xcite and the reported damping coefficient , @xmath32 , of the bm has a large variation from @xmath41 to @xmath42 kg m@xmath13s@xmath14  @xcite depending on the models . for a numerical calculation
, we set the damping in the range of @xmath43 kg m@xmath13s@xmath14 .
it has been reported that the guinea pig ohcs are able to change its length up to @xmath44 when the transmembrane potential is varied .
this corresponds to a displacement of @xmath45 m for the ohcs of @xmath46 m at the high - frequency end  @xcite .
hence , in the regime of high frequency of @xmath47khz , the adjustable parameter @xmath9 is roughly estimated about @xmath48 m/s . by considering the ohc as a capacitor and using the observed membrane potential , membrane capacitance  @xcite , and the scale of the ohc  @xcite , we estimate the value of @xmath3 to be about @xmath48 kg s@xmath49 .
the response of the bm to the sound wave can be then obtained straightforwardly using eqs .
( [ full ] ) and  ( [ active ] ) . in the following calculations , we set @xmath50 and the characteristic frequency ( cf ) , @xmath51 . it will be shown later that these relations arise from the conditions of the hopf bifurcation . in the present calculations ,
the noise width of the distribution is chosen to be zero db .
the compressive and dynamical responses of the bm are obtained as shown in fig .
[ compressive ] and fig .
[ dyn ] , respectively .
[ compressive ] demonstrates that the response of the bm is non - linear and compressive at the cf , while the response is linear at other frequencies .
this compressive response of the bm agrees well with the observations  @xcite as expected .
[ dyn ] shows the time course of the bm response , reproducing the experimental observation that the onset time of the response is shorter than the offset time , although the onset time is somewhat larger than the experimental values  @xcite .
it is important to notice that the velocity of the bm increases continuously as long as the stimulation is being turned on if the braking mechanism is not included @xmath52 .
hence , the braking mechanism introduced in our model is essential to explain the experimental observation at a sufficiently strong stimulation .
a dependence of the bm velocity on @xmath9 is plotted in fig .
the results clearly show that lower critical velocity produces smaller response at the same stimulation , thus producing a supercompressive behavior as observed in experiment  @xcite .
this is the effect of the braking mechanism , which is significantly different from the hopf bifurcation model .
note that the saturation of the bm response disappears if the braking mechanism is not considered .
, the brownian fluctuation is suppressed , while in the region of @xmath53 , the brownian fluctuation is amplified . for the plots ,
the intensity of the thermal noise is @xmath54pa , and the strength of the fluctuation is @xmath54db .
( a ) @xmath55kgm@xmath13s@xmath14 , @xmath56 m/s .
( b ) @xmath57kgm@xmath13s@xmath14 , @xmath22 m/s . ]    khz , @xmath58khz , both strengths of the stimulation are @xmath11db , the response is measured at the place where the @xmath59khz , @xmath60m@xmath16/s@xmath16 and @xmath23kgm@xmath13s@xmath14 . ]
the gain of the oscillator is defined as the ratio of the displacement of the bm to the stimulation , @xmath61 using eq .
( [ full ] ) , we obtain the gain , which is a scaling function of the stimulation when the stimulation is not too large , @xmath62 .
this result agrees well with the experiment  @xcite and the result of the hopf bifurcation model  @xcite , @xmath63 .
indeed our model includes the hopf model naturally . to show this , we expand the active force , eq .
( [ active ] ) , up to the lowest nonlinear term when @xmath64 is small .
( [ full ] ) is then approximately written as @xmath65 in the fourier spectra space , this directly corresponds to the hopf equation  @xcite with the following bifurcation conditions , @xmath66 hence our energy depot model with a braking mechanism reduces to the hopf bifurcation model in the limit of weak stimulation .
one of the most important results of our model is an amplification of the thermal noise itself in the absence of an external stimulation . at the bifurcation point or when @xmath67 , the thermal noise is well suppressed .
however , when @xmath68 , so that too much of the internal energy is converted into the kinetic energy , the thermal noise can be largely amplified as shown in fig .  [ vc ] . here , we set the mean average of the noise at @xmath54pa and the width at @xmath54db .
since it can be generally assumed that the oscillators are distributed in the vicinity of the bifurcation point , the noise may be amplified incoherently in the absence of the input signal . however , when a weak input signal with the same cf is introduced , it may phase - lock to the already existing amplified noise , thus enhancing the selectivity as discussed in ref .  @xcite . because it is believed that the sbmo is crucial for understanding the spontaneous otoacoustic emission  @xcite , our model may provide a clue to explore this interesting and important phenomenon .
[ dist ] shows the nonlinear distortion of the response of the bm at the position where the cf is @xmath69khz .
the stimulation contains two tunes , @xmath70khz and @xmath58khz .
the strengths both are @xmath11db .
our model shows clearly the distortions at @xmath71 , @xmath72 in the response of the bm .
in this paper , we propose a theoretical scheme for the active and passive response of the bm based on a concept of the energy depot model with a braking mechanism . according to the experimental observations ,
the ohc is assumed to play a role of the energy depot by supplying an active force to the bm and also by reabsorbing an excessive kinetic energy like the regenerative - brake system in automobiles and electric vehicles  @xcite .
it is shown that a single equation of motion , eq  ( [ full ] ) , can produce all the essential passive and active behaviors . moreover
, our model shows that the brownian noise can be sufficiently amplified under a certain condition , thus leading to the sbmo .
although we applied our model only to the mammalian cochlea , it is equally possible to be applied to the nonmammalian vertebrates  @xcite because the only required physical characteristics are a supply of the active force and a self - adaptation . in this sense
, we believe that the concept of the energy depot model with a braking mechanism is quite universal and can be applied to any underdamped biological systems .
this work is supported by the korea science and engineering foundation ( kosef ) ( r01 - 2006 - 000 - 10083 - 0 ) .
furness and c.  m. hackney , _ the structure and composition of the stereociliary bundle of vertebrate hair cells _ , in _ vertebrate hair cells _ ,
r.  a. eatock , r.  r. fay and a.  n. popper ( springer , new york , 2006 ) , chapter .  3 , pp .
high auditory sensitivity , sharp frequency selectivity , and otoacoustic emissions are signatures of active amplification of the cochlea . 
 the human ear can also detect very large amplitude sound without being damaged as long as the exposed time is not too long . 
 the outer hair cells are believed as the best candidate for the active force generator of the mammalian cochlea . in this paper 
 , we propose a new model for the basilar membrane oscillation which successfully describes both the active and the protective mechanisms by employing an energy depot concept and a critical velocity of the basilar membrane . 
 one of the main results is that thermal noise in the absence of external stimulation can be amplified leading to the spontaneous basilar membrane oscillation . 
 the compressive response of the basilar membrane at the characteristic frequency and the dynamic response to the stimulation are consistent with the experimental results as expected . 
 our model also shows the nonlinear distortion of the response of the basilar membrane .