A Martinez

F Leblanc

J Y Chaufray

R Modolo

O Witasse

Y Dong

T Hara

J Halekas

R Lillis

J Mcfadden

F Eparvier

L Leclercq

J Luhmann

S Curry

D Titov

B Jakosky

Influence of Extreme Ultraviolet irradiance variations on the precipitating ion flux from MAVEN observations

We study the influence of the solar EUV flux intensity on the precipitating ion fluxes as seen by MAVEN/SWIA, an energy and angular ion spectrometer. We defined three periods with significantly different EUV flux intensity (1.6 and 3.2 times the lowest EUV intensity) and compare the precipitating ion flux measured by MAVEN/SWIA during each period. At low energy [30-650] eV, we find that the median (average) precipitating ion flux during the medium and low EUV periods are respectively 1.7 (2.1) and 3 (3.5) times more intense than the flux during the high EUV period. At high energy eV, a similar trend in the intensity of the precipitating ion flux is observed but with an increase by 50% (46%) and 70% (79%) respectively. A larger EUV flux does therefore not seem to favour heavy ion precipitation into Mars' atmosphere, contrary to modelling prediction and overall expectations.

I.

Introduction

Mars' upper atmosphere is constantly bombarded by energetic ions from either the solar wind or from Mars' exosphere (planetary pickup ions) [START_REF] Lillis | Characterizing Atmospheric Escape from Mars Today and Through Time, with MAVEN[END_REF]. Planetary ions created by the ionization of exospheric neutral particles can be accelerated by the motional solar wind electric field. Some of these ions escape [START_REF] Brain | Atmospheric escape from unmagnetized bodies[END_REF], but a significant portion can impact Mars' atmosphere, leading to collisional cascade in Mars' thermosphere and to the ejection of atmospheric particles into the exosphere [START_REF] Luhmann | Dayside pickup oxygen ion precipitation at Venus and Mars: Spatial distributions, energy deposition and consequences[END_REF]. This process, named "atmospheric sputtering" [START_REF] Johnson | Plasma-induced sputtering of an atmosphere[END_REF], could have induced a significant atmospheric loss along Mars' history [START_REF] Luhmann | Evolutionary impact of sputtering of the Martian atmosphere by O + pickup ions[END_REF][START_REF] Leblanc | On Mars' Atmospheric Sputtering after MAVEN first Martian year of Measurements[END_REF][START_REF] Leblanc | Sputtering of the Martian atmosphere by solar wind pick-up ions[END_REF][START_REF] Leblanc | Role of molecules in pick-up ion sputtering of the Martian atmosphere[END_REF].

Atmospheric escape induced by sputtering at present epoch is expected to be small compared to other mechanisms [START_REF] Leblanc | On the origins of Mars' exospheric nonthermal oxygen component as observed by MAVEN and modeled by HELIOSARES[END_REF] and so difficult to measure. However, since the main driver of sputtering is ion precipitation [START_REF] Johnson | The sputtering of an oxygen thermosphere by energetic O +[END_REF][START_REF] Wang | Modeling of the O + pickup ion sputtering efficiency dependence on solar wind conditions for the Martian atmosphere[END_REF]2015), it is crucial to constrain the dependence of the precipitating ion flux on present solar wind conditions. While many studies have been carried out on the influence of the Extreme-Ultraviolet/Ultraviolet (EUV/UV) flux on the Martian environment as well as on ion escape [START_REF] Modolo | Influence of the solar EUV flux on the Martian plasma environment[END_REF][START_REF] Ma | Ion escape fluxes from Mars[END_REF][START_REF] Terada | Atmosphere and water loss from early Mars under extreme solar wind and extreme ultraviolet conditions[END_REF][START_REF] Edberg | Plamas boundary variability at Mars as observed by Mars Global Surveyor and Mars Express[END_REF]Lundin et al, 2008Lundin et al, ,2013;;Ramstad et al., 2015;[START_REF] Dong | Seasonal variability of Martian ion escape through the plume and tail from MAVEN observations[END_REF], the relation between ion precipitation and EUV flux has only been studied in theoretical works [START_REF] Chaufray | Mars solar wind interaction: Formation of the Martian corona and atmospheric loss to space[END_REF][START_REF] Leblanc | Role of molecules in pick-up ion sputtering of the Martian atmosphere[END_REF][START_REF] Wang | Modeling of the O + pickup ion sputtering efficiency dependence on solar wind conditions for the Martian atmosphere[END_REF][START_REF] Wang | Statistical studies on Mars atmospheric sputtering by precipitating pickup O + : Preparation for the MAVEN mission[END_REF] and briefly discussed in [START_REF] Ramstad | Ion distributions in the vicinity of Mars: Signatures of heating and acceleration processes[END_REF] We therefore describe here the first empirical study on the influence of the EUV flux on ion precipitation by analyzing MAVEN measurements [START_REF] Jakosky | The Mars Atmosphere and Volatile Evolution (MAVEN) Mission[END_REF]. In section II, we describe the set of data used in this work. In section III, we carefully analyzed the various potential solar wind drivers to extract three sets of measurements obtained during different EUV flux intensity but for similar solar wind conditions and geographical coverage. In section IV, we summarize and discuss the main results of this study and conclude in section V.

II. Data set used for this analysis

We use measurements performed by the Solar Wind Ion Analyzer (SWIA) [START_REF] Halekas | The Solar Wind Ion Analyzer for MAVEN[END_REF], the Magnetometer (MAG) (Connerney et al., 2015a;2015b) and the Solar Extreme Ultraviolet Monitor (EUVM) [START_REF] Eparvier | The Solar Extreme Ultraviolet Monitor for MAVEN[END_REF]. The MAVEN/SWIA is an energy and angular ion spectrometer covering an energy range between 25eV/q and 25keV/q with 48 energy steps logarithmically spaced, a field of view (FOV) of 360°x90° on 64 angular bins and 4s time resolution. We based our study on MAVEN/SWIA despite the lack of mass resolution because MAVEN/STATIC (which has mass resolution) has much more restricted coverage of the precipitating flux. The MAVEN/EUVM measures the solar irradiance in three bands from the soft X-ray to the EUV range (in three spectral bands 0.1-7nm, 17-22nm and 121-122nm) with a temporal resolution of 1s. The solar wind density, speed, and IMF vector are measured by MAVEN/SWIA and MAVEN/MAG and averaged on an orbit-by-orbit basis in order to characterize the solar wind conditions (Halekas et al., 2017). We then reconstruct the Mars Solar Electric (MSE) coordinate system in which the X-axis is toward the Sun, the Z-axis points along the solar wind motional electric field and the Y-axis completing the right hand system [START_REF] Fedorov | Structure of the Martian wake[END_REF].

In order to reconstruct the precipitating ion flux, we follow the method described in [START_REF] Leblanc | Mars heavy ion precipitating flux as measured by Mars Atmosphere and Volatile EvolutioN[END_REF] and [START_REF] Martinez | Variability of precipitating ion fluxes during the September 2017 event at Mars[END_REF]. We select all MAVEN measurements performed between 200 and 350km. Within such altitude range, any ion which velocity direction is at less than 75° with respect to the local nadir direction has a very large probability to impact Mars' atmosphere. Therefore, to reconstruct the precipitating flux, we sum all measurements of SWIA anodes which FOV is at less than 75° from the local zenith direction. Moreover, in order to exclude reconstructed precipitating flux with poor coverage, we only consider measurements during which the total FOV of SWIA anodes is more than 65% of the 75° solid angle cone centered on the zenith direction.

In this study, to characterize the EUV flux during the declining solar cycle from medium to minimum

III. Upstream solar wind conditions and planetary coverage

To investigate the Solar EUV dependence of the precipitating ion flux, we first need to reconstruct the solar wind parameters. [START_REF] Lillis | Characterizing Atmospheric Escape from Mars Today and Through Time, with MAVEN[END_REF], [START_REF] Edberg | Plamas boundary variability at Mars as observed by Mars Global Surveyor and Mars Express[END_REF] and [START_REF] Ramstad | Solar wind-and EUV-dependent models for the shapes of the Martian plasma boundaries based on Mars Express measurements[END_REF] suggested that the dynamic pressure compresses Mars' magnetosphere and favors the acceleration and precipitation of the planetary pickup ions. [START_REF] Martinez | Variability of precipitating ion fluxes during the September 2017 event at Mars[END_REF] and Hara et al. (2017a) showed that the precipitating ion flux can be organized with respect to the orientation of the solar wind convective electric field. Also, the presence of crustal magnetic fields [START_REF] Acuña | Global distribution of crustal magnetization discovered by the Mars Global Surveyor MAG/ER Experiment[END_REF][START_REF] Connerney | Tectonic implications of Mars crustal magnetism[END_REF] can also influence locally the precipitating ion flux, as shown by Hara et al. (2017b). For each orbit, during which the precipitating flux was measured, we therefore used the part of MAVEN orbit in the solar wind to infer the solar wind. We use such solar wind measurements prior to our precipitating flux measurement if the time difference between the two measurements is less than 4 hours and 30 minutes (the MAVEN orbital period). We assume that the solar wind conditions do not change significantly between MAVEN measurements of the solar wind and MAVEN measurements of the precipitating flux (in average 2 hours and 20 minutes).

In order to verify the validity of this assumption, we compare the solar wind density and speed measurements from Mars Express(MEX)/ASPERA-3 (Barabash et al., 2006) With such criteria, we did not identify any measurement for the low EUV period and ASPERA-3 measurements for 4.2% and 8.0% of MAVEN precipitating flux measurements for the medium and high EUV periods (see also the supplementary materials, Table S.1). The set of measurement comparisons displayed in Figure 1 for the high EUV period, show that the solar wind speeds as measured by SWIA and ASPERA-3 are usually in very good agreement (MAVEN: 403 ± 12km/s and MEX: 396 ± 12km/s, for 85 measurements). Contrary to the solar wind speed, the measurements of the density display a relatively poor agreement (Figure 1b) but still acceptable in an average and within the one-sigma dispersion (MAVEN: 3.31±0.35𝑐𝑚 -3 and MEX: 3.07±0.31𝑐𝑚 -3 , for 85 measurements).

The approach to select the set of measurements used for our study follows the method developed in [START_REF] Dong | Seasonal variability of Martian ion escape through the plume and tail from MAVEN observations[END_REF]. As in [START_REF] Dong | Seasonal variability of Martian ion escape through the plume and tail from MAVEN observations[END_REF], we restrict the set of precipitating measurements to those obtained during solar wind conditions with dynamic pressure 𝑃 𝑑𝑦𝑛 = 𝑚 𝑝 𝑛 𝑆𝑊 𝑉 𝑆𝑊 2 , (where m p is the mass of the proton, 𝑛 𝑆𝑊 and 𝑉 𝑠𝑤 the solar wind density and velocity) between 0.3 and 2.6nPa

(moderate solar wind dynamic pressure) and with IMF strength between 2.2 and 6.7nT. We also restrict our sample of precipitation measurements to those obtained in an SZA interval between 70° and 130°. Furthermore, in order to avoid the potential influence of the crustal fields [START_REF] Leblanc | On the origins of Mars' exospheric nonthermal oxygen component as observed by MAVEN and modeled by HELIOSARES[END_REF]Hara et al. 2017b), we also only consider measurements performed when the average magnetic field between 200 and 350km is less than 60nT. For each set of solar wind parameters, we calculate the solar wind dynamic pressure, the norm of the solar wind motional electric field defined where 𝑚 𝑂 is the mass of an 𝑂 + ion, 𝜃 𝑐𝑜𝑛𝑒 is the angle between the IMF direction and the MSO X-axis and q is the electron charge value.

as 𝐸 = |𝑉 ⃗ 𝑆𝑊 × 𝐵 ⃗ 𝐼𝑀𝐹 |
We also define the MSE angle as the anticlockwise angle between the vector formed by the latitude and the longitude in MSE of the position where the ion precipitation is being measured and the East MSE direction (MSE longitude equal to +180° and latitude equal to 0°) [START_REF] Martinez | Variability of precipitating ion fluxes during the September 2017 event at Mars[END_REF]. In order to demonstrate that the selected sets of measurement for the three EUV periods are similar in terms of solar wind conditions and planetary geographic coverage (MSO and MSE frames, see also the supplementary materials, Figures S.1 and S.2), we calculate for each parameter, the area (green part in Figure 2.e and 2.f) common to the three distributions. The ratio between this common area and each of the three areas provides a measure of the overlap between the three sets of measurement for a given solar wind parameter. In Table 2, for 10 solar parameters, we provide the minimum percentage of the distribution areas in common to the three periods as well as the first and third quartile, the mean and 1-sigma standard deviation of each parameter.

Figure 2 displays the distribution of the values of each of these parameters for the three EUV periods and illustrates our method to determine the similarity between these sets of measurement. As an example, in the case of the gyroradius (Figure 2.e), at least 83% of each sets of measurement were performed for similar values of this parameter. This percentage decreases to 62% for the solar wind speed (Figure 2.f). On the other hand, the three EUV periods are clearly distinct (Figure 2.a) and correspond to a null common area (Table 2).

High Medium Low

Common coverage (%)

EUV Flux 𝜇 ± 𝜎 0.42 ± 0.08 0.21 ± 0.04 0.13 ± 0.02 0.0 If we now only consider the medium and low periods, for all parameters listed in Table 2, the distribution area in common to these two sets of measurement represents more than 75% of each distribution. According to these percentages, we conclude that the three sets of precipitating ion measurements were obtained under similar solar wind conditions.

To further demonstrate this conclusion, considering the solar wind speed parameter with the less similar distributions between the three periods, we reduce each sample so that the solar wind speed measured during each period was restricted to values between 370 and 500km/s. In that case, the percentage increases to 81%. We then calculate the values of the precipitating ion flux with these reduced samples (see also the supplementary material, Table S.2) and found that all measured precipitating flux values are equivalent for the two different samples except during the medium period when the flux values differ slightly more than one sigma. In another words, the reduction of the sampling improves the similarity between the three sets of measurement and does not change significantly the measured precipitating ion flux

IV. Results and Analysis

Our results show that the average ion precipitation decreases when the EUV flux increases (see Table 1 and Figure 3). The energy dependence of the differential flux is similar from one period to the other, but we can note that the increase in precipitating ion flux is more significant for energies below 1 to 3 keV than above. This conclusion is not consistent with studies modelling the precipitating ion flux for different solar EUV fluxes [START_REF] Chaufray | Mars solar wind interaction: Formation of the Martian corona and atmospheric loss to space[END_REF] which showed that the flux of O + which re-impacts the atmosphere should be smaller at solar minimum than at solar maximum.

However, they also concluded that the differences between the two periods were smaller than predicted by [START_REF] Luhmann | Evolutionary impact of sputtering of the Martian atmosphere by O + pickup ions[END_REF] using simple gas dynamic description of Mars' interaction with the solar wind. These authors explained this difference by the deceleration of the solar wind due to the mass-loading [START_REF] Dubinin | Mass-loading near Mars[END_REF] which happens further away from Mars at high solar activity and reduces therefore the probability of accelerated pickup ions to re-impact Mars' atmosphere.

Mass-loading depends to first order on the ionization rate of the Martian exosphere. The exospheric density at a given altitude also depends on the EUV/UV flux which heats the Martian thermosphere whereas the ionization rate directly depends on the solar radiation flux in the wavelength range below 91nm. [START_REF] Forbes | Solar flux variability of Mars' exosphere densities and temperatures[END_REF] found a very good correlation (0.96) between the long-term EUV flux (proxy F10.7) and the Martian exospheric density at 390km in altitude. Seasonal variations of the exosphere might also significantly impact the density at a given altitude as studied in [START_REF] Bhattacharyya | A strong seasonal dependence in the Martian hydrogen exosphere[END_REF] and [START_REF] Chaufray | Variability of the hydrogen in the martian upper atmosphere as simulated by a 3D atmosphere-exosphere coupling[END_REF], as well as the position of the bow shock (Halekas et al, 2017).

A peak of exospheric hydrogen density during the Martian dust season was reported by [START_REF] Chaffin | Unexpected variability of Martian hydrogen escape[END_REF], [START_REF] Bhattacharyya | A strong seasonal dependence in the Martian hydrogen exosphere[END_REF] and Halekas (2017) between Ls=200° and 300°. Therefore, an increase in mass-loading of the solar wind might be related to this seasonal variability but would only explain the decrease of the precipitating flux during the high EUV period. During the low and medium periods, there are no significant change in season so that the difference in precipitating flux cannot be associated to seasonal variability.

As originally suggested by [START_REF] Johnson | Sputter contribution to the atmospheric corona on Mars[END_REF], an increase of the EUV/UV flux should increase the number of pickup exospheric ions but might also load more efficiently the solar wind and reduce the percentage of these ions able to re-impact the atmosphere. What our study seems to suggest is that this effect might be much more efficient than shown in [START_REF] Chaufray | Mars solar wind interaction: Formation of the Martian corona and atmospheric loss to space[END_REF], perhaps due to the limited spatial resolution of their simulation at this time (~300km).

Another potential effect is the MSO latitude of the crustal magnetic fields which is solar dependent. [START_REF] Brain | Variability of the altitude of the Martian sheath[END_REF] showed that the Magnetic Pile-Up Boundary altitude changes significantly with season because of the latitude variation of the main crustal field structure (centered at 180° GEO East longitude and -50° GEO latitude) along one Martian year. Such effect might be actually due to an increase exospheric hydrogen density rather due to the crustal fields latitudes according to [START_REF] Chaufray | Variability of the hydrogen in the martian upper atmosphere as simulated by a 3D atmosphere-exosphere coupling[END_REF]. Actually, [START_REF] Hall | Annual variations in the Martian bow shock location as observed by the Mars Express mission[END_REF] shows that an increase in the EUV flux is associated with a position of the bow shock further from Mars, suggesting also an influence of the exosphere on the induced magnetosphere. In any case, we cannot exclude that part of the variation in precipitating flux described here might be due to the crustal field, as actually shown by Hara et al. (2017b). We calculate the average position of the main crustal field structure for the three sets of measurements and found -47 ± 17°, -45 ± 16°, and -46 ± 17° latitude MSO and 27 ± 106°, -20 ± 107° and -3 ± 84° longitude MSO for the low, medium and high EUV periods. Therefore, the relative variation of the precipitating ion flux between the three periods selected in our paper might be also related to the variation of the crustal field latitudinal positions. In order to test this possibility, we further reduce our set of measurements in order to only use measurements of the precipitating ion flux performed when the main crustal field structure was at a SZA larger than 120°. The reconstructed differential precipitating ion fluxes display very similar relative variation showing, here again, that the main driver for this variation should be the EUV flux. [START_REF] Dong | Seasonal variability of Martian ion escape through the plume and tail from MAVEN observations[END_REF] found that the total escape rate of oxygen ions with 𝐸 𝑖 ≥6 eV increases by 1.5 when the EUV flux increases by a similar factor. Our analysis suggested that an increased EUV flux induces a decrease of the ion precipitation. In the case of the high EUV period (similar to that in Dong et al

  with those measured by MAVEN. Starting from 𝑛 𝑀𝐴𝑉𝐸𝑁/𝑆𝑊 𝑎𝑛𝑑 𝑉 𝑀𝐴𝑉𝐸𝑁/𝑆𝑊 the solar wind density and speed as measured by MAVEN/SWIA before a measurement of the precipitating flux below 350 km, we determine 𝑛 𝑀𝐸𝑋/𝑆𝑊 𝑎𝑛𝑑 𝑉 𝑀𝐸𝑋/𝑆𝑊 , as measured by MEX/ASPERA-3(Barabash et al., 2006) when MAVEN's altitude was within the time interval during which SWIA measured the precipitating flux. We only consider ASPERA-3 measurements if the measured density was larger than 0.01cm -3 and the quality flag larger than 0.6.

Figure 1 :

 1 Figure 1: Comparison between the measured solar wind parameters by MAVEN/SWIA (blue) and

Figure 2 :

 2 Figure 2: Distribution of the set of measured solar wind parameters for the high EUV period

  (2017)), for the ions with 𝐸 𝑖 ≥25 eV, we have an integrated precipitating flux equal to 1.3 × 10 22 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠. 𝑠 -1 which should be compared to the estimate by Dong et al. (2017) of an escape rate of 1.8 × 10 24 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠. 𝑠 -1 at 0.2R Mars in altitude. Therefore, the precipitating flux represents only 1% of the escaping flux in the energy range considered in this paper.

Figure 3 :

 3 Figure 3: Precipitating ion differential energy spectra as measured by MAVEN/SWIA during high EUV

[Date begin- Date End] (dd/mm/yyyy)

	22nm. We sort the MAVEN data from December 2014 to July 2018 into three periods: high EUV
	(From 2 December 2014 to 10 March 2015) with an average EUV flux of 0.42mW.m -2 , medium EUV
	(From 4 June 2015 to 22 October 2015, from 23 December 2015 to 1 June 2016, and from 4
	December 2016 to 20 July 2017) with an average EUV flux of 0.21mW.m -2 and low EUV (from 28
	November 2017 to 31 July 2018) with an average EUV flux of 0.13mW.m -2 . These three intervals
	represent several different Mars seasons. Table 1 summarizes the main characteristics of each
	period.
		High	Medium	Low
		[02/12/2014-10/03/2015]	[04/06/2015-22/10/2015] [23/12/2015-01/06/2016] [04/12/2016-30/07/2017]	[28/11/2017 31/07/2018]
	Heliocentric distance (AU)	[1.38-1.43]	[1.54-1.66] [1.39-1.63] [1.49-1.66]	[1.40-1.65]
	Solar Longitude (LS) °	[244.3°-305.2°] (northern hemisphere winter)	[352.5°-57.9°] [85.1°-161.5°] [273.5°-35.7°] (one Martian year)	[93.4°-221.1°] (northern hemisphere summer and autumn)
	Mean EUV flux
	(𝒎𝑾. 𝒎 -𝟐 ) from channel A of	0.42	0.21	0.13
	MAVEN/EUVM
	Average Precipitating
	ion flux (𝒆𝑽. 𝒄𝒎 -𝟐 . 𝒔𝒓 -𝟏 . 𝒔 -𝟏 )	1.18 (±0.11) × 10 7	2.49 (±0.12) × 10 7	4.05 (±0.33) × 10 7
	[30,650] eV
	Median Precipitating ion
	flux (𝒆𝑽. 𝒄𝒎 -𝟐 . 𝒔𝒓 -𝟏 . 𝒔 -𝟏 )	0.62 × 10 7	1.07 × 10 7	1.83 × 10 7
	[30,650] eV
	Average Precipitating
	ion flux (𝒆𝑽. 𝒄𝒎 -𝟐 . 𝒔𝒓 -𝟏 𝒔 -𝟏 )	15.9 (±0.7) × 10 7	23.9 (±0.7) × 10 7	26.9 (±1.1) × 10 7
	[650,25000] eV
	Median Precipitating ion
	flux (𝒆𝑽. 𝒄𝒎 -𝟐 . 𝒔𝒓 -𝟏 𝒔 -𝟏 )	11.0 × 10 7	16.1 × 10 7	19.7 × 10 7
	[650,25000] eV
	Number of measurements	425	1205	374
	activity, we use the MAVEN/EUVM channel A which measures solar irradiance between 17 and

Table 1 :

 1 Summary

of each period of our study. In brackets, the 1-sigma standard deviation (uncertainty) of the precipitating ion flux values.

  where B IMF is the interplanetary magnetic field (IMF), the Alfvén Mach number 𝑀 𝐴 = 𝑉 𝑆𝑊 𝑉 𝐴 ⁄ where V A is the Alfvén speed, the solar wind flux defined as 𝐹 𝑆𝑊 = 𝑛 𝑆𝑊 𝑉 𝑠𝑤 and the pickup 𝑂 + gyroradius in the solar wind 𝑅 𝑔 = 𝑚 𝑂 𝑉 𝑆𝑊 sin 𝜃 𝑐𝑜𝑛𝑒 𝑞𝐵 𝐼𝑀𝐹

Table 2 :

 2 Mean

µ, standard deviation , first quartile Q25 and third quartile Q75 of each solar parameter distribution for the three sets of precipitating flux measurement. The last column provides the percentage of the area in common between the 3 distributions.

Acknowledgements

This work was supported by the DIM ACAV and the ESA/ESTEC faculty. This work was also supported by CNES "Système Solaire" program and by the "Programme National de Planétologie" and "Programme National Soleil-Terre". This work is also part of HELIOSARES Project supported by the ANR (ANR-09-BLAN-0223), ANR MARMITE (ANR-13-BS05-0012-02) and ANR TEMPETE (ANR-17-CE31-

We study the precipitating ion flux from MAVEN instruments over more than three years of data divided into three intervals of solar EUV/UV intensity: between 2 December 2014 and 10 March 2015 (Ls from 244.3° to 305.2°), between 4 June 2015 and 1 June 2017 (two Ls periods: from 273.5° to 57.9° and from 85.1° to 161.5°) and between 28 November 2017 and 31 July 2018 (Ls from 93.4° to 221.1°). By constraining the solar wind conditions, we show that these three samples were obtained under similar solar wind conditions and cover similar regions in a MSE frame.

We reconstruct the measured precipitating ion flux by MAVEN/SWIA. The increase of the EUV flux intensity from 0.13 to 0.42mW.m -2 is clearly associated with a decrease of the precipitating ion flux from 18.3 to 6.2 × 10 6 eV. cm -2 . sr -1 . s -1 (by almost a factor 3) for the energy range between 30 and 650 eV and a decrease by 18% to 44% at higher energies ([650-25000] eV). This somewhat surprising result, in contradiction with previous theoretical studies [START_REF] Chaufray | Mars solar wind interaction: Formation of the Martian corona and atmospheric loss to space[END_REF], might illustrate the complex relation between the variations of the exosphere as induced by the solar EUV/UV flux and the solar wind interaction with Mars. [START_REF] Dong | Seasonal variability of Martian ion escape through the plume and tail from MAVEN observations[END_REF] reported an almost linear dependence of the ion escape on the EUV/UV flux, whereas we have found an almost linear anti-correlation between the EUV flux and the precipitating ion flux. In others words, even if the EUV flux increase the ionization rate in the exosphere, the probability of a pickup ion to escape or to precipitate into the atmosphere should also be changed, illustrating a much more complex dependence of the fate of Mars' planetary ions on the solar conditions than anticipated.


Pascal Martin
email: pascal.martin@curie.fr

J Hudspeth

Dr A J Hudspeth

Mechanical frequency tuning by sensory hair cells, the receptors and amplifiers of the inner ear

Keywords: auditory system, cochlea, hair bundle, Hopf bifurcation, transduction, traveling wave 2

Mechanical Frequency Tuning by Sensory Hair Cells, the

Receptors and Amplifiers of the Inner Ear

Introduction

Imagine a strain gauge that reliably detects deflections of sub-nanometer magnitude. It responds from static deflection to periodic stimulation at frequencies approaching 200 kHz. Its dynamic range is outstanding, encompassing twelve orders of magnitude in input power. Tuned to specific input frequencies, the device can exhibit a quality factor as great as 1000. Less than 10 mm in diameter, it functions at high humidity-even under water-and is insensitive both to vibration and to electromagnetic interference. The system requires no cryogenic cooling; in fact, it generally operates near 310 K. This device is not a state-of-the art marvel of microfabrication, but rather the cochlea, or mammalian hearing organ (Figure 1a). Although no single cochlea meets all of the specifications noted above, all cochleae-including our own-are similar in development, structure, and operation. In fact, the essential features of our ears greatly antedate the mammalian line of evolution, so the ears of fishes, amphibians, and reptiles (including birds) share many features with those of mammals (1). This article reviews the operation of the cochlea and its mechanoreceptorssensory hair cells-with particular emphasis on frequency tuning, which is essential for our capacity to distinguish sound sources and interpret spoken words on the basis of their acoustic spectra. Naturally occurring sounds vary widely in waveform and duration. Some sounds, such as a clap of thunder or the crash of a falling object, are transient impulses.

However, most of the acoustic signals in which we are especially interested-especially those of verbal communication-are periodic in nature and last for tens to thousands of cycles. These characteristics imply that sounds can usefully be detected by a resonant system that accumulates energy across many cycles and thereby registers even very weak stimuli. We shall, in fact, analyze the ear's operation in these terms in the principal parts of this article.

While vibrating in a fluid whose viscosity would seem to preclude resonance, the cochlea nonetheless conducts a real-time frequency analysis of its inputs and represents each frequency component with high resolution. Frequency tuning stems both from the physical characteristics of the ear's components and from the emergent properties of the ear's "active process", which reflect the fact that hair cells operate far from thermodynamic equilibrium to mechanically amplify their inputs.

Analyzing the features of sound

The frequencies present in a sound are resolved in the auditory periphery by mechanisms antecedent to neural coding (2). This step occurs in two ways that are not mutually exclusive. In some auditory systems, different frequency components are separated along the receptor organ's length on the basis of the physical properties of the conductive medium. The most refined version of this approach is that of the mammalian cochlea, in which mechanical traveling waves peak at frequency-dependent positions [START_REF] Von Békésy | Experiments in Hearing[END_REF]. The mechanically sensitive hair cells at each position are then excited by only a specific band of frequencies and in turn respond best to those frequencies. The most effective frequency is termed the best or characteristic frequency. Although extrinsic filtering of the input delivered to the hair cells helps establish a frequency map along the cochlea, we shall see that the hair cells themselves sharpen the map by providing active mechanical feedback to the cochlear traveling waves. Hair-cell tuning is thus the result of both extrinsic and intrinsic mechanisms.

In simpler hearing organs, acoustic energy is distributed more-or-less equally to all the hair cells, but those cells nonetheless differ in their individual sensitivities to sounds of different frequency. Some hair cells are broadly tuned by a tuning fork-like mechanism in their mechanically sensitive organelles [START_REF] Aranyosi | [END_REF]5). A hair cell can also display electrical resonance of its membrane; here the number, conductance, and kinetic properties of Ca 2+ and K + channels determine the frequency at which a given cell's membrane potential resonates (6)(7)(8)(9)(10). This electrical tuning mechanism appears to be ubiquitous in nonmammalian vertebrates but is limited to low frequencies, up to about 1 kHz.

For each of these tuning mechanisms, hair cells are systematically arrayed along a single axis as a function of frequency. This phenomenon is termed "tonotopy" or "pitch by place." Although the relation between position and characteristic frequency is approximately linear in hearing organs with relatively few hair cells, the common pattern in more elaborate organs is a logarithmic mapping: each tenfold increment in frequency is represented by a roughly equal distance along the organ and a comparable number of hair cells. An interesting exception occurs in the cochleae of bats, which greatly overrepresent the narrow range of frequencies at which the animals vocalize and to which their sonar is most sensitive (11,12). In contrast, most ears operate over a broad frequency range (13); in humans, hearing spans three orders of magnitude, from 20 Hz to 20 kHz. Because each hearing organ is endowed with only a few thousand hair cells, it can devote only a limited number of sensory cells to the detection of a given sound frequency. As discussed below, the relatively small number of hair cells may limit the signal-to-noise ratio of sound detection (14,15).

Mechanical and hydrodynamic features of the cochlea

Although the anatomy and operation of the cochlea are quite complex, extensive review articles document the details (16,17). The pressure changes associated with sounds are captured by the external ear and funneled into the ear canal. These pressure changes set the thin eardrum into oscillatory motion, which is propagated through the three tiny bones of the middle ear. The last of the bones, the stapes, undergoes a piston-like movement at the oval window, one of the two elastic diaphragms opening into the boneencased cochlea, and thus engenders oscillatory pressure changes there (Figure 1a).

Owing to both the areal difference between the large eardrum and the tiny oval window and the system of levers represented by the bones, the apparatus of the middle ear largely matches the acoustic impedance of the air to the input impedance of the liquidfilled cochlea. This feature minimizes reflections and ensures that most of the incoming energy is transduced into electrical signals. However, optimal impedance matching occurs only at a specific frequency, about 2 kHz in humans; the middle ear thus acts as a band-pass filter (18). Movement of the stapes produces a pressure difference transverse to the basilar membrane, an elastic band that spirals within the cochlea and supports the receptive hair cells. As a consequence, the basilar membrane undergoes an oscillatory motion at the frequency or frequencies of stimulation. In the short cochleae of some reptiles, the entire basilar membrane moves as a unit and every hair cell receives a comparable stimulus. A key feature of the mammalian cochlea is that the basilar membrane displays gradients in stiffness and mass along its length [START_REF] Von Békésy | Experiments in Hearing[END_REF]19). Such a basilar membrane undergoes a more complex motion: a traveling wave originates at the base, gradually increases in amplitude as it propagates toward the apex, peaks sharply at a specific position, and then dissipates (20)(21)(22)(23)(24). Most importantly, the position of the peak is frequency-dependent: high frequencies excite the cochlear base most strongly, whereas low frequencies stimulate the apex. Intermediate frequencies are represented in monotonic order between the two extremes. The cochlea therefore acts as a real-time frequency analyzer that distributes the energy from any acoustic stimulus to specific hair cells on the basis of the signal's frequency content.

Structure and function of sensory hair cells

Oscillatory stimuli within a cochlea excite mechanically sensitive hair cells, which number about 16,000 in each human cochlea [START_REF] Retzius | Das Gehörorgan der Wirbelthiere. II. Das Gehörorgan der Reptilien, der Vögel und der Säugethiere 26[END_REF]. The unique and essential component of a hair cell-and the source of its peculiar name-is its mechanosensitive organelle, the hair bundle (26). This structure has nothing to do with actual hair, but corresponds to a tuft of microscopic, cylindrical processes emanating from the cell's flattened top surface (Figure 1b). Each of these processes is termed a stereocilium, a name meaning "stiff hair." The lengths of stereocilia and the heights of the corresponding hair bundles vary widely, from less than 1 µm to more than 50 µm. This variation correlates with the frequency at which the hair cell responds best: the shortest bundles operate at frequencies exceeding 100 kHz and the longest at those below 100 Hz. The tonotopic map in auditory organs is associated with systematic variations of hair-bundle morphology, with an inverse relation between the height and the number of stereocilia (27)(28)(29). Hair cells with higher characteristic frequencies are endowed with stiffer hair bundles (30,31).

The stereocilia in every hair bundle stand in rows of increasing height (Figure 1c), and deflecting the bundle toward its tall edge-defined as the positive direction-excites the hair cell (32,33). A movement in the opposite direction is inhibitory, whereas an orthogonal deflection has little or no effect. The conversion of hair-bundle deflections into electrical signals within the hair cells is conceptually simple (16,34). Along the axis of sensitivity, each short stereocilium is attached to the side of tallest adjacent stereocilium by a 150 nm-long protein strand termed the tip link (Figure 1d) (35). As a result of this oblique connection, the shearing motion that occurs between the two stereocilia when they move in the positive direction increases the tension in the tip link. That tension is communicated to a few mechanically sensitive channels at the link's lower insertion, which open to allow K + and Ca 2+ ions to flow into the hair cell and thus to depolarize it (36,37). The cell then increases its release of the synaptic transmitter glutamate, which excites the nerve fibers in contact with the hair cell (38)(39)(40).

The mechanical properties of tip links vary along the cochlea's tonotopic axis. The links are stiffer and bear more tension at the cochlear base than at the apex, and there appears to be a smooth gradient between (31). Although differences in the proteins themselves might account for this gradient, each constituent is represented by only one known gene and alternative mRNA splicing or post-translational modifications seem unlikely to be able to produce the range of observed properties. An interesting possibility is that tip links of a uniform molecular composition exhibit distinct properties as a result of strain hardening (41). Using an optical trap to extend individual molecules of protocadherin 15, one of the two proteins constituting the tip link (42), evokes the behavior expected for entropic stiffness: as a molecule's extension increases, its tension grows superlinearly (43). Each tip link's tension is set by a molecular motor at its upper insertion.

In at least some hair bundles, numerous myosin 1c molecules constitute this motor (44)(45)(46)(47); myosin 7a molecules might also contribute (48,49). The number of myosin molecules associated with each tip link or the force that each molecule can produce might rise with increasing characteristic frequency, resulting in increased tension and therefore greater stiffness. As noted below, this arrangement likely contributes to the tuning of hair bundles to specific frequencies.

The unitary conductance of the transduction channels, which is mediated in part by the pore-forming property of transmembrane channel-like proteins 1 and 2 (TMC1 and TMC2) (50)(51)(52), increases from the apex to the base of the cochlea (53-55). Each tip link is associated with an increasing number of TMC1 molecules along the tonotopic axis and a channel at a given cochlear location is endowed with multiple conductance states (55).

The tonotopic gradient in channel conductance might actually result from a variable number of identical channels gating cooperatively within a transduction element. In any case, the magnitude of the transduction current associated with one tip link increases with the characteristic frequency of the hair cell. Because the magnitude of the transduction current acts as a control parameter of active motility by the hair cell (56)(57)(58)(59), this feature is probably essential to modulate the frequency at which the active process boosts the input to a given hair cell.

Gating of mechanoelectrical-transduction channels

Every hair bundle acts as a mechanosensitive antenna. Deflection of a hair bundle elicits an electrical response that may be monitored by using a microelectrode to record the change in transmembrane potential of the hair cell. This response is graded with the amplitude of stimulation, but saturates at a magnitude of a few tens of millivolts (32). A more revealing technique uses a voltage-clamp circuit, which implements negative feedback to determine the ionic current that flows into a stimulated cell at a fixed potential.

Normalizing the result to the maximal value provides a measure of the channel open probability 𝑃𝑃 o , which is related to deflection 𝑋𝑋 of the hair bundle from its resting position by a Boltzmann equation (60):

𝑃𝑃 o = 1/[1 + 𝑒𝑒 -(𝑋𝑋-𝑋𝑋 0 ) 𝛿𝛿 ⁄ ]. [1]
The parameter 𝑋𝑋 0 is the deflection at which half the channels are open. Because its value dictates the operating point 𝑃𝑃 o (𝑋𝑋 = 0) of the transducer when there is no stimulus, this parameter will hereafter be called the set-point deflection. The characteristic lengthscale 𝛿𝛿 sets the width and maximal slope of the sigmoidal relation (61), and thus the operating range and mechanosensitivity of the hair bundle, respectively. . This condition is necessary to ensure sensitivity to small hair-bundle deflections. The tension results from an active pulling force 𝐹𝐹 mot R exerted by myosin molecular motors on the tip links (64). The motor force is balanced by the tension in the gating springs, which extend by a distance 𝑋𝑋 R . It follows that

𝑋𝑋 0 = Δ𝐸𝐸 ∅ 𝑍𝑍 -𝑋𝑋 𝑅𝑅 . [2]
The resting bundle is accordingly displaced in the negative direction by a distance 𝑋𝑋 SP = 𝐹𝐹 mot R /𝐾𝐾 𝑆𝑆𝑆𝑆 from the position at which it would reside with no tension in the spring. Here 𝐾𝐾 𝑆𝑆𝑆𝑆 represents the stiffness associated with the stereociliary pivots at which the stereociliary insertions flex. Each of these pivots consists of a few dozen actin filaments (65,66) that are reinforced by additional proteins (46).

Open probability and adaptation of cochlear hair cells

The mammalian cochlea is endowed with two types of hair cells that serve different Although the phase of the input is lost, auditory nerve fibers project to the auditory cortex according to a tonotopic map that informs the brain about the frequency of the stimulus.

The open probability 𝑃𝑃 o R = 0.5 is the worst insofar as electrical integration is concerned, for the Boltzmann relation between the input and the output (Equation 1) is antisymmetric about that point and a sinusoidal input yields no steady-state electrical

response. An open probability of one-half is ideal, however, for a response proportional to the transduction current: the relation between displacement and open probability-and thus current-is steepest there. This condition is optimal for amplification of basilarmembrane vibrations by electromotility of outer hair cells, a process whereby the cell body contracts or elongates in response to depolarization or hyperpolarization, respectively (74). As discussed below, the mechanical properties of hair bundles relevant to mechanical amplification are also optimal at an open probability of one-half (75). Thus, incompatible physical requirements for sensing and amplifying high-frequency sounds, exemplified here by the open probability of the transduction channels, may explain why the mammalian cochlea has established a division of labor between the inner, sensing and outer, amplifying hair cells.

Owing to its high mechanical sensitivity, a hair bundle's operating range is narrow:

the channels' open probability varies between 5 % and 95 % over a range of deflections 6𝛿𝛿 ≈ 100 nm, less than the diameter of an individual stereocilium. Even a modest force could deflect the bundle enough to saturate its response. To prevent this, natural selection developed an adaptation process that maintains high responsiveness (76)(77)(78).

Adaptation shifts the relation 𝑃𝑃 o (𝑋𝑋) along the displacement axis to continuously reset the operating point at steady state to the resting value 𝑃𝑃 o R , corresponding to a change in the set-point deflection 𝑋𝑋 0 . The two contributions to 𝑋𝑋 0 (Equation 2) have led to two broadand not mutually exclusive-classes of adaptation models. In the first class, "motorbased-models," adaptation occurs by an adjustment of the position at which the upper end of each tip link is anchored (78,79). This repositioning changes the extension of the gating spring from the resting value 𝑋𝑋 𝑅𝑅 . More specifically, the tip link's insertion is dragged downward by tension in the link but pulled upward by molecular motors, whose activity is regulated by the intracellular concentration of Ca 2+ (80). When the two forces are unbalanced, the motors move until force balance is restored and the motors stall. As a consequence of motor movements, adaptation translates the set-point deflection along the axis of displacement by a distance 𝑋𝑋 A :

𝑋𝑋 0 = Δ𝐸𝐸 ∅ 𝑍𝑍 -𝑋𝑋 𝑅𝑅 + 𝑋𝑋 𝐴𝐴 . [3]
In the second class of models, "Ca 2+ -reclosure models", the set-point deflection 𝑋𝑋 0 changes owing to alterations in the intrinsic energy difference Δ𝐸𝐸 ∅ . Within this framework, adaptation occurs when Ca 2+ binds to the channel, or to a closely associated protein, and stabilizes the channel's closed state (81,82). A positive deflection of the hair bundle increases the Ca 2+ influx into the hair bundle, resulting in an increase of Δ𝐸𝐸 ∅ and thus 𝑋𝑋 0 , and evokes channel closure. Conversely, adaptation to a negative bundle deflection occurs when a decreased influx of Ca 2+ destabilizes the closed state of the channel.

Although adaptation clearly relies on Ca 2+ feedback for most hair cells, the dependence of adaptation on Ca 2+ in mammalian cochlear hair cells has been debated (83,84). The lipid environment of the transduction channels might instead regulate or even mediate adaptation (85,86), but the underlying mechanism remains elusive. As discussed below, all the manifestations of active hair-bundle motility, which are interpreted as mechanical correlates of adaptation, depend on Ca 2+ .

Mechanical consequences of channel gating

Although electrical measurements show the open probability of a hair cell's transduction channels as a function of the hair-bundle deflection, mechanical measurements are even more revealing of how transduction occurs. An individual hair bundle can be studied in vitro by attaching to it the tip of a fine, horizontal, cylindrical glass fiber typically 100 µm in length and 0.5 µm in diameter (87,88). Imaging the fiber's tip onto a photodiode pair then allows measurement of movements with a nanometer precision and a frequency response to about 1 kHz. Moreover, displacement of the fiber's base with a piezoelectric stimulator applies to the bundle a force that can be computed after calibration of the fiber.

Through the use of feedback between bundle and fiber positions, an investigator can also impose more complex stimuli on a bundle (15,89,90).

Because of thermal flickering of the transduction channels between their open and closed states, the tension in a gating spring is a stochastic variable. Its mean value

𝑇𝑇 GS = 𝐾𝐾 GS (𝑋𝑋 + 𝑋𝑋 R -𝐷𝐷)𝑃𝑃 o + 𝐾𝐾 GS (𝑋𝑋 + 𝑋𝑋 R )(1 -𝑃𝑃 o ) = 𝐾𝐾 GS (𝑋𝑋 + 𝑋𝑋 R -𝐷𝐷𝑃𝑃 𝑜𝑜 ) [4]
depends on the channels' open probability 𝑃𝑃 𝑜𝑜 (𝑋𝑋) given by Equation 1, and thus varies nonlinearly with the hair-bundle deflection 𝑋𝑋. We recall that 𝐾𝐾 GS is the gating-spring stiffness, 𝑋𝑋 R the gating-spring extension at rest, and 𝐷𝐷 the gating swing. The force 𝐹𝐹 required to deflect the hair bundle by a distance 𝑋𝑋 is then described by the sum of two contributions 𝐹𝐹 = 𝐾𝐾 GS (𝑋𝑋 + 𝑋𝑋 R -𝐷𝐷𝑃𝑃 𝑜𝑜 ) + 𝐾𝐾 SP (𝑋𝑋 -𝑋𝑋 SP ), [5] corresponding to the tension 𝑇𝑇 GS in the gating springs, a nonlinear function of 𝑋𝑋, and a linear elastic restoring force from the stereociliary pivots of stiffness 𝐾𝐾 SP and equilibrium position 𝑋𝑋 SP . The mechanical relaxation associated with channel gating-the result of the gating swing 𝐷𝐷-is of profound importance, for it introduces a mechanical nonlinearity with multiple consequences.

The most striking consequence of channel gating is its contribution to the stiffness 𝐾𝐾 HB = 𝐹𝐹′ of a hair bundle. Here and in the following, a prime denotes differentiation with respect to an appropriate variable, in this case the bundle's deflection 𝑋𝑋. From Eqs. 1 and 5,

𝐾𝐾 HB = 𝐾𝐾 𝐺𝐺𝑆𝑆 (1 -𝐷𝐷𝑃𝑃 o ′ ) + 𝐾𝐾 SP = 𝐾𝐾 ∞ - 𝑁𝑁𝑍𝑍 2
𝑘𝑘 𝐵𝐵 𝑇𝑇 𝑃𝑃 𝑜𝑜 (1 -𝑃𝑃 𝑜𝑜 ). [6] We see that channel gating effectively softens the hair bundle, a phenomenon called gating compliance (91). Gating compliance reduces the bundle's stiffness by a maximal amount Δ𝐾𝐾 = 𝑁𝑁𝑍𝑍 2 /(4𝑘𝑘 B 𝑇𝑇) from the asymptotic value 𝐾𝐾 ∞ = 𝐾𝐾 𝐺𝐺𝑆𝑆 + 𝐾𝐾 SP that is measured at large negative (𝑃𝑃 𝑜𝑜 ≅ 0) or positive (𝑃𝑃 𝑜𝑜 ≅ 1) deflections. If the gating force 𝑍𝑍 is sufficiently great, the hair-bundle stiffness can actually become negative for values of the open probability near one half (92). Such behavior inevitably leads to mechanical instability, which natural selection has exploited to engineer the hair cell's active process.

Channel gating has another remarkable consequence: it engenders an increase in the effective friction of a hair bundle, called the gating friction (75,93,94). Because of the finite activation kinetics of the transduction channels, thermal equilibration of the channel's open probability is not instantaneous but occurs on a timescale 𝜏𝜏 C = 10 µs -1 ms. As a result, when the bundle moves in response to a time-varying stimulus, channel-gating forces arise with a delay and are therefore dissipative. Gating friction is associated with a friction coefficient 𝜆𝜆 𝐶𝐶 ≅ Δ𝐾𝐾 • 𝜏𝜏 𝐶𝐶 that can be large enough to dominate the viscous drag on micrometer-sized hair bundles and thus to limit the kinetics of bundle movements. Perhaps to compensate for this effect, hair cells that respond to higher frequencies are endowed with faster transduction channels (95) that impose less gating friction.

Gating compliance and gating friction, which are intimately related to the mechanosensitivity of transduction channels, impose a reciprocal relation between stimulus forces and channel gating. External forces affect the open probability of the channels by changing the elastic energy stored in the gating springs; in return, channel gating changes the extension of the gating springs, producing internal forces-gating forces-that affect both the stiffness and the friction of the hair bundle. This framework can be generalized to any system comprising bistable elastic elements, as has been proposed for instance for muscle fibers (96)(97)(98).

Resonant mechanical tuning from active hair-bundle motility

The inner ears of vertebrates operate in a liquid environment in which hydrodynamic damping is inescapable. For a mechanically passive hair bundle of height 5 µm, the bundle's stiffness 𝜆𝜆𝑋𝑋 ̇= -𝑘𝑘𝑋𝑋 + 𝐹𝐹 𝐴𝐴 + 𝐹𝐹, [7] in which 𝑋𝑋 represents a small hair-bundle deflection with respect to a fixed point, 𝜆𝜆 is the friction coefficient associated with the bundle's movement, and 𝑘𝑘 is the bundle's stiffness.

𝑘𝑘 ≈ 2 mN•m -1 , viscous-drag coefficient 𝜆𝜆 ≈100 nN•s•m -1 ,
𝐹𝐹 denotes an externally applied force, especially that owing to auditory stimulation. Here and below, overdots represent temporal derivatives of the corresponding orders. To drive oscillations, the active force 𝐹𝐹 𝐴𝐴 must provide feedback on the hair-bundle position 𝑋𝑋. We write to linear order 𝜏𝜏 𝐴𝐴 𝐹𝐹 𝐴𝐴 ̇= 𝑘𝑘 � 𝑋𝑋 -𝐹𝐹 𝐴𝐴 , [8] in which τA is the time constant of the force feedback and 𝑘𝑘 � represents the stiffness of the elastic element that mediates this feedback. We shall see below how these two parameters can be related to the timescale of adaptation and to gating-spring stiffness, respectively.

The two equations may be combined into a single, second-order relation:

𝑚𝑚 𝐸𝐸𝐸𝐸𝐸𝐸 𝑋𝑋 ̈= -𝜆𝜆 𝐸𝐸𝐸𝐸𝐸𝐸 𝑋𝑋 ̇-𝑘𝑘 𝐸𝐸𝐸𝐸𝐸𝐸 𝑋𝑋 + 𝐹𝐹 � . [9]
This representation maps the dynamical system to a classic harmonic resonator with effective mass 𝑚𝑚 𝐸𝐸𝐸𝐸𝐸𝐸 = 𝜆𝜆𝜏𝜏 𝐴𝐴 , drag coefficient 𝜆𝜆 𝐸𝐸𝐸𝐸𝐸𝐸 = 𝜆𝜆 + 𝑘𝑘𝜏𝜏 𝐴𝐴 , and stiffness 𝑘𝑘 𝐸𝐸𝐸𝐸𝐸𝐸 = 𝑘𝑘 -𝑘𝑘 � . In the ensuing analysis, this formulation offers the didactic advantage of delineating how the components of specific implementations contribute to resonant frequency tuning. Note that the effective force

𝐹𝐹 � = 𝐹𝐹 + 𝜏𝜏 𝐴𝐴 𝐹𝐹 ̇ [10]
applied to the system depends both on the actual force 𝐹𝐹(𝑡𝑡) applied to the hair bundle and on its time derivative. During sinusoidal stimulation 𝐹𝐹(𝑡𝑡) = 𝐹𝐹 � 𝑒𝑒 -𝑖𝑖𝑖𝑖𝑖𝑖 + 𝐹𝐹 � * 𝑒𝑒 +𝑖𝑖𝑖𝑖𝑖𝑖 , the effective force 𝐹𝐹 � (𝑡𝑡) remains sinusoidal but with an altered phase and amplitude with respect to 𝐹𝐹(𝑡𝑡). Because Equation 9is linear, the system responds at the frequency of the stimulus, with 𝑋𝑋(𝑡𝑡) = 𝑋𝑋 � 𝑒𝑒 -𝑖𝑖𝑖𝑖𝑖𝑖 + 𝑋𝑋 � * 𝑒𝑒 +𝑖𝑖𝑖𝑖𝑖𝑖 .

Although we have neglected the bundle's actual mass in our description of hairbundle mechanics (Equation 7), the interplay between hair-bundle mechanics and delayed force feedback endows the hair bundle with effective inertia. Force feedback also affects both the bundle's apparent friction coefficient and its stiffness. When the bundle's stiffness is positive (𝑘𝑘 > 0), force feedback increases friction. However, as described earlier, a defining property of mechanosensitive ion channels is gating compliance: channel gating produces a mechanical relaxation that effectively softens the gating springs (Equation 6), an effect that can be strong enough to result in negative stiffness of the hair bundle. When a hair bundle operates at a position of negative stiffness (𝑘𝑘 < 0), force feedback reduces friction on the hair bundle.

A key feature of the active system in the absence of an external force (𝐹𝐹 = 0) is that stability vanishes when 𝜆𝜆 EFF = 𝜆𝜆 + 𝑘𝑘𝜏𝜏 A = 0, the condition for a Hopf bifurcation.

Spontaneous oscillations then emerge at the natural frequency 𝜔𝜔 0 = �𝑘𝑘 EFF 𝑚𝑚 EFF ⁄ , which can be expressed as

𝜔𝜔 0 = ��𝑘𝑘 -𝑘𝑘 � � (𝜆𝜆𝜏𝜏 A ) ⁄ . [11]
When operating on the stable side of the bifurcation (𝜆𝜆 EFF > 0) and when stimulated at a frequency 𝜔𝜔 ≅ 𝜔𝜔 0 , the system becomes highly resonant in response to an external sinusoidal force 𝐹𝐹 if the effective friction 𝜆𝜆 EFF becomes small enough. The quality factor then obeys 𝑄𝑄 = �𝑘𝑘 EFF 𝑚𝑚 EFF 𝜆𝜆 EFF = ��𝑘𝑘 -𝑘𝑘 � �𝜆𝜆𝜏𝜏 A / � (𝜆𝜆 + 𝑘𝑘𝜏𝜏 A ) ≫ 1. Precisely at the Hopf bifurcation, Equation 9indicates that the complex impedance 𝐴𝐴(𝜔𝜔)

= 𝐹𝐹 � (𝜔𝜔)/𝑋𝑋 � (𝜔𝜔) = k 𝐸𝐸𝐸𝐸𝐸𝐸 [1 -(𝜔𝜔 𝜔𝜔 0 ⁄ ) 2 ]/(1 -𝑖𝑖𝜔𝜔𝜏𝜏 A )
of the critical oscillator vanishes at the oscillator's natural frequency 𝜔𝜔 = 𝜔𝜔 0 . Both the quality factor and linear sensitivity 𝜒𝜒 = 1/�𝐴𝐴 ̃� to the stimulus then formally diverge and the magnitude of the response falls under the control of nonlinearities that are discussed below (Equation 20).

As in any harmonic resonance, high sensitivity to sinusoidal stimulation at resonance is associated with sharp frequency selectivity, a property that is obviously advantageous for sound detection. The system is actively tuned at a natural frequency 𝜔𝜔 0 (Equation 11) that appears as the inverse geometric mean of two timescales: the characteristic time 𝜏𝜏 A of the active force feedback and the mechanical response time 𝜏𝜏 MEC = 𝜆𝜆/(𝑘𝑘 -𝑘𝑘 � ) to small step forces. Frequency tuning is an emergent property of the interplay between passive hair-bundle mechanics and active force feedback. The natural frequency depends on parameters that control the bundle's stiffness and friction as well as on the timescale of the active feedback.

The imaginary part 𝐴𝐴 𝑖𝑖 of the complex impedance 𝐴𝐴(𝜔𝜔) = 𝐹𝐹 � (𝜔𝜔)/𝑋𝑋 � (𝜔𝜔) betrays the mechanical activity of the system. From Equation 9, we write

𝐴𝐴 𝑖𝑖 = 𝑘𝑘 𝐸𝐸𝐸𝐸𝐸𝐸 𝜔𝜔𝜏𝜏 A ��1 -� 𝑖𝑖 𝑖𝑖 0 � 2 � - 1 𝑖𝑖 0 𝜏𝜏 A 𝑄𝑄 � /[1 + (𝜔𝜔𝜏𝜏 A ) 2 ]. [12]
As the system traverses a Hopf bifurcation (𝑄𝑄 → +∞), 𝐴𝐴 𝑖𝑖 changes sign at 𝜔𝜔 = 𝜔𝜔 0 : 𝐴𝐴 𝑖𝑖 > 0 for stimulation at a frequency below the system's natural frequency (𝜔𝜔 < 𝜔𝜔 0 ) and conversely 𝐴𝐴 𝑖𝑖 < 0 for stimulation above the natural frequency (𝜔𝜔 > 𝜔𝜔 0 ). This behavior is generic for Hopf oscillators (115). Positive values of 𝐴𝐴 𝑖𝑖 correspond to active energy pumping into the surrounding fluid-negative dissipation. A passive harmonic oscillator must instead dissipate energy (𝐴𝐴 𝑖𝑖 < 0) at all frequencies. As long as the timescale of the active force feedback is great enough that 𝜔𝜔 0 𝜏𝜏 A 𝑄𝑄 > 1, corresponding to 𝜏𝜏 A > 𝜆𝜆/�𝑘𝑘 � �, active energy pumping occurs even when the system (Eqs. 7-8) operates some distance from the Hopf bifurcation, though over the narrower range of frequencies 𝜔𝜔 < 𝜔𝜔 0 �1 -1/(𝜔𝜔 0 𝜏𝜏 A 𝑄𝑄). This situation is relevant to the cochlea, in which energy is pumped into the traveling wave basal to the position where the amplitude of vibration peaks (115,116). The local response of the cochlea to weak sounds of varying frequency can actually be described by that of phenomenological oscillators obeying Equation 9, in which the stimulus 𝐹𝐹 � (𝑡𝑡) corresponds-remarkably enough-to a combination of the pressure difference transverse to the basilar membrane and its weighted time derivative (117, 118).

The phase advance of the force 𝐹𝐹 � driving the effective harmonic oscillator with respect to the phase of the actual external force 𝐹𝐹 again bespeaks the active nature of the force feedback provided by the adaptation process.

Active force feedback from adaptation

How is active force feedback (Equation 8) actually implemented by a hair cell? As discussed above, the hair bundle is endowed with a dynamic biasing mechanismadaptation-that continuously resets the bundle's open probability to the resting value 𝑃𝑃 𝑜𝑜 R to prevent saturation and preserve sensitivity to weak, time-varying stimuli. Without loss of generality, we assume below that the operating point is at 𝑃𝑃 𝑜𝑜 R = 0.5, corresponding to maximal displacement sensitivity. A minimal description of the interplay between overdamped, but nonlinear, hair-bundle mechanics and adaptation can be written as 𝜆𝜆𝑋𝑋 ̇= -𝐾𝐾 GS [𝑋𝑋 -𝐷𝐷(𝑃𝑃 𝑜𝑜 -0.5)] -𝐾𝐾 SP 𝑋𝑋 + 𝐹𝐹, [13] 𝜏𝜏𝑋𝑋 ̇0 = 𝑋𝑋 -𝑋𝑋 0 , [14] in which 𝑃𝑃 𝑜𝑜 obeys Equation 1, 𝐾𝐾 GS and 𝐾𝐾 SP are the respective contributions of the gating springs and stereociliary pivots to hair-bundle stiffness, and 𝜏𝜏 is the timescale of adaptation. In this instance, adaptation is described by a relaxation process of the set-point deflection 𝑋𝑋 0 toward the bundle's deflection 𝑋𝑋. In a steady state, 𝑋𝑋 = 𝑋𝑋 0 = 0 and 𝑃𝑃 𝑜𝑜 = 0.5. Through linear expansion about this fixed point, the nonlinear model described by Equations 13-14 can be mapped to the generic linear model for active hair-bundle motility (Eqs. [7][8] with

𝑘𝑘 = 𝐾𝐾 GS (1 -𝐷𝐷𝑃𝑃 o ′ ) + 𝐾𝐾 SP , 𝑘𝑘 � = -𝐾𝐾 GS 𝐷𝐷𝑃𝑃 𝑜𝑜 ′ , 𝐹𝐹 A = -𝐾𝐾 GS 𝐷𝐷𝑃𝑃 o ′ 𝑋𝑋 o = -𝑁𝑁𝑍𝑍𝑃𝑃 o ′ 𝑋𝑋 o ,
𝜆𝜆 A 𝑋𝑋 ̇A = +𝐾𝐾 GS (𝑋𝑋 -𝑋𝑋 A -𝐷𝐷𝑃𝑃 o ) -𝐹𝐹 M , [16]
with 𝑃𝑃 𝑜𝑜 (𝑋𝑋) = 1/{1 + Λexp[-(X -X A )/δ]} and Λ = exp[Δ𝐸𝐸 ∅ /(k 𝐵𝐵 𝑇𝑇)] a constant that sets the channel's open probability when there is no tension in the tip links. The motor dynamics (Equation 16) is described by a linear relation of slope 𝜆𝜆 A between the tension

𝑇𝑇 GS = 𝐾𝐾 GS (𝑋𝑋 -𝑋𝑋 A -𝐷𝐷𝑃𝑃 o )
in the gating springs and the motor velocity 𝑋𝑋 ̇A; the motors stall when gating-spring tension balances the force 𝐹𝐹 M produced by the motors. To account for Ca 2+ effects on the operating point of the transducer (82) and on the kinetics of adaptation (78), we write that the motor force depends on the channels' open probability as

𝐹𝐹 M = 𝐹𝐹 max (1 -𝑆𝑆𝑃𝑃 o ), [17]
in which 𝑆𝑆 represents the strength of Ca 2+ feedback (75). In this formulation, the motor force is maximal when the channels are closed and the Ca 2+ concentration at the motor site is low, but decreases when the channels open and allow an influx of Ca 2+ . Note that the strength of Ca 2+ feedback is expected to increase with the extracellular Ca 2+ concentration and at hyperpolarizing transmembrane potentials, both of which augment the Ca 2+ influx through an open transduction channel and can be controlled experimentally (57,119,120).

Again through linear expansion about a fixed point, the nonlinear motor-based model described by Eqs. 15-17 can be mapped to the generic linear model for active hairbundle motility (Eqs. 7-8). We then identify the stiffness 𝑘𝑘 = 𝐾𝐾 𝐺𝐺𝑆𝑆 (1 -𝐷𝐷𝑃𝑃 o ′ ) + 𝐾𝐾 SP of the hair bundle and the stiffness 𝑘𝑘 � = 𝐾𝐾 GS (1 -𝐷𝐷𝑃𝑃 o ′ ) of the gating springs, both of which include gating compliance (Equation 6). The active force 𝐹𝐹 A = 𝑘𝑘 � 𝑋𝑋 A stems from a tension change in the gating springs of stiffness 𝑘𝑘 � as the result of the movement 𝑋𝑋 A of the motors; the time constant of the ensuing force feedback, or equivalently of adaptation, is

𝜏𝜏 A = 𝜆𝜆 A /[𝐾𝐾 𝐺𝐺𝑆𝑆 (1 -𝐷𝐷𝑃𝑃 o ′ ) + 𝑆𝑆𝐹𝐹 max 𝑃𝑃 o ′ ] . [18]
By controlling the value of 𝑃𝑃 o ′ , the transducer's operating point affects the timescale of adaptation in remarkable ways (57). At an operating point near 𝑃𝑃 o = 0.5, where 𝑃𝑃 o ′ peaks, strong Ca 2+ feedback (𝑆𝑆𝐹𝐹 max > 𝐾𝐾 GS 𝐷𝐷) reduces the time constant; adaptation can consequently be fast, occurring on timescales of a millisecond or less. Under conditions of weak Ca 2+ feedback (𝑆𝑆𝐹𝐹 max < 𝐾𝐾 GS 𝐷𝐷), gating compliance instead prevails and adaptation becomes quite slow (𝜏𝜏 A > 100 ms). At operating points where 𝑃𝑃 o ′ is small, 𝑃𝑃 o ≅ 0 or 𝑃𝑃 o ≅ 1, the timescale of adaptation is simply 𝜏𝜏 A = 𝜆𝜆 A 𝐾𝐾 GS ⁄ , which lies in the range of tens of milliseconds.

The requirement for negative stiffness (𝑘𝑘 < 0) to situate the hair bundle near a Hopf bifurcation implies that gating compliance is strong enough to impose 𝑘𝑘 � < 0. However, the Ca 2+ -feedback strength must then be great enough to keep 𝜏𝜏 A > 0, but not too large to maintain the system at a stable fixed point (𝜆𝜆 + 𝑘𝑘𝜏𝜏 ≥ 0). The hair bundle cannot oscillate spontaneously in the absence of Ca 2+ feedback (𝑆𝑆 = 0). We note that groups of molecular motors under elastic loading, embodied here by myosin motors pulling on tip links, can in principle oscillate collectively even with no Ca 2+ feedback (121-123); this is not the case in the hair cell (58). At the Hopf bifurcation, spontaneous oscillations occur at a frequency that grows with the strength 𝑆𝑆 of Ca 2+ feedback and the maximal force 𝐹𝐹 max that the motors can produce (Eqs. 11 and 18). In keeping with this reasoning, the conductance of the transduction channels and the tip-link tension increase with the natural frequency of hair cells along the tonotopic axis of the cochlea (31,53,124).

In the alternative formulation, the Ca 2+ -reclosure model of adaptation postulates that Motors remain necessary to provide tension in the tip links and to set an operating point at rest within the steep region of the sigmoidal relation 𝑃𝑃 o (𝑋𝑋). However, the motors in this instance do not move on the short timescales associated with auditory frequencies.

Under these assumptions, the equation of motion for the hair bundle is given by Equation 13. To describe adaptation (125), we assume that the set-point deflection 𝑋𝑋 0 (𝐶𝐶) is a function of the Ca 2+ concentration 𝐶𝐶 at the binding site; the channels' open probability still obeys Equation 1. We further assume that the kinetics of adaptation is limited by diffusion from the channel's pore to the Ca 2+ -binding site and thus by the timescale 𝜏𝜏 of Ca 2+ -concentration changes there. We write As with the motor model, a Hopf bifurcation can occur if the gating compliance is strong enough to mediate negative bundle stiffness (𝑘𝑘 < 0). In addition, the Ca 2+ sensitivity 𝜇𝜇 must be large enough to impose 𝑘𝑘 EFF = 𝑘𝑘 -𝑘𝑘 � > 0, but not too large to maintain the system at criticality (𝜆𝜆 + 𝑘𝑘𝜏𝜏 A ≅ 0). Note that other formulations of the Ca 2+ -reclosure model have been proposed; for example, the kinetics of adaptation might be limited by the rate constant for Ca 2+ binding (81). The uncertainty about this issue might be resolved by the identification of the Ca 2+ -binding molecule, perhaps the protein CIB2 (126, 127).

𝜏𝜏𝐶𝐶 ̇= 𝐶𝐶 0 + 𝐶𝐶 1 𝑃𝑃 o -𝐶𝐶, [ 19
This analysis emphasizes the dynamic interplay between negative stiffness resulting from passive gating forces produced by the transduction channels and delayed force feedback provided by myosin motors and Ca 2+ binding. By actively reducing the friction on a hair bundle and by effectively increasing the bundle's mass, this interplay can bring an overdamped hair bundle near a Hopf bifurcation, where the bundle becomes highly resonant in response to sinusoidal stimuli near the natural frequency of the oscillatory instability. This frequency is an emergent property that depends on the passive mechanical properties of the bundle as well as the parameters that condition the kinetics and magnitude of the active force feedback. Multiple parameters can be tuned to adjust the characteristic frequency at which the active bundle responds most sensitively to an incoming sound. It is thus no wonder that the number and height of stereocilia within a hair bundle (27)(28)(29), the stiffness and tension of the tip links (31), the conductance and activation kinetics of the transduction channels (128, 129), and the concentration of Ca 2+ buffers in hair cells (130) display gradients along auditory organs.

The auditory nonlinearity

The most striking manifestations of the ear's complexity are the signs of nonlinearity. The ear does not operate as a high-fidelity sound receiver, but instead introduces distortions-"phantom tones"-that modify the acoustic input (131). The sound that we perceive is consequently not precisely the sound to which we are exposed! The properties of the auditory nonlinearity have been extensively studied through psychophysical experiments (132). Some distortions are expected, of course, from the saturating nonlinearity imposed by the sigmoidal relation between the hair bundle's deflection and the open probability of the transduction channels (Equation 1) and from the ensuing mechanical nonlinearity in the force-displacement relation of the hair bundle (Equation 5) (133). However, this static nonlinearity alone does not explain the remarkable properties of auditory distortions.

Instead, the properties of two-tone distortions in hearing are consistent with an "essential nonlinearity" for which there is no stimulus weak enough to elicit a linear response (132,134). Thus the problem is not to understand saturation at high levels, which is expected, but rather why the ear remains nonlinear even at very low levels of stimulation. What is the basis of the ear's peculiar nonlinearity?

The generic nonlinearity associated with any active dynamical system operating near a Hopf bifurcation-a critical oscillator-provides a powerful and economic means to interpret the most salient properties of the cochlea and of two-tone distortions (115, 135). As discussed above in the context of active hair-bundle motility, the complex impedance 𝐴𝐴(𝜔𝜔) = 𝐹𝐹 � (𝜔𝜔)/𝑋𝑋 � (𝜔𝜔) of a critical oscillator vanishes in response to sinusoidal stimulation at the oscillator's natural frequency. To leading order of a general series expansion (136), operation near a Hopf bifurcation mandates that the relation in the Fourier domain between the response 𝑋𝑋 � and the force 𝐹𝐹 � adopt the cubic form

𝐹𝐹 � ≅ 𝐴𝐴𝑋𝑋 � + 𝐵𝐵�𝑋𝑋 � � 2 𝑋𝑋 � , [20]
in which 𝐵𝐵 is a complex coefficient that describes the magnitude of the nonlinear term. At the Hopf bifurcation and for stimulation at the natural frequency 𝜔𝜔 0 , the response �𝑋𝑋 � � ∝ �𝐹𝐹 � � 1/3 and the sensitivity 𝜒𝜒 ∝ �𝐹𝐹 � � -2/3 display characteristic power-law behaviors. At a given force amplitude, the relation between the frequency and sensitivity displays a peak centered at 𝜔𝜔 0 with a width Δ𝜔𝜔 ∝ �𝐹𝐹 � � 2/3 that increases with the magnitude of the force and is inversely related to the sensitivity (115, 137, 138).

A critical oscillator is thus endowed with automatic gain control and frequency selectivity: the weakest stimuli are detected with highest sensitivity and sharpest frequency selectivity, but the sensitivity and selectivity deteriorate as the magnitude of the stimulus rises. These remarkable properties recapitulate those observed experimentally both for a single hair bundle (139) and for the basilar membrane (140). As a result of nonlinear amplification, forces that elicit vibrations beyond a given threshold can be much weaker than in the passive case, greatly expanding the dynamic range. However, these benefits come at a cost: the nonlinearity associated with the Hopf bifurcation (Equation 20) distorts the response to a two-tone stimulus when the stimulus frequencies As a result of noise, there is no true essential nonlinearity: the response always starts to grow linearly at some low stimulus level before displaying the one-third power law expected for a critical oscillator. This behavior is actually observed in experiments, both for single hair cells (139) and for the basilar membrane (149).

𝑓𝑓
Mechanical coupling between identical noisy oscillators provides a means of achieving greater sensitivity to low stimulus levels and thus of increasing the signal-tonoise ratio: coupling results in synchronization that effectively reduces the noise intensity, approaching more closely the behavior of a critical oscillator (14,15). The larger the number of oscillators in a cluster, the more sensitive and frequency-selective the detection and the greater the quality factor of the resonance. This property is relevant to bats, some of which mobilize nearly half of the cochlea, corresponding to some 2,000 outer hair cells, for echolocation near a single frequency. The quality factor of the resonance accordingly reaches values exceeding 1,000 (104). Most mammals, including humans, distribute their hair cells more uniformly along the tonotopic map and thus devote a much smaller number of cells, probably about 100, to the detection of any given frequency. Tuning is correspondingly much broader than in bats, with quality factors of 10-20 (150). In owing to the limited number of hair cells available, there is a tradeoff between frequency selectivity at a particular frequency and the range of frequencies over which the organ can operate with high frequency selectivity.

Conclusion

Although we now have a good grasp of the activity of individual hair cells, we understand less well the subtle operation of the mammalian cochlea. Lodged in the hardest bone of the body, the cochlea is a small, delicate, metabolically vulnerable organ whose operation is readily compromised by experimental intervention. Because movements of the relevant structures are of nanometer dimensions, complex interferometric apparatus is required to make measurements. The organ is largely transparent, so the optical signals are weak and even the best apparatus cannot detect some movements. Moreover, it is difficult to sustain the active features of the cochlea under in vitro conditions. Most importantly, we have only a rudimentary understanding of the emergent properties associated with the activity of multiple cochlear hair cells that are coupled through a complex mechanical and hydrodynamic environment.

Progress in our understanding of the cochlea will likely emerge from the interplay of two approaches. The first is improvement in the techniques for measuring cochlear motions. Optical coherence tomography has recently revolutionized such recordings and should improve further as it is adapted for the special needs of auditory physiology. The second is theoretical modeling on the basis of physical principles. For more than 50 years, mathematical approaches have been of critical importance in understanding the propagation of cochlear traveling waves. More recently, bifurcation theory has illuminated the influence of tension in a gating spring of stiffness K GS (maroon). An external force F displaces the bundle a distance X by acting on the gating-spring stiffness K GS as well as on the stereociliary-pivot stiffness K SP (green) and the bundle's viscous drag of coefficient λ (brown). Active force feedback on the bundle's displacement may proceed through two adaptation mechanisms that are not mutually exclusive. In the motor model, myosin molecules (red) move a distance X a to restore force balance between the motors' stall force F a and tension in the gating spring; the velocity of motion X ̇a is inversely related to the friction coefficient λa (gray). In the Ca 2+ -reclosure model, Ca 2+ that enters the stereocilium through transduction channels acts to reclose them.

  𝐷𝐷 and the tension by 𝑍𝑍. At thermal equilibrium, the energy difference Δ𝐸𝐸(𝑋𝑋) = -𝑍𝑍(𝑋𝑋 -𝑋𝑋 0 ) between the open and closed states of the channel sets the channel's open probability 𝑃𝑃 o at position 𝑋𝑋. The open probability obeys Equation 1, from which one can identify 𝛿𝛿 = 𝑘𝑘 B 𝑇𝑇/𝑍𝑍, in which 𝑘𝑘 B is the Boltzmann constant and 𝑇𝑇 is temperature. Therefore, the greater the gating force 𝑍𝑍, the larger is the transduction current in response to a given deflection of the hair bundle: the parameter 𝑍𝑍 controls the mechanosensitivity of the channel. Note that channel opening is stochastic, rather than deterministic: each channel flickers back-and-forth between its two states hundreds to millions of times a second. The set-point deflection 𝑋𝑋 0 reflects the effects of two countervailing forces at the transduction channel. In the absence of any tension in the gating spring, the channel would remain shut at nearly all times: the energy difference ΔE ∅ ≈ 10 𝑘𝑘 B 𝑇𝑇 between the open and closed states of the channel is large enough that thermal agitation would rarely provide the energy needed to effect opening (63). With the hair bundle at rest, however, each gating spring normally bears enough tension to open a significant fraction of the channels, resulting in an operating point at rest 𝑃𝑃 o R = 𝑃𝑃 o (𝑋𝑋 = 0) that lies within the steep region of the sigmoidal relation 𝑃𝑃 o (𝑋𝑋)

  functions and are associated with distinct patterns of open probability 𝑃𝑃 o R at the resting position (67, 68). Inner hair cells, which transduce acoustic stimuli into electrical activity of auditory nerve fibers, characteristically display open probabilities near 0.2 (69, 70). In contrast, mammalian outer hair cells, which send little information to the brain but serve instead as mechanical amplifiers of input signals to the inner hair cells, operate near an open probability of 0.5 (71-73). An open probability 𝑃𝑃 o R = 0.2 makes sense in light of the strategy that the inner cells use to represent high-frequency stimuli. This operating point ensures that the cells rectify sinusoidal inputs to generate a steady-state potential change at frequencies that are low-pass filtered by the cell membrane, typically beyond 3 kHz.

  and mass 𝑚𝑚 ≈ 50•10 -15 kg yield a quality factor 𝑄𝑄 = √𝑘𝑘𝑚𝑚/𝜆𝜆 = 0.1. These values are typical for vestibular hair cells or low-frequency cochlear hair cells with characteristic frequencies in the kilohertz range. At the highest frequencies audible by any mammal, which are about a hundredfold as great, the corresponding bundles are a fifth as tall and are expected to be approximately one hundredfold as stiff(30,31). Assuming that the bundle's mass and friction coefficient vary only weakly along the tonotopic axis of auditory organs (99), the quality factor likely remains below unity: most passive hair bundles are overdamped (but see (100-102)). Inertial loading of the hair bundle with an accessory structure such as the tectorial membrane in the mammalian cochlea (103) can promote moderate mechanical resonance, but cannot account for the sharp tuning observed in vivo at the level of the basilar membrane and auditory nerve, with quality factors that can reach 1000 in some bats (104). These considerations pose a challenge: how might strong resonance occur under such highly damped conditions?The hair bundle is mechanically active(16, 34, 105). By mobilizing energy resources of biochemical origin, a hair cell can power active movements of its hair bundle to boost the bundle's performance as a mechanoelectrical transducer. Active hair-bundle motility displays several manifestations. A quiescent hair bundle can respond to a force step with a twitch (106)-a transient movement with a directionality opposite that of the applied force-or by developing a net pulling force on an attached fiber (107), a phenomenon analogous to the stretch activation of muscle. These properties are advantageous for sensitive detection of step stimuli, such as those evoked by abrupt head or fluid movements in vestibular or lateral-line organs, respectively. Hair bundles can also oscillate spontaneously(58,87, 108), a phenomenon that likely underlies spontaneous otoacoustic emission, the unprovoked emanation of sound from an ear (109).Active motility provides a decisive functional advantage for hearing, for an oscillatory hair bundle amplifies periodic inputs near the natural frequency of spontaneous oscillations, thus enhancing the sensitivity and sharpening the frequency selectivity of auditory detection (108). The extra-and intracellular Ca 2+ concentrations, the membrane potential at rest, the stiffness of an elastic load, and constant external forces have been identified as possible control parameters of hair-bundle activity. Varying one of these parameters elicits transitions between one form of motility and another(57,89, 110, 111); individual bundles can thus operate in multiple mechanosensory modes. Of particular importance for the physics of hearing, transitions between quiescence and spontaneous oscillation position the hair bundle near a Hopf bifurcation(112, 113). At this operating point, the hair bundle shows signatures of the generic properties of any dynamical system operating near a Hopf bifurcation-a critical oscillator: amplification with a specific form of compressive nonlinearity and sharp frequency tuning, both of which depend on the magnitude of stimulation (Equation20).A common strategy in considering active hair-bundle motility is to present the experimentally observed phenomena, then to encapsulate them in a physical model. Here we proceed in the opposite direction, starting from a minimal abstract model and subsequently incorporating details about its biological implementation. We hope by the means to demonstrate the generic features of hair-cell activity and to emphasize that the experimental data might reflect any of several mechanisms, or indeed a combination of them.The dissipation of power during spontaneous activity and the power gain during amplification (108), associated with violation of the fluctuation-dissipation theorem (114), indicate that the hair bundle is driven out of equilibrium by an Internal energy source that produces an active mechanical force 𝐹𝐹 𝐴𝐴 on the bundle. Neglecting mass in view of the overdamped mechanics of a passive hair bundle, we write a linear equation of motion to describe the bundle's movements:

  and 𝜏𝜏 A = 𝜏𝜏. We see that adaptation provides delayed force feedback on the hair bundle by responding to tension changes in the gating springs of stiffness 𝑘𝑘 � , effectively producing the active force 𝐹𝐹 A given by the channel-gating force associated with a change in the set-point deflection 𝑋𝑋 o . The requirement of negative stiffness (𝑘𝑘 < 0) to bring the system near a Hopf bifurcation (𝜆𝜆 + 𝑘𝑘𝜏𝜏 ≅ 0) imposes that gating compliance be sufficiently great. Equation 14 captures the phenomenology of adaptation, but not its biological mechanism. As introduced earlier, two classes of models have been proposed to describe adaptation: those based on motor activity and on Ca 2+ -mediated channel reclosure. In the motor model, changes in the set-point deflection 𝑋𝑋 o result from movements 𝑋𝑋 A of myosin motors along the actin cores of the stereocilia, modulating tension in the tip links. Within this framework, we write(57,58,75) 𝜆𝜆𝑋𝑋 ̇= -𝐾𝐾 GS (𝑋𝑋 -𝑋𝑋 A -𝐷𝐷𝑃𝑃 o ) -𝐾𝐾 SP 𝑋𝑋 + 𝐹𝐹,[15]

Ca

  2+ affects the intrinsic energy difference Δ𝐸𝐸 ∅ of the transduction channel between its closed and open states by binding to the channel or an associated protein and stabilizing the closed state. As in the motor-based model, this mechanism changes the set-point deflection 𝑋𝑋 0 and shifts the relation 𝑃𝑃 o (𝑋𝑋) in the direction of the hair-bundle deflection 𝑋𝑋.

  ] in which 𝐶𝐶 0 and 𝐶𝐶 0 + 𝐶𝐶 1 are the steady-state Ca 2+ concentrations when the channels are closed (𝑃𝑃 o = 0) and open (𝑃𝑃 o = 1), respectively. Linearization about a fixed point brings this nonlinear Ca 2+ -reclosure model (Equations 13 and 19) to the form of the generic linear model (Equations 7-8) of active hair-bundle motility with 𝑘𝑘 = 𝐾𝐾 GS (1 -𝐷𝐷𝑃𝑃 o ′ ) + 𝐾𝐾 SP , 𝑘𝑘 � = -[𝜇𝜇/(1 + 𝜇𝜇)]𝐾𝐾 GS 𝐷𝐷𝑃𝑃 o ′ , 𝐹𝐹 A = -𝜇𝜇𝐾𝐾 GS 𝐷𝐷𝐶𝐶/𝐶𝐶 1 , and 𝜏𝜏 A = 𝜏𝜏/(1 + 𝜇𝜇). The dimensionless parameter 𝜇𝜇 = 𝐶𝐶 1 𝑃𝑃 o ′ 𝑋𝑋 0 ′ characterizes the Ca 2+ sensitivity of the transducer.

Figures: Figure 1

 1 Figures:

  Mechanoelectrical-transduction apparatus of the mammalian cochlea. (a) A drawing of the human cochlea (modified from reference 25) shows nearly three tapered helical turns. The green rectangle at the right represents an area corresponding to that in panel (b). (b) A scanning electron micrograph of a mouse's cochlea portrays the surface of the sensory epithelium, including the top surfaces and hair bundles of inner and outer hair cells. Deflecting the top of any hair bundle to the right (arrow), away from the central axis of the cochlear helix, excites the associated hair cell. (c) The heart-shaped top surface of an outer hair cell bears approximately 70 upright stereocilia in three rows that increase in each channel has two states-open and closed-and its molecular gate stochastically flickers between them under the influence of the tension in an associated gating spring of stiffness κ. Opening of a channel reduces the extension of and the tension in the gating spring by the gating swing D and the gating force Z = κD, respectively. Each hair bundle is endowed with N parallel transduction elements that together contribute a stiffness K GS = Nκ. (c) The external force F required to deflect a hair bundle by an amount X deviates from linear, Hookean behavior over the narrow range of positions in which the channels' open probability changes significantly. For large negative or positive stimuli, the curve's slope K ∞ = K GS + K SP is the sum of the stiffness contributions K GS from the grating springs and K SP from the stereociliary pivots. The slope declines for intermediate values and can even become negative if the gating force is sufficiently great. Under force-clamp conditions (thin horizontal line), this behavior destabilizes the bundle's position X = X 0 at which P o = 0.5 (open circle) and the bundle becomes bistable, residing at either of two stable fixed points (solid circles). (d) The hair bundle's stiffness as a function of displacement is given by the slope of the preceding curve. The reduction of stiffness associated with channel opening and closing-the gating compliance-can reduce the stiffness until it becomes negative (dashed line). (e) An effective mechanical description of the hair bundle depicts a channel (pink) that opens or closes through distance D under

  1 and 𝑓𝑓 2 fall within the active bandwidth Δ𝜔𝜔 of the nonlinear amplifier. The one-third power law �𝑋𝑋 � � ∝ �𝐹𝐹 � � 1/3between the local input and the response, which is intimately coupled with frequency selectivity, provides the essential nonlinearity needed to account for twotone interference in hearing (134, 135, 141). Distortion products are observed in mechanical vibrations of the basilar membrane and can be strong enough to be detected in the ear canal as sounds called distortion-product otoacoustic emissions. For weak sound stimuli, the strongest of these emissions, which occurs at the same frequency 2𝑓𝑓 1 -This discrepancy might be resolved by taking into account viscoelastic longitudinal coupling between oscillators tuned to different frequencies (14, 145, 146) and noise inherent to hair cells (75, 147). Coupling must result in a collective response that is nontrivially related to the local nonlinear properties of the individual oscillators. Near the hearing threshold, the ear responds to mechanical vibrations with amplitudes that are so low that they become comparable to-or even smaller than-thermal fluctuations of the ear's constituents. The Hopf bifurcation, however, is a mathematical property of deterministic dynamical systems; its generic properties can only approximate the behavior of biological systems. Noise blurs the Hopf bifurcation by destroying the phase coherence of weak oscillations and imposes non-vanishing values of the

	Physical limits to auditory sensitivity
	𝑓𝑓 2 as the dominant phantom tone that is perceived, is actually used as a non-invasive
	hearing test for newborns.
	Auditory detection by critical oscillators provides a physical principle to apprehend
	key features of the complex phenomenology evinced by the cochlea (115), and more
	generally by the active hearing of many vertebrates and some insects (142). Within this
	framework, sensitivity, frequency selectivity, and nonlinearity are intimately related and
	interdependent. In a minimal description of the mammalian cochlea, the organ is reduced
	to a set of critical oscillators with natural frequencies distributed along a tonotopic axis;
	fluid coupling between the oscillators results in active travelling waves that recapitulate
	the most salient features of experimental observations (143, 144). However, contrary to
	the experimental observation (140), in current models of the cochlea based on critical

oscillators resonances are very sharp at low stimulus levels, as should be the case if the response is dominated by the local response of the oscillator driven at its natural frequency.

oscillator's complex impedance. An active resonance to weak stimuli can still occur, but with noise-limited values of the quality factor

(75, 147)

. A critical oscillator with added noise effectively behaves as a noiseless oscillator operating on the stable side of a Hopf bifurcation, with parameters renormalized by the noise intensity and nonlinearities (148).

Acknowledgments

The authors are grateful to the members of their research groups for comments on the manuscript. PM is supported by the French National Research Agency, from grants ANR-16-CE13-0015, ANR-11-LABX-0038, and ANR-10-IDEX-0001-02. AJH is an Investigator of Howard Hughes Medical Institute.

the nonlinear behavior of active hair cells. As new data emerge, theoretical analysis must accommodate the characteristic features of cochlear hair cells: profound nonlinearity, stochasticity owing to a noisy environment, and the out-of-equilibrium operation associated with the ear's remarkable active process.

Disclosure statement

The authors are not aware of any affiliations, memberships, relationships, funding, or financial holdings that might affect the objectivity of this review.