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the avian and mammalian ears have similarities and differences . the innervation pattern of avian tall hair cells resembles that of mammalian inner hair cells in having primarily afferent innervation , suggesting their role as the primary sensor . in contrast , the innervation pattern of short hair cells in the avian ear is exclusively efferent and resembles that of outer hair cells , suggesting the role of motor element in the cochlear amplifier @xcite . since bending of hair bundles ( including tilting motion of the apical surface of hair cells @xcite ) is the only motile response in short hair cells , such a physiological role requires that the hair bundles of short hair cells exert force between the tectorial membrane and the basilar papilla . there are marked differences . one such difference is the absence of pillar cells and the tunnel of corti in the avian ear . in addition , the presence of electrical tuning @xcite and absence of somatic motility in avian hair cells make them apart . yet another is the morphology of the tectorial membrane ( tm ) . is it true that the difference in the motile element of the cochlear amplifier is the reason for the structural difference ? here this issue is examined by focusing upon the morphology of the tectorial membrane . the stiffness of the avian tm would be low compared with the mammalian one , being devoid of collagen @xcite . it shows perforations reminiscent of swiss cheese in electron micrographs due to fixation instead of continuous low density mass as observed in unfixed preparations @xcite . since collagen is largely responsible for the mechanical anisotropy of the mammalian tm @xcite , its absence would make the material properties of the avian tm very different from the mammalian one , in which stiffness gradient and the significance of the resonance frequency has been documented @xcite the avian tm is tapered toward the abneural side ( fig . [ fig : anatomy ] ) . the steepness of tapering increases toward the basal end with higher best frequencies @xcite . in contrast to the flat and featureless surface of the mammalian tectorial membrane @xcite , the avian tm has an elaborate structure on the side facing the basilar papilla . a thin veil - like structure descends to the basilar papilla , surrounding each hair cell @xcite . a dome like recess above each hair cell and each hair bundle makes contact with the tectorial membrane at the dome @xcite . what is the functional significance of these structural features ?    like outer hair cells in the mammalian ear , short hair cells are located nearly above the middle of the basilar membrane , the movement of which in the up - down direction @xcite would effectively transmit bending stress to the short hair cells hair bundles . in contrast , tall hair cells , which functions as the sensor , are located away from the center of the basilar membrane , at a location seemingly unsuitable to sense the motion of the basilar membrane directly . thus it is likely short hair cells are involved in amplification by a hair bundle active process @xcite , transmitting the mechanical vibration of the basilar membrane to the hair bundles of tall hair cells . the problem is how that can happen ? to address this question , it requires to identify the vibrational mode , which involves the basilar membrane , the basilar papilla , and the tm . since reported measurements are limited to the basilar membrane with mssbauer technique @xcite , we need to assume how other elements could move . since not much else is known for certain , the models considered herein are largely based on morphological observations and rather speculative .    since hair cell bundles are stimulated by shear , it would be logical to assume that shear motion between the basilar papilla and the tm stimulates the hair bundles of tall hair cells , similar to the mammalian ear . however , there are clear differences . the orientation of hair bundles in the avian ear is not as uniform as in mammalian cochlea . although the orientation is relatively uniform in the basal region , it forms domain structure in a more apical region @xcite . in the following , we examine the mode of motion that is consistent with the bundle alignment in the basal area , then consider a possible mechanism for domain formation in more apical region . electron micrographs show that hair bundles are oriented primarily in the direction orthogonal to the longitudinal axis of the basilar papilla , particularly in the basal higher frequency region . this orientation is consistent with the shear between the tectorial membrane and the basilar papilla , analogous to the mammalian ear ( fig . [ fig : modes ] a ) .    however , there are appreciable differences between the two . the avian ear does not have pillar cells or a tunnel of corti . thus it may not have well - defined pivoting points for creating shear from the up - down motion of the basilar membrane . in addition , the tectorial membrane ( tm ) is tapered , making it thicker on top of tall hair cells . the tm is attached to a wall , where it is thickest , nearer to tall hair cells . how can such a morphology be consistent with the assumption that short hair cells serve as the cochlear amplifier ? to explain this puzzle , charles steele suggested that the tm should have resonance near the best frequency @xcite . indeed , resonance would make it work . however , a question still remains . why should it tapered toward short hair cells ? the avian tm works despite its shape ? is there an advantage of this morphology ? since so many factors remain uncharacterized , it would be advantageous to examine the simplest possible model to obtain a conceptual understanding of the issue rather than more complex ones , which would require many more details . for this reason , two - mass models are used here for describing the tm ( fig . [ fig : modes ] ) . one has a smaller mass @xmath0 , located at the top of short hair cells and another with a larger mass @xmath1 , located at the top of tall hair cells . the small mass @xmath0 is connected to the larger mass @xmath1 with a spring with stiffness @xmath2 and the larger mass @xmath1 is connected with the wall by another spring with stiffness @xmath3 . let @xmath4 the displacement of the bigger mass from its equilibrium position , and that of smaller one be @xmath5 . let @xmath6 be viscous drag on mass @xmath1 and @xmath7 be negative drag on mass @xmath0 to mimic the amplifying role of short hair cells ( fig . [ fig : modes ] a ) . the set of equations is given by , @xmath8 \\ \left(m\frac{d^2}{dt^2}+\eta_2\frac{d}{dt}+k\right)x - k(x - x)=0 , \label{eq : motion}\end{aligned}\ ] ] where @xmath9 $ ] is a periodic external force of angular frequency @xmath10 applied to mass @xmath0 , which is located just above the center of the basilar membrane . here it is assumed that up - down motion of the basilar membrane results in this external force . following steele @xcite , let @xmath11 be the resonance frequency of the tectorial membrane , i.e. @xmath12 and seek a steady state solution .        under this assumption , the force applied to the mass @xmath1 is plotted for a special case of @xmath13 ( fig . [ fig : drag ] ) . for a given value of @xmath1 , this force is larger for smaller values of the reduced drag @xmath14 where the ratio @xmath15 is smaller . since the reduced drag is defined by @xmath16 , this parameter is smaller for higher frequency @xmath11 . thus the plot shows that the force is maximal for smaller ratio @xmath15 at larger frequencies . this result is consistent with the experimental finding that the sharpness of tapering of the tm increases from the apical to the basal end @xcite . one can argue that the force relevant to the transduction of tall hair cells would be better related to drag @xmath17 rather than the force @xmath18 . that does not change the optimal mass ratio because drag is represented by @xmath19 , maximizing at the same mass ratio . the condition for resonance is not very strict because those peaks exist for @xmath20 . * domain structure *    while hair bundles in the basal area of the basilar papilla are aligned perpendicular to the longitudinal axis of the papilla , those in more apical area are aligned in a patchwork fashion , particularly in the middle between neural and abneural ends . how can this pattern be explained ? suppose the motion of the tectorial membrane relative to the basilar papilla is perpendicular to the surface of the basilar papilla ( fig . [ fig : modes ] b ) , how do hair bundles bend ? each hair bundles is attached to the tectorial membrane at the edge of a dome - like recess @xcite . the fibrous veils , which descend from the tectorial membrane and reach the basilar papilla , surround each hair cell at its margins . some parts of the seemingly extendable veils form triangles with hair bundles and parts of the basilar papilla @xcite , which is much denser and therefore expected to be much stiffer . this structure could couple elastic elongation and shortening of the veils , resulting in a bending of the hair bundles .    since bending of a hair bundle results in local displacement of the tm , bending of the neighboring hair bundles in the same direction is energetically favorable if those hair bundles are oriented in the same direction . for this reason , up - down motion of the tm favors domain formation in hair bundle orientations . the size of domains is limited to even out those local deformations of tm to make the average motion in the up - down direction . + * equations of motion *    in this mode of motion , we assume for simplicity , that two masses do not interact directly , but only indirectly being mediated by the basilar papilla , which pivots at the supporting point ( fig . [ fig : modes ] b ) . let @xmath21 the distance of the large mass @xmath1 from the pivoting point , and @xmath22 that of the smaller mass @xmath0 . notice that @xmath23 and for that reason , the torque is more effectively produced by the force applied by the mass @xmath0 . the equations of motion would be given by , @xmath24,\end{aligned}\ ] ] where @xmath25 is the moment of inertia of the basilar papilla , @xmath26 the angle of rotation of the basilar papilla around its pivoting point and @xmath27 s ( j=1,2,3 ) are either positive or negative , depending on whether they have damping or amplifying effect . the quantity @xmath28 is the perpendicular displacement of the mass @xmath0 , and @xmath29 that of the mass @xmath1 . here @xmath30 and @xmath31 represent elasticity due to hair bundles and veils . the quantity @xmath32 represents the elastic elements not associated with hair bundles . since this is a three body problem , which may not have a periodic solution in general , we examine a special case . since we are interesting in the mass ratio @xmath15 , we regard the basilar papilla as mediating energy transfer between the smaller mass @xmath0 and the bigger mass @xmath1 , letting @xmath33 . the longer lever arm of the hair bundles of short hair cells have is advantageous for short hair cells to function as an amplifier . under this condition , the force applied to the mass @xmath1 potted against the mass ratio @xmath15 has a maximum for a given @xmath14 , similar to the sliding mode ( fig . [ fig : jump ] ) . we assumed that short hair cells provide negative damping whereas tall hair cells do not . this requires that active force generation at short hair cells is greater than in tall hair cells . in addition , the source of viscous drag is important . up - down motion of the tectorial membrane with respect to the basilar papilla requires volume changes in the gap between the tectorial membrane and the basilar papilla , even though changes are small . since the soft and possibly porous structure of the tectorial membrane may not allow fluid movement at auditory frequencies , the fluid in the gap is likely go through veils surrounds each hair cells , creating an extra viscous drag . in general , such drag would reduce the energy efficiency . however , it could be beneficial in this particular situation if it makes the drag of tall hair cells net positive and that of short hair cell net negative . the modes of motion described above are not mutually exclusive and are likely coexist . the observation that basal hair bundles are well aligned than more apical bundles and that basal hair bundles have flatter appearance than more apical ones @xcite could indicate a relative significance of the sliding mode at higher frequencies . it is likely that in more apical area the jumping mode , which is consistent with domain formation , is dominant . the tapering of the tm is a feature optimizing the sensitivity of tall hair cells in the presence of active motion of the hair bundles of short hair cells . one could argue that studying the modes of motion in the avian inner ear could be premature : first of all , very few physiological data are available . the only data on the vibration of the basilar membrane was published in 1987 by gummer at al .  using the mssbauer effect @xcite , showing traveling wave and that the amplitude is somewhat smaller compared with the mammalian counterpart . the mechanical properties of the basilar papilla nor of the tm have not been measured . moreover , studying the modes of motion may not be easy . even in the mammalian cochlea , which has been studied much more extensively , the details are becoming clearer only recently . nonetheless the avian ear would be an interesting subject for a couple of reasons . one such reason is that the avian basilar papilla has some resemblance to the mammalian ear , such as tonotopic organization and differentiated roles of two types of hair cells , and yet it has much simpler morphology , such as the absence of pillar cells and the tunnel of corti besides different tm morphologies , possibly suggesting simpler modes of motion . in addition , its cochlear amplifier must depend solely on an active process in hair bundles , which may be based on fast adaptation @xcite and possibly , in part , avian prestin @xcite . also the electrical properties of chick hair cells @xcite are quite different from mammalian hair cells in having an intrinsic electrical tuning mechanism that likely in part shapes the mechanical components , whether they are the hair bundle or prestin @xcite . for this reason , the mechanical motion in the avian ear does not have to be as sharply tuned as in the mammalian one . it is possible that the relatively simple structure of the avian ear could possibly be advantageous for identifying the modes of vibration .    assuming a local tm resonance @xcite is attractive in view of its role in the mammalian cochlea @xcite . for the jumping mode of motion , it would be reasonable because hair bundles , which have systematic gradient @xcite , would make a significant contribution to the stiffness . how realistic is the assumption of resonance for the sliding mode of motion ? could young s modulus , density , and the dimension of the tm be compatible with the assumption of resonance from the basal end to the apical end ? can a gradient of the mechanical properties of the tm exist ? these issues are perhaps experimentally testable . the difference between short and tall hair cells is another requirement for our models models presented here . specifically , our models require that the net effect of tall hair cells should be drag and that of short hair cells should be negative drag . this requires substantial difference in the magnitude of active force production in the two hair cell types . indeed , some difference in bending responses , both voltage - induced and not , in these hair cell types has been observed @xcite . however , it remains unclear whether or not the differences are large enough to satisfy our requirement . there are other possible modes of motion that are not considered here . the degree of simplification of the present models in describing the tm is obvious . in addition , there are other factors to consider . for example , does the basilar membrane move together with the basilar papilla as a solid body or can they possibly have a difference in the phase as well as amplitude ? photomicrographs show an appreciable space between the basilar papilla with embedded hair cells and the basilar membrane @xcite . in addition , these two largely parallel structures appear to have separate anchoring points on the neural side . this feature could possibly allow an external force to drive basilar papilla more than the tectorial membrane . choe y , magnasco mo , hudspeth aj ( 1998 ) a model for amplification of hair - bundle motion by cyclical binding of ca@xmath34 to mechanoelectrical - transduction channel . proc natl acad sci usa 95:1532115326                      jones gp , lukashkina va , russell ij , elliott sj , lukashkin an ( 2013 ) frequency - dependent properties of the tectorial membrane facilitate energy transmission and amplification in the cochlea . biophys j 104:13571366                    steele cr ( 1997 ) three - dimensional mechanical modeling of the cochlea . in : lewis er , long gr , lyon rf , narins pm , steele cr , hecht - poinar e ( eds ) diversity in auditory mechanics , singapore : world scientific , pp .
while the mammalian- and the avian inner ears have well defined tonotopic organizations as well as hair cells specialized for motile and sensing roles , the structural organization of the avian ear is different from its mammalian cochlear counterpart . 
 presumably this difference stems from the difference in the way motile hair cells function . 
 short hair cells , whose role is considered analogous to mammalian outer hair cells , presumably depends on their hair bundles , and not motility of their cell body , in providing the motile elements of the cochlear amplifier . 
 this report focuses on the role of the avian tectorial membrane , specifically by addressing the question , `` why is the avian tectorial membrane tapered from the neural to the abneural direction ? '' 
 address = otolaryngology & head and neck surgery , stanford university , stanford , ca 93405 , usa , altaddress = nidcd , nih , bethesda , md 20892 , usa    address = otolaryngology & head and neck surgery , stanford university , stanford , ca 93405 , usa , altaddress = molecular & cellular physiology , stanford university , stanford , ca 93405 , usa