31vocal

Simple motor gestures for birdsong Tim Gardner, G. Cecchi, M. Magnasco1 and R. Laje and Gabriel B. Mindlin2 1Center for Studies in Physics and Biology, The Rockefeller University, 1230 York Ave, New York, NY 10021 USA2 Departamento de F'isica, FCEN, UBA Ciudad Universitaria, Pab. I (1428), Buenos Aires Argentina We present a model of sound production in a songbirds vocal organ, and find that much of the complexityof the song of the canary (Serinus canaria) can be produced from simple time variations in forcing functions. The starts, stops and pauses between syllables, as well as variation in pitch and timbre are inherent in themechanics and can often be expressed through smooth variation in the frequency and relative phase of two driving parameters. PACS numbers: Human language and the song of many bird species areboth learned by juveniles through experience. The ontogeny of learned song begins with early food begging calls, contin-ues through more complex babblings known as sub-song, to eventually reach an adult form of song which is first plastic,and later stereotyped [1].This development depends on auditory experience. If a juvenile is deafened and unable to hear itsown sound production, or if a juvenile is isolated from the experience of any adult song model to copy, abnormal forms ofsong develop.The time-course of development and the dependence on auditory experience are similar to aspects of humanspeech. This suggests that there may be common principles of learning and memory underlying human speech and the songof birds[2]. A variety of current experimental approaches are directed towards a better understanding of how song percep-tion and production are represented in the brain, and ultimately how experience shapes the neural changes which es-tablish these representations[3] [4]. However, the broad extent of neural activity involved in song presents an increasingchallenge for interpretation[5]. In order to understand the link between brain activity and song, a physical understanding ofthe process of sound production in the songbird's vocal organ is essential. In the following, we demonstrate a simple mech-anism which can account for the acoustic structure of many song elements. The physical model we describe involves aminimum number of parameters, and can account for a wide range of observed birdsong elements. Fitting of the model'sparameters in order to reproduce recorded songs, suggests that the acoustic complexity of individual elements in the song ofcanaries arises from simple modifications of a very generic gesture in lung pressure and vocal fold tension. The vocal organ of songbirds, known as the syrinx, has beenthe subject of a long history of interesting studies. The existence of membranes which vibrate being capable of generat-ing sound waves was established by Ruppel in 1933 [6]. At present, the precise mechanisms of sound production in thesyrinx are still a matter of debate. (see [7] and [8]) Current experiments suggest that the syrinx generates sound primar-ily through oscillation of the lateral labia - tissue folds which open and close the air passage from the bronchi to the tra-chea [9]. Through direct videography of a phonating syrinx, xflfl aflfl bflfl Tra che a -> fl fl b\Gamma ro nch us-> fl fl a.flfl c.flfl Mflfl tracheaflfl bronchusflfl Kflfl fl flD,D b5 labiumflexternumfl fl labiumflinternum b1 b2 b3 b4 1 mm One c ycle of lab ial osc illat ion 2 FIG. 1: Illustration of the syrinx, terms of the model, and labial dy-namics. Panel a. illustrates the songbird syrinx which has a phonatory labium at the top of each bronchus. Panel b. illustrates ourssumption of the shape of the labia during a cycle of oscillation. Panel c. is a diagram of the terms in our model. Here we havedrawn only one side of the symmetric pair of labia in the model.) In each figure, darker shading indicates higher pressure according toeqn. (0.3). Larsen and Goller give evidence that these labia function ina manner homologous to the human vocal folds[10]. In support of this, Fee et. al. have demonstrated that the transitionfrom periodic to chaotic vibrations in the Zebra finch syrinx can be modeled by a classic two-mass model of human vocalfold oscillation[11][12]. In contrast to these aperiodic sounds, in what follows, we examine vocalizations composed of lo-cally periodic waveforms, typical of canaries and many other species. The primary element of the canary lexicon is the syl-lable, a 15 - 300ms vocalization. of which the canary, Serinus canaria, typically has a few dozen types. One of the simplest models to account for the transfer of the 2 Pressure [kPa] Hopf flbifurcation labia oscillating Ten sio n p er u nit are a [N /cm ] 1600 Hz 1200 Hz 0\Delta 2 4\Theta 6\Lambda 8\Xi labiafls\Pi table fl0\Sigma 5\Upsilon 10 15 20 8\Phi 00 Hz 3 FIG. 2: Illustration of sound production as a function of \Psi ff\Omega andfiffifl The left boundary of the shaded region indicates the critical values of pressure and tension at which self-oscillations are in-duced. The dark bands in this figure are numerically calculated contours of iso-fundamental frequency. Higher Spectral content isschematically indicated by darker shading. (No vocal tract filtering is applied.) ffl i`j fl j'j'^_*,*aess_oeo/*ffiAE , OEO/i^ fl j"!$#&%('")*'*oe*ss"oe+*ffi, ,OE AE i-j fl j&.o/!$#,%/'")*'*oe , ss_oeo/*ae0 , 12i3j fl j'j4j&.*^')*'*oe , 5768i-j fl j"9'oeo/* and: 6 iO/j fl j';"oeo/* . With these, and \Psi ff\Omega ranging from 0 to 3 !&\Psi :(as experimentally observed), and fi from zero to 8 =?ss"oe+* , , the model generates frequencies within canary range, and with oscilla-tions in the range of j_?(* to ;'^"j"?(* , consistent with experimental evidence[9, 12]. kinetic energy of air to vocal fold oscillations was introducedby Titze [13]. It is built upon experimental observation that the human vocal folds support both lateral oscillations and a up-ward propagating surface wave [13], and successfully predicts the fundamental frequencies of voiced sounds in terms of sen-sible physiological parameters. There is no direct evidence in birds, but this "flapping" mode of oscillation is consistent withrecent videography of the bird's syrinx during song, and in the isolated syrinx[10][12]. As illustrated in figure 1, the oppos-ing labia have a convergent profile when they move away from each other, and a more planar profile when they move towardseach other. This results in a greater pressure on the labia during the opening phase, and an overall gain in energy in eachcycle of oscillation. This mechanism does not depend on the geometrical details of the folds, but the calculations are eas-ier with a simplifying geometric hypothesis. Approximating the labia shape with straight edges as illustrated in fig. 1, theassumption of a flapping mode can be written in terms of a phenomenological constant @ as follows:ACBDAFEHGJIffiG @ K IKML (1) N B N EHG2IPO @ K IKML (2) The upper edge of the labium is denoted A , the lower edge N ,and the center of the labium IRQ Following Titze, who uses a fl fl fl 0S fl 6 Fre qun cy kHz TT ime 200 ms P K cU anary syllables model syllables FIG. 3: Three natural syllables, three artificial syllables and theircorresponding control trajectories. The elements are illustrated in time-frequency format, which was computed through sliding, over-lapped Fourier transforms. (Pixel grey level represents the power of sound on a logarithmic scale.) Parameters are drawn with respect tofigure 2. Arrows indicate the direction of traversal over the curve. In each case, the trajectory begins at the minimum pressure value, andtraverse the ellipse in a counter-clockwise direction. As discussed in the text, the parameterization of time in each ellipse is chosen tocause a slowing of traversal rate near the point of maximum pressure. For each syllable, the vocal tract filter has the following pa-rameters: VXWYi[Z fl j"oeo/*8",V AE i]. fl j"oeo/*^"$.YW`i fl ._^"oe+*^"&. AE i fl 94oeo/*8" and abiJj fl c,fl This filter leads to a transient emphasis of the power ofthe second and third harmonics as indicated by the arrows. A further point can be illustrated with this figure. For fast syllables, sound pro-duction at a particular point \Psi ff\Omega and fi " is not equivalent to the steady state sound production. Were it equivalent, syllable one would bealmost symmetric in time. In effect, there is some overlap between the time scale of equilibration of the oscillation and the time scale ofthe fastest syllables. (phenomenologically modified) Bernoulli equation [13], thepressure averaged over the surface of the labia can be written as a function of bronchial pressure dfe and labial position: dhg B d e'ikj OlAnm N_o Q (3) Substituting equations for N and A into that for d g , and equat-ing this averaged pressure with the driving force of a damped 3 100ms iPK Canary song Model song Control Parameters 0 5 kHz Time FIG. 4: Synthetic signals generated by slow modulations of pressureand tension. (\Psi p\Omega and fi are taken from a brief sample of the positions of two identical, spring-coupled pendulums subject to a harmonicrescaling of time. \Psi ff\Omega ranges between zero and ;4!&\Psi : . fi ranges between zero and q4=?ss"oe+* , ) On the bottom of the figure, a consec-utive sequence of syllables from natural canary song is included for comparison with the synthetic signals. harmonic oscillator, we have:rtsIffiG2uwvIffiG2uffix i vI ozy G2---I^B dhe A E O N E G~"" @ vI ICG N EHG @ vI Q (4) The parameters r , - , u and uCx describe the mass, resti-tution constant and coefficients of a non-linear dissipation all per unit area. The nonlinear dissipation term i vI o y is in-troduced ad-hoc so that the variable I can only take values between precise boundaries, mimicking collisions [14].In this model, we make the assumption that the bird controls vocalizations through the bronchial pressure d e and thelabial elasticity - . We assume that the elasticity term corresponds to the concerted activity of one or more syringealmuscles which change the radial stiffness of the labia. Presumably, this term can be related to the activity of the musclesmost correlated with changes in the fundamental frequency of the sound [15]. In what follows, we refer to this term astension. The behavior of the system for different parameters of pres-sure and tension is displayed in Figure 2, where the left boundary of the shaded region indicates the critical values at whichself-oscillations are induced. As the pressure d e is increased, the equilibrium state loses its stability in a Hopf bifurcationand energy is transferred into the labia (there is an effective negative dissipation). Near the boundary, the oscillationshave very little spectral content. As pressure is increased to points deeper in the region of oscillation, spectral content ofthe waveform increases monotonically. The final feature of the model is the filtering of the soundproduced by the oscillations through the acoustic properties of a cavity whose dimensions approximate the trachea and beak of a canary. The aperture of the beak alters frequencycontent of the sound in a manner similar to that observed experimentally [16] [17]. For each simulation illustrated in thispaper, beak aperture was fixed. The filter was constructed by approximating trachea and beak by two tubes of lengths "A,A* ," x and areas C'C^A* and C' x . The input pressure dRD^ (proportional to the time derivative of the flow) generates a wave which ispartially reflected and partially transmitted to the second tube that models the tract at the interface between the first tubes[18]. The reflection coefficient E^E,A**G* xL'B i C'?A* O C' x o m i C'C^A* G C' x o , and the transmission coefficient L A*7G* xL^B j O E^E,A**G* x . At the inter-face between the second tube the wave is again partially reflected and partially transmitted. Calling A i Lzo (N e4i Lzo ) the for-ward (backward) wave in the first tube and N g i Lzo (L/ e4i Lzo ) the forward (backward) wave in the second tube, the equationsaccounting for the boundary conditions are [19] A i Lzo B d D^+i Lzo G N e"i L O @ A* o (5)N e"i Lzo B E^ A*7G* x A i L O @ A* o G L A*7G* x L/ e'i L O @ x o (6)N g i Lzo B L A**G* x A i L O @ A* o G E^ A*7G* x L/ e'i L O @ x o (7)L/"e i Lzo BN^N' N g i L O @ x o*NG (8) where N' accounts for the reflection coefficient of the interfacebetween the third tube and the atmosphere (no losses beingN'2BO""O j ), and the time @"D^ is the time it takes a sound wave totravel "R'D^ . Given an estimation of physical parameters consistent with experimental measurements (see values in the cap-tion of Figure 2), we find that the simulations produced by the model have a frequency range and spectral content compara-ble to natural vocalizations. Given a physically based model with few parameters, whatkind of control of d e and - might the bird employ to generate the elements of its song? The oscillations that are establishedin a typical vocalization are in the order of one or two R^S^S'T^S, while a syllable duration is between j4T,&U""Y"U* and Z' T,&T,,U""Z^U* . Withthis separation of time scales, If the paths in parameter space are swept slowly, at each time L+Z. the system will behave ba-sically as expected in a steady condition with dfe i L+Z.4o B d Z.e and- i L+Z."o BIJ- Z. . Given the simplicity of the parameter spacedefined in figure 2, this implies that the time course of - will essentially trace out the time course of the fundamental fre-quency of the syllable. Pressure will control not only sound amplitude but also the relative strength of the higher harmon-ics. For example, an up sweep syllable like the second one dis-played in Figure 3 requires that the control trajectory begin at a point of low pressure and tension and enter the region ofoscillation pictured in figure 2 by increasing pressure. While inside the region of oscillation, tension must be incrementedto produce the up sweep in frequency. Finally, at a position of high - NG the pressure must be reduced to end the vocal-ization by crossing out of the region of oscillation. In accord with this simple example,we found that many canary syllablescould be approximated by excursions as simple as harmonic oscillations of pressure and tension. Specifically, we gener-ated elliptical excursions in pressure and tension which had 4 the following form: dhe B dfI. G C'$d-A,L/_a* U*Fic'a,fi Lzo G a, D^ o (9) -c^Bd^- I. G~e^ d-A,L/_a* U*Fie,a,Ri Lzo G a,ng* o Q (10) For the figures in this paper, the parameterization of time iseither linear, or else involves a slight slowing for high pressure. ( va,fi Lzo B L/ , or va,hi LzoC^l' j O~l^Ml/n^n'Ml/o""ngr^r's^s's,n'E,t^kr'r^t,7u""+u*$t, NG for constantsa, E NG and y" Q ) We want to emphasize that this parameterization of time is the same for both pressure and tension. This meansthat the time-scale and complexity of the two control parameters are equivalent. In this form, an ellipse in the space ofpressure and tension characterizes the identity of each syllable. Figure 3 illustrates the similarity between three ellipti-cal control trajectories and three notes chosen from a single bird's song. For this figure, the fitting procedure was qualita-tive in nature. The range of - could be estimated from the frequency contours of figure 2, and the phase and amplitudeof the ellipse were chosen to reproduce the qualitative shape of the sonogram. Work is in progress to build an algorithmicfitting procedure. The elliptical paths in parameter space described above arecharacteristic of the solutions of a wide range of coupled oscillators. This suggests that the structure of some canary songs,determined by the ordering of a sequence of syllables, may be modeled by the slow dynamics of changing phase relation-ships in the oscillations controlling d e and ---Q To illustrate this hypothesis, figure 4 demonstrates the response of the model toparameter oscillations characterized by two oscillators whose frequency and relative phase are slowly changing. The result-ing sequence of vocalizations demonstrates elements and transitions qualitatively similar to a sequence of syllables mea-sured experimentally. Specifically, the starts, stops and pauses between syllables, as well as variation in pitch and timbre arereproduced by the simple oscillations of d and ---Q In this work we found that the spectral and temporal di-versity of many canary vocalizations can be modeled by the response of a non-linear equation to simple forcing functions.One particular implication of our model is that the control of syringeal tension does not need to be any more detailed thanthe recorded pressure in a particular air sac which is known to be smooth for most canary syllables [20][21]. For somesongs, simple non self intersecting cycles in the d e and - parameter space were able to reproduce the data. These cyclescould be approximated by ellipses, which ultimately suggests that the key variable to be controlled in order to build a song isthe phase difference between simple oscillations in bronchial pressure and vocal fold tension.Acknowledgements We thank Fernando Nottebohm, Albert Libchaber, JimHudspeth, and David Vicario for their comments on the manuscript. This work was partially funded by the Alexander Mauro Fellowship, UBA, CONICET, and Fundacion Antor-chas. [1] Nottebohm F, Nottebohm ME, Crane L., Behav. Neural Biol,46,445-71 (1986). [2] Brainard M. S., Doupe A. J.,Nature Reviews 1 31-40 (2000).[3] Yu A.C., Margoliash D., Science 273, 1871-5 (1996). [4] Ribeiro S., Cecchi G.A., Magnasco M., Mello C.V., Neuron 21,359-371 (1998). [5] Nottebohm F. The anatomy and timing of vocal learning inbirds. In The Design of Animal Communication,( ed. Hauser M., Konishi M.) pp. 37-62 (MIT Press, 2000).[6] Ruppel W., J. Ornithol. 81 433-522 (1933) [7] Greenewalt, C. H. Bird Song: Acoustics and Physiology(Smithsonian Institution Press, Washington D.C. 1968). [8] Fletcher N.H. J. Theor. Biol 135, 455-481 (1988).[9] Goller F, Larsen ON. Proc. Natl. Acad. Sci. USA 94, 14787-91 (1997).[10] Larsen ON, Goller F., Proc. R. Soc. Lond. B 266, 1609-1615 (1999).[11] Ishizaka K., Flanagan J.L. The Bell System Technical Journal 51, 1233-1268 (1972).[12] Fee M.S., Shraiman B., Pesaran B., Mitra P.P. 395, Nature 67- 71 (1998).[13] Titze I.R. J. Acoust. Soc. Am. 83, 1536-1552 (1988). [14] The system z'z^ iijz.C!?a z^ ,A,#$OE z^ , z'z.liA`!R'A' z^ is a classic ex-ample of the theory of relaxation oscillations, and describes a bounded motion of the variable z. . Writing z.A^iA"A~ and#[iA*!R'A' z^ , we obtain a standard system that reads z'A~AEiC,# ,z'#?iE`!R'A'kA~E'#Y"OEC^#E^!I`E"I' t, # , . Written as a second order oscillator, it reads as I^A~/#L'A'I"A~DH!E^OEJz'A~(# E"I' t,nN~ z'A~nO` , . The cubic term guarantees a pre-cise bounded motion of A~ . See V.I Arnold, V. S. Afrajmovich, Yu. S. Ilyashenko and L. P. Shilnikov Bifurcation Theory andCatastrophe Theory, Springer (Berlin) 1999. [15] Suthers, R. A. J. Neurobiology 33 , 632-652 (1997).[16] Hoese WJ, Podos J, Boetticher NC, Nowicki S, J. Exp. Biol. 203, 1845-1855 (2000).[17] R. Suthers and F. Goller, ?Motor correlates of vocal diversity in songbirds?, in Current ornitology, volume 14, edited by V.Nolan Jr. et al. Plenum Press, New York, 1997. [18] Titze I.R. Principles of Voice Production (Prentice Hall, 1993).[19] M. A. Trevisan, M. C. Eguia and G. B. Mindlin, Phys. Rev. E 63 026216 (2001)[20] Calder W.A. Comp. Biochem. Physiol. 32, 251-258 (1970 ). [21] Hartley RS, Suthers RA. J. Comp Physiol 165,15-26 (1989).

Generated automatically from the PDF file.

[up [PDF Document][Top]