nonlinear dynamics and the synthesis of zebra finch ...

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International Journal of Bifurcation and Chaos, Vol. 22, No. 10 (2012) 1250235 (8 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127412502355

Int. J. Bifurcation Chaos 2012.22. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 03/31/17. For personal use only.

NONLINEAR DYNAMICS AND THE SYNTHESIS OF ZEBRA FINCH SONG YONATAN SANZ PERL∗ , EZEQUIEL M. ARNEODO∗ , ANA AMADOR† and GABRIEL B. MINDLIN∗ ∗Physics Department, Universidad de Buenos Aires, Ciudad Universitaria Pab I, Buenos Aires, BsAs 1428, Argentina †Organismal Biology and Anatomy, University of Chicago, IL 60637, USA Received December 23, 2010; Revised September 23, 2011 Behavior emerges as the interaction between a nervous system, a peripheral biomechanical device and the environment. In birdsong production, this observation is particularly important: songbirds are an adequate animal model to unveil how brain structures reconfigure themselves during learning of a complex behavior as song. Therefore, it is important to understand which features of behavior are controlled by independent tuning of neurophysiological parameters, and which are constrained by the biomechanics of the peripheral vocal organ. In this work, we show that many of the acoustic features in the Zebra finch song are in fact conditioned by the biomechanics involved. Keywords: Bifurcation; birdsong; biophysics.

1. Introduction Singing behavior in songbirds is a suitable animal model system for studying learned vocal behavior (e.g. [Doupe & Kuhl, 1999]). In order to gain a thorough understanding of central neural control mechanisms of song production and learning, it is necessary to unravel how the peripheral systems translate neural instructions into the full range of complex acoustic behavior. Vocalizations produced by birds include sound features that span a wide range from near tonal quality to broad-band noisy signals with high temporal complexity in amplitude and frequency modulation. This range of acoustic features is generated by an interaction of central neural control, peripheral motor systems and morphological properties of the sound generating structures and vocal cavities. In songbirds, a well-characterized motor pathway controls respiration, two independent sound generators in the vocal organ and upper

vocal tract structures. All of these interact to give rise to remarkable acoustic diversity often within the vocal repertoire of a single species [Zeigler & Marler, 2008]. The avian vocal organ, known as the syrinx, is located in the junction between trachea and bronchii. It consists of a pair of soft tissues on each side which resemble the human vocal folds and is a highly nonlinear mechanical device. Nonlinear systems can display complex solutions even when driven by simple parametric instructions, but the extent to which acoustic complexity emerges directly from complex behavior of the vibrating structures as opposed to specific complex neural activation patterns is still largely unknown. In the most widely studied songbird, the Zebra finch (Taeniopygia guttata), song consists of two to eight different syllables, which are composed of spectrally rich sounds with well defined fundamental frequency, noisy elements and tonal sounds. Acoustically, the Zebra finch song is one of the most

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challenging ones presented by oscine birds (group of vocal learners that accounts for 40% of the known bird species). In this work we will use nonlinear tools in order to show that many of the acoustic features of the Zebra finch song are constrained by the biomechanics of the vocal organ. Moreover, we will show that the most relevant characteristics are not due to the details of the biomechanics but arise from the bifurcations of the system that define the initiation of the vocalizations. The work is organized as follows. In the first section, we review some basic aspects of the physics and physiological control of birdsong production. The second section is divided in the analysis of the sound source dynamics, and the properties of the filter that mediates between the sound sources and the emitted sound. Section 3 contains our conclusions.

2. The Physics of Birdsong Production The avian vocal organ is a versatile organ with variations across species. In oscine birds the syrinx is located at the junction of the primary bronchii and the trachea, where free moveable connective tissue membranes, the labia, are set in oscillatory motion through an energy exchange from the airstream propelled from the air sacs. The rapid membrane oscillations generate air pressure perturbations, sound waves that travel through the trachea and beak before leaving the bird under the form of complex songs. The syrinx is controlled by a set of muscles (six pairs, in the case of oscine birds). Although a complete picture is still missing, large ventral muscles (syringealis ventralis) are believed to modulate the frequency of the vocalizations by causing with their contraction a stretching of the oscillating labia [Goller & Suthers, 1996]. The dorsal muscles are associated with active closing of the syringeal lumen. A mathematical model to account for the production of sound in the syrinx was proposed in [Gardner et al., 2001; Laje et al., 2002], following Titze’s flapping model for human vocal folds dynamics [Titze, 1988]. The model assumes that for high enough values of the airflow, soft pieces of tissue labia start to oscillate with a wave-like motion. In order to describe this wave, it is assumed that two basic displacement modes are active: a lateral displacement of the tissues and a flapping-like motion

Fig. 1. Frontal section of the labia showing the meaning of each variable of the model given by Eqs. (1)–(3). Adapted from [Amador & Mindlin, 2008].

responsible for an out-of-phase oscillation of the top and bottom of the labia. If the mean position of each labium is represented by the variable x (see Fig. 1), we assume its dynamics obeys the following dynamical system [Amador & Mindlin, 2008]:   dx   = y,    dt   (1) 1 dy  = (−k(x)x − b(y)y   dt m      − cx2 y + d + a p ). 0

lab av

In the second equation of the two-dimensional dynamical system above, the first term represents a restitution force, the second one a dissipation, the third term a nonlinear dissipation responsible for bounding the labial motion, and the fourth term a gating (independent of labial position or velocity). The position-dependent nonlinear dissipation serves to model collisions between the labia or with the containing walls, either one bounding their motions. The restitution and dissipation terms are allowed to be nonlinear (k(x) = k1 + k2 x2 , b(y) = b1 + b2 y 2 ). Finally, the last term describes the effect of the interlabial pressure. In order to express it in terms of the dynamical variables, a kinematic description of these modes is carried out. The half separation between the lower and upper edges of the labia (a1 and a2 ) are written as a1 = a01 + x + τ y, (2) a2 = a02 + x − τ y,

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where a01 , a02 are the half separations at the resting state, and τ stands for the time it takes the wave propagating along the labium to go through half its vertical size. In this way, the average pressure pav between the labia can be written as a fraction of the air sac pressure (ps ), as a2 δa + 2τ y = ps pav = ps 1 − (3) a1 a01 + x + τ y where δa = a01 −a02 (see [Amador & Mindlin, 2008] for a detailed explanation). Assuming that the coefficients in the restitution term of system (1) are proportional to the tension of the ventral syringeal muscles (the rationale behind this assumption being that their contraction stretches the labia) this model was subjected to direct test, by driving it with time dependent parameters obtained experimentally. Using EMG data from electrodes implanted on ventral syringeal muscles to obtain a time dependent parameterization of the restitution (k1 ), and air sac pressure data

for ps , we were able to synthesize a realistic song in two different species with very different acoustical properties [Mindlin et al., 2003; Sitt et al., 2010].

2.1. Sound source dynamics The model given by Eqs. (1)–(3) presents a variety of dynamical regimes as parameters are varied. The bifurcation diagram, in terms of (ps , k1 ) is displayed in Fig. 2. In the shaded region, three fixed points exist. The curves defining its border correspond to saddle-node bifurcations, where a pair of fixed points, one stable and one unstable, collide. The solid lines represent Hopf bifurcation curves, where fixed points lose stability against periodic solutions. The point at which a Hopf curve meets tangentially a saddle-node curve is a Takens–Bogdanov bifurcation point: a linear singularity with two zero eigenvalues. The complete set of qualitatively different solutions found in its vicinity is displayed in the figure.

Fig. 2. Bifurcation diagram of the biomechanical model in the (ps , k1 ) parameter space. The integration of the model for each region is shown in the corresponding insets. In this set of parameters, a Takens–Bogdanov bifurcation occurs, where a saddle-node bifurcation is touched tangentially by a Hopf bifurcation and a homoclinic bifurcation. Path A passes through a SNILC bifurcation and path B through a Hopf bifurcation. The dot at the intersection between the Hopf line and one of the two saddle-node curves is the point at which a Takens–Bogdanov bifurcation takes place. The dark line in the transition 5–2 indicates where SNILC bifurcation occurs. 1250235-3

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In order to generate a syllable, a bird has to enter into the region of parameter space where stable oscillations take place (region 2 in Fig. 2). Within that region, the bird might achieve changes in the syllable’s acoustic features. For example, by changing k1 during the vocalization, the syllable’s fundamental frequency is changed. The generation of syllables with different spectral contents occurs, according to the model, when the bird modulates the physiological instructions in order to follow paths as those indicated in Fig. 2. The path for high values of k1 (path B in Fig. 2) generates tonal sounds, whereas the path for low k1 values gives rise to harmonic stacks (path A in Fig. 2). Notice that in the first case, the oscillations are born in a Hopf bifurcation, with a well defined frequency and zero amplitude. In path A, the oscillations are born with zero frequency and high amplitude (Saddle Node in Limit Cycle bifurcation, SNILC). These pulse-like solutions were observed directly using high-speed video recordings and flow measurements [Jensen et al., 2007]. Strong evidence for this mechanism was obtained by a systematic study of the relationship between fundamental frequency and spectral content in many Zebra finch individuals from different colonies (i.e. subjected to different learning experiences). The spectral content of a segment of song is quantified via the Spectral Content Index (SCI). It reflects how energy is distributed about the fundamental frequency, having its absolute minimum for pure tones, where its value is 1. By plotting the SCI of syllable fragments as a function of its fundamental frequency, we found a clear relationship between these two acoustic features. The same relationship is expected if low fundamental frequency sounds arise in SNILC bifurcations. Figure 3 displays the data and the prediction of the model. This observation is an example of a set of acoustical features that cannot be independently controlled by the nervous system. They arise as simple neural instructions are executed (get into an oscillatory region of the parameter space of the biomechanical peripheral device), and it is the nature of the bifurcation that determines a pack of acoustic features to be present in the uttered sound. Notice that in previous efforts [Laje et al., 2002; Mindlin et al., 2003], the only nonlinearity taken into account was the phenomenologically introduced nonlinear dissipation. Those models present a Hopf bifurcation and lead to tonal sounds.

Fig. 3. The spectral content of the syllables and sound frequency are correlated. The spectral content is quantified by an index called SCI which measures the distribution of the harmonic and their relative weights over the complete range of frequencies. Each syllable analyzed is represented by a point. Different point types correspond to different birds. The dotted line is obtained through a numerical experiment where only ps was changed generating sounds of frequency smaller than 1.5 kHz. The solid line was computed changing ps and a d0 , which physiologically corresponds to the activity of dorsal syringeal muscles.

2.2. Sound filtering The time evolution of the fundamental frequency during a vocalization is one of the most notorious acoustic features of birdsong. Compared to human vocalizations, birdsong has richer fundamental frequency variation and poorer vocal tract dynamics. Human voiced sounds are delicately modulated by a tract that is controlled through subtle labial, lips and tongue movements. Yet, birds do make an active use of their vocal tract. In some species, the distensible oral cavity is actually used to produce nearly tonal sounds [Riede et al., 2006]. The fundamental frequency typically changes during a vocalization. In order for its peaks to coincide with it, the vocal tract has to be adjusted dynamically. This effect allows to generate a more tonal sound, and probably an important sound amplitude enhancement. The tract of songbirds is constituted by a trachea, an oropharingeal–esophageal cavity and a beak. To model the tract, it is a good approximation to consider a closed-open tube (the length of the Zebra finch tract is approximately 3.4 cm). If Pi (t) stands for the pressure at the input of the tube, then

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Pi (t) = αx(t) − rPi (t − τ ),

(4)

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where αx(t) is the contribution to the pressure fluctuations by the modulated flow at the input of the tract, r is the reflection coefficient at the opposing end of the tube, and τ = 2L/c, with c the sound’s velocity [Arneodo & Mindlin, 2009]. In previous efforts [Laje et al., 2002; Mindlin et al., 2003], we reduced our model of the tract to a series of tubes. In recent work, Fletcher and coworkers noticed that some birds modify the geometry of their tract in a way coordinated with frequency modulations in the song [Fletcher et al., 2006]. They conjectured (and later measured) temporal changes in part of their vocal tract that are compatible with stressing sounds with time dependent spectral features. As illustrated in Fig. 4, the trachea is connected through a glottis to the extensible oropharingeal– esophageal cavity (OEC) [Fletcher et al., 2006]. The response of this cavity to the pressure fluctuations at the end of the trachea can be well approximated by that of a Helmholtz resonator. Fletcher and coworkers studied the tract (approximated as a tube connected to a resonator) using impedances, which allowed them to study the dependence of the resonances with the parameters of the filter. In order to compute the time evolution

of the sound source-vocal tract system, it is convenient to write also the dynamical equations of the pressure fluctuations in such a resonator, and solve the whole system of differential equations describing the behavior of the source, the trachea and the OEC. In acoustics, it is common to write an analog electronic computational model to describe a system of filters. The acoustic pressure is represented by an electric potential, while the volume flow by the electric current [Fletcher, 1992]. In this framework, short constrictions are inductors and cavities (smaller than the wavelengths) are well represented by capacitors. The equations for the equivalent circuit of the post-tracheal part of the vocal tract of Fig. 4 read:   di1   = Ω1 ,   dt        1 1 1 dΩ1   Ω1 =− i1 − Rh +   dt Lg Ch Lb Lg      Rb Rh 1 1 dVext (5) + i3 − +  L C L L L dt g g g  h b      Rh   Vext , +    Lg Lb       Lg Rb 1 di3   = − Ω1 − i3 + Vext ,  dt Lb Lb Lb where the relationships between the components of the electric analog and the acoustic elements are displayed in Table 1 [Kinsler et al., 1982], with Ω1 defined as the time derivative of i1 . Then the equations in terms of the features of the vocal tract, using the parameters of Table 1. Acoustic and electrical analogues. Electrical Impedance L Resistance

Fig. 4. (Top) Diagram of the vocal tract system of the bird: trachea, glottis, OEC and beak. Pi (t) stands for the pressure at the input of the tube and Po (t) stands for the pressure at the output of the trachea. (Bottom) The equivalent circuit of the post-tracheal part of the vocal tract. 1250235-5

R Capacitance C

Acoustic Inertance ρ l M= 0 S Resistance ρ0 ck2 2π Compliance V C= ρ0 c2 R=

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Table 1 read   di1   = Ω1 ,   dt        Sg c2 Sg ck2 dΩ1   = − G + i − Ω1  1   dt lg Vh 2π lg      Sg c2 c2 k4 GSg + i3 −  lg Vh 4π 2 lg       Sg dP0 GSg ck2    + Po , +   lg ρ0 dt lg 2πρ0       Gck2 G Glg di3     dt = − S Ω1 − 2π i3 + ρ Po , g 0

(6)

where ρ0 is the air density; lg and Sg are the length and the area of the glottis; Vh is the volume of the OEC; G is the gape parameter and represents the constriction made by the beak and the tongue combined, in such a way that the inertive term for the beak is Lb = ρG0 [Fletcher et al., 2006]. With these filters, it is possible then to advance beyond fundamental frequency dynamics (controlled by syringeal intrinsic muscles), and synthesize realistic sounds. Approximating the trachea by a close-open tube we can explain the ubiquitous presence of frequencies at about 2.5 kHz and 7.5 kHz in Zebra finch vocalizations. Interestingly, it is neither of these two frequencies that is the dominant one in the final output, but the first resonance of

the OEC. This frequency strongly depends on the volume of the OEC and the glottal area. Putting together the mathematical models for the sound generator and vocal cavities, it is possible to fit the parameters in order to synthesize a realistic Zebra finch song. Figure 5 illustrates a typical result. In the top panel we show the sonogram of a recorded song. The vertical axis represents frequency and the horizontal axis represents time. The bottom panel displays the sonogram of the corresponding synthetic song. The fitting of the source parameters was carried out as in [Sitt et al., 2010]. Briefly, the physical model accounting for the labial dynamics was taken to a standard form and adimensionalized. The parameters of this simplified model were adjusted so that solutions of given fundamental frequencies would present the right spectral content, according to the experimental results obtained for this species [Sitt et al., 2008]. Then, paths in parameter space were reconstructed so that the synthetic sounds would present time dependent acoustic features similar to those in the recordings. In order to do so, the song (sampled at 44.1 kHz) is decomposed into a sequence of sound segments. These segments are approximately 20 ms long, which is sufficiently short to avoid large variation in the physiological instructions but long enough to compute the observable quantities used in our description. For each segment, the parameters of the source are chosen from a list generated with the model. That list

Fig. 5. Sonogram of a typical Zebra finch song. The top panel shows the recorded song of a freely behaving bird and the bottom panel shows the corresponding synthetic song. Note the diversity of acoustical elements that can be produced within the song, from noisy to almost tonal sound, and also including harmonic stacks (second and last syllables) that are very characteristic of Zebra finch song. 1250235-6

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associates, for each pair of source parameters, the values of (w, SCI) (i.e. the fundamental frequency, and the spectral content index) of the synthesized sound. The selection of parameters in the list is such that the difference in frequency and spectral content between the synthetic sound and the recorded segment is minimized. As to the tract parameters, the tracheal length was fixed at 3.4 cm while OEC volume and glottal length were varied to match the position of the third peak in the range (1 kHz–8 kHz).


3. Conclusions and Future Work Over the last years, we have carried out a research program focusing on the physics of birdsong production. The purpose of this line of work is to unveil which features of this complex, learned behavior are constrained by the nonlinear nature of the peripheral device which stands between the nervous system and the environment. Are all acoustic features learned, and independently controlled by the nervous system? In this work, we show an example of the subtle interplay between the nervous system and the biomechanics: some aspects of the song are determined by the bifurcation taking place when sound is started. We have complemented this theoretical program with the experimental search of the parameters which, according to the model, drive the syrinx. By running our models with the experimentally recorded instructions we validated our hypothesis. Moreover, we are building biomimetic devices which integrate the equations describing our model, while the physiological instructions sent by the nervous system to the syrinx are recorded [Sitt et al., 2010]. It is for this reason that in this work, we explicitly write the equations that rule the dynamics of pressure fluctuations in an OEC. Most of the studies on vocalizations use impedances, which is not the best strategy if one is after online integration of the model. In the coming years, nonlinear dynamics might play an important role in the study of birdsong. In this work, we discussed the acoustic features of the Zebra finch song. This species generates complex sounds, but relatively poor songs, built from small numbers of syllables arranged in stereotyped motives. Other species, like domestic canaries (Serinus canaria), present richer songs. Recently, the air sac pressure used by canaries to utter different syllables were systematically studied, and they were

found to be well approximated by the subharmonics of a simple nonlinear system [Alonso et al., 2009; Alliende et al., 2010; Trevisan et al., 2006]. Where in the neural architecture this dynamics is generated is unknown, but whoever is familiar with the delicate structure that exists between the different subharmonic solutions of a driven dynamical system cannot but be amazed to find this order in biological data. This raises another challenging problem for dynamics: how does low dimensional, yet nontrivial dynamics emerge out of the ensembles of thousands of neurons constituting the neural motor pathway of songbirds? Recent advances in nonlinear dynamics might provide us with answers in this search [Ott & Antonsen, 2008, 2009; Alonso et al., 2010], but certainly the delicate balance between low dimensionality and complexity that seems to be present in the physiological gestures used by songbirds during singing might be an interesting field to explore the collective behavior of large sets of nonlinear units.

Acknowledgments We thank T. Riede and F. Goller for useful comments. This work was partially funded by NIH, CONICET and UBA.

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