Eur. Phys. J. B (2014) 87: 300 DOI: 10.1140/epjb/e2014-50566-5

Low dimensional dynamics in birdsong production

Ana Amador and Gabriel B. Mindlin

Eur. Phys. J. B (2014) 87: 300DOI: 10.1140/epjb/e2014-50566-5

Colloquium

THE EUROPEANPHYSICAL JOURNAL B

Low dimensional dynamics in birdsong production

Ana Amadora and Gabriel B. Mindlin

Department Physics, FCEyN, University of Buenos Aires and IFIBA-CONICET, 1428 Buenos Aires, Argentina

Received 20 August 2014 / Received in final form 11 November 2014Published online 10 December 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014

Abstract. The way in which information about behavior is represented at different levels of the motor path-way, remains among the fundamental unresolved problems of motor coding and sensorimotor integration.Insight into this matter is essential for understanding complex learned behaviors such as speech or bird-song. A major challenge in motor coding has been to identify an appropriate framework for characterizingbehavior. In this work we discuss a novel approach linking biomechanics and neurophysiology to exploremotor control of songbirds. We present a model of song production based on gestures that can be relatedto physiological parameters that the birds can control. This physical model for the vocal structures allowsa reduction in the dimensionality of the behavior, being a powerful approach for studying sensorimotorintegration. Our results also show how dynamical systems models can provide insight into neurophysio-logical analysis of vocal motor control. In particular, our work challenges the actual understanding of howthe motor pathway of the songbird systems works and proposes a novel perspective to study neural codingfor song production.

1 Introduction

Birdsong is a complex behavior, which emerges out of theinteraction between a nervous system, a peripheral biome-chanical device, and the environment. At least for approx-imately forty percent of the known species, some degree oflearning is involved in the process of acquiring the species-specific vocalizations. This task is quite rare in the animalkingdom and so birdsong has emerged as an importantanimal model for studying how a nervous system recon-figures itself during the acquisition of a complex behaviorthrough learning [1–3].

Between the nervous system and the actual songstands a peripheral system, including the respiratory sys-tem, the vocal tract and the sound source. The last oneis a complex biomechanical device called the syrinx. Thisis a bipartite structure between the bronchi and the tra-chea that holds two pairs of internal labia, which mod-ulate the airflow generating sound waves. The configura-tion of this device can be controlled by the activation ofspecific muscles, and those changes are ultimately trans-duced in acoustic modulations of the uttered sounds. Sincethe generation of sound requires also establishing airflowbetween the internal labia, a bird has to exquisitely co-ordinate respiratory and syringeal muscles to provide theuttered sounds with specific acoustic features [4].

Those physiological instructions in charge of control-ling the syrinx are the output of a nervous system, whichdedicates well-defined neural nuclei to their production.

a e-mail: anita@df.uba.ar

Each nucleus is set of a few thousands of interconnectedneurons, and the sets are also connected among them-selves, constituting a structure known as the song system.A variety of techniques have been applied to unveil howthe different parts of the song system end up generatingthe necessary motor patterns in control of the syrinx. De-spite important efforts in the last decades, the problemhas shown to be quite elusive.

The problem seems discouragingly complex and yet,in this work we describe three instances where low di-mensional nonlinear dynamics allows unveiling importantaspects of birdsong production. First, we show that manyacoustic features present in birdsong arise when relativelysimple physiological instructions drive a simple model thatcaptures the essential dynamics of the syrinx. We testedthis model by exposing both anaesthetized and sleepingbirds to replicas of their own songs, and compared theresponses of neurons selective to the bird’s own song tothe synthetic stimuli generated with a low dimensionalnonlinear model. The similarity in the neural responsesallowed us to build confidence on the pertinence of a lowdimensional description of the biomechanics involved inbirdsong production. Afterwards, we move one step closerto the nervous system, and show that those relatively sim-ple physiological gestures needed to drive the syrinx canalso be generated as the solutions of a low dimensional dy-namical system representing a simple neural architecture.

The work is organized as follows. In Section 2 wedescribe the biomechanics of birdsong production, andthe mechanisms by which relatively simple physiologi-cal instructions (air sac pressure and labial tension) are

Page 2 of 8 Eur. Phys. J. B (2014) 87: 300

a b

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Fig. 1. Typical zebra finch song and vocal production model.(a) The top panel shows the spectrogram of the sound wave(middle panel) corresponding to a zebra finch utterance. Thepressure pattern used to generate this song is shown in thelower panel, where each expiratory pulse (pressure above greenline) defines a syllable indicated with letters in the top panel.The grey area indicates inspiratory pulses. (b) The syrinx ispositioned at the base of the trachea, followed by the glottis,the oro-esophageal cavity (OEC) and the beak. All these struc-tures form the vocal tract that acts as a filter of the sound wavegenerated at the syrinx. The syringeal labia are modeled as anonlinear oscillator.

transduced into complex sounds. Section 3 describes airsac pressure dynamics in the case of canaries. We showthat pressure patterns can be obtained as a the result ofthe nonlinear competition between two time scales, anddescribe experiments that we performed to manipulateone of these time scales in order to test our prediction.In Section 4 we show that the same neural architectureused to generate the previously described physiological in-structions is capable of generating the labial tension in adifferent species. In Section 5 we discuss these results.

2 Generating acoustically complex signalswith reasonably simple instructions

Song spectrograms have been the most extended tool fordescribing singing behavior and relating neural recordingsto singing (see top panel of Fig. 1a).

These are obtained by performing a time-frequencyanalysis of the sound signal. This linear technique allowsinferring, graphically, several acoustic features of the song,as the fundamental frequency, the harmonic content of thetime trace, or the relative emphasis of frequencies (colorcoded in the sonogram). It is natural to ask whether thebird has to code (and eventually learn) all those featuresindependently, or if the nonlinear nature of the avian vo-cal organ constrains those features. Ultimately, it is thedynamics of the structures – principally the dynamics ofthe intrinsic syringeal musculature controlling the tensionof the syringeal labia and of the thoracic and abdominalmusculature controlling air sac pressure – that producesthe airflow fluctuations which after being filtered by theupper vocal structures generates the sound.

In an extensive development over many years, we havetested the hypothesis that much of the acoustical complex-ity present in oscine song was the result of relatively simpletrajectories in a low dimensional physiological parameterspace of pressure and tension driving a highly nonlinearbiomechanical device (e.g., [5–10]). This hypothesis hasbeen tested in several species with very different typesof songs, both spectral and temporal wise: rufous-collaredsparrow (Zonotrichia capensis), canary (Serinus canaria),northern cardinal (Cardinalis cardinalis) brown thrash-ers (Toxostoma rufum) and zebra finch (Taeniopygia gut-tata). Moreover, the nonlinear nature of the sound sourceallows fundamental frequency modulation through pres-sure modulation. This fact was a prediction of the modeland was tested experimentally in a suboscine species, thegreat kiskadee (Pitangus sulphuratus) [11]. Before this re-markable result, it was assumed that frequency and pres-sure were two physiological parameters controlled inde-pendently by the central nervous system. The nonlinearmodel allowed a more integrative perspective.

Similarly to human speech, birdsong is typically gener-ated during expiration (see Fig. 1a). During song produc-tion a stereotyped pattern of expiratory air sac pressurepulses, alternate with brief, deep inspirations, called mini-breaths (shaded region in the lower panel of Fig. 1a). Inthis way, the respiratory activity determines the coarsetemporal structure of the song (syllable sequence). There-fore, a model for birdsong production requires inspectingthe dynamics of syringeal labia during those expiratorypulses.

In order to generate a dynamical model for the vocalsource, we propose that the labia support two modes ofvibration: an upward propagating wave and a lateral dis-placement around their midpoint positions, as suggestedby videography of the folds during phonation [12]. Thesemodes are coordinated in such a way that energy is gainedfrom the airflow in each cycle, making auto-sustained os-cillations possible. In order to do so, the labia presenta convergent profile when moving away from each other,and a more planar profile during approaching. When thelabia present a convergent profile, the average pressurebetween the labia is close to the air sac pressure. Whenthe labia are approaching, the average pressure betweenthem is similar to the atmospheric pressure. In this way,the pressure is high when the labia are moving away fromeach other, and low when the labia are approaching eachother. This allows to overcome dissipative forces and totransfer energy from the airflow to the labia [4,5,13].

The dynamical description of this mechanism requireswriting Newton’s equations for a labium [4,5]. If x standsfor the midpoint position of a labium of unitary mass, theyread:

x = y

y = −k(x)x − βy − γx2y + aavpav (1)

where the first term of the second equation represents the(nonlinear) elastic restitution (k(x) = k1 + k2x

2) [14], thesecond one the linear dissipation, the third one a nonlin-ear dissipation accounting for the existence of boundaries

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Fig. 2. Bifurcation diagram of birdsong production model.Qualitative differences in the dynamics are displayed in thenumbered regions. The physical model (a) and its normal form(b) exhibit analogous bifurcation diagrams. The integrationof the model for each region is shown in the numbered insetsin (c). In this set of parameters, a Takens-Bogdanov bifurcationoccurs, where a saddle-node bifurcation (brown line) is touchedtangentially by a Hopf bifurcation (blue line) and a homoclinicbifurcation (dashed green line). A SNILC bifurcation occursbetween region 2 to 5, and a Hopf bifurcation between regions 1and 2.

for the oscillations, and pav is the spatial average of theinter labial pressure. This last term is responsible for theenergy transfer from the airflow to the labium and it canbe written in terms of the sub syringeal pressure psub,the midpoint position of the labium x, its velocity y, theresting positions of the edges of the labium (x01 and x02),and a parameter describing the time τ that takes the wavepropagating upward in the labia to cover half its length:

pav = psub(x01 − x02 + 2τ y)/(x01 + x + τ y). (2)

These equations allow describing the dynamics of thesound source that can be explored in a two-parameterspace. In Figure 2a we show the bifurcation diagram ofthe system moving parameters (psub, k1). If the parame-ters’ values are in region 1, only one attracting fixed pointexists. This corresponds to non-oscillating labia, i.e., silentregime. In region 2 we find that the fixed point is unstableagainst a limit cycle. Oscillations of the labia correspondto sound production. Regions 3, 4, and 5, bounded by thesaddle node curves (full brown line in Fig. 2) that convergeto the cusp bifurcation point, have three fixed points. Be-tween regions 2 and 5, a saddle node in a limit cycle(SNILC) bifurcation takes place, and between regions 1and 2 a Hopf bifurcation (full red line in Fig. 2) [15]. Thedashed green line indicates a homoclinic bifurcation.

This simple model captures the physiological mecha-nism of sound initiation: when the bird intends to vocalize,it increases air sac pressure to pass through a threshold inorder to generate auto-sustained oscillations that wouldmodulate the airflow generating sound (see recorded airsac pressure signal and sound wave in Fig. 1a). In themodel, this could happen in two distinctive dynamicalways: through a Hopf or a SNILC bifurcation. The formergenerates tonal sounds: sinusoidal oscillations, born with

a defined frequency and zero amplitude. The latter gener-ates oscillations born with infinite period and spectrallyrich: the saddle node remnant generates a spike-like wave,providing a substantial amount of energy to the harmon-ics. This rich variety of sounds covers a lot of the birdsongdiversity.

The other physiological feature that is described bythe model is that for increasing values of the parame-ter describing the lineal restitution (k1), the oscillationswill present higher fundamental frequencies. This param-eter is related to the activity of syringeal muscles thatincrease the tension of syringeal labia. It has been shownthat increasing muscle activity results in increasing valuesof fundamental frequency [16].

Another way of controlling the fundamental frequencyis generating the oscillations through the SNILC bifur-cation, where frequency values depend on the distancefrom the bifurcation point. In the model, this can be con-trolled not only by the restitution parameter but also bythe pressure parameter, which provides a dynamical ex-planation for fundamental frequency modulation throughair sac pressure [11,14,17].

In order to minimize the number of parameters in-volved in the description we performed a reduction of themodel to a normal form: a minimal mathematical descrip-tion of the dynamics capable of presenting the same bi-furcation diagram as the original model [15,18]. We de-rived a standard equation which presents the bifurcationdiagram of the physical model. It is a two dimensionaldynamical system presenting a cusp and a Hopf bifurca-tion, obtained as a third order expansion around a Takens-Bogdanov linear singularity [18], which reads:

x = y

y = −α(t)γ2 − β(t)γ2x − γ2x3 − γx2y

+ γ2x2 − γxy. (3)

Comparing the bifurcation diagram of the physical modeldefined by equation (1) (see Fig. 2a) with that of the nor-mal form defined by equation (3) (see Fig. 2b), a propermapping of air sac pressure and tension into the unfold-ing parameters allows to recover the qualitative dynamics.In reference [19] the relationship was explicitly computed;the basic operations involved were (i) a translation of thevalues of (psub, k1) where a Takens Bogdanov bifurcationtakes place in the physical model to (α, β) = (0, 0), (ii) amultiplicative scaling, and (iii) a rotation of π. The nor-mal form equations describe dynamical behavior in theregime of interactions between Hopf and a saddle node inlimit cycle (SNILC) bifurcations. For zebra finches, mostphonatory oscillations start in SNILC bifurcations, as theair sac pressure is increased (going from region 5 to re-gion 2 in the parameter space shown in Fig. 2). Such dy-namical constraints help to define the universe of zebrafinch sounds. For example, there is a tight relationshipbetween spectral content of a sound and its fundamentalfrequency: the lower the frequency, the higher the spec-tral content [20]. Small changes in parameters can produceradically different changes in sound output, depending onwhere those movements occur in the bifurcation space.

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So far, we have described the sound source, whichsits at the base of the vocal tract (see Fig. 1b). In or-der to reproduce realistic sounds, we modeled the tracheaas a tube and then its output was used to excite the oro-esophageal cavity (OEC) which is modeled as a Helmholtzresonator [10,21], and its output will be our synthetic song.

In this way it is possible to synthesize songs with a lowdimensional mathematical model whose parameters areeasy to interpret in terms of physiological and anatomicalobservations. The time-dependent parameters can changesmoothly in order to reproduce the acoustic properties ofthe recorded song that the model for song production at-tempts to describe. These time traces represent “motorcoordinates”, and they potentially provide a more nat-ural bridge between the activity of the central nervoussystem and behavior than do sound spectrograms [8]. Wetested this hypothesis by generating a synthetic copy ofthe recorded birdsong (e.g., Fig. 3). To go beyond thestriking similarity between synthetic and natural songs, weused a highly accurate neurophysiological measurement inorder to assess the biological relevance of the low dimen-sional model. We recorded neural responses to these au-ditory stimuli in song motor nuclei, demonstrating strongpositive results [8].

3 The nature of the instructions: the caseof pressure for canary song

As we have discussed above, a central aspect of the mo-tor control of birdsong production is the capacity to gen-erate diverse respiratory rhythms, which determine thecoarse temporal pattern of song. The neural mechanismsthat underlie this diversity of respiratory gestures and theresulting acoustic syllables are largely unknown.

The neural network responsible for the generation ofthe gestures involved in birdsong has been well described.It includes two forebrain nuclei: HVC (used as a propername) and the robust nucleus of the arcopallium (RA)that generates the coordinated motor patterns that helpto shape the instructions driving the muscles controllingrespiration, the vocal organ and upper vocal tract struc-tures. However, it has been debated to what degree thistelencephalic motor program contains direct instructionsfor detailed patterns such as the various timescales of thebehavioral output [22]. One model, based in studies per-formed in zebra finches, proposes a direct control by thetelencephalic song control area, such that all timescalespresent in the song arise directly from the output signalof this neural nucleus. Support for this model was derivedfrom observations of sparsely coding output neurons inHVC as well as experiments in which song was stretchedby cooling of HVC [23,24]. On the other hand, in the caseof canaries, an interactive model has been proposed, wheremotor instructions emerge from nonlinear interaction be-tween timescales of different components of the motor con-trol network [7,25]. Two pieces of evidence point towardsthis integrative picture. One of them is the morphology ofthe pressure patterns used to generate the different sylla-bles used in canary song. Figure 4a illustrates a typical

Fig. 3. The output of a low dimensional model for song pro-duction is able to reproduce a complex natural behavior. Analgorithmic procedure allows to obtain synthetic copy of a ze-bra finch song by fitting two acoustic features of the recordedsong. The top panel shows the sound wave and spectrogramof the recorded song and the lower panel corresponds to thesynthetic song.

pressure pattern during canary song production [9,26].The different parts of this temporal pattern can be ob-tained as a minimally complex neural architecture thatis driven by simple instructions. Namely, integrating thefollowing dynamical system:

x = λ (−x + S(ρx + ax − by)),y = λ (−y + S(ρy + cx − dy)),

with S(x) = (1 + e−x/x0)−1, (4)

where x stands for the activity of a population of excita-tory units, y for the activity of a population of inhibitoryneurons, and ρx(t), ρy(t) for the inputs to those popu-lations. The constants a, b, c and d describe the archi-tecture of the array. The top panel of Figure 4b shows acaricature of this model, with interconnected nucleus ofexcitatory and inhibitory populations. A good intuitionon the dynamics of this system can be gained by inspect-ing the different behavior displayed by this system understationary values of the input parameters and ρx, ρy [27].Figure 4b shows the bifurcation diagram of the system.Note the similarity between Figures 4b and 2.

In order to generate different patterns, the nature ofthe driving signal has to experience some changes, typi-cally the amplitude and frequency of the forcing. In thisway, different shapes in the time series can be generated asa unique neural architecture driven by slightly different in-structions. An example of this is shown in Figure 4c whereall the pressure patterns corresponding to a canary song

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Fig. 4. Respiratory patterns during canary song. (a) Recordedair sac pressure during singing. The colored lines indicate dif-ferent syllable types. (b) Diagram of neural nuclei of excitatoryand inhibitory populations and the bifurcation diagram of thesystem. Note the similarity between this figure and Figures 2aand 2b (the numbers indicate the same dynamical regime ofFig. 2). (c) Synthetic pressure patterns obtained integratingthe dynamical systems described by equation (4) with differ-ent values of (ρx, ρy), shown as colored dots and lines in thebifurcation diagram of (b).

are generated using the model described by equation (4),using different combinations of (ρx, ρy). The colored dotsand lines in the bifurcation diagram of Figure 4b corre-spond to a particular pressure pattern in Figure 4c.

Since any neural architecture is intrinsically nonlinear,even slight changes in the driving instructions can leadto qualitatively different patterns. In this way, differentpatterns have been interpreted as sub-harmonic solutionsof simple instructions [7,9].

The other evidence suggesting the integration of differ-ent time scales in the generation of the pressure patternsused in canary song comes from the results obtained asone of the time scales is manipulated [25]. It has been wellestablished that the thermal manipulation of HVC affectsthe timing of the song [24]. This experiment showed thatat least one of the pertinent time scales could be perturbedby thermally tempering with that telencephalic nucleus.

A simple integrative model, which can also reproducethe shapes of the pressure temporal patterns, predicts thata small change in the frequency of an average instructioncoming from HVC may provoke a drastic change in thetemporal pattern of resulting syllables. This facilitates acontrol mechanism where simple neural instructions inter-acting with downstream neural architecture can result incomplex rhythms in the output patterns. Manipulation ofthe telencephalic timescale through local cooling results inthe predicted effects of initial stretching and then “break-ing” of syllables when the cooling range is extended. Thesenovel “syllable-breaking” patterns can be interpreted interms of bifurcations of the model [25].

In the paradigm of nonlinear interaction, HVC, or acircuit including HVC, does play an important role inthe generation of the respiratory patterns. This paradigmchallenges the idea of a “look up table” linking burstsof HVC activity with brief segments of song, with HVC

bursts determining a unique timescale in the birdsongsystem[22]. We do not propose a specific location in theneural architecture of the songbird system for the secondtimescale necessary to explain the shape and rate of themeasured pressure patterns and their breaking. Instead,we show that a minimal computational model (two neu-ral populations: interconnected excitatory and inhibitoryneural nuclei) is capable of transducing very simple in-structions into the specific patterns measured in canaries.

4 The nature of the instructions: the caseof tension in finches

The initial computational models of birdsong productionsuggested that very simple gestures for pressure and labialtension could account for many of the features found insong. For example, the nature of the frequency modula-tion was thought to be controlled by the phase differencebetween those gestures, which could be as simple as har-monic fluctuations [5]. The actual measurement of the airsac pressure in canaries unveiled a subtler scenario: dif-ferent syllables required different pressure patterns. Yet,the diversity of gestures was not a capricious collectionof shapes: they can be obtained as the different solutionspresented when a minimal neural architecture is driven bysimple driving patterns [9,25]. Is this also the case for thepatterns corresponding to the tension?

This issue is slightly more difficult to test. The smoothtime series that is measured by a pressure transducer con-nected through a cannula to an air sac has no parallelwhen it comes to estimating labial tension. The songbirdsyrinx is a complex anatomical structure composed bycartilaginous rings and half-rings, muscles and labia. Sixpairs of syringeal muscles provide mechanical control inthe syrinx, four of which have both insertion sites on thesyrinx (intrinsic muscles), being attached to the cartilagi-nous structures. The phonating labia are attached to theinternal part of modified cartilaginous half-rings. In thisway, specific syringeal muscles would contract to move thecartilage that would ultimately modify the properties ofthe labia. Moreover, very little is known about the histol-ogy or elastic properties of the labia in songbirds [28]. Thecloser one gets to measure a proxy for the labial tensionis to measure the electrical activity of the syringeal mus-cles innervating the syrinx. Unfortunately, the relation-ship between the EMG of the different syringeal musclesand the actual physiological parameters in the model isstill not completely unveiled. In other words, there is nounique relationship between EMG activity of individualsyringeal muscles and the parameters controlled by them.Moreover, it is not yet solved how the different musclessynergistically control the acoustic properties.

Despite these difficulties, it was recently shown that areconstruction of the gestures necessary to drive a simplebiomechanical model to generate realistic song was plausi-ble. In order to validate the reconstruction procedure twostrategies were followed. First, the reconstructed air sacpressure was compared with the experimentally recorded

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Fig. 5. Simple motor gestures generate complex behavior.Recorded zebra finch song (a) is well reproduced by the outputof a low dimensional model (b) when the parameters are in thevicinity of bifurcations. Panel (c) shows the temporal evolutionof one of the two parameters of equation (3), namely, β(t) thatcan be related to the tension of the syringeal labia.

one, obtaining a high correlation between the timetraces [10]. More spectacularly, synthetic song producedwith the model, when driven by the reconstructed phys-iological gestures, was played to sleeping birds while theactivity of neurons selective to the bird’s own song wasrecorded [8]. Despite the extraordinary selectivity of theseneurons to the bird’s own song [29,30], the syntheticsong was capable of eliciting responses. Thereby, we per-formed electrophysiological experiments in order to obtaina highly accurate biological measurement to validate thebirdsong production model.

Assuming the biomechanical model described in Sec-tion 2 (a model capable of starting oscillatory behav-ior through a SNILC bifurcation), we created syntheticversions of the songs that our test birds sang. Time-dependent parameters of the model describing the labialdynamics were reconstructed to account for the time-dependent acoustic properties of the sound. We used analgorithmic procedure to reconstruct unique functions forthe air-sac pressure (α(t)) and the tension of syringeallabia (β(t)) [10]. The result of the procedure for one songis illustrated in Figure 5, showing that many features ob-served in the spectrogram of the recorded song (Fig. 5a)were also present in the synthesized song (Fig. 5b). Re-markably, relatively simple time traces of reconstructedpressure and tension arose from fitting the bird’s song(see Fig. 5c for an example of reconstructed tension). Infact, song was described by the sequence of these pressure-tension trajectories, which we call gestures, with gestureonsets and offsets defined as discontinuities in either thepressure or tension functions.

For the songs reported in reference [8], we found thatmost of the parameters could be approximated well byfractions of sine functions, exponential decays, constantsor a combinations of these. Most remarkably, those pat-terns can also be found when a unique neural architecture

10

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Fig. 6. Tension of the syringeal labia of zebra finches generatedby a simple neural architecture. (a) Bifurcation diagram of thedynamical system described by equation (4). Simple parame-ter explorations near bifurcation regions generate a diversityof tension gestures that could be used to generate syntheticsong (b). The colored lines and numbers show the correspon-dence between the exploration in parameter space and the tem-poral evolution of the system’s output.

is driven by essentially the same simple instruction: a kickfollowed by a slow decay. In Figure 6a we display the bi-furcation diagram of a low dimensional dynamical systemruling the average dynamics of a population of excitatoryunits coupled to a population of inhibitory ones (the samesystem described with Eq. (4)). The lines represent bi-furcation curves: curves in parameter space that separateparameter regions where the system presents qualitativelydifferent behaviors. The three curves ending in arrows rep-resent trajectories in the parameter space of the driving ofexcitatory and inhibitory populations (ρx, ρy). The lowerpanel of Figure 6b shows the solutions generated by theneural network under the driving showed by the curves A,B and C in Figure 6a. No quantitative calculation is re-ally needed in order to recognize that the solutions cor-respond to the reconstructed patterns (as the one shownin Fig. 5c). The same neural architecture (an excitatoryand an inhibitory population of neurons), and similar tem-poral patterns driving it (kicks with exponential decays)were necessary in order to generate the different tensionpatterns.

5 A common theme: variability around lineardegeneracies

There is a hierarchy of complexity in this phenomenon.The sounds are complex indeed, with rich and subtle re-lationships between different acoustic properties like fun-damental frequency and spectral content. Indeed, most ofthe acoustic features of birdsong come in package. Theorigin of this is the nonlinear nature of the vocal organ,which bounds the spectral relationships between the ut-tered sounds. Moreover, these do not depend on the de-tails of the biomechanics, but on the basic family (“nor-mal form”) of dynamical system the model of the soundsource belongs to. Because of this, it is quite straight-forward to generate a diversity of songs using the samebiomechanical model, even modeling songs belonging tobirds of very distant groups (e.g. oscines and suboscines).

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Moreover, many acoustic features are found both in birdsand humans, which share the basic biomechanical princi-ples of the sound sources, but details of musculature andstructure are indeed very different [17,28].

There is structure at the level of the instructions driv-ing the syrinx itself. In canaries, the different pressure pat-terns used to utter different syllables can be obtained bydriving a basic neural architecture with very simple timeseries. It is remarkable that those simple driving instruc-tions are transduced into specific patterns: small oscilla-tions mounted on a DC level, period solutions followedby a period doubled ones, or larger pulses consisting ofrapid oscillations followed by a slowly decaying pulse (seeFig. 4c). Even more surprising: those patterns deform aspredicted by the dynamical model when one of the timescales involved is modified using an experimental setup tocontrol temperature [25].

To complete the picture, in zebra finches, for which theother key physiological parameter driving the syrinx hasbeen reconstructed from theoretical means, the labial ten-sion can also be generated as the same basic neural archi-tecture is driven by very simple time series. The exponen-tial decays, the rapid increases, the oscillations followedby exponential decays, the constant frequencies followedby a small decaying tail: all the features reconstructed inthe process of attempting to synthesize realistic sounds,could be found by driving a simple neural architecture.

These three instances point to a unique strategy: thetransduction of somewhat simple time series into richerones. Probably high in the song system (where high istaken as a synonym of distant to the peripheral vocal de-vice) the instructions are coded in very simple terms, andrichness is gained as the instructions are integrated intoother systems. After all, the bird first breathes, then vo-calizes, and finally learns to sing. It is therefore likely thatthe final outcome of the songbird’s vocal system is theintegration of those systems.

In the three scales described in this manuscript, thediversity of gestures can be achieved easily, due to thefact that the system was operating in the neighborhoodof a linear singularity [31]. From the point of view of themodeling, this allows us to change rapidly between differ-ent solutions. Interesting enough, the regions in parameterspace that converge in a linear singularity are open re-gions of the parameter space. This allows us to have boththe benefits of flexibility and robustness, two of the mostremarkable ingredients of behavior.

6 Conclusions

In this paper we analyzed the generation of birdsong froma dynamical perspective. Much of the complexity of thesounds could be tracked to the nonlinear nature of theavian vocal organ. Specifically, many acoustic featuresended up emerging in “packages” controlled by the un-derlying dynamics of the biomechanics. A typical exampleis the relationship between spectral content and funda-mental frequency in the song of the zebra finch. Further-more, nonlinear dynamics seems to play a role beyond

the periphery: in the canary song, the diversity of sylla-bles requires respiratory patterns that are sub-harmonicresponses of a simple dynamical system driven by simpleinstructions. Suggestively, the dynamical system can betrivially interpreted as the one ruling the dynamics of asimple neural architecture, overwhelmingly present in theneural system needed for birdsong production. Remark-ably, the same dynamical system, driven by simple timeseries, also reproduces the tension gestures that have beenreconstructed in the song of zebra finches.

We are not yet capable of identifying where in thesong system (i.e., the specific neural architecture neededto generate song), but the idea that the physiological in-structions emerge out of the interaction between differentlevels of complexity is already a breakthrough. Birds havedeveloped song, building on a pre-existing system: the res-piratory one. Even non-learners have developed the abil-ity to coordinate their respiratory activity with the propermorphological modifications of the air pathways, control-ling constrictions that allowed the modulation of airflowand therefore, the production of song. Oscine birds, whichconstitute approximately 40 percent of the known birdspecies, have built on top of this structure a sophisticatedset of neural nuclei that help to learn the physiologicalpatterns controlling the syrinx and the respiration. Theintegrated circuit then is a complicated neural structurethat includes brainstem nuclei that project to the neuronsin charge of controlling respiration and syringeal muscle,as well as to the telencephalic nuclei which integrate thesesignals with auditory and sensory information, to finallyproject back to the brainstem. This integrated anatomyis consistent with our description of progressively com-plex instructions interacting nonlinearly, and challenge awidespread view of telencephalic nuclei (HVC) in charge ofa look up table where all the behavior is coded in brief (ap-proximately 10 ms) commands. The bias towards stressingthe role of telencephalic nuclei in the system is also a resultof the relative simplicity to perform neural recordings inthis region, compared to other parts of the song system. Itis one of the great challenges in the field to overcome thisdifficulty, in order to unveil the relative role of differentparts of the song system in the generation and learning ofthis complex and rich behavior.

We would like to thank Santiago Boari for comments to thismanuscript. This work describes research partially funded byCONICET, ANCyT, UBA and NIH through RO1-DC-012859.

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