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- Abstract
- I. INTRODUCTION
- II. MODEL AND ANALYSIS METHOD
- III. RESULTS
- IV. DISCUSSION AND CONCLUSION
- References

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Phys Rev E Stat Nonlin Soft Matter Phys. Author manuscript; available in PMC 2010 October 13.

Published in final edited form as:

Phys Rev E Stat Nonlin Soft Matter Phys. 2008 June; 77(6 Pt 1): 061922.

Published online 2008 June 27. PMCID: PMC2954056

NIHMSID: NIHMS224498

Department of Surgery, Division of Otolaryngology Head and Neck Surgery, University of Wisconsin Medical School, Madison, Wisconsin 53792-7375, USA

The effect of glottal aerodynamics in producing the nonlinear characteristics of voice is investigated by comparing the outputs of the asymmetric composite model and the two-mass model. The two-mass model assumes the glottal airflow to be laminar, nonviscous, and incompressible. In this model, when the asymmetric factor is decreased from 0.65 to 0.35, only 1:1 and 1:2 modes are detectable. However, with the same parameters, four vibratory modes (1:1, 1:2, 2:4, 2:6) are found in the asymmetric composite model using the Navier-Stokes equations to describe the complex aerodynamics in the glottis. Moreover, the amplitude of the waveform is modulated by a small-amplitude noiselike series. The nonlinear detection method reveals that this noiselike modulation is not random, but rather it is deterministic chaos. This result agrees with the phenomenon often seen in voice, in which the voice signal is strongly periodic but modulated by a small-amplitude chaotic component. The only difference between the two-mass model and the composite model is in their descriptions of glottal airflow. Therefore, the complex aerodynamic characteristics of glottal airflow could be important in generating the nonlinear dynamic behavior of voice production, including bifurcation and a small-amplitude chaotic component obscured by strong periodicity.

The bifurcation and chaos phenomena related with nonlinear systems are widely observed in physical and biological systems, such as electronic circuits [1], superconductors [2], neurology [3], and cardiology [4]. Nonlinear dynamic theory provides deeper insight into these complex natural phenomena. The voice of a vertebrate is generated by a highly nonlinear oscillator, comprised of the vocal folds and glottal air-flow. Rich nonlinear phenomena, including subharmonics, diplophonic, double period doubling bifurcation, chaos, and spatiotemporal chaos, have been observed in birdsong [5–7], human voice [8], the cries of newborn infants [9], excised larynx experiments [10], and the calls of mammals [11,12]. Knowledge of the nonlinear components of voice is expected to be useful for understanding voice production [13], improving the realistic synthesis of voice [14], and providing potential clinical applications in assessment and treatment of voice diseases [15]. Therefore, the nonlinear dynamics in voice production have recently received considerable attention.

Several models have been used to investigate the generation of nonlinear dynamics in voice [13,14,16–21]. Irregular vibration patterns were initially detected in a two-mass model with a sufficiently large left-right tension imbalance [13]. Bifurcations and chaos were then predicted by a symmetric vocal fold model with tissue parameters obviously deviating from the normal values [16]. Irregular vocal fold oscillations were also found in the continuous model with soft tissue [20] and pathological phonation [21]. These studies suggest that abnormalities in vocal tissue are important causes of irregular vocal fold vibration. The introduction of random perturbations into a vocal fold model has been found to decrease the threshold value necessary for a system to produce irregular vocal fold vibration [17]. Recently, by considering the source-filter acoustic interaction, delayed feedback due to reflected sound in the vocal tract was shown to contribute to extremely rich dynamics, such as period doubling bifurcation and nonperiodic oscillation [18]. These studies explain the cause of rough voice production and irregular vibrations of vocal folds that are typical of many vocal diseases.

Glottal airflow plays an important role in voice production because vocal fold vibrations are generated by the fluid-tissue interaction. It is known that real fluid is very complex, and various dynamical behaviors can be found in real fluid, including asymmetry, vortices, turbulence [22,23], bifurcation [24], and spatiotemporal chaos [25,26]. These phenomena are also components of glottal airflow [27–35]. These complexities of glottal airflow could be responsible for variation in the phonation threshold pressure [36] and for the asymmetric vibration of the vocal fold [37]. However, this causal linkage was not considered in most of the previous studies of irregular vocal fold vibration because the airflow was often simplified in accordance with Bernoulli’s law and the laminar, nonviscous, and incompressible assumption. Few works have studied the influence of the complex spatiotemporal behaviors of glottal airflow on the vocal fold vibration, particularly irregular vibrations.

In this study, we simulated and analyzed the contribution of the complexity of glottal airflow on irregular vocal fold vibration. The Navier-Stokes (NS) equations were used to predict the spatiotemporal behavior of glottal aerodynamics, and the tissue mechanics of the vocal folds were simplified to an asymmetric two-mass model. The interaction of the vocal fold and the glottal airflow generates self-oscillation. Using this model, we studied the various oscillatory patterns of vocal folds. To investigate the effect of airflow on vocal fold vibration, the dynamic behaviors of the vocal folds when driven by an airflow description based on the NS equations and when driven by an airflow description based on Bernoulli’s law were compared. The nonlinear detection method proposed by Barahona and Poon [4,38,39], which has the ability to detect the nonlinearity in a short, strong periodic series, was used to analyze the nonlinear characteristics of vocal fold vibration.

An asymmetric composite model, developed from the composite model [37], was employed to predict the vocal fold vibrations induced by complex airflow. The asymmetric two-mass model [13,40] was used to describe the tissue mechanics of vocal folds with unilateral laryngeal paralysis,

$$\frac{d{\mathbf{x}}_{\alpha}}{dt}=\mathbf{v}({\mathbf{x}}_{\alpha},{\mathit{\mu}}_{t\alpha})+{\mathbf{F}}_{\alpha}.$$

(1)

The displacement vector **x*** _{α}* is written as
${\mathbf{x}}_{\alpha}={[\begin{array}{cccc}{x}_{1\alpha},& {v}_{1\alpha}& {x}_{2\alpha}& {v}_{2\alpha}\end{array}]}^{T}$. The velocity vectors

$$\mathbf{v}({\mathbf{x}}_{\alpha},{\mathit{\mu}}_{t\alpha})=\left(\begin{array}{c}{v}_{1\alpha}\\ \frac{1}{{m}_{1\alpha}}\left(-{r}_{1\alpha}{v}_{1\alpha}-{k}_{1\alpha}{x}_{1\alpha}-\mathrm{\Theta}(-{a}_{1\alpha}){c}_{1\alpha}\frac{{a}_{1\alpha}}{2L}-{k}_{c\alpha}({x}_{1\alpha}-{x}_{2\alpha})\right)\\ {v}_{2\alpha}\\ \frac{1}{{m}_{2\alpha}}\left(-{r}_{2\alpha}{v}_{2\alpha}-{k}_{2\alpha}{x}_{2\alpha}-\mathrm{\Theta}(-{a}_{2\alpha}){c}_{2\alpha}\frac{{a}_{2\alpha}}{2L}-{k}_{c\alpha}({x}_{2\alpha}-{x}_{1\alpha})\right)\end{array}\right),$$

(2)

where *x _{iα}* is the displacement of mass

$$\mathrm{\Theta}(x)=\{\begin{array}{ll}tanh(50x/{x}_{0}),\hfill & x>0\hfill \\ 0,\hfill & x\le 0\hfill \end{array}.$$

(3)

Unilateral laryngeal paralysis is modeled as follows:

$${m}_{il}={m}_{ir}/Q,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{k}_{il}={Qk}_{ir},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{k}_{cl}={Qk}_{cr},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{c}_{il}={Qc}_{ir},$$

(4)

where *Q* is the asymmetric factor. The other parameters of the left vocal fold are identical with those of the right vocal fold. Table I presents the lumped-mass parameters used in the simulation.

The intraglottal airflow force vector **F*** _{α}* applied to the vocal fold surface provides the driving force for the vocal fold vibration and can be defined as

$${\mathbf{F}}_{\alpha}={\left[0,(L/{m}_{1\alpha}){\int}_{0}^{{d}_{1\alpha}}{p}_{\alpha}(y)dy,0,0\right]}^{T},$$

(5)

where *d*_{1}* _{α}* is the thickness of mass

$$\xb7\mathbf{V}=\mathbf{0},$$

(6)

$$\rho \frac{\mathbf{V}t+\rho \mathbf{V}\xb7\mathbf{V}=\mu {2}^{\mathbf{V}}}{}$$

(7)

where *ρ*=1.2 kg/m^{3} is the flow density, *μ*=1.8 ×10^{−5} N s/m^{2} is the viscosity, **V** is the velocity vector, and *p* is the pressure of flow. The glottal length is 0.3 cm. The prephonatory superior and inferior glottal widths are both 0.178 57 mm. A 2-cm inlet duct and an 8-cm outlet duct represent the trachea and the vocal tract, respectively. Their widths are both 2 cm. Figure 1 illustrates the structure of the airflow channel around the glottis. The no-slip condition was applied to both the tracheal wall and the vocal tract wall. The specified pressure condition was set at the inlet of the sub-glottal tube to drive the model, and the zero pressure boundary condition was set at the outlet of the supraglottal tube. The moving wall condition was applied to the fluid-solid interaction boundary. Using the given boundary condition, the flow pressure *p* can be solved using the NS equations.

In our simulation, the NS equations and the two-mass vibration model were fully coupled and simultaneously solved in every iteration step. The values of the glottal displacements and the velocities solved from the two-mass model were applied to the glottal wall to define the boundary condition. Simultaneously, the airflow pressure solved from NS equations was applied as the driving force for the two-mass model. By alternating between solving the NS equations and solving the asymmetric two-mass equation, a self-oscillating solution was obtained from the iterative process. The NS equations were solved by using the FLOTRAN CFD analysis of the finite-element software ANSYS. A custom program was developed in ANSYS Parametric Design Language to solve the two-mass equation with the fourth-order Runge-Kutta routine and manage the airflow-tissue interaction. In the time domain, both the NS equations and the two-mass equation were integrated with a time step of 50 *μ*s. Before the above model was used in this study, we halved the grid spacing, reduced the time step and the convergence criteria by a factor of 10, and reran the simulation. The solution is essentially the same as before. The tests verified that this model is independent of the grid size, time step, and convergence criteria [37,41,42]. The aerodynamic outputs of this model were also compared with previous static model simulations and experimental measurements [27–30] in a symmetric situation (*Q*=1). It was found that glottal aerodynamics, such as the pressure distribution, Coanda effect, and others, can be correctly predicted by this model. More details can be found in our previous paper [37].

It has been suggested that the strength of nonlinearity in the data can be measured by comparing nonlinear and linear predictability [43], and this concept has been applied to quantify the amount of nonlinearity in the vocalizations of macaque screams, piglet screams, and dog barks [44]. Based on a similar concept, the nonlinear dynamics detection method proposed by Barahona and Poon [38] was used to analyze the nonlinearity of the model output in this study. This nonlinearity detection method is superior when applied to short time series, even when the series is heavily contaminated with noise or in the presence of strong periodicity. As a result, it has been successfully employed to detect chaos in the beat-to-beat interval series of electrocardiograms [4] and in human speech [45]. This method is based on the comparison of the one-step-ahead prediction error of the linear and the nonlinear model. Its procedure is as follows:

For time series *x _{n}* (

$${x}_{n}^{\prime}={a}_{0}+{a}_{1}{x}_{n-1}+{a}_{2}{x}_{n-2}++{a}_{\kappa}{x}_{n-\kappa}+{a}_{\kappa +1}{x}_{n-1}^{2}+{a}_{\kappa +2}{x}_{n-1}{x}_{n-2}++{a}_{M-1}{x}_{n-\kappa}^{d}.$$

(8)

The coefficient *a _{m}* is estimated through a Gram-Schmidt procedure, and the standard deviations of the one-step-ahead prediction error are calculated,

$$\epsilon {(\kappa ,d)}^{2}\frac{{\displaystyle \sum _{n=1}^{N}}{[{x}_{n}^{\prime}(\kappa ,d)-{x}_{n}]}^{2}}{{\displaystyle \sum _{n=1}^{N}}{[{x}_{n}-\overline{x}]}^{2}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\text{with}\phantom{\rule{0.16667em}{0ex}}\overline{x}=\frac{1}{N}\sum _{n=1}^{N}{x}_{n}.$$

(9)

For each data series, the best linear model (*d*=1) and the best nonlinear model (*d*>1) are obtained, respectively, by searching for *κ*^{lin} and *κ*^{nl}, which minimizes

$$C(r)=log\epsilon (r)+r/N,$$

(10)

where *r* [1,*M*] is the number of polynomial terms of the truncated Volterra expansions from a certain pair {*κ*,*d*}. The minimum *C* values corresponding with the linear and nonlinear model are *C*^{lin} and *C*^{nl}, respectively. If the best nonlinear model is significantly more predictive than the best linear model (*C*^{lin}>*C*^{nl}), it indicates that the original series is nonlinear. Otherwise, it may be inferred that the original series is not chaotic or the chaotic component is too weak to be detected.

During phonation, the high subglottal pressure pushes the left and right vocal folds apart, and air escapes from the lungs through the glottis. High-velocity airflow decreases the glottal airflow pressure, which causes the vocal folds to close. The glottal airflow provides the energy for the vocal fold vibration, so it has an important effect on vocal fold vibration and voice production. Figure 2 presents the spatiotemporal pattern of the downstream glottal airflow with *Q*=0.64 and *P _{s}* =1.0 kPa. The velocity value is detected at the line

Figure 3 plots the maxima of *x*_{1}* _{α}* as a function of the asymmetric factor

(Color online) The maxima of *x*_{1}_{α} as a function of the asymmetric factor *Q* obtained from the composite vocal fold model, where the subglottal pressure is 1.0 kPa. (a) The left vocal fold; (b) the right vocal fold. Both vocal folds exhibit rich **...**

(Color online) The waveforms of vocal fold vibration obtained from the composite model. (a) *Q*=0.64 (1:1 mode); (b) *Q*=0.50 (1:2 mode); (c) *Q*=0.45 (2:4 mode); (d) *Q*=0.36 (2:6 mode), where *m:n* mode is defined as the different vibration patterns with the **...**

In the following discussion, the different vibration patterns are labeled with the ratio of the number of the maxima of *x*_{1}* _{l}* and

(Color online) Detection nonlinear dynamics of the maximum series of *x*_{1}_{α} from the composite model with *Q*=0.64. (a), (b) The next-maximum maps of *x*_{1}_{l} and *x*_{1}_{r}. (c), (d) The *C*(*r*) value displayed as a function of the number of polynomial terms. In **...**

At *Q*=0.50, the peak number of the waveform of *x*_{1}* _{r}* is almost twice the peak number of

(Color online) Detection nonlinear dynamics of the maximum series of *x*_{1}_{α} from the composite model with *Q*=0.50. (a), (b) The next-maximum maps of *x*_{1}_{l} and *x*_{1}_{r}. (c), (d) The *C*(*r*) value displayed as a function of the number of polynomial terms. The **...**

More complex dynamic behaviors can also be observed in the composite vocal fold model with the decreasing of asymmetric factor *Q*. At *Q*=0.45, there are two peaks in the wave-form of *x*_{1}* _{l}* and four peaks in the waveform of

(Color online) Detection nonlinear dynamics of the maximum series of *x*_{1}_{α} from the composite model with *Q*=0.45. (a), (b) The next-maximum maps of *x*_{1}_{l} and *x*_{1}_{r}. (c), (d) The *C*(*r*) value displayed as a function of the number of polynomial terms. The **...**

In addition, Steinecke and Herzel found that two different modes could coexist in an asymmetric two-mass model [13]. Moreover, the borderline between the basin attractions of both modes is smooth; therefore, rather weak perturbations may induce abrupt jumps to other regimes. In this study, besides the four typical 1:1, 1:2, 2:4, and 2:6 modes, we also found a transient mode in which the system switches from one mode to another mode. For example, at *Q*=0.56, the system sometimes showed a 1:1 mode, and sometimes showed a 1:2 mode. The transient mode might be the result of jumps between two coexisting modes due to aerodynamic perturbations.

The above simulation based on the asymmetric composite model showed rich dynamic behaviors. In order to investigate the role of glottal airflow in generating these dynamic behaviors, we also simulated the vocal fold vibration using the asymmetric two-mass model, where the glottal aerodynamics is simplified to Bernoulli’s relationship. Figure 8 plots the maxima of *x*_{1}* _{α}* as a function of the asymmetric factor

(Color online) The maxima of *x*_{1}_{α} as a function of the asymmetric factor *Q* obtained from the asymmetric two-mass model, where the subglottal pressure is 1.0 kPa (*P*_{s} =0.01). (a) The left vocal fold; (b) the right vocal fold. When 0.476< **...**

(Color online) The waveforms of vocal fold vibrations obtained from the asymmetric two-mass model. (a) *Q*=0.6 (1:1 mode); (b) *Q*=0.45 (1:2 mode).

In addition, we can see that the waveform predicted by the two-mass model is strictly periodic, which also diverges from the results predicted by the composite model. The strict periodicity can easily be displayed by the next-maxima plots. Figure 10(a) presents the next-maxima plots of the right vocal fold at *Q*=0.45. It can be observed that the trajectories are concentrated in two individual points, but are not separated into two distinct domains. The trajectory alternates between the two points. The vocal folds follow a strict double periodic oscillation. The nonlinear dynamics detection method shows that the nonlinear polynomial and linear polynomial are similarly predictive [see Fig. 10(b)], which con-firms that the double period series from the two-mass model is regular and periodic, rather than chaotic.

(Color online) Nonlinear analysis on the maximum series of *x*_{1}_{r} from the asymmetric two-mass model with *Q* =0.45. (a) The next-maximum maps; (b) the nonlinear detection.

The only difference between the two-mass model and the composite model is in their descriptions of glottal airflow. In the two-mass model, the glottal airflow is assumed to be laminar, nonviscous, and incompressible. The various dynamical behaviors of glottal airflow are ignored. However, in a composite model, the NS equations include a more vivid and accurate description of the airflow than does Bernoulli’s law. The NS equations can predict complex spatiotemporal behaviors of airflow. We believe that the different dynamic behaviors predicted by the two different models are caused by this discrepancy between glottal airflow descriptions. The rich dynamic behaviors of the composite model could be related to the aerodynamic characteristics of glottal airflow. Therefore, the above comparison indicates that the complexity of glottal airflow could be one possible reason for the chaotic component in the voice production system.

Previous voice analyses have shown that the voices produced by mammals and birds have significant nonlinear characteristics. In the past, these nonlinearities in voice were usually explained by vocal fold asymmetry [13], the source-filter acoustic interaction [19], delayed feedback due to reflected sound in the vocal tract [18], turbulence noise disturbance [17], abnormal biomechanical parameters of vocal fold tissue [16], and extremely high subglottal pressure [10,16]. These theories successfully explain the rough voice that is often related to the pathological conditions of the vocal folds.

However, more general nonlinear phenomena have not been thoroughly explained by these previous studies. It is known that although a normal voice is strongly periodic, small undulations in amplitude still exist. Traditionally, this disturbance of the voice amplitude is treated as a random noise and described by the shimmer parameters. Recently, nonlinear dynamic analysis of the speech signal shows that these nonlinear characteristics can be found even in normal voice with strong periodicity [8,44,45]. These findings indicate that the small-amplitude disturbances in a periodically healthy voice may not be random noise, and may instead be a deterministic signal. This concept is still arguable [49] because the strong periodic characteristics of voice signal often obscure these random undulations, thereby decreasing the reliability of the analysis results of nonlinear dynamic methods. Moreover, the mechanism of production of the random undulation in a periodic voice signal is not completely clear.

Titze has successfully explained the low-frequency voice perturbation as the results of the random nature of motor neuron firing [50,51]. It is known that the sustained muscle contractions are made up of a large number of individual twitches of groups of muscle fibers. A single twitch usually lasts only a fraction of a second. The interval between two twitches varies with the neural signals to fire the twitch. The neurological fluctuations are perceived as unsteadiness in pitch and loudness [50,51]. Besides the random neuron firing, this study indicates that the glottal airflow could be another source of voice perturbation. We found that the output of a composite model is strongly periodic. However, its amplitude value is disturbed by a noiselike disturbance; moreover, the amplitude of this noiselike disturbance is much smaller than the amplitude of the vocal fold vibration. For example, at *Q*=0.64, the noiselike disturbance amplitude makes up about 1% and 8% of the peak-to-peak amplitude of the left and right vocal folds, respectively. The nonlinear detection method confirms that the noiselike disturbance is not random, but rather it represents deterministic chaos. This simulation suggests that the vocal fold vibration can be considered a periodic signal modulated by a small-amplitude chaotic series. This result is reflected in a similar phenomenon often seen in voice. Although this phenomenon cannot be predicted by the asymmetric two-mass model, it can be observed in the composite model. Therefore, we can say that the small-amplitude chaotic component present in strong periodic voice production systems might be related to the complexities of glottal aerodynamics.

In addition, we also found that in a two-mass model that ignores the complexity of glottal airflow, only two vibratory modes (1:1 mode and 1:2 mode) can be observed within the asymmetric factor range 0.35≤*Q*≤0.65. However, using a composite model that considers complex glottal airflow, the simulation showed more modes within the same asymmetric factor range, including 1:1 mode, 1:2 mode, 2:4 mode, and 2:6 mode. It is known that the vocal fold tissue and glottal airflow interact with each other during voice production. This suggests that the effect of the airflow force on the vocal fold dynamics is not the corruption of observations by errors that are independent of the dynamics, but a feedback process wherein the system is perturbed by a small random amount at each time step, just like the effect of dynamic noise on a dynamical system [47,48,55]. Previous studies found that the dynamic disturbance could significantly shift the structure of the bifurcation diagram [17,47,48,52–55]. Consequently, this rich dynamic phenomenon in the composite model could also be related to the complex aerodynamics of glottal air-flow.

In a previous study, it was reported that the disrupting effect of noise becomes more significant when the system is close to the bifurcation point corresponding to the onset of sustained oscillations [55]. In this study, our findings based on 1.0 kPa subglottal pressure are obtained well inside the domain of sustained oscillations. When the subglottal pressure is close to the onset value of sustained oscillation, the lower subglottal pressure brings the vocal fold system near to its bifurcation point, which could create a significant chaos-noise effect [55]. On the other hand, the lower subglottal pressure indicates smaller aerodynamic noise and more regular glottal airflow. As a result, it can be expected that sub-glottal pressure is an important parameter that influences the chaotic component of a voice production system, which might be of interest for further study.

In summary, this study found that glottal aerodynamics could greatly contribute to the generation of the nonlinear dynamic behaviors of voice production, such as bifurcation and a small-amplitude chaotic component obscured by strong periodicity. These results might be useful in understanding the dynamic mechanisms of voice production and in improving the quality of speech synthesis.

This study was supported by NIH Grants No. 1-RO1DC006019 and No. 1-RO1DC05522 from the National Institute of Deafness and other Communication Disorders.

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