# Voice Simulation: The Next Generation
## 3 Results
### Low-Dimensional Tissue Modeling  
Motion of the vocal folds may be considered as a composition of two principal motions: an oscillation in the lateral direction (perpendicular to the airflow), which opens and closes the glottis, plus a surface wave that propagates in the direction of the airflow [6; 7]. The first component may be described by a second order lumped representation of a mechanical oscillator:  
\[m\ddot{x}+h(x)\dot{x}+kx=p_{g}(x,\dot{x}) \tag{8}\]  
where \(m\) is the vocal fold mass per unit area of the medial surface, \(k\) is its stiffness, \(x\) is the tissue displacement from its rest position, \(h(x)\) is a positive function that accounts for all energy dissipation effects due to tissue deformation, collision between the opposite vocal folds and other factors, and \(p_{g}(x,\dot{x})\) is the glottal pressure. The surface-wave component may be added through a simple kinematic description in d'Alembert's form  
\[\xi(y,t)=x\Big{(}t-\frac{y}{c}\Big{)} \tag{9}\]  
where \(\xi\) is the lateral displacement of the tissues at any position \(y\) along the glottal channel, measured from its midpoint, and \(c\) is the surface wave speed [7; 8]. The Van der Pol model is obtained by combining the above equations, assuming that the approximation \(x(t-y/c)\approx x(t)-(y/c)\dot{x}(t)\) holds, and that the supraglottal pressure \(p_{e}\) is negligible. Then, we obtain  
\[m\ddot{x}+h(x)\dot{x}+kx=\frac{p_{s}}{v_{\text{char}}}\dot{x} \tag{10}\]  
where \(v_{\text{char}}\) is a characteristic glottal coefficient [9]. Defining further a quadratic energy dissipation function \(h(x)=r+sx^{2}\), and using a suitable transformation of variables, a van der Pol's type equation is obtained [10]:  
\[\ddot{x}-\beta\Big{(}\lambda-x^{2}\Big{)}\dot{x}+x=0 \tag{11}\]  
While this equation does not include source-filter interaction, it allows for the application of an extensive theory already available on van der Pol oscillators, e.g., [11] that partially applies to phonation. In fact, this well-known equation has been applied to characterize a number of systems in case of human physiology, from the heartbeat [12] to Parkinsonian tremor [13], and also to the vocalization of songbirds, e.g., [14]. For instance, the theory tells us that a Hopf bifurcation occurs at \(\lambda=0\), which implies \(p_{s}=rv_{\text{char}}\). This value of \(p_{s}\) is known as the phonation threshold pressure and it represents a measure of the effort required to produce voice [15]. Depending on the shape of the dissipation function \(h(x)\), the bifurcation may assume the subcritical form and cause an oscillation hysteresis phenomenon [16] with different threshold values for oscillation onset vs. offset. However, the assumption that the supraglottal pressure is negligible does not hold if the larynx canal (the epilarynx tube) is narrow and the mucosal wave velocity is large [7]. Then, self-sustained vocal fold oscillation is also dependent on the airway configuration, also known as source-filter interaction. Figure 2 shows a bifurcation diagram for voice onset and voice offset based on van der Pol's equation. The predicted hysteresis is an important finding in vocal fold mechanics. It guides further exploration with higher dimensional models that include the fluid transport and wave propagation in the entire airway. Asymmetries between the right and left vocal folds may be simulated by considering a pair of coupled van der Pol oscillator of the form in Equation (10), e.g.,  
\[ \begin{split}\ddot{x}-\beta\big{(}\lambda-x^{2}\big{)}\dot{x}+(1- \Delta)x=\alpha\left(\dot{x}^{\prime}-\dot{x}\right)\\ \ddot{x}^{\prime}-\beta\Big{(}\lambda-x^{\prime 2}\big{)}\dot{x}^{ \prime}+(1+\Delta)x^{\prime}=\alpha\left(\dot{x}-\dot{x}^{\prime}\right)\end{split}  \tag{12}\]  
where \(x,x^{\prime}\) are the displacements of the right and left vocal folds, respectively, \(-1<\Delta<1\) is a stiffness asymmetry term, and \(\alpha\) is an aerodynamic coupling coefficient [18]. Figure 3 shows a bifurcation diagram.