The results of the experiments provide evidence for various parameters of the EGG waveform which are suitable for decribing laryngeal activity on the linguistic and paralinguistic layers of human communication. In the discussions of the experiments some physiological correlates and interdependencies between the EGG signal parameters and glottal activity were suggested, but were not proved. Although a direct validation of the experimental results is not possible (or at least complicated even for clear cases) or ethically inappropiate, verification can be achieved using a computer model of vocal fold vibration. In agreement with the spirit of the research institution at which this study has been carried out (Institut für Maschinelle Sprachverarbeitung), computer simulation was established as the main method for the scientific verification of our hypotheses.

The model of vocal fold vibration is used here to provide the representation of the contact area of the folds. The foundations of the simulation are:

- the two-mass model of the vocal folds (Ishizaka & Flanagan, 1972)

- the contact area and the electroglottogram model as proposed by Childers et al. (1986)

The relations to the body-cover model of the vocal folds (Titze, 1994; Story & Titze, 1995) and other models of the contact area (Titze, 1989) are also discussed.

23. Introduction

The oscillations of the vocal folds are caused by the acting together of the following types of forces: the aerodynamic forces related to airflow, the mechanical forces related to the tension of the muscular structures and the damping forces. The function of the larynx during normal phonation is usually modelled as shown in Fig.39.

Figure 39. The functional model of the vocal folds including: a model of the airflow though the glottis, a mechanical model of the vocal folds and the input impedance of the vocal tract (see Gubrynowicz, 1997:3)

In the simplest case, each fold is considered to be a simple harmonic oscillator which moves, when the aerodynamic force Fm acts in phase with movement velocity. This force depends on the glottal airflow which characterized by pressure, volume flow velocity and glottal area. Newton's law of motion for the oscillator has the following form (Ishizaka & Flanagan, 1972):

              (20)

where m denotes the mass of the fold, r is its stiffness and s(x) is the viscous damping of the fold (Titze, 1994:91). Displacement and velocity are denoted as x and x' respectively. F is the deflecting force and t denotes the time.

The stiffness and damping parameters depend on the vocal folds' displacement, and the properties of the tissue and will be defined later. As follows from Fig. 39, the aerodynamic force Fm depends on the air pressure P and acts perpendicularly to the folds' tissue surface. If for example, the air pressure P in the glottis grows in the supraglottal direction and the vocal folds are moving apart, then Fm will act in the same direction the tissue is moving. Energy will be transferred to the tissue by the air. This energy overcomes the viscous damping of the tissue, the folds are pushed apart and the glottis opens. Based on Bernoulli's energy law (section 23.1) the air pressure acting on the medial surfaces of the folds can be approximated as (Titze, 1994:92)

            (21)

where a1 is the cross-sectional area of the glottis at the glottal input (subglottal), a2 is the cross-sectional area at the glottal output (supraglottal). Ps and P1 denote the sub- and supraglottal pressures. This means that the aerodynamic force depends on the geometrical shape of the glottis and on the difference in pressure below and above the glottis (transglottal pressure). As follows from (21), the driving force depends mostly on the inlet pressure of the vocal tract. The vocal fold movements are related to the vocal tract air column and its configuration as depicted in Fig.39. The supraglottal pressure depends on the inertia of the air column as well as the acceleration of the air flow:

             (22)

where r is the air density, L is the length of the vocal tract, a is the cross-sectional area of the air column (vocal tract) and U denotes the volume velocity of the airflow. This explains the suction of the vocal folds - as the glottis closes, the inertia of the air column causes negative pressure above the folds, boosting the closing of the glottis. An important aspect of this mechanism is the delay of the rise and fall of the supraglottal pressure with respect to the opening and closing of the glottis. This effect can be modelled by the simple one-mass model of the vocal folds, which was introduced by Flanagan and Landgraf (1968). A computer model of equation (22) was designed using the electrical analogy of the acousto-mechanical model. In correspondence to this analogy, an equivalent electrical circuit was built, where the driving pressure is represented by the voltage source and the glottis is made up of resistant and inductive elements representing the damping and the stiffness of the tissue as well as the inertance (inertia of the air column). The current flow in the circuit is equivalent to the air volume velocity.

In the one-mass model only lateral movements of the oscillating mass were permitted. The glottal area and volume velocity waveforms which were generated by the model, were similar to those observed in natural human voicing. Nevertheless, some effects were not properly modelled. Repeated observation of vocal folds movement (see section 5) has demonstrated that tissue displacement is rather non-uniform and that the upper and lower portions of the vocal folds move out of phase. This fact cannot be properly represented by the one-mass model in which the folds move like a solid bar. Hence, a more complex model, a two mass model of vocal fold vibration had to be constructed and tested.

Depending on the initial configuration of the glottis, the properties of the tissue and the air pressure, the upper and lower edges of the vocal folds move in a non-uniform manner. Typically, the bottom edge leads the movement, reaching an appropriate position (in the direction of the overall movement) before the upper edge does. This causes a change in the glottis shape during each cycle of the movement, making it convergent during opening and divergent during closing. It was observed (Scherer & Titze, 1983) that this creates a rise in the driving pressure and its asymmetry, which allows a more effective use of the aerodynamic force. The alternation of the convergent and divergent glottis configuration can lead to self-sustained oscillation of the vocal folds (Ishizaka & Matsuidara, 1972; Titze, 1988). This effect can be modelled by a two-mass model.

A schematic representation of the two-mass approximation of the vocal folds is presented in Fig. 40.

Figure 40. The two-mass approximation of the vocal folds in longitudinal and superior views (from Ishizaka & Flanagan, 1972:1236).

The following approximations are assumed:

• the folds are symmetrical (both folds have the same properties)
• the trachea, the lungs, as well as vocal tract inlet are of constant size and are represented by cylindrical tubes
• the glottis is represented by the constriction between these tubes, which is also cylindrical. Its size depends on the folds' displacement
• the subglottal contraction occurs over the distance lc
• the supraglottal expansion occurs over the distance le
• the vocal fold comprises an upper and lower part, each of them represented by a simple mechanical oscillator described by mass, stiffness (spring) and damping (m, s, r)
• both parts of the fold may only move laterally, the lateral displacements of both masses are denoted by x1, x2, respectively
• the masses are coupled by a linear spring of the stiffness kc
• the volume velocity of the airflow across the glottis Ug does not change
• the cross-sectional areas of the glottis are related to the initial, neutral positions of the folds (A g01, Ag02 describe the initial cross-sectional area at the height of the first and second mass) and to the displacements of both masses: