## 3 Results and Discussion
### Fundamental Frequency and Related Perturbation Parameters  
Other properties of the pathological vocal fold, such as natural frequency, probably have some differences from the normal vocal fold. To determine the effect of differing muscle activation on phonatory frequency and perturbation parameters, we fixed the activity of two muscles at a certain value and varied the value of the other to create plots of frequency and related perturbation parameters versus muscle activity (Figure 7). Figure 7a shows that the CT muscle is most highly correlated with fundamental frequency. Similar to normal people, increasing CT muscle activity will always increase  
Figure 7: Effect of CT, TA, LC activity on fundamental frequency and perturbation parameters. (**a**) Mean fundamental frequency. (**b**) Mean jitter (variation in frequency). (**c**) Mean shimmer (variation in amplitude). \(F_{0}\) in people with vocal fold polyps. However, increasing TA and LC muscle activity only causes a decrease in \(F_{0}\). The deformation of the polyp counteracts the increase in stiffness. Activation of the TA and LC muscle under this condition may only shorten the vocal fold and reduce the stiffness. These examples illustrate the antagonistic roles that the CT and adductor muscles play in phonation onset parameters. As pointed out in previous work, these muscle activities could be interpreted as a requirement for stability in the laryngeal framework. When the vocal folds are being tensed, restraining forces may be needed to counteract gross movement of the larynx. In our pathological model, the control range of the vocal fold on frequency may be wider than that seen in the normal physiological situation. The reason is that we amplified the influence of muscle activation on vocal fold vibration. Our model aims to shed light on the effect of muscle activation on frequency to help find adapted muscle activities for different phonation situations. In fact, it should be noted that the actual muscle activation range of polyp vocal folds is much smaller than that of normal vocal folds. Hence, there will be a decrease in the upper frequency range for actual pathological vocal folds. Variations in frequency (jitter) and amplitude (shimmer) have been commonly used to describe pathological voice quality. For our model, jitter is calculated by the average absolute difference between consecutive periods, divided by the average period, as  
\[Jitter=\frac{\frac{1}{N-1}\sum_{i=1}^{N-1}\left|T_{i}-T_{i-1}\right|}{\frac{1 }{N}\sum_{i=1}^{N}T_{i}} \tag{25}\]  
where \(T_{i}\) is the extracted period length, and \(N\) is the number of periods [26]. The mean value of jitter during phonation is depicted in Figure 7b. Overall, there was not a strong correlation with activity of any muscle. For large values of \(a_{\text{CT}}\), the jitter was high, reflective of aperiodic vibration. For small values of \(a_{\text{TA}}\), a decrease in \(a_{\text{TA}}\) will generally cause an increase in jitter. Shimmer is the cycle-to-cycle variation in amplitude. Similar to jitter, shimmer is measured as the average absolute difference between the amplitudes of consecutive periods, divided by the average amplitude:  
\[Shimmer=\frac{\frac{1}{N-1}\sum_{i=1}^{N-1}\left|A_{i}-A_{i-1}\right|}{\frac{ 1}{N}\sum_{i=1}^{N}A_{i}} \tag{26}\]  
where \(A_{i}\) is the extracted amplitude and \(N\) is the number of extracted periods [26]. Figure 7c shows the mean value of shimmer on phonation. Shimmer seems to have a strong link with CT and LC activity. The waveform of \(U_{\text{A}}\) is similar to the waveform for shimmer. In general, increasing \(a_{\text{CT}}\) will cause an initial increase in shimmer, followed by a decrease. Shimmer is high when \(a_{\text{CT}}\) ranges from 0.3 to 0.6. When \(a_{\text{CT}}\) is between 0.3 and 0.4, shimmer is at its maximum. For LC muscle, shimmer increases monotonically when \(a_{\text{LC}}<0.5\). In particular, for \(a_{\text{LC}}<0.2\), the value of shimmer is much higher than at other activity levels.