Figure 1

(Color online) Bifurcation diagram for (a) the physical and (b) the normal form models. For the physical model, the diagram is displayed in the air sac pressure and the tension parameter space $(p_s,k_1)$ and for the normal form model in the $(\alpha ,\beta )$ parameter space. The existence of neighboring regions as 1 and 2, and regions 5 and 2 is the mathematical condition for a model to be capable of displaying both tonal and spectrally rich sounds with well-defined fundamental frequencies. Typical portraits of the $\left(x,y\right)$ phase spaces for each region are shown in the corresponding lower insets. In both upper panels, continuous thick line represents the Hopf bifurcation, dashed line the homoclinic bifurcation, and dotted blue lines the saddle-node bifurcations. Paths A and H represent the different ways to start oscillation of the labia by crossing the Hopf (H) and the SNILC (A) bifurcation lines, thus generating respectably either tonal or spectrally rich sounds.Reuse & Permissions

Figure 2

(Upper panel) Natural song. (Middle panel) Synthetic song generated by the computational model with ad hoc parameters. (Bottom panel) Physiologically driven electronic syrinx implemented in a DSP. $L=1.9\text{cm}$, $r=-0.9$. Arrows indicate the segments used to fit.Reuse & Permissions

Figure 3

Simultaneous experimental recordings while the bird was singing of (a) sound pressure, (b) air sac pressure, (c) right $vS$ EMG, and (d) left $vS$ EMG.Reuse & Permissions

Figure 4

(Color online) Extraction of the driving parameters of the model from the smoothed EMG activities of the $vS$ muscles. Rectification of the EMG signal and smoothing by low-pass filtering of the (a) right and (c) left $vS$ activities. In both panels, phonation time and $\alpha ={\alpha }_{\text{on}}$ are indicated with a black line; also no-phonation and $\alpha ={\alpha }_{\text{off}}$ are indicated with a green (light gray) line. The driving parameter ${\beta }_i^{\text{proposed}}$ is shown at times where $\alpha ={\alpha }_{\text{on}}$ in panels (b) and (d).Reuse & Permissions

Figure 5

(Color online) Activity as a function of the pitch, $f_0$. In the left panel (a), the pitch of the low-frequency syllables $\left(f_0<1.8\text{kHz}\right)$ was plotted as a function of the left-side ventral muscle activity. In (b), the pitch of those syllables was plotted as a function of the right ventral activity. In both panels, the pitch of the high-frequency syllables was plotted as a function of the right ventral activity.Reuse & Permissions