Mathematical Modelling of Sound Production in Birds

In this thesis, the physics of birds phonation is discussed using a two-mass models approach and the theory of nonlinear dynamics. Two-mass models of the human larynx (rescaled two-mass model and trapezoidal model) have been adapted to the dimension of the avian syrinx to study pressure onset, control of harmonic overtones and “registers” of the sound radiated by the birds vocal organ (syrinx) in the absence of source-tract coupling. Our simulations are a first step towards more realistic modelling of the syrinx. A detailed bifurcation analysis of the trapezoidal model confirms that the geometry and the rest position of the syrinx can influence the harmonic spectra drastically, suggests possible mechanisms involved in the production of rich-harmonic spectra during inspiration and is used to describe quantitatively the contribution of syringeal muscles. The latter is implemented in the model by means of driving time-dependent parameters controlling the labia rest position and frequency modulation. The main focus of the present work is on: • mathematical modelling of the avian syrinx to study the effects of small dimensions on the pressure onset and the effects of the syrinx geometry and rest position on the harmonic spectra of the sound produced at the vibrating source. • bifurcation analysis of the trapezoidal model. Results are employed in the implementation of syringeal muscles by means of time-dependent parameters and contribute to explore possible mechanisms involved in the production of more rich harmonic spectra, in particular regarding the inspiratory part of the vocalization of a ring dove (coo). • model implementation of time-dependent parameters to study quantitatively the contribution of the syringeal muscles on the labia rest areas and control of the fundamental frequency. In Chapter 3, we analyze two symmetric two-mass models of the avian syrinx. Our first model (rescaled two-mass model) applies to songbirds and is a rescaled version of the well-known human two-mass model. Our second model (trapezoidal model) introduces a smoother geometry and is used to simulate the ring dove (Streptopelia risoria) syrinx. Simulations show that both models exhibit self-sustained vibrations and that the intensity of harmonics depends strongly on the configuration of the syrinx. The rescaled two-mass model does not present instabilities. The trapezoidal model, however, displays coexisting limit-cycles that represent vibrations with, and without collisions at the same pressure. Register-like transitions are accompanied by subharmonics and deterministic chaos. In the rescaled two-mass model almost pure tones are found near the onset of vibrations and strong harmonics appear at higher pressures due to collisions. For a small upper mass and a rectangular geometry, collisions leading to strong harmonics can be avoided only near the phonation onset. At higher pressures counteracting forces would be required to diminish collisions. We hypothesize that the avoidance of strong collisions in song birds might be achieved by the medial tympaniform membranes (MTM) that are continuous with the inner vibrating labia. In our model of the ring dove syrinx no collisions occur at default parameters. Consequently, harmonics are fairly weak. The smoother configuration and equal upper and lower masses counteract collisions even at relatively high pressures. This is presumably due to a stronger effect of the subsyringeal pressure acting on both masses. Our simulations reveal that the configuration of the syrinx influences the intensity of overtones. Therefore, the amount of energy in the harmonics could also be controlled by syringeal muscles that directly affect the configuration of the syrinx. In Chapter 4 we aim to study the contribution of different control parameters in the coo of the ring dove at the syrinx level. The biomechanics of the syrinx, in fact, is very complex and not well understood. The neuromuscular control of vocalisation in birds requires a complicated and precisely coordinated motor control of the vocal organ (i.e. the syrinx), the respiratory system and upper vocal tract. We designed and implemented a quantitative biomechanical syrinx model, i.e., a slightly modified version of the trapezoidal model that is driven by physiological control parameters and includes a muscle model. Our simple nonlinear model reproduces the coo, including the inspiratory note, with remarkable accuracy and suggests once more that harmonic content of song can be controlled by the geometry and rest position of the syrinx. Furthermore, by systematically switching off control parameters, we demonstrate how they affect amplitude and frequency modulation and we generate new experimentally testable hypotheses. Independent control of amplitude and frequency is not possible with the simple syringeal morphology of the ring dove. We speculate that songbirds evolved a syrinx design that uncouples the control of different sound parameters and allows for independent control. This evolutionary key innovation provides an additional explanation for the rapid