⚙️ Physical Model

Important

This documentation is under construction.

Detailed description of the physical model and numerical solution of the pacakge.

Syrinx

Modeled as a nonlinear ha

(1)\[\frac{dx^2}{dt^2} = \gamma^2[-\alpha (t)-\beta (t) x + x^2 - x^3] - \gamma (1+ x )x\frac{dx}{dy}\]

(2)\[\begin{split}\begin{align}\label{syrinx_edos} & \frac{dx}{dt} = y\nonumber\\ & \frac{dy}{dt} = \gamma^2[-\alpha (t)-\beta (t) x + x^2 - x^3] - \gamma (1+ x )xy \end{align}\end{split}\]

Here, is the air-sac pressure and is the labial tension.

[1, 2, 3, 4, 5]

Trache

Modeled as a tube with one end open, the one coneecting with the syrinx, and the other closed, the one connected to the OEC.

(3)\[\begin{split}\begin{gather}\label{trache_edos} p_i (t) = A y(t) + p_{back}\left( t - \frac{L}{c} \right) \\ p_{back} (t) = -r p_i\left( t - \frac{L}{c} \right)\\ p_{out} (t) = (1-r)p_i\left( t - \frac{L}{c} \right)\\ \frac{d p_{out}}{dt} (t) = \frac{p_{out}(t)- p_{out}(t-dt)}{dt} \end{gather}\end{split}\]

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Oropharingeal- Esophageal Cavity (OEC)

Modeled as a Helmholtz resonator.

(4)\[\begin{split}\begin{align}\label{OEC_edos} \frac{d}{dt} p_g &= p_g'= dp_g \nonumber\\ \frac{d}{dt} dp_g &= dp_g'= -\frac{1}{C_{OEC} M_g}p_g - R_{OEC} \left( \frac{1}{M_b} + \frac{1}{M_g}\right) dp_g \nonumber\\ & \qquad \qquad + \frac{1}{M_g}\left( \frac{1}{C_{OEC} } + \frac{R_{OEC} R_b}{ M_b}\right) p_b + \frac{1}{M_g}\frac{dp_{out} }{dt} + \frac{R_{OEC}}{M_g M_b } p_{out} \nonumber\\ \frac{d}{dt} p_b &= p_b' = - \frac{M_g}{M_b} p_g' - \frac{R_b}{M_b} p_b + \frac{1}{M_b} p_{out} %\\ & \color{myblue} \frac{d\vec{p}}{dt} = \vec{h}(\vec{p}, p_{out}, p_{out}'), \quad \vec{p} = (p_g, p_g', p_b) \end{align}\end{split}\]

[6, 7]

Minimization Problem

(5)\[\begin{split}\begin{equation}\label{opt_general} \begin{aligned} \underset{ \gamma \in \mathbb{R},\; \alpha,\beta\in \mathbb{R}^n}{\text{min}} &\qquad ||\hat{SCI}_{real} - \hat{SCI}_{synt} ( \gamma,\alpha,\beta)||_2 + || (\hat{FF}_{real} - \hat{FF}_{synt}(\gamma,\alpha,\beta)||_2 \\ & \qquad \qquad - corr(FC_{real},FC_{synt}(\gamma, \alpha, \beta)) \\ \text { subject to } & \qquad \gamma \in \Omega_\gamma, \quad \beta \in \Omega_\beta , \quad \alpha \in \Omega_\alpha \end{aligned} \end{equation}\end{split}\]

Numerial Solution

References

[1]

Rodrigo Laje, Timothy J. Gardner, and Gabriel B. Mindlin. Neuromuscular control of vocalizations in birdsong: a model. Phys. Rev. E, 65:051921, May 2002. URL: https://link.aps.org/doi/10.1103/PhysRevE.65.051921, doi:10.1103/PhysRevE.65.051921.

[2]

Franz Goller and Roderick A. Suthers. Role of syringeal muscles in controlling the phonology of bird song. https://doi.org/10.1152/jn.1996.76.1.287, 76:287–300, 1996. URL: https://journals.physiology.org/doi/10.1152/jn.1996.76.1.287, doi:10.1152/JN.1996.76.1.287.

[3]

Ole Næsbye Larsen and Franz Goller. Direct observation of syringeal muscle function in songbirds and a parrot. Journal of Experimental Biology, 205:25–35, 1 2002. URL: https://journals.biologists.com/jeb/article/205/1/25/33066/Direct-observation-of-syringeal-muscle-function-in, doi:10.1242/JEB.205.1.25.

[4]

Gabriel B. Mindlin & Daniel Margoliash Ana Amador, Yonatan Sanz Perl. Elemental gesture dynamics are encoded by song premotor cortical neurons. Nature, 495(7439):59–64, Mar 2013. URL: https://doi.org/10.1038/nature11967, doi:10.1038/nature11967.

[5]

Tim Gardner, G. Cecchi, M. Magnasco, R. Laje, and Gabriel B. Mindlin. Simple motor gestures for birdsongs. Physical Review Letters, 87:208101, 10 2001. URL: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.87.208101, doi:10.1103/PhysRevLett.87.208101.

[6]

Yonatan Sanz Perl, Ezequiel M. Arneodo, Ana Amador, Franz Goller, and Gabriel B. Mindlin. Reconstruction of physiological instructions from zebra finch song. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 84:051909, 11 2011. URL: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.84.051909, doi:10.1103/PHYSREVE.84.051909/FIGURES/8/MEDIUM.

[7]

Neville H. Fletcher, Tobias Riede, and Roderick A. Suthers. Model for vocalization by a bird with distensible vocal cavity and open beak. The Journal of the Acoustical Society of America, 119:1005, 1 2006. URL: https://asa.scitation.org/doi/abs/10.1121/1.2159434, doi:10.1121/1.2159434.