Birds make a variety of vocalizations and calls, all of which serve different purposes. Some sounds warn of impending danger, others act as commands, and several even keep track of juvenile family members. In contrast, birds use “songs” almost exclusively for mating; this is especially true of species that breed in areas with low visibility—like dense shrubbery or open meadows with tall grass—where sound is the most reliable signal type. During a minisymposium presentation at the 2019 SIAM Conference on Applications of Dynamical Systems, currently taking place in Snowbird, Utah, Eve Armstrong of the University of Pennsylvania presented a geometric dynamical systems tool for analyzing birdsong in cowbirds.
The cowbird song is comprised of three short introductory notes followed by a final high-frequency whistle, collectively lasting slightly just over one second. The timbre of a male’s song is crucial to attracting a mate. “Female songbirds exhibit distinct preferences for songs of particular males,” Armstrong said. This is also true of females raised in isolation, as they independently rank a set of songs in identical order. Female cowbirds display their proclivities fairly overtly by posturing; when they hear a song they like, they prepare to mate via a copulation solicitation display (CSD). However, these preferences dissolve with the presence of lesions in the female sound circuit, specifically the HVC — a nucleus in songbirds’ brains that contributes to the learning and production of birdsong.
“To date, all acoustic analysis tools have failed to uncover any metric for this female song preference,” Armstrong said. This is unsurprising, since vocal production is a complex nonlinear phenomenon but most tools rely on linear spectral analysis and prematurely linearize the signal. To combat this dearth of effective methods, Armstrong proposes a dynamical systems “recipe” for analyzing songs that involves embedding a scalar time series of acoustic wave pressure into an attractor trajectory in reconstructed phase space. The trajectory is trained to reproduce an authentic birdsong, thus yielding a synthesized replication song. She assumes that a nonlinear dynamical system creates the acoustic time pressure series, which one can utilize to reconstruct the geometric structure of these unknown dynamics in a new phase space. Armstrong then uses the resulting orbit in space as training data to resynthesize the birdsongs. “The orbit remains stable for several orbits, then noisily transitions to another one in a different plane with a different shape before ending up in a third orbit, also with a slightly different tilting,” she said. “The idea here is not to linearize anything, which is why we’re starting with the pressure time series.”
To determine how best to employ the orbit as a tool for song reconstruction and manipulation, Armstrong decided on the following three-part plan:
- Write an empirical model treating the time-delayed vectors as variables
- Train an optimization procedure to learn the model parameters required to fit the output to the song
- Play the synthesized songs to both lesioned and non-lesioned female cowbirds.
She first created an empirical model wherein each variable is written in a Taylor series expansion taken to the second order. After estimating the parameters, Armstrong found that her model was able to generate the three short notes of the catbird song more accurately than the final whistle, the replication of which sounds a bit grainier than it should (see Figure 1).
Figure 1. A comparison of the true song versus the synthesized product. The three introductory notes are nearly identical, though slight variation occurs in the final whistle.
With the model in place, Armstrong wanted to see if the female cowbirds responded at all to the synthesized signals during this year’s mating season (cowbird mating season begins in late April and typically lasts about six weeks). Thus far, her model has yielded the following results in ascertaining a classification scheme for song synthesis:
- Convergence on a solution for the parameters requires at least two time delays
- Female cowbirds are responding to songs created with polynomial degree 2
- Removing first-degree terms does not appear to impact the birds’ responses.
However, removing second-degree terms significantly degrades song timbre, which causes a substantial decrease in female response. Armstrong presented her findings for three different birds (“green,” “yellow,” and “red”) in a chart that included results for 10 different songs, including three negative controls (see Figure 2). “The yellow and red birds have more nuanced response,” Armstrong said. “They can recognize the best and worst true song and they have some kind of opinion about the synthesis.” She ultimately concludes that a three-dimensional model of degree 2 captures something—but not everything—of significance to the female cowbirds.
Figure 2. A detailed comparison of three birds' reactions to 10 different songs (three of which were negative controls).
In the future, Armstrong hopes to officially identify a metric that scales with female cowbirds’ CSD ratings of songs. She also seeks to study evolution of the “strategy” of one male cowbird over the course of a single mating season; further investigate the disparity between the dynamics of the birdsong’s three underlying notes versus the final whistle; and determine whether HVC-lesioned birds and non-lesioned control birds will respond differently. Additional plans include testing female responses to synthesized songs created by higher-dimensional phase spaces and/or higher-degree polynomial models. “We’d like to create a whole menu to play back to the birds,” Armstrong said.
Armstrong intends to eventually establish a detailed mapping between her geometric framework and an existing dynamical model of the syrinx. If this mapping is indeed possible, it could represent a step towards a general formula for the use of phase space reconstruction to infer an underlying dynamical system.
||Lina Sorg is the associate editor of SIAM News.