Biomedical Engineering Reference
In-Depth Information
the brown thrasher, these muscles rotates bronchial cartilages towards the
syringeal lumen [Goller and Suthers 1996a].
The role of the muscles used to actively close the syrinx by adducting labia
can be taken into account by an additional term
f
0
in our simple model, (4.5).
With this additional term, the model reads
x
=
y,
y
=
cx
2
y
−
kx
+(
β
−
b
)
y
−
−
f
0
.
(5.10)
The association of the parameter
f
0
with the activity of the muscles that
actively adduct the labia can be seen by considering the equilibrium points
of the system, also known as
fixed points
. The fixed points
x
of a system are
found by setting all time derivatives to zero, that is, in our system,
f
k
x
=
−
.
(5.11)
By varying the value of
f
0
, we can move the fixed point either towards the
position representing the labia being pushed against each other or towards
the position representing the labia being pushed against the walls. At these
positions, oscillations are prevented.
This gesture can eventually completely adduct the labia. However, the
muscle used here is briefly active on the side used to vocalize, prior to the
emission of sound. Owing to this additional gesture, the pressure at which
the labial oscillations effectively start is larger than that necessary if only
the mechanisms described in the previous section are active. In this way, the
bird delays somewhat the beginning of the oscillations. It also happens for
this species that the end of the labial oscillations is anticipated by the active
closing of the syringeal lumen through the dorsal muscles. It is worth pointing
out that the value of the force necessary to “hold” the oscillations depends
on the value of the restitution coe
cient of the tissue (given by the activity
of the siringealis ventralis muscle). In our simple model, the conditions under
which no oscillations take place can be written as
>k
(
β
−
b
)
|
f
0
|
.
(5.12)
c
That is, if the labia are vibrating at a higher frequency (higher
k
), a larger
force from the dorsal muscles is needed in order to prevent the oscillations.
This is a precise prediction of the physical model described so far, and has
been validated by experimental measurements [Mindlin et al. 2003].
We are now going to work with these physical models in order to recon-
struct the dynamical character of the control parameters needed to repro-
duce artificially the syllables of a song. The result of this exercise is displayed
in Fig. 5.7, in which the natural and artificial songs of a chingolo sparrow
(
Zonotrichia capensis
) are displayed. In order to build the artificial song, it
