Biomedical Engineering Reference
In-Depth Information
0.45
0.9
(a)
(b)
0.4
0.8
0.35
0.7
0.3
0.6
0.25
0.5
0.2
0.4
0.15
0.3
0.1
0.2
0.05
0.1
0
5
10
15
20
25
30
0
2
4
6
8
10
12
14
16
18
Male note number
Female note number
Fig. 9.2.
Average time interval between notes as a function of note position in
a typical hornero duet. (
a
) Males show a clear tendency to increase the note pro-
duction rate as the duet develops. (
b
) Females, on the other hand, may increase,
decrease or nonmonotonically vary the note production rate as the duet develops.
Each trace was shifted vertically for clarity
This structure is very closely related to the one we described at the begin-
ning of the chapter. The organization of the rotation numbers as a function
of the forcing period is known as the “devil's staircase”, a complex structure
where the steps are rational numbers, as displayed in Fig. 9.3b. The steps in
the devil's staircase show universal features, regardless of the details of the
forced system (which can be either a nonlinear oscillator or an excitable sys-
tem) and the nature of the periodic forcing (a harmonic function, a sequence
of impulses, etc.).
It is possible to interpret the existence of a staircase in the duets as a
consequence of the male's notes acting as a unidirectional forcing on the
female's motor control through an auditory pathway. Since the notes sung by
the female manage to lock properly, both the respiratory and the syringeal
gestures would constitute a nonlinear system such that when it was forced it
would show a devil's staircase (it would be an excitable system or a relaxation
oscillator), suggesting that at some level the corresponding motor neurons are
closely connected.
9.2.3 Test Duets
In order to check the plausibility of the proposed mechanism, we can generate
synthetic duets from a theoretical, physical model of the syrinx such as the
one proposed in Chap. 4, reading
b
)
x
+
k
(
t
)
x
+
cx
2
x
=0
,
x
−
(
p
(
t
)
−
(9.2)
where
p
(
t
)
b
stands for the difference between the bronchial pressure and the
linear dissipation,
k
(
t
) for the stiffness of the syringeal labia and
c
stands for
−
