Biomedical Engineering Reference
In-Depth Information
increasing from around 5 Hz up to 20 Hz. We call
p
(
t
)and
k
(
t
) the respiratory
and syringeal gestures, respectively.
Analogously, we can generate female notes by driving (9.2) with respira-
tory and syringeal gestures
p
(
t
)and
k
(
t
), generated this time by a nonlinear
oscillator forced, in turn, by the male gestures. We are not making any as-
sumptions about the biological substrate of these nonlinear oscillators. We
do know that the cyclic gestures generating the syllables of the females have
to emerge out of some circuitry which involves nonlinear units. We conjec-
ture that this circuitry has to be affected by the auditory signals from the
duetting partner. Before attempting to build a biological model for this cir-
cuitry, we shall assume that the cyclic gestures are generated by the simplest
conceivable nonlinear oscillator, forced by a temporal function of increasing
frequency. The paradigmatic model chosen to emulate the cyclic instructions
driving the syrinx of the female was
u
= 75(
u − u
3
+
A
cos(
ω
(
t
)
t
)+0
.
5
− v
)
,
(9.3)
v
=6(
u
+0
.
7
−
0
.
8
v
)
,
(9.4)
with
ω
(
t
) varying from around 5 Hz up to 20 Hz. The cyclic instructions used
to drive the syrinx of the female were taken as
p
(
t
)=
u
v
and
k
(
t
)=
u
+
v
,in
order to generate realistic-looking syllables. The synthetic duet is displayed
in Fig. 9.1b.
OK, but what if the staircase in Fig. 9.3a (the actual duets) was just
coincidence? In order to check that it was not an effect of a simple mis-
match between frequencies changing independently, surrogate random duets
were analyzed. Surrogate duets were assembled by first taking two randomly
chosen duets, and then eliminating the male notes from the first duet and
the female notes from the second duet. Surrogate duets were then subjected
to the same analysis as performed on the original duets. Results are shown
in Fig. 9.3c, where the staircase structure is lost. For a quantitative mea-
sure of the staircase structure, we defined
φ
as the average overlap between
steps, using the steps of 1
/
2, 1
/
3and1
/
4 (the only steps for which we know
their length approximately know their length). The ratio of the length of the
overlap to the average step length was computed between the 1
/
2and1
/
3
steps, and the same was done between the 1
/
3and1
/
4 steps. The two ratios
were then averaged to give
φ
. In the case of the surrogate duets (Fig. 9.3c),
the average overlap
φ
is almost 70%, while in the case of the original duets
(Fig. 9.3a)
φ
is less than 20%.
The moral of this tale is that behind the appealing rhythm of the hornero
duets, there is more of a complex nonlinear circuitry than “musical talent”
(unless one wishes to explore the possibility of musical talent being related
to this mechanism!). In any case, just as the acoustic features of the notes
(as shown in the spectra) were in some cases affected by nonlinear effects (on
the timescale of thousands of Hz), nonlinearities can also play an important
role on the timescale of the syllables.
−
