Biomedical Engineering Reference
In-Depth Information
a factor of 10 V. The voltage
V
b
is just
V
a
. By means of another multiplier,
we construct
V
c
=
v
2
/
(10 V), which is in turn multiplied by
w
in order to
generate the function
V
d
=
wv
2
/
(100 V
2
).
In the third layer of the circuit, the external function
β
(
t
)
−
b
emulating the
cyclic changes in bronchial pressure above the linear dissipation is multiplied
by the voltage
w
, generating the voltage
V
e
=
w
(
β
(
t
)
−
−
b
)
/
(10 V), which,
after an inverting amplifier, gives rise to
V
f
=
b
)
/
(10 V). Finally,
the voltages
V
b
,
V
d
and
V
f
are fed into the integrator at the top left through
R
1
,
R
2
and
R
3
, respectively. Therefore,
−
w
(
β
(
t
)
−
dt .
wv
2
R
2
×
1
C
v
(
t
)
R
1
×
w
(
β
(
t
)
−
b
)
w
=
−
−
10 V
+
100 V
2
−
(7.23)
R
3
×
10 V
As a result, calling
V
x
=
−v
and
V
y
=
w/
(
RC
), we obtain
V
x
=
V
y
,
(7.24)
(
t
)
RR
1
C
2
×
1
β
(
t
)
−
b
V
y
=
−
V
x
V
y
+
10 V
V
x
−
10V
V
y
,
(7.25)
100 V
2
R
3
C
×
R
2
C
×
which have the same form as (7.1), our simple model. We chose
C
=0
.
1
µ
F,
R
1
=10Ω,
R
2
=33Ω,
R
3
=10kΩand
R
=10kΩ.
The integrator that we have described can be driven with our forcing
functions
β
(
t
)and
(
t
), which emulate the time-varying air sac pressure and
the tension of the ventral muscles, respectively. In the implementation that
we are discussing, the amplitudes of the forcing functions can be in the order
of volts, and the frequencies in the range 1-10 Hz. For small values of the
function
β
(
t
), the voltage
v
basically follows the forcing, but beyond a critical
value of
β
(
t
) the voltage
v
displays large-amplitude oscillations, at a frequency
much larger than the forcing frequency. This activity represents the oscillation
of the labia in our problem. At this point, we say, in terms of the bifurcation
diagrams in Chap. 5, that the values of
β
(
t
) take the system into the region
of oscillations. For the values of capacitances and resistances chosen, these
rapid oscillations have frequencies in the order of kHz.
How can we visualize our integration? In a lab, a simple oscilloscope can
be used. But for a simple application such as this one, and for the ranges
of frequencies involved, the data can be downloaded to a PC by sending
v
to the input of a commercial loudspeaker, the output of which is sent to a
commercial PC sound card, and the signal can be recorded by any sound
software. The free software
snd
, running under a Linux operating system,
and several others programs running under Windows, are good options for
analyzing the voltage time series obtained from our circuit, as well as for
processing the signals and computing their spectral properties.
In Fig. 7.4, we show the sonogram of a “composition” generated by the
device. In order to generate this audio file, four different recordings were
made. Different values of the frequency of the forcing parameters
β
(
t
)and
