168
A. Ben-Tal
R
q
P m
P L (t)
k s
P A
V A
Fig. 10.4 A simple model of the lungs. The lungs is modeled as a single container with a moving
plate. The plate is connected to a spring with a constant k s . The pressure inside the container is P A
and the volume of the container is V A . The pressure outside is P m (assumed to be constant), the air
flow is q and the resistance to flow is R. The pleural pressure, P L (t), is a given function of time.
Figure adapted from [3]
V A (t) = V A (0)e
−
E
R t
+ e
−
E
R t
t
0
1
R
(P m − P L (τ )) e
E
R τ dτ
(10.9)
If P L (t) is periodic then V A (t) in steady state is periodic too.
Another example of forced oscillations can be found in a model of heart rate control (Fig. 10.5) [5]. The heart period is controlled by the parasympathetic nerve which
itself is affected by the baroreceptors, central respiratory drive and stretch receptors.
There are nine equations in this model and the exact details are not essential. Therefore, the equations are not given here (see [5] for the details). The important thing
to note in Fig. 10.5 is that the signal A(t) (the central respiratory drive, Fig. 10.5a)
is an input to the model. This signal is a simplification of the bursting signal shown
in Fig. 10.2. In principle, had we known more about the operation of the respiratory
neural circuitry and how it responds to feedback signals, we could have closed the
loop on respiration and create an autonomous model for the short-term control of the
cardio-respiratory system.
10.3 Alternative Transformations
The modeling examples illustrated in Sect. 10.2 demonstrated that for modeling convenience, intrinsic oscillations in an autonomous system can be replaced by forced
oscillations, creating non-autonomous systems. Inspired by the modeling examples
we suggest that, in special cases, the process can be reversed. That is, the nonautonomous systems can be transformed to “native” autonomous. We show two
examples of how this can be done. Clearly there are many other examples to be
found.
A. Ben-Tal
R
q
P m
P L (t)
k s
P A
V A
Fig. 10.4 A simple model of the lungs. The lungs is modeled as a single container with a moving
plate. The plate is connected to a spring with a constant k s . The pressure inside the container is P A
and the volume of the container is V A . The pressure outside is P m (assumed to be constant), the air
flow is q and the resistance to flow is R. The pleural pressure, P L (t), is a given function of time.
Figure adapted from [3]
V A (t) = V A (0)e
−
E
R t
+ e
−
E
R t
t
0
1
R
(P m − P L (τ )) e
E
R τ dτ
(10.9)
If P L (t) is periodic then V A (t) in steady state is periodic too.
Another example of forced oscillations can be found in a model of heart rate control (Fig. 10.5) [5]. The heart period is controlled by the parasympathetic nerve which
itself is affected by the baroreceptors, central respiratory drive and stretch receptors.
There are nine equations in this model and the exact details are not essential. Therefore, the equations are not given here (see [5] for the details). The important thing
to note in Fig. 10.5 is that the signal A(t) (the central respiratory drive, Fig. 10.5a)
is an input to the model. This signal is a simplification of the bursting signal shown
in Fig. 10.2. In principle, had we known more about the operation of the respiratory
neural circuitry and how it responds to feedback signals, we could have closed the
loop on respiration and create an autonomous model for the short-term control of the
cardio-respiratory system.
10.3 Alternative Transformations
The modeling examples illustrated in Sect. 10.2 demonstrated that for modeling convenience, intrinsic oscillations in an autonomous system can be replaced by forced
oscillations, creating non-autonomous systems. Inspired by the modeling examples
we suggest that, in special cases, the process can be reversed. That is, the nonautonomous systems can be transformed to “native” autonomous. We show two
examples of how this can be done. Clearly there are many other examples to be
found.