Biomedical Engineering Reference
In-Depth Information
Once in the neighborhood of
f
(
u
),
u
∼
0and
v
slowly decreases, since
v
=
ku
and
u>
0. Notice that the system will evolve so that it “sticks” to
the branch of positive slope of the nullcline: if, owing to the decrease in
v
,the
system finds itself below the nullcline, the direction of the force will now be
pointing towards the nullcline. The system evolves while trying to keep
v
−
0.
A qualitative change occurs when the system reaches the minimum of
f
(
u
).
Now
v
is smaller than
f
(
u
), and therefore
u
will decrease rapidly. But there
is no longer a branch of the nullcline in the region of the phase space with
u>
0 to stop the dynamics. The system evolves in time in such a way that
u
decreases, until it reaches the branch with positive slope of
f
(
u
)for
u<
0.
Once again, in the neighborhood of
f
(
u
), the system will cause
v
to increase,
following
v
=
∼
ku
with
u<
0. Notice that the oscillations that arise alternate
rapid jumps with slowly varying time evolutions, as displayed in Fig. 4.2
(bottom trace). These oscillations are known as
relaxation
oscillations.
What we have learned from this system can help us to understand the
dynamics of (4.5), since the van der Pol equation can be mapped onto our
equations. In order to do so, we have to define
u ≡ x
and
y ≡ v − cu
3
/
3+
(
β
−
b
)
u
.
The choice of all the forces mentioned so far in building our model is not
arbitrary: we have in mind the processes that will be relevant when we try
to understand the operation of the syrinx. And we have discussed almost all
the elements that we need.
−
4.3 Oscillations in the Syrinx
4.3.1 Forces Acting on the Labia
Let us remember that the syrinx is a bipartite device. At each junction be-
tween the bronchi and the trachea, there is a pair of labia whose dynamics we
want to analyze. It is not di
cult to accept that a muscle tissue, after being
stretched or compressed, will recover owing to a restitution force. Neither is it
di
cult to accept that the system has friction (actually, it would be di
cult
to accept the contrary!). Finally, each labium has a bounded space to move
in. When the labia move away from each other, allowing an airflow through,
they meet the cartilaginous tubes to which they are attached. Later, within
the same oscillation, the labia approach each other until they eventually col-
lide. Consequently, it is also natural to expect a nonlinear dissipation force
like the one described earlier. The most di
cult problem to be solved in order
to establish a description of the dynamics of the syrinx in the framework of
our discussion in the last section is to understand the origin of what we have
called the “external force”. That force is responsible for avoiding the decay
of the oscillations due to the dissipation terms.
