Biomedical Engineering Reference
In-Depth Information
from one attractor to another, keeping the system on a trajectory that is unstable
in the absence of control, or combinations of the two. One example of a bio-
medical problem that naturally involves control theory, but also pushes its cur-
rent limits, is the prevention of cardiac arrhythmias in humans (see Part III,
chapter 3.3, by Glass.) Here one has a spatially extended system large enough to
support complex spatiotemporal activity, though the desired behavior is a sim-
ple, regular heartbeat. There is some reason to hope that nonlinear dynamics
models will provide useful descriptions of cardiac electrodynamics and new
ideas for suppressing instabilities associated with certain types of arrhythmias.
Recent work has focused on the onset of alternans (period doubling) in paced
cardiac tissue (31) and the manipulation or destruction of spiral waves in excit-
able media models (1).
The world of nonlinear dynamical systems is full of complex structures and
surprising behavior. There is now a well-developed language for characterizing
all sorts of attractors and bifurcations as parameters are varied. The classifica-
tion schemes will (probably) never be complete, however, and studies of sys-
tems as complex as living tissues and biological networks (metabolic, genetic,
immunological, neuronal, ecological) are highly likely to uncover new mathe-
matical structures. Systems with strongly stochastic elements or many interact-
ing variables will require further connections to be made between nonlinear
dynamics proper and statistical mechanics. As indicated by many of the chapters
in this volume, all of these concepts can and should be brought to bear in the
study of biomedical systems.
6.
NOTES
1. If the equations of motion contain a second derivative of
x
1
, say, the
above form is recovered by defining
x
2
=
x
and writing
x
wherever the second
2
derivative of
x
1
appeared in the equations.
2. The coefficients are the external parameters designated by
p
above.
3. It is possible, however, for there to be an initial increase in some vari-
ables before the ultimate relaxation toward the origin occurs. This can happen
when the eigenvectors associated with significantly different modes are
non-
normal
(not perpendicular to each other in state space). See (32) for a discus-
sion of this effect and presentation of several examples.
4. Strictly speaking, this is a bit of a misnomer, as an exponent should not
be a dimensionful quantity. The physically relevant quantity is the Lyapunov
exponent M multiplied by some characteristic time in the system.
5. Note that at points where the trajectory appears to cross itself it must be
really separated in the third dimension since the future behavior is unique once
an initial point in state space is given.
