Biomedical Engineering Reference
In-Depth Information
might well expect physical theory to play a significant role, to the logic of neural
networks or gene regulation, where the role of physical/mathematical modeling
may be less obvious.
The establishment of a connection between physical theory and biomedical
observations generally involves a combination of fundamental physical reason-
ing and
a posteriori
model validation. The field of nonlinear dynamics is of cru-
cial importance for both purposes. It provides both the techniques for analyzing
the equations of motion that emerge from the physical theory and a useful lan-
guage for framing questions and guiding the process of model validation (as
noted by Shalizi, Part II, chapter 1, this volume). Because the concept of a
nonlinear dynamical system is rich enough to encompass an extremely broad
range of processes in which the future configurations of a system are determined
by its past configurations, the methods of analysis developed in the field are
useful in a huge variety of contexts.
To appreciate the validity of a particular research result involving the appli-
cation of nonlinear dynamical theory and properly interpret the specific conclu-
sions, it is important to grasp the broad conceptual basis of the work. The
purpose of this chapter is to explain the meaning and crucial consequences of
nonlinearity so as to provide an operational understanding of the principles un-
derlying the modeling discussed in other chapters. (Almost all of the chapters in
the present volume rely on techniques and approaches whose roots lie in the
development of nonlinear dynamics as a discipline. The chapters by Subrama-
nian and Narang [Part III, chapter 2.2], Lubkin [Part III, chapter 3.1], Tabak
[Part III, chapter 5.2], Solé [Part III, chapter 6.2], and Segel [Part III, chapter
4.1] all make direct reference to nonlinear dynamical models of precisely the
sort discussed here.) Along the way, certain fundamental terms will be defined
and illustrated with examples, but the reader interested in the details of the
mathematics will have to look elsewhere. Two excellent textbooks that do not
require familiarity with mathematical concepts beyond basic calculus are (28)
and (3). For treatments of more advanced topics, a good place to start is (25).
Nonlinear dynamics enters the biomedical literature in at least three ways.
First, there are cases in which experimental data on the temporal evolution of
one or more quantities are collected and analyzed using techniques grounded in
nonlinear dynamical theory, with minimal assumptions about the underlying
equations governing the process that produced the data. That is, one seeks to
discover correlations in the data that might guide the development of a mathe-
matical model rather than guess the model first and compare it to the data. (See
the chapter by Shalizi [Part II, chapter 1] in this volume for a discussion of time-
series analysis.) Second, there are cases in which symmetry arguments and
nonlinear dynamical theory can be used to argue that a certain simplified model
should capture the important features of a given system, so that a phenomenol-
ogical model can be constructed and studied over a broad parameter range. Of-
ten this leads to models that behave qualitatively differently in different regions
