Biomedical Engineering Reference
In-Depth Information
the modeling of cardiac electrodynamics, which is being approached from sev-
eral directions simultaneously. The construction of PDE models based on de-
tailed understanding of the physiology of cardiac tissue (26), the construction
and simulation of PDE models reproducing realistic patterns of spatiotemporal
activity (10,34), and the construction of ODE models or discrete maps reproduc-
ing bifurcations observed in small pieces of cardiac tissue (31), are all being
brought to bear in a grand attempt to explain the onset and characteristics of
arrhythmias from simple period doubling to fibrillation.
The second way in which complexity can enter the world of nonlinear dy-
namics is through the sheer abundance of distinct variables and the logical, or
causal, relations among them. Even without taking into account the spatial dis-
tribution of concentrations of molecules, for example, the immune system or the
metabolic network in a cell can exhibit surprising behavior due to nonlinear in-
teractions among the concentrations of distinct types of molecules. The nature of
these connections, and in particular the topology of the network representing
them, has become a central theme in current research. As yet, rather little is un-
derstood about the dynamical processes that occur within such large networks
containing many feedback loops. Our purpose here is just to illustrate the way in
which nonlinear dynamics becomes the natural language for discussing the be-
havior of complex systems of this sort.
Consider the problem of modeling the regulatory network that governs gene
expression in a cell. At its most basic level, the cell can be thought of as a dy-
namical system of interacting biomolecules produced through the mechanisms
of gene expression. In this picture, the future chemistry of a cell is determined
by which genes are expressed at any given moment. The products of transcrip-
tion and translation of genes interact in extraordinarily complicated ways and act
back on the processes of transcription and translation so as to influence which
genes are expressed at a later time. To model this system the nonlinear dynami-
cist might begin by defining the system variables to be the levels of expression
of each gene. Thus the life history of a cell becomes a trajectory through a state
space of dimension equal to the number of genes in the network.
Modeling of the detailed interactions among all of the proteins and nucleic
acids in the cell would make for a horrifically complicated mathematical system,
from which it would be very difficult to glean any useful insights. Instead, one
can hope (and perhaps expect) that many features of the state space trajectories
are universal, i.e., that they do not depend on the details of the interactions. One
is then led to devise models that retain the general logical structure of genetic
regulatory networks but are defined by interactions simple enough to be effi-
ciently simulated and studied analytically. One approach, pioneered in the con-
text of genetic regulatory networks by Kauffman (14,15) is to assume that gene
expression level is a Boolean variable and that the logical relations among dif-
ferent genes' activities are essentially random. As it turns out, the behavior of
