Biomedical Engineering Reference
In-Depth Information
qualitative behaviors: stable and unstable fixed
points, periodic, almost periodic or chaotic behav-
iors. This fact makes RNN applicable for model-
ing some cognitive functions such as associative
memories, unsupervised learning, self-organizing
maps and temporal reasoning.
The mathematical models of neural networks
arise both from the modeling of some behaviors
of biological structures or from the necessity of
Artificial Intelligence to consider some structures
which solve certain tasks. None of these two cases
has as primary aim stability aspects and a “good”
qualitative behavior. On the other hand, these
properties are necessary and therefore important
for the network to achieve its functional purpose
that may be defined as “global pattern formation”.
It thus follows that any AI device, in particular a
neural network, has to be checked
a posteriori
(i.e.
after the functional design) for its properties as a
dynamical system, and this analysis is performed
on its mathematical model.
A common feature of various RNN (automatic
classifiers, associative memories, cellular neural
networks) is that they are all
nonlinear dynami-
cal systems with multiple equilibria
. In fact it is
exactly the equilibria multiplicity that gives to
all AI devices their computational and learning
capabilities. As pointed out in various reference
topics, satisfactory operation of a neural network
(as well as of other AI devices) requires its evolu-
tion towards those equilibria that are significant
in the application.
Let us remark here that if a system has sev-
eral isolated equilibria this does not mean that
all these equilibria are stable - they may be also
unstable. This fact leads to the necessity of a
qualitative analysis of the system's properties.
Since there are important both the local stability
of each equilibrium point and also (or more) the
global behavior of the entire network, we shall
discuss here RNN within the frameworks of the
Stability Theory
and the
Theory of Systems with
Several Equilibria.
The chapter is organized as follow. The
Back-
ground
section starts with a presentation of RNN
from the point of view of those dynamic properties
(specific to the systems with several equilibria),
which make them desirable for modeling the as-
sociative memories. Next, there are provided the
definitions and the basic results of the
Theory of
Systems with Several Equilibria
, discussing these
tools related to the
Artificial Intelligence
domain
requirements. The main section consists of two
parts. In the first part it is presented the basic tool
for analyzing the desired qualitative properties
for RNN as systems with multiple equilibria. The
second part deals with the effect of time-delays on
the dynamics of RNN. Moreover, one will consider
here the time-delay RNN under forcing stimuli
that have to be “reproduced”(synchronization).
The chapter ends with
Conclusions
and comments
on
Future trends
and
Future research directions
.
Additional reading
is finally suggested.
bACkGROUND
A.
The state space of RNN may display stable and
unstable fixed points, periodic, almost periodic or
even chaotic behaviors. A concise survey of these
behaviors and their link to the activity patterns
of obvious importance to neuroscience may be
found in (Vogels, Rajan & Abbott, 2005). From
the above mentioned behaviors, the fixed-point
dynamics means that the system evolves from an
initial state toward a state (a
stable fixed-point
equilibrium
of the system) in which the variables
of the system do not change over the time. If
that
stable fixed-point
is used to retain a specific
pattern then, given a distorted or noisy pattern
of it as an initial condition, the system evolution
will be such that the stable equilibrium point will
be eventually attained to. This process is called
the
retrieving of the stored pattern
. Since an as-
sociative memory has to retain several different
patterns, the system which models it has to have
several stable equilibrium points
. More general,
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