Complex Neuro-Cognitive Systems - Foundations on Natural and Artificial Computation - page 3

3 Complex Systems
The suggestion to integrate bottom-up, large-scale brain simulations and top-
down theories such as BICAs to progress in neuro-cognition research has been
made from time to time. However, the predicted success has not appeared what
is apparently due to the fact that both approaches have restrictions which cannot
be overcome even by integration. Actually, both analysis methods are applicable
only to a limited class of systems, the (near-)decomposable systems, as shown
elsewhere [1,2]. There I have argued that the subjects of study of cognitive and
computational neuroscience - cognitive systems that realise functions localised
in neural circuits of the brain - are not members of this class. They are instances
of complex systems which resist the usual reductionist analyses.
The study of complex systems originated during the last three decades or so
from the interplay of disciplines such as physics, mathematics, biology, economy,
engineering, and computer science. There is still no generally accepted definition
of complexity, despite a multitude of proposed approaches (e.g. [8,9,10]).
Important is the following distinction: we must differentiate between systems
that are complex and those which are merely complicated [11, p.511]:
A complicated system is composed of a large number of interacting com-
ponents. Importantly, the properties of such a system can be accurately
predicted from a knowledge of the properties of each of its components and
a complete enumeration of their interactions. In other words, a compli-
cated system is exactly the sum of its parts. Complex, on the other hand,
is a term reserved for systems that display properties that are not pre-
dictable from a complete description of their components, and that are
generally considered to be qualitatively different from the sum of their
parts.
Editorial. Nature Biotechnology, 1999
From complexity theory we know that complex phenomena can be produced
from the interaction of rather simple components. Well-understood examples
are artificial neural networks and cellular automata. These are compositionally
complex systems, and it is indeed feasible to predict their behaviour from the
knowledge of the properties of the components and their interactions. This is
completely different in the case of complex systems whose behaviour emerges
in an unpredictable way. The question then arises: how should complex systems
be studied, and which methods of investigation are available? In the following, I
will consider the method of nonlinear dynamical analysis . Other methods include
relational modelling [12,13] and quantum theories [14].
4 Nonlinear Dynamical Analysis
4.1 Terminology
To apply a nonlinear dynamical analysis to a complex system means to specify
its temporal evolution. The state space (or phase space ) of the system consists of