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From: park@usceast.UUCP (Kihong Park)
Newsgroups: comp.ai
Subject: Re: Chaos and AI
Message-ID: <3147@usceast.UUCP>
Date: 16 Mar 90 05:24:25 GMT
References: <6925@cps3xx.UUCP> <3142@usceast.UUCP> <1480@watserv1.waterloo.edu>
Organization: University of South Carolina, Columbia
Lines: 58

In article <1480@watserv1.waterloo.edu> ssingh@watserv1.waterloo.edu ($anjay "lock-on" $ingh - Indy Studies) writes:
>There was an article awhile back in Discover called "The Body Chaotic."
>It mentioned that the EEGs of epileptics during a seizure become far
>more periodic and regular as whole groups of neurons begin firing in
>sync across the brain. A normal person's EEG shows little trace of
>periodicity. It suggests that a nice chaotic "buzz" is the normal
>state of affairs.

This is an interesting valid point. But, remember when one is talking about
the dynamics of the brain itself, "periodicity" has to be interpreted in a
slightly different way.

The brain is a huge chunk of very many interacting modules rather than a
one-module entity. Even from a macroscopic viewpoint, physiologists have known
for a long time that different parts of the brain have distinguishable
functionalities. It is to be expected that within such discernible 
macro-modules, many more micro-modules will exist.

If we assume, just for the sake of argument, that low-level modular structures
consist on the order of thousands of neurons, then these "elementary"
cohesive units may function abiding by the "periodicity principle". Of course,
the time interval within which this regularity manifests itself may very
well be rather short.

Such modules will interact in a complex fashion whereby one module's short-time
behavior will be a function of its surrounding modules. Hence, viewed from
"above", the emerging activity pattern(even subglobally), may be very complex.
Indeed, it would be exceptional if the global activity pattern of a system
composed of very many modules would show apparent regularity.

If the information processing done by the system is of a "simple" kind, then
global regularity may be the norm. But for complex automata such as the
brain, low-level, local regularity coupled with higher-level "irregularity"
is to be expected. We know Class 4 cellular automata(Wolfram) are capable
of universal computation. Empirical observation says that the time-evolution
of these machines shows "regular" local structures, but globally these local
regularities may interact in a complicated way.

Last argument: from an information-theoretic point-of-view, a system's global
behavior being apparently "patternful", i.e., easy to discern simple
organization principles, implies that the entropy of the system on the
average is low. Very low entropy systems cannot perform complex information
processing tasks. Of course, the other extreme is also detrimental. 
For example, unimodular Hopfield nets are only good at memorizing things in a 
context-sensitive fashion. Abstracted as CAs, they converge to fixed points 
or limit cyles, and hence are low-entropy dynamical systems.

>BTW, usually wherever there is chaos, fractals are lurking nearby. In the
>excitement about chaos, fractals seem to have faded into the woodwork. Has
>anyone seen or done work which tries to tie fractals, chaos, and NNs together
>into a biologically plausible model? 

Indeed, chaos and fractals are close-knit subjects. The dynamics of chaotic
systems tend to topologically equal cantor sets, a fractal object. But, this
may be beside the point when talking about the relation between chaos and
neural nets. I have heard that astronomers have identified fractal distributions
in the universe. Are fractal distributions or connectivity patterns
observable in the brain?