More responses: Chaos

[Pankaj Mehra]

First, a small note:
	In his response (sent directly to the list), Albert Boulanger
	makes a connection between annealing and ergodicity. He then
	describes an optimization procedure based on slowly constraining
	the dynamics of a search process. He claims that the algorithm
	has better convergence than annealing.

RESPONSES RECEIVED
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From: B344DSL at UTARLG.UTA.EDU (???)

On the usefulness of chaos in neural models (particularly biologically-
related ones), I recommend the following two articles:

1. Skarda, C. & Freeman, W. J. (1987).  How brains make chaos to make sense
of the world.  Behavioral and Brain Sciences 10: 161-195.  (This is about
the olfactory system, with some speculative generalizations to other mamm-
alian sensory systems.)

2. Mpitsos, G. J., Burton, R. M., Creech, H. C., & Soinila, S. O. (1988).
Evidence for chaos in spike trains of neurons that generate rhythmic motor
patterns.  Brain Research Bulletin 21: 529-538.  (This is about motor systems
in mollusks.)

Both of these authors see chaos as a useful biological device for promoting
behavioral variability.  In the Skarda-Freeman article, there are several
commentaries by other people that deal with this issue, including commentaries
by Grossberg and by myself that suggest that chaos is not the only possible
method for achieving this goal of variability.  (There are also responses
to the commentaries by the author.)  Both Freeman and Mpitsos have pursued
this theme in several other articles.

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From: pluto at cs.UCSD.EDU (Mark Plutowski)

With regards to:
3a. Is chaos a precise quantitative way of stating one's ignorance of
    the dynamics of the process being modeled/controlled?

Formally, probability theory (in particular, the theory of stochastic
processes) is a precise quantitative way of stating one's ignorance of
the dynamics of a process.  (Say, by approximation of the empirical
data by a sequence of processes which converge to the data
probabilistically.)

According to my knowledge, a chaotic attractor gives a concise
representation which can describe processes which are complicated
enough to appear (behaviorally) indistinguishable from random
processes.  In other words, the benefit of a chaotic model is in its
richness of descriptive capability, and its concise formulation.

Perhaps chaotic attractors could be employed to provide deterministic
models which give a plausible account of the uncertainties inherent in
the probabilistic models of our empirical data.  Whereas the
probabilistic models can predict population behavior, or, the
asymptotic behavior of a single element of the population over time, an
equally accurate chaotic model may as well be able to predict the
behavior of a single element of the population over a finite range of
time.

I anticipate that the real challenge would be in inferring a chaotic
attractor model which can give a testable hypothesis to tell us
something that we cannot ascertain by population statistics or
asymptotic arguments. Without this latter condition, the chaotic model
may be just an unnecessary generalization - why use a chaotic model if
more well-understood formalisms are sufficient?  However, if it is used
to provided a model which is formally convenient, then, the benefit is
not necessarily due to descriptive capability, and so, the resulting
models need not necessarily give a mechanistic model of the empirical
data (and hence, potentially reducing the predictive capability of the
models.)

Of course, this is all IMHO, due to my expectation that noise inherent
in our best experimental data will tend to cause researchers to first
fit the data with stochastic processes and classical dynamical systems,
and then try to refine these models by using deterministic chaotic
models.

Perhaps someone better informed on the use of these models could
comment on whether this perspective holds water?  It may be the case
that I am missing out on capabilities of chaotic models which are not
apparent in the popular literature.

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END OF RESPONSES