Copy Link
Add to Bookmark
Report
Neuron Digest Volume 05 Number 52
Neuron Digest Monday, 4 Dec 1989 Volume 5 : Issue 52
Today's Topics:
Re: Data Complexity
Re: Data Complexity
Re: Data Complexity
Re: Data Complexity
Re: Kolmogorov Complexity references (was Data Complexity)
Re: Kolmogorov Complexity references (was Data Complexity)
Neural Nets in Gene Recognition
Re: Neural Nets in Gene Recognition
genetic algorithm simulators
Re: genetic algorithm simulators
Send submissions, questions, address maintenance and requests for old issues to
"neuron-request@hplabs.hp.com" or "{any backbone,uunet}!hplabs!neuron-request"
Use "ftp" to get old issues from hplpm.hpl.hp.com (15.255.176.205).
------------------------------------------------------------
Subject: Re: Data Complexity
From: ted@nmsu.edu (Ted Dunning)
Organization: NMSU Computer Science
Date: 16 Oct 89 17:38:56 +0000
ivan bach is very enthusiastic, but relatively uninformed about recent
developments in determining the complexity of finite sequences.
he also has an annoying tendency to post news that is more than 80
columns.
[[Editor's Note: Ivan's article was reformatted for Digest inclusion. PM]]
here is a partial commentary on his posting,
In article <4542@imagen.UUCP> ib@apolling (Ivan N. Bach) writes:
Path: opus!lanl!cmcl2!nrl-cmf!think!ames!amdcad!sun!imagen!daemon
From: ib@apolling (Ivan N. Bach)
Claude Shannon, founder of the theory of information, ... He
decided to call this measure "entropy," because he realized that
his formula:
H = -K x sum of (pi x log pi) for all i=1 to i=a
shannon actually qualified this measure extensively. it is only valid
if successive symbols are independent, and only valid for stationary
ergodic sequences. none of these qualifications apply in most
biological systems.
Lila Gatlin showed that Claude Shannon's formula could be used to
calculate the entropy of a randomly chosen base in the DNA chain of
bases in the nucleus of a living cell. If all four types of bases
in a DNA chain are equiprobable, the entropy of a randomly chosen
base in the DNA chain is equal to: ... = 2 bits
actuallly lila gatlin showed that you can misuse shannon's formula to
get an upper bound on the amount of information in a dna chain.
Computer programmers know that they have to use at least 2 bits to
Cencode 4 different values.
real computer programmers know that this is not true.
If all bases in a DNA chain are not equiprobable, the probability
of a certain type of base occurring at a random position in a DNA
chain is proportional to the number of bases of its type in the
... Therefore, the entropy of a randomly chosen base in the
DNA chain of "Micrococcus lysodeikticus" is equal to:
= 1.87 bits
this is entirely wrong. see below.
Lila Gatlin came to the conclusion that the entropy of a complex
system is equal to the sum of entropies of its components.
lila is again mistaken, unless what actually was said was that the
entropy of a system composed of _independent_ parts is the sum of the
entropies of its components. of course _complex_ systems generally do
not allow such naive decomposition.
[ arithmetic maunderings about bits and blueprints deleted ]
[ fuzzy comparison of a turing machine to protein synthesis ]
4. Instead of constructing a full-blown neural network, we could
specify a "blueprint" for contructing that network. I think that
the amount of information in the blueprint can always be smaller
than the amount of information in the network if there is any
redundancy or repetition in the structure of the network.
actually the blueprint must be less than or equal to the network.
otherwise, we would just use the network as the blueprint. see below
for a reasonable definition of information content.
5. If you use a certain "alphaneural net, you should use each type
of component with the same frequency if you want achieve the
maximum information capacity.
of course maximum information _content_ (or capacity) would also be
obtained by making all components completely independent. this is the
same as recommending that people use RAM for their neural net. you
_don't_ want a system with maximum content. you want a system that
encodes and carries meaning. these are very different things.
shannon's formulae are simple enough to be misused extensively by all
kinds of people. such misuse does not mean that they are any more
applicable than they were in 1949. in particular, if you compute the
entropy of english text, you over-estimate the information content by
a factor of 4 or more. for instance, this posting has an entropy of
4.6 bits per letter, while english is generally accepted to have an
information content of 1 bit per letter or less. the difference is
entirely due to the assumption of independence between letters.
an even more dramatic example is a sequence formed by the regular
expression {abcd}*. if we take any 100 characters from a random
location in such a sequence, and compute the entropy for the sequence,
we will find that we apparently have 200 bits of information. in
fact, though, we have only 2 bits of information because once we know
the first letter in the sequence, we know all the letters.
a much more sound approach to the complexity of sequences was proposed
in the 60's by various soviet mathematicians, and was popularized for
use with chaotic systems by v.i. arnold in the 70's. this model
defines the complexity of a finite string to be the size of the
smallest computer program which can be used to recreate the sequence.
in arnold's work, the sequence was the itinerary of a finite map, but
this does not really matter if we are starting with a sequence from a
finite alphabet.
obviously, the upper bound on the complexity of a finite sequence is
the sequence size itself, since we can easily write a program for, say,
a ram machine which builds a list containing the sequence in the same
number of steps as the original sequence is long.
the complexity of one of our 100 character substrings of {abcd}* is
much less, since we can write a very simple program which produces
one of these substrings.
in practice, information content of a complex sequence can be
estimated by building relatively simple models for inter-symbol
dependencies and then measuring the entropy of the errors the model
makes in predicting the next symbol. this is nearly equivalent to the
approach used by arnold and co, since if you can't build a predictor
at all, then the error stream becomes equivalent to the original
sequence.
ted@nmsu.edu
Dem Dichter war so wohl daheime
In Schildas teurem Eichenhain!
Dort wob ich meine zarten Reime
Aus Veilchenduft und Mondenschein
------------------------------
Subject: Re: Data Complexity
From: park@usceast.UUCP (Kihong Park)
Organization: University of South Carolina, Columbia
Date: 18 Oct 89 15:14:04 +0000
In article <TED.89Oct16113856@aigyptos.nmsu.edu> you write:
>ivan bach is very enthusiastic, but relatively uninformed about recent
>developments in determining the complexity of finite sequences.
I agree with your comments regarding the outdatedness and errors of Ivan's
statements. As to describing the information content of a sequence over a
finite alphabet set, the algorithmic information complexity measure as
developed independently by Kolmogorov, Chaitin, and Solomonov can be
fruitfully applied here.
Basically, they define the complexity of a sequence as the size of the
minimum program which faithfully generates the sequence, if run on some
universal turing machine. Thus, for a DNA sequence to encode maximum
information, i.e., be maximally patternless, its algorithmic complexity
must nearly equal the length of the sequence.
As to Ivan's comments about the applicability of entropy to represent the
efficiency of information coding in a network, many workers in the field
have realized that for a given task, a more or less 'adequate' network size
should be strived for. One of the principal motivations for doing this is
the empirical observation that if a network size is 'too large' compared to
the complexity of the task at hand, then it may generate solutions which do
not 'generalize' well, given the opportunity to do so.
What 'generalization' means is a different problem altogether, but what
this observation shows us is that one has to constrain the degree of
freedom of a network to entice it to capture the most compact
representation. This constitutes so to say, the 'minimal program'. Thus,
the problem at hand is knowing the optimal size of a network for a given
task. I do not see an easy way how entropies can help us in solving this
correspondence problem.
> Dem Dichter war so wohl daheime
> In Schildas teurem Eichenhain!
> Dort wob ich meine zarten Reime
> Aus Veilchenduft und Mondenschein
It's interesting to notice that some people equate our profession to that
of poets. Is this a revival of romanticism ?
------------------------------
Subject: Re: Data Complexity
From: demers@beowulf.ucsd.edu (David E Demers)
Organization: EE/CS Dept. U.C. San Diego
Date: 18 Oct 89 20:37:44 +0000
In previous article ted@nmsu.edu (Ted Dunning) writes:
>ivan bach is very enthusiastic, but relatively uninformed about recent
>developments in determining the complexity of finite sequences.
Agreed.
[discussion of Bach's posting + critical commentary deleted]
>shannon's formulae are simple enough to be misused extensively by all
>kinds of people. such misuse does not mean that they are any more
>applicable than they were in 1949.
Despite the misuse, the basic concept is very valuable.
>a much more sound approach to the complexity of sequences was proposed
>in the 60's by various soviet mathematicians, and was popularized for
>use with chaotic systems by v.i. arnold in the 70's.
This work goes by the name of "Kolmogorov Complexity", usually.
Solomonoff, Kolmogorov, Chaitin did much of the pioneering work in the
early 60's, from independent reasons - examining the notion of randomness
[Kolmogorov] and inference [Solomonoff].
>obviously, the upper bound on the complexity of a finite sequence is
>the sequence size itself,
[[... See Ted's article for description. -PM]]
The above is a very good quick description of the concept. It essentially
quantifies Occam's Razor in modeling. Computational learning theory takes
much of these ideas; Valiant, et al make use of the same principle.
I apologize for once again posting with the 'F' key before gathering
citations. Walt Savitch just gave a talk last week on Kolmogorov
Complexity for the Cognitive Science seminar here at UCSD; I will scare up
a summary and some references and post soon for those interested in
pursuing the idea in more depth.
Dave DeMers demers@cs.ucsd.edu
Dept. of Computer Science & Engineering demers@inls1.ucsd.edu
UCSD {other non-Internet mailpaths
La Jolla, CA 92093 according to normal syntax...}
------------------------------
Subject: Re: Data Complexity
From: andrew@dtg.nsc.com (Lord Snooty @ The Giant Poisoned Electric Head )
Organization: National Semiconductor, Santa Clara
Date: 23 Oct 89 01:58:24 +0000
I guess the entropy calcs use the p.ln(p) expression, as per Shannon.. but
as per Hofstadter, what about the complexity of the context or "frame"?
What tells you which jukebox will play your record? If you could give da
Vinci a binary dump of a set of Postscript graphics files describing his
drawings, could he work out that they were his? - or would you have to
provide him with the appropriate "jukebox" (PC + OS + etc)?
Of course, the correct answer is the latter one. So my question is: have
you included the complexity of the decoding apparatus in your entropy
estimates?
...........................................................................
Andrew Palfreyman and the 'q' is silent,
andrew@dtg.nsc.com as in banana time sucks
------------------------------
Subject: Re: Kolmogorov Complexity references (was Data Complexity)
From: demers@beowulf.ucsd.edu (David E Demers)
Organization: EE/CS Dept. U.C. San Diego
Date: 28 Oct 89 17:20:47 +0000
I recently promised to provide some follow up references on Kolmogorov
Complexity, following a thread discussing how one might measure
"complexity" of a model, such as a neural net.
For Kolmogorov Complexity in general, the best place to start would be with
an introduction to the notions of Kolmogorov Complexity and its application
to a number of different problems which can be found in:
Li, M. and Paul Vitanyi, Kolmogorov Complexity and Its Applications, in
Handbook of Theoretical Computer Science, (J. van Leeuwen, Ed.)
North-Holland, 1989.
A preliminary version of the same work is in the Proceedings of the 3d IEEE
Structure in Complexity Theory Conference, 1988.
The same authors have a book out by Addison-Wesley, An Introduction to
Kolmogorov Complexity and its Applications (1989).
For original work, see the references in the bibliography of the above
book. Let me point out the following which are probably most important:
Kolmogorov, AN, "Three approaches to the quantitative definition of
information" Problems in Information Transmission 1, 1-7, 1965;
Shannon & Weaver, The Mathematical Theory of Communication, U. of Illinois
Press, 1949 (for basics of information theory)
Kolmogorov, A.N. "Logical Basis for information theory and probability
theory", IEEE Trans. on Info. Theory ?? 1968 (sorry)
Solomonoff, R.J., "A formal theory of inductive inference", Information &
Control 30, 1-22 and 224-254 (1964).
Chaitin, G.J., "Randomness and Mathematical Proof" Scientific American, 232
(May 1975), pp47-52
Chaitin, G.J., "Algorithmic Information Theory", IBM J. Res. Dev. 21,
350-359 (1977)
Chaitin, G.J., Information, Randomness and Incompleteness. (World
Scientific Pub, 1987)
other surveys are:
Zvonkin, A.K. and L.A. Levin, "The complexity of finite objects and the
development of the concepts of information and randomness by the means of
the Theory of Algorithms", Russ. Math. Survey 25, 83-124 (1970)
Schnorr, C.P., "A survey of the theory of random sequences", in Basic
Problems in Methodology and Linguistics (Butts, R.E. Ed.) 1977, pp.
193-210.
Application of Kolmogorov Complexity, or similar ideas, can be seen in work
of Valiant, Gold, Rissanen & others (maximum likelihood, maximum
entropy...). I might mention
Rissanen, J., "Modeling by the shortest data description", Automatica 14,
465-471 (1978)
for a practical suggestion for statistical modeling - minimum description
length - which has two components, one essentially computing the size of
the model and the other computing the size of the data when encoded using
the theory.
APPLICATIONS:
Can state and prove Goedel's incompleteness theorem using Kolmogorov
Complexity,
Can derive a prime-number theorem, # primes less than n is on the order of
n/ (log n (log log n)^2) ,
can state and prove the pumping lemma for regular languages in a couple of
lines,
and so on...
In the neural net world, Tenorio & Lee used a variant of Rissanen's minimum
description length along with a stochastic growth method for determining
when to add new nodes to a network (TR-EE 89-30, June 1989, Purdue
University School of Electrical Engineering)
Hope this is of interest!
Dave DeMers
Dept. of Computer Science & Engineering C-014
and Institute of Non-Linear Science
UCSD
La Jolla, CA 92093
demers@cs.ucsd.edu
demers@inls1.ucsd.edu
------------------------------
Subject: Re: Kolmogorov Complexity references (was Data Complexity)
From: munnari.oz.au!bruce!lloyd@uunet.uu.net (lloyd allison)
Organization: Monash Uni. Computer Science, Australia
Date: 30 Oct 89 01:28:15 +0000
try also
J. Rissanen Stochastic Complexity
and
C. S. Wallace and P. R. Freeman Estimation and inference by compact coding
Journal of the Royal Statistical Society (series B) V 39 No3 1987
pages 223-239 and 252-265 (Rissanen)
and 240-265 Wallace and Freeman
also C. S. Wallace and D. M. Boulton An information Measure for
Classification Computer Journal 11(2) 185-194 1968
------------------------------
Subject: Neural Nets in Gene Recognition
From: eesnyder@boulder.Colorado.EDU (Eric E. Snyder)
Organization: University of Colorado, Boulder
Date: 06 Nov 89 03:11:15 +0000
I am looking for some references on the application of neural nets to
recognition of genes in DNA sequences. I have heard second-hand of some
research at LANL but Medline does not seem to be well informed on the
matter....
Thanx,
- -------------------------------------------------------------------------
TTGATTGCTAAACACTGGGCGGCGAATCAGGGTTGGGATCTGAACAAAGACGGTCAGATTCAGTTCGTACTGCTG
Eric E. Snyder
Department of Biochemistry Proctoscopy recapitulates
University of Colorado, Boulder hagiography.
Boulder, Colorado 80309
LeuIleAlaLysHisTrpAlaAlaAsnGlnGlyTrpAspLeuAsnLysAspGlyGlnIleGlnPheValLeuLeu
- -------------------------------------------------------------------------
------------------------------
Subject: Re: Neural Nets in Gene Recognition
From: eesnyder@boulder.Colorado.EDU (Eric E. Snyder)
Organization: University of Colorado, Boulder
Date: 09 Nov 89 15:32:07 +0000
Thanks for the several replies providing references on the subject.
Several more people requested that I forward the information I recieved on
the subject. Our mailer bounced several of these replies so I'll just post
it here (the comp.ai people have probably seen it before):
*************************************************************************
From: ohsu-hcx!spackmank@cse.ogc.edu (Dr. Kent Spackman)
Subject: connectionist protein structure
The two articles I mentioned are:
Holley, L.H.; Karplus, M. Protein structure prediction with a neural
network. Proceeding of National Academy of Science, USA; 1989;
86: 152-156.
Qian, Ning; Sejnowski, Terrence J. Predicting the secondary structure
of globular proteins using neural network models. J Mol Biol; 1988;
202: 865-884.
I have an article that will be published in the proceedings of the
Symposium on Computer Applications in Medical Care, in Washington, D.C., in
November, entitled: "Evaluation of Neural Network Performance by ROC
analysis: Examples from the Biotechnology Domain". Authors are M.L.
Meistrell and myself.
Kent A. Spackman, MD PhD
Biomedical Information Communication Center (BICC)
Oregon Health Sciences University
3181 SW Sam Jackson Park Road
Portland, OR 97201-3098
From: Lambert.Wixson@MAPS.CS.CMU.EDU
Subject: DNA,RNA, etc.
Holley and Karplus, Proceedings of the National Academy of Science 86,
152-156 (89).
- ----
From: mv10801@uc.msc.umn.edu
Subject: Re: applications to DNA, RNA and proteins
George Wilcox (mf12801@sc.msc.umn.edu) does work on predicting protein
tertiary structure using large backprop nets.
- --Jonathan Marshall
Center for Research in Learning, Perception, and Cognition
205 Elliott Hall, Univ. of Minnesota, Minneapolis, MN 55455
- ----
>From munnari!cluster.cs.su.OZ.AU!ray@uunet.UU.NET Fri Sep 29 23:40:55 1989
Subject: applications to DNA, RNA and proteins
Borman, Stu "Neural Network Applications In Chemistry Begin to Appear",
C&E News, April 24 1989, pp 24-28.
Thornton, Janet "The shape of things to come?" Nature, Vol. 335 (1st
September 1988), pp 10-11.
You probably know about the Qian and Sejnowski paper already.
The Thornton "paper" is a fast overview with a sentence or two comparing
Q&S's work with other work.
Borman's C&E piece is fairly superficial, but it mentions some other people
who have played with this stuff, including Bryngelson and Hopfield, Holley
and Karplus (who apparantly have published in Proc. Nat. Acad. Sci., 86(1),
152 (1989)) and Liebman.
The 1990 Spring Symposium at Stanford (March 27-29, 1990) will have a
session on "Artificial Intelligence and Molecular Biology". The CFP lists
Neural Networks (very broad-minded of them!), so it might be worth a look
when it comes around.
From: "Evan W. Steeg" <steeg@ai.toronto.edu>
Subject: NNets and macromolecules
There is a fair amount of work on applying neural networks to
questions involving DNA, RNA, and proteins. The two major
types of application are:
1) Using neural networks to predict conformation (secondary
structure and/or tertiary structure) of molecules from their
sequence (primary structure).
2) Using nets to find regularities, patterns, etc. in the sequence
itself, e.g. find coding regions, search for homologies between
sequences, etc.
The two areas are not disjoint -- one might look for alpha-helix
"signals" in a protein sequence as part of a structure prediction
method, for example.
I did my M.Sc. on "Neural Network Algorithms for RNA Secondary
Structure Prediction", basically using a modified Hopfield-Tank
(Mean Field Theory) network to perform an energy minimization
search for optimal structures. A technical report and journal
paper will be out soon. I'm currently working on applications
of nets to protein structure prediction. (Reference below).
Qian and Sejnowski used a feed-forward net to predict local
secondary structure of proteins. (Reference above).
At least two other groups repeated and extended the Qian &
Sejnowski experiments. One was Karplus et al (ref. above)
and the other was Cotterill et al in Denmark. (Discussed in
a poster at the Fourth International Symposium on Artificial
Intelligence Systems, Trento, Italy Sept. 1988).
Finally, a group in Minnesota used a supercomputer and back-prop
to try to find regularities in the 2-d distance matrices (distances
between alpha-carbon atoms in a protein structure). An interim
report on this work was discussed at the IJCNN-88 (Wash. DC) conference.
(Sorry, I don't recall the names, but the two researchers were
at the Minnesota Supercomputer Center, I believe.)
As for the numerous research efforts in finding signals and patterns
in sequences, I don't have these references handy. But the work
of Lapedes of Los Alamos comes to mind as an interesting bit
of work.
Refs:
E.W. Steeg.
Neural Network Algorithms for the Prediction of RNA Secondary
Structure.
M.Sc. Thesis, Computer Science Dept., University of Toronto,
Toronto, Ontario, Canada, 1988.
Evan W. Steeg (416) 978-7321 steeg@ai.toronto.edu (CSnet,UUCP,Bitnet)
Dept of Computer Science steeg@ai.utoronto (other Bitnet)
University of Toronto, steeg@ai.toronto.cdn (EAN X.400)
Toronto, Canada M5S 1A4 {seismo,watmath}!ai.toronto.edu!steeg
- -----
From: pastor@PRC.Unisys.COM (Jon Pastor)
Subject: Re: applications to DNA, RNA and proteins
@article(nakata85a,
Author="K. Nakata and M. Kanehisa and D. DeLisi",
Title="Prediction of splice junctions in mRNA sequences",
Journal="Nucleic Acids Research",
Year="1985",
Volume="13",
Number="",
Month="",
Pages="5327--5340",
Note="",
Annote="")
@article(stormo82a,
Author="G.D. Stormo and T.D. Schneider and L.M. Gold ",
Title="Characterization of translational initiation sites in E. coli",
Journal="Nucleic Acids Research",
Year="1982",
Volume="10",
Number="",
Month="",
Pages="2971--2996",
Note="",
Annote="")
@article(stormo82b,
Author="G.D. Stormo and T.D. Schneider and L.M. Gold and A. Ehrenfeucht",
Title="Use of the `perceptron' algorithm to distinguish translational initiation sites in E. coli",
Journal="Nucleic Acids Research",
Year="1982",
Volume="10",
Number="",
Month="",
Pages="2997--3010",
Note="",
Annote="")
In addition, there is going to be (I think) a paper by Alan Lapedes, from
Los Alamos, in a forthcoming book published by the Santa Fe Institute;
my group also has a paper in this book, which is how I know about Lapedes'
submission. I am going to try to contact the editor to see if I can get a
preprint; if so, I'll let you know. I didn't attend the meeting at which
Lapedes presented his paper, but I'm told that he was looking for splice
junctions.
- ----
From: ff%FRLRI61.BITNET@CUNYVM.CUNY.EDU (Francoise Fogelman)
Subject: proteins
We have done some work on the prediction of secondary structures
of proteins. This was presented at a NATO meeting (Les Arcs, march 1989)
and will be published in the proceedings.
F. Fogelman
LRI Bat 490
Universite de Paris Sud
91405 ORSAY cedex FRANCE
Tel 33 1 69 41 63 69
e-mail: ff@lri.lri.fr
- ----
The book "Evolution, Learning and Cognition", the article
"Learning to Predict the Secondary Structure of Globular Proteins"
by N. Qian & T. J. Sejnowski.
------------------------------
Subject: genetic algorithm simulators
From: schmid@spica.cbmvax (Bob Schmid)
Organization: /usr2/schmid/.organization
Date: 02 Oct 89 17:20:28 +0000
I see in David Goldberg's book, _Genetic Algorithms_, mention of public
domain GA programs (page 71) with some "bells & whistles". Code by Booker
and DeJong (1985), and Grefenstette (1984) is specifically mentioned.
Since neural-nets seem to be prime targets of such algorithms, I was
wondering, does anyone out there know of sources (by ftp, uucp or email)
for these or any other GA public domain source code?
Any leads will be most appreciated!
- -Bob
R. Schmid
<schmid@cbmvax.cbm.commodore.com>
<uunet!cbmvax!schmid>
------------------------------
Subject: Re: genetic algorithm simulators
From: thearlin@vdsvax.crd.ge.com (Thearling Kurt H)
Organization: General Electric CRD, Schenectady, NY
Date: 02 Oct 89 19:15:43 +0000
>[ ... ] does anyone out there know of sources (by ftp, uucp or
>email) for these or any other GA public domain source code?
Here are a couple sources. The Grefenstette code (GENESIS) is availabe
from him via email (gref@aic.nrl.navy.mil). Rick Riolo also has a
subroutine package available (CFS-C) and his email address is
Rick_Riolo@um.cc.umich.edu. He prefers to have you send him a tape and he
will return it to you with the code. Contact him for the details. There is
also a system that I just came accross called "whales and plankton," which,
according to its author, "is a program which models a small ecology and is
a study of genetic algorithms." It was posted to comp.sources.misc and is
available via anonymous ftp from xanth.cs.odu.edu [128.82.8.1]. It is
located in Volume7 and is named whpl.
I don't know what the Booker and DeJong code is but I would be interested
in finding more about it. If anybody knows where to get it, please let me
know.
Also, is the code included in Goldberg's book available from somewhere? I
don't really want to type it in if I can get it over the net. If someone
has already typed it it, could you send it to me?
kurt
Kurt Thearling General Electric CRD
thearlin@vdsvax.crd.ge.com Bldg. KW, Room C301
uunet!vdsvax.crd.ge.com!thearlin P.O. Box 8
kurt@bach.csg.uiuc.edu Schenectady, NY 12301
------------------------------
End of Neurons Digest
*********************
