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<DIV><FONT face=Arial><STRONG>To all, </STRONG></FONT></DIV>
<DIV><FONT face=Arial><STRONG></STRONG></FONT> </DIV>
<DIV align=justify><FONT face=Arial><STRONG>There's been some interest expressed
in our coupled neuron, "MRM" map, some studies</STRONG></FONT></DIV>
<DIV align=justify><FONT face=Arial><STRONG>of which were published
</STRONG><FONT face=Arial><STRONG>in Physica D in 1989. It might be fun for
others to explore its behavior since</STRONG></FONT>
<DIV align=justify><FONT face=Arial><STRONG><EM>most of its potential
phenomenology has remained unexplored
unreported)</EM> </STRONG></FONT></DIV>
<DIV align=justify><FONT face=Arial size=2></FONT> </DIV></FONT></DIV>
<DIV align=justify><FONT face=Arial><STRONG>You can find it at </STRONG><A
href="http://www.cieloinstitute.org"><STRONG>www.cieloinstitute.org</STRONG></A><STRONG> where
you can use the <U>coupled logistics</U></STRONG></FONT></DIV>
<DIV align=justify><FONT face=Arial><STRONG>button and find the <U>zipped
downloadable program</U> and a data file. </STRONG></FONT></DIV>
<DIV align=justify><STRONG><FONT face=Arial>The </FONT><FONT
face=Arial> data and graphics can be saved for
analyses.</FONT></STRONG></DIV>
<DIV align=justify><FONT face=Arial><STRONG>. </STRONG></DIV>
<DIV align=justify><FONT face=Arial><STRONG></STRONG></FONT> </DIV>
<DIV align=justify><FONT face=Arial><STRONG>It runs on most common platforms.
The published paper is also available at the site.
</STRONG></FONT></DIV>
<DIV
align=justify><STRONG>
</STRONG></DIV></FONT>
<DIV align=justify><FONT face=Arial><STRONG>The combined one and two
dimensional dynamical system </STRONG></FONT><FONT face=Arial><STRONG>models the
output of coupled neurons.</STRONG></FONT></DIV>
<DIV align=justify><FONT face=Arial><STRONG>The output of one modulates the
parameter of the other. </STRONG></DIV>
<DIV align=justify><FONT face=Arial><STRONG>In its rather
interesting two dimensional, quasi-symmetrical parameter space
(illustrated in the
paper ) </STRONG></FONT></FONT></DIV>
<DIV align=justify><STRONG><FONT face=Arial>moving along the diagonal generates
most if not all known one dimensional dynamics. O</FONT><FONT face=Arial>ff
diagonal exploration </FONT></STRONG></DIV>
<DIV align=justify><FONT face=Arial><STRONG>yields all or most if not all known
in two dimensions (Hopf, tangent, quasi-periodic, period
adding etc bifurcations, </STRONG></FONT></DIV>
<DIV align=justify><FONT face=Arial><STRONG>Arnold's tongues in a
dissipative system, etc. The output of the transitional parameter space
between </STRONG></FONT></DIV>
<DIV align=justify><FONT face=Arial><STRONG>one and two dimensions is
interesting and complex (and relatively unexplored). With some
patience, it can be very rich.</STRONG></FONT></DIV>
<DIV align=justify><STRONG><FONT face=Arial>The hierarchical self
similarity in some of the phase space objects is striking amd reminiscent
of the "fractional scaling" reported</FONT></STRONG></DIV>
<DIV align=justify><STRONG><FONT face=Arial>in some brain data.
</FONT></STRONG> </DIV>
<DIV align=justify><FONT face=Arial><STRONG></STRONG></FONT> </DIV>
<DIV align=justify><FONT face=Arial><STRONG>The system
is </STRONG></FONT><FONT face=Arial><STRONG>operated by four convenient
parameter "handles" and programmed in very close</STRONG></FONT></DIV>
<DIV align=justify><FONT face=Arial><STRONG>to machine language so that things
run relatively quickly. </STRONG></FONT></DIV>
<DIV align=justify><FONT face=Arial><STRONG></STRONG></FONT> </DIV>
<DIV align=justify><FONT face=Arial><STRONG>Cheers</STRONG></FONT></DIV>
<DIV align=justify><STRONG><FONT face=Arial></FONT></STRONG> </DIV>
<DIV align=justify><STRONG><FONT face=Arial>Arnold </FONT></STRONG></DIV>
<DIV align=justify><FONT face=Arial><STRONG></STRONG></FONT> </DIV>
<DIV align=justify>
<DIV><FONT face=Arial>
<DIV><FONT face=Arial>
<P class=MsoNormal style="MARGIN-BOTTOM: 12pt"><STRONG><SPAN>Arnold J. Mandell,
M.D.<BR>Cielo Institute, Asheville, NC<BR></SPAN><U><SPAN
style="FONT-SIZE: 10pt; COLOR: blue">www.cieloinstitute.org</SPAN></U><BR><U><SPAN
style="FONT-SIZE: 10pt; COLOR: blue"><A
href="">ajmandell@charter.net</A></SPAN></U><BR></STRONG><SPAN><STRONG>Clinical
Professor of Psychiatry and Human Behavior, Emory University,<BR>Adjunct
Professor of Mathematical Sciences, FAU, Boca Raton, FL<BR>Founding Chairman and
Professor Emeritus of Psychiatry (Neuroscience,<BR>Pharmacology, Mathematics)
UCSD, La Jolla, CA<BR>MacArthur Prize Fellow Laureate in Theoretical
Neuroscience<BR>Humboldt Prize Fellow Laureate in Dynamical Systems (Bremen
University). </STRONG></SPAN></P></FONT></DIV>
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