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<p class="ydp76ba64cbMsoNormal"><span style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">Iman,</span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"> Thank
you for the information. I could only find a power point
attachment of a simulation in your post. I did not
find a scientific publication where you compared the phase differences changes
between an original EEG recording and a ICA reconstruction after removing one
or more components. Please re-send your
study. Also please give the citation to
any of your publications or other’s publications where phase differences were
compared between the original EEG recording and post ICA reconstruction. It will be interesting to see if you found
similar changes like in the study by Montefusco-Siegmund et al or by Georges
Otte or even in the example pre vs post data files that you can download from
the internet. I am assuming that you
have downloaded the EEG data and then used a JTFA like the Hilbert transform or
even the FFT cross-spectrum to prove to yourself that the phase differences between
the original and the ICA reconstruction have not been preserved.</span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"> </span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">As for the mathematics concerning reconstruction from a
lower dimensional matrix to a higher dimensional matrix where there are no
simple linear transforms I refer you to Taken’s theorem where “<span style="color: rgb(34, 34, 34);">The reconstruction preserves the
properties of the dynamical system that do not change under smooth coordinate
changes, but it does not preserve the geometric shape of structures in phase
space.</span>” Also, in standard
differential geometry math courses the issue of lower dimensional manifold mapping
to higher dimensional manifolds shows a loss of information in all cases. Also, commonsense operates here where one tries
to reconstruct 19 channels of EEG using only 15 or 16 or 17 ICA components
hence a loss of information.</span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);"> </span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);">Finally, the brain is
not a total chaotic organ. As
demonstrated by many scientists (e.g., Nunez; Walter Freeman; Roberto-Pascual
Marqui; E. Roy John; Joel Lubar; etc) coherence and phase differences are well behaved
and highly reproducible within and between subjects. Coherence and phase are dependent on the
number and strength of connections between groups of neurons. Here is a URL to a study that tested Paul
Nunez’s two-compartmental model of Coherence and Phase Differences and found
that these measures vary as a function of distance and packing density:</span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);"><a href="http://www.appliedneuroscience.com/TWO-COMPARTMENTAL_MODEL_EEG_COHERENCE.pdf" rel="nofollow" target="_blank">http://www.appliedneuroscience.com/TWO-COMPARTMENTAL_MODEL_EEG_COHERENCE.pdf</a></span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);"> </span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);">Here is a url to a
study that used EEG LORETA correlations to replicate Diffusion Tensor Imaging
measures of connectivity in the brain:</span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);"><a href="http://www.appliedneuroscience.com/DTI-ThatcherHumanBrainmapping.pdf" rel="nofollow" target="_blank">http://www.appliedneuroscience.com/DTI-ThatcherHumanBrainmapping.pdf</a></span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);"> </span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);">Here is a url to a
study that measured phase lock and phase shift duration from birth to about 16
years of age in 458 and where phase differences were stable and well
behaved: http://www.appliedneuroscience.com/PhaseresetDevelopment.pdf</span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);"> </span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);">If you do a search of
the National Library of Medicine database (Pubmed) using the search terms “EEG coherence”
you will find 2,874 citations. There is
huge consistency in this vast literature which would be impossible if the brain
was totally chaotic.</span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);"> </span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);">Best regards,</span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);"> </span></p>
<p class="ydp76ba64cbMsoNormal"><span style="font-size: 12pt; line-height: 107%; font-family: "Times New Roman", serif; color: rgb(34, 34, 34);">Robert</span></p>
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