[Eeglablist] IC figures

[Slater, Jeremy D]

Very nice explanation. One minor but important note is that it is not quite the situation that "each ICA component [is] a signal generated inside the brain". Most likely are. But any other sufficiently strong non-brain electrical signal that may get time-locked to your EP stimulus, such as eye-blink, can be separated out as an ICA component as well. One can take advantage of this using ICA as a filter, but it's probably not a bad idea to be aware that not everything that contributes to scalp surface electrical topography necessarily originates in the central nervous system ;-)
 
Regards,
 
Jeremy
 
Jeremy D. Slater, MD
Director, Texas Comprehensive Epilepsy Program
University of Texas - Houston Medical School
6431 Fannin Street
MSB 7.100
Houston, TX 77030
 
Phone: 713-500-7106
Fax: 713-500-7120

________________________________

From: eeglablist-bounces at sccn.ucsd.edu on behalf of Germán Gómez-Herrero
Sent: Fri 11/23/2007 5:20 AM
To: 'Nikolai Novitski'; eeglablist at sccn.ucsd.edu
Subject: Re: [Eeglablist] IC figures



Hi Nikolai,

Imagine that in an ERP or EEG experiment we have only 2 brain populations
(P1,P2) that are active and whose bioelectrical activity are producing most
of the variance in the scalp EEG. Let us call s1(t) and s2(t) the electrical
activation patterns that are being generated at P1 and P2, respectively.
Then, if we call x1(t) and x2(t) the signals acquired in two scalp
electrodes we can write (based on the quasistatic approximation of brain
volume conduction):

EQUATION 1:

x1(t) = a11*s1(t)+a12*s2(t)
x2(t) = a21*s1(t)+a22*s2(t)

where a11,a12,a21,a22 are just some scalar values modeling the electrical
transfer from the locations P1,P2 to the electrodes locations (i.e. the
volume conduction effects).

Then, if we further assume that s1(t) and s2(t) are statistically
independent from each other then, we can use ICA to estimate a (randomly
scaled version of) the transfer coefficients a11,a12,a21,a22 as well as a
(randomly scaled version of the) source activations s1(t) and s2(t) based on
only on the observed scalp signals x1(t) and x2(t), that is:

[a11/k, a12/k, a21/k, a22/k, k*s1(t), k*s2(t)] = ICA(x1(t),x2(t))

Where k is an unknown scaling factor. Since the brain generators P1,P2 are
independent it makes sense to study each of them separately. Thanks to ICA
we know all the variables involved in the system of equations above
(EQUATION 1) except for the factor k. Then to study the contribution to the
EEG of the first ICA component (s1(t)) we can just set to zero s2(t) and see
that EQUATION 1 becomes:

EQUATION 2:
x1(t) = (a11/k)*(k*s1(t)) = a11*s1(t)
x2(t) = (a21/k)*(k*s1(t)) = a21*s1(t)

So this means that the scalp EEG at any electrode is just a scaled version
of the source activation computed by ICA. Therefore, each independent
component is identified by its (randomly scaled) activation k*s1(t) and the
scaling factors for each electrode a11/k,a21/k. EEGLAB uses k*s1(t) to plot
the spectrum and the component ERP (if epoched). Therefore the component
spectrum plotted in EEGLAB does not correspond to any scalp location, it is
just the spectrum of a randomly scaled version of the signal that is
actually being generated inside the brain, in the brain population P1.

When EEGLAB plots the scalp topography of component 1 it actually plots the
values (a11/k),(a21/k). Note from EQUATION 2 that those values would be the
values of the actual scalp potentials only when s1(t)=1/k. In what time
instant does that happen? We can't know since the scaling factor k is
unknown and therefore we can't know when s1(t) is going to take an unknown
value :). However, note that that the ratio between the potentials at
different scalp locations is constant at ANY time:

x1(t)/x2(t) = [(a11/k)*(k*s1(t))] / [(a21/k)*(k*s1(t))]   =  a11/a21

These relative values x1(t)/x2(t) conceptually tell us whether component 1
was generated in an brain area closer to electrode x1 or closer to electrode
x2. Thus, they are all we need for localizing in the brain the population P1
using any of the inverse methods available in the literature (e.g. LORETA or
just your own intuition). Then to your question that at what time the scalp
distribution is plotted you can say that the relative values of that scalp
distribution x1(t)/x2(t) are the same for ANY time but the actual values
x1(t),x2(t) are arbitrary and do not correspond to any certain time instant.

So summarizing, you have to understand each ICA component as a signal
generated inside the brain. Imagine that s1(t) would be a sinusoid generated
somewhere in the temporal cortex, imagine also that electrode x1 is located
in the temporal lobe and x2 in the occipital lobe. Then, component 1 has a
single spectrum (an impulse at the frequency of the sinusoid). If only
component 1 would be active, the signals acquired in the scalp would be just
scaled sinusoids and so their spectrums would also be just scaled impulses
at the frequency of that sinusoid. Furthermore, the fact that
(a11/k)/(a21/k) is quite large tells you that component 1 might be located
much closer to electrode x1 than to electrode x2, i.e. it is located in the
temporal cortex. Note that we know this without knowing the actual values of
a11 and a21 (i.e. we do not know k) but just from the fact that
(a11/k)/(a21/k)=a11/a21 is large.

My explanations above quite simplistic and discard many important issues but
I think they capture the main idea. Probably other EEGLAB users or
developers will tell you more.

Hope that helps,
Germán

---------------------------------------------------------------------
Germán Gómez-Herrero
M. Sc., Researcher
Tampere University of Technology
P.O. Box 553, FI-33101, Tampere, Finland
Phone:   +358 3 3115 4519
Mobile:  +358 40 5011256
Fax:     +358 3 3115 4989
http://www.cs.tut.fi/~gomezher/index.htm

From: eeglablist-bounces at sccn.ucsd.edu
[mailto:eeglablist-bounces at sccn.ucsd.edu] On Behalf Of Nikolai Novitski
Sent: Thursday, November 22, 2007 4:52 PM
To: eeglablist at sccn.ucsd.edu
Subject: [Eeglablist] IC figures

Dear all,
 
I have a very basic question about ICA.
When you plot component's properties (Plot>Component Properties ) , you get
a picture with component scalp distribution, component ERP (if epoched) and
component spectrum. My question is: at what time point the map is plotted
and at what scalp location the ERP is plotted? And how they are determined?
 
Thank you
Nikolai Novitski 


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