EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.
EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.
EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.
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Listing of the Relevant <strong>Connectivity</strong> Equations. All of the equations<br />
below can be evaluated using a h<strong>and</strong> calculator <strong>and</strong> the equations can<br />
be easily programmed by a competent programmer. See the sections<br />
above for details. The goal is to help develop st<strong>and</strong>ardization <strong>and</strong><br />
simplification for the implementation of <strong>EEG</strong> connectivity measures:<br />
1- Pearson Product Correlation Coefficient<br />
r<br />
=<br />
∑ XY − ∑ X∑<br />
∑ ∑<br />
2<br />
2<br />
2<br />
2<br />
[ N X − ( X ) ][ N Y − ( Y ) ]<br />
∑<br />
N<br />
Y<br />
∑<br />
2- The cospectrum <strong>and</strong> quadspectrum (see section 9):<br />
a(xf 1 ) = cosine coefficient for the frequency (f 1 ) for channel X<br />
b(xf 1 ) = sine coefficient for the frequency (f 1 ) for channel X<br />
u(yf 2 ) = cosine coefficient for the frequency (f 2 ) for channel Y<br />
v(yf 2 ) = sine coefficient for the frequency (f 2 ) for channel Y<br />
The cospectrum <strong>and</strong> quadspectrum are algebraically defined as:<br />
Cospectrum (f 1 ,f 2 ) = a(xf 1 ) u(yf 2 ) + b(xf 1 ) v(yf 2 )<br />
Quadspectrum (f 1 ,f 2 ) = a(xf 1 ) v(yf 2 ) – b(xf 1 ) u(yf 2 )<br />
3- Auto-spectrum<br />
F(x) = (a 2 (x) + b 2 (x))<br />
4- The cross-spectrum amplitude:<br />
2<br />
2<br />
=<br />
(( a(<br />
x)<br />
u(<br />
y)<br />
+ b(<br />
x)<br />
v(<br />
y))<br />
+ ( a(<br />
x)<br />
v(<br />
y)<br />
− b(<br />
x)<br />
u(<br />
y))<br />
5- Coherence<br />
Coh (f) =<br />
(<br />
∑<br />
N<br />
( a(<br />
x)<br />
u(<br />
y)<br />
+ b(<br />
x)<br />
v(<br />
y)))<br />
∑<br />
N<br />
( a(<br />
x)<br />
2<br />
2<br />
+ b(<br />
x)<br />
2<br />
+ (<br />
)<br />
∑<br />
N<br />
∑<br />
N<br />
( a(<br />
x)<br />
v(<br />
y)<br />
− b(<br />
x)<br />
u(<br />
y)))<br />
u(<br />
y)<br />
2<br />
+ v(<br />
y)<br />
2<br />
)<br />
2