EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.

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Listing of the Relevant Connectivity Equations. All of the equations below can be evaluated using a hand calculator and the equations can be easily programmed by a competent programmer. See the sections above for details. The goal is to help develop standardization and simplification for the implementation of EEG connectivity measures: 1- Pearson Product Correlation Coefficient r = ∑ XY − ∑ X∑ ∑ ∑ 2 2 2 2 [ N X − ( X ) ][ N Y − ( Y ) ] ∑ N Y ∑ 2- The cospectrum and quadspectrum (see section 9): a(xf 1 ) = cosine coefficient for the frequency (f 1 ) for channel X b(xf 1 ) = sine coefficient for the frequency (f 1 ) for channel X u(yf 2 ) = cosine coefficient for the frequency (f 2 ) for channel Y v(yf 2 ) = sine coefficient for the frequency (f 2 ) for channel Y The cospectrum and quadspectrum are algebraically defined as: Cospectrum (f 1 ,f 2 ) = a(xf 1 ) u(yf 2 ) + b(xf 1 ) v(yf 2 ) Quadspectrum (f 1 ,f 2 ) = a(xf 1 ) v(yf 2 ) – b(xf 1 ) u(yf 2 ) 3- Auto-spectrum F(x) = (a 2 (x) + b 2 (x)) 4- The cross-spectrum amplitude: 2 2 = (( a( x) u( y) + b( x) v( y)) + ( a( x) v( y) − b( x) u( y)) 5- Coherence Coh (f) = ( ∑ N ( a( x) u( y) + b( x) v( y))) ∑ N ( a( x) 2 2 + b( x) 2 + ( ) ∑ N ∑ N ( a( x) v( y) − b( x) u( y))) u( y) 2 + v( y) 2 ) 2
6- Phase Delay or phase difference ( a( x) v( y) − b( x) u( y)) N Phase difference (f) = Arctan ∑ ( a( x) u( y) + b( x) v( y)) ∑ N 7- Auto Channel Cross-Frequency Coherence (f 1 ,f 2 ) (ACC) after complex demodulation: ( ACC = ∑ N ( a( x' f ) u( x' ' f ) + b( x' f ) v( x' ' f ))) 1 2 ∑ N ( a( x' f ) 1 1 2 2 + b( x' f ) 2 2 2 + ( ) ∑ N ∑ N ( a( x' f ) v( x' ' f ) − b( x' f ) u( x' ' f ))) u( x' ' f ) 2 1 2 2 + v( x' ' f ) 2 2 ) 1 2 2 8- Cross Channel Cross-Frequency Coherence (f 1 ,f 2 ) (CCC) for channels X and Y after complex demodulation ( CCC = ∑ N ( a( x' f ) u( y' f ) + b( x' f ) v( y' f ))) 1 2 ∑ N ( a( x' f ) 1 1 2 2 + b( x' f ) 2 2 2 + ( ) ∑ N ∑ N ( a( x' f ) v( y' f ) − b( x' f ) u( y' f ))) u( y' f ) 2 1 2 2 + v( y' f ) 2 2 ) 1 2 2 9- Auto Channel Cross-Frequency phase difference (f 1 , f 2 ) (ACP) after complex demodulation ACP = Arc tan ∑ N ∑ N ( a( x' f ) v( x' ' f ) − b( x' f ) u( x' ' f )) 1 ( a( x' f ) u( x'' f ) + b( x' f ) v( x'' f )) 1 2 2 1 1 2 2 10- Cross-Channel Cross-Frequency Phase Difference (f 1 ,f 2 ) (CCP) after complex demodulation CCP = Arc tan ∑ N ∑ N (( a( x' f ) v( y' f ) − b( x' f ) u( y' f )) 1 (( a( x' f ) u( y' f ) + b( x' f ) v( y' f )) 1 2 2 1 1 2 2
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(unpublished manuscript - Version 1
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29- Definition of the Cross Channel
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possible and this is why the cospec
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is because there are relatively lon
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Fig. 1 - Illustration of volume con
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Fig. 3- Example of cospectrum (in-p
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coefficient is a valid and importan
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Eq. 1- r = ∑ N ∑ N ( X ( X −
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Fig. 5 - Comparisons between the Br
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to a Pearson product-moment correla
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constitute the basis for all spectr
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The frequency analysis of a time se
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That is, the cross-spectrum power i
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Demo). 11- Third Compute Coherence
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100 90 80 70 60 50 40 30 20 10 0 0
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frame” of consciousness (Thatcher
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Fig. 9 - Various degrees and types
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Fig. 10 - Illustrations of phase re
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when the 1 st derivative of the pha
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Fig. 13 shows an example of two 10
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Figure 15 - The x-axis are differen
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Fig - 16. Top is coherence (y-axis)
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Coherence inflation is defined as a
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time over which the average is comp
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Nonetheless, “instantaneous” ph
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Example of conventional digital EEG
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score; 2- Simplification by providi
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the brain. The integration of QEEG
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Page 65 and 66: Fig. 22 - Illustration of phase str
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Page 69 and 70: Fig. 24 - Top is filtered EEG at 25
Page 71 and 72: Fig. 26 - Top is the spontaneous EE
Page 73 and 74: 27- How to compute Auto-channel Cro
Page 75 and 76: 30- Cross-Channel Cross-Frequency C
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Page 81 and 82: www.appliedneuroscience.com 37 - Co
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Page 85 and 86: Fig. 33 - Example of phase differen
Page 87 and 88: Fig. 35 - Example of 4 different fr
Page 89 and 90: Fig. 37 - Example of a measure of c
Page 91 and 92: Fig. 39- Illustration of how cross-
Page 93 and 94: Fig. 41- Example of cross-frequency
Page 95 and 96: Gray, C.M., Konig, P., Engle, A.K.
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Page 99 and 100: 35 - 48, 2007. Thatcher, R.W., Bive
Page 101 and 102: Cannon, R., Lubar, J., Sokhadze, E.
Page 103 and 104: Lubar, J., Congedo, M. and Askew, J
Page 105 and 106: Yoo, S. S. et al. Increasing cortic
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