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

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Fig. 4 – The correlation coefficient (and coherence) includes at least two possible couplings and mixtures of these two types of coupling: 1- where neuron A influences neuron B and vice versa and, 2- where a third neuron ‘C’ influences neuron A and neuron B and there is no connection between A and B. Comodulation omits the standard ‘C’ possibility and is limited to where neuron A influences neuron B and vice versa. The limitation of the term “comodulation” is that without partial correlation analyses or path analyses it is not possible to omit coupling number 2 which means that the term comodulation can be misleading unless these additional analyses are conducted. As discussed by Pikovsky et al (2003) the term modulation is complicated and it is possible for there to be modulation without synchronization and synchronization without modulation. As stated by Pikovsky et al (2003, p. 77) “Generally, modulation without synchronization is observed when a force affects oscillations, but cannot adjust their frequency.” Without further analyses to determine this distinction it is best to simply refer to amplitude or power correlation. The distinguishing characteristic of the application of the Pearson product correlation coefficient is the computation of the time course of the normalized covariance of spectra over an interval of time:
Eq. 1- r = ∑ N ∑ N ( X ( X − − X )( Y − Y ) X ) 2 ∑ N ( Y − Y ) 2 or the computationally simpler equation that one can compute more easily using a hand calculator: Eq. 2 - r = ( N N∑ XY − ∑ X∑Y 2 2 2 ∑ X − ( ∑ X ) )( N∑Y − ( ∑ Y ) 2 ) For example, if one computes the FFT over 1 second epochs for a 60 second recording period, i.e., N = 60, then the number of degrees of freedom in the spectral correlation coefficient (SCC) for channels X and Y = 60 – 1 = 59. For 59 degrees of freedom then a correlation of 0.258 or higher is statistically significant at P < .05. This is a valid and commonly used connectivity measure, however, it is important to remember that the correlation coefficient includes volume conduction + network connectivity, i.e., they are inextricably confounded. This is because the correlation coefficient omits phase difference and involves the “in-phase” or autospectral values and therefore volume conduction can not be separated and eliminated. This makes it more difficult to know if factors such as the number and strength of connections are what are changing due to experimental control or is it the “volume conduction” that is changing? As explained in section 8, coherence using complex numbers and phase differences separate volume conduction from network dynamics and automatically solve this problem. Another method of applying the Pearson Product correlation was developed by Lexicor, Inc. in the 1990s. This method computes the correlation between EEG spectra measured from two different locations and uses the individual spectral bin values within a frequency band. For example, if there are five frequency bins in the alpha frequency band (i.e., 8Hz, 9Hz, 10Hz, 11Hz and 12Hz), then N = 5 and the number of degrees of freedom = N – 1 = 4. When the degrees of freedom = 4 then a correlation coefficient of 0.961 or higher is necessary in order to achieve statistical significance at P < .05. Equations 1 and 2 are used to calculate the Pearson product correlation in both instances.
Page 1 and 2: (unpublished manuscript - Version 1
Page 3 and 4: 29- Definition of the Cross Channel
Page 5 and 6: possible and this is why the cospec
Page 7 and 8: is because there are relatively lon
Page 9 and 10: Fig. 1 - Illustration of volume con
Page 11 and 12: Fig. 3- Example of cospectrum (in-p
Page 13: coefficient is a valid and importan
Page 17 and 18: Fig. 5 - Comparisons between the Br
Page 19 and 20: to a Pearson product-moment correla
Page 21 and 22: constitute the basis for all spectr
Page 23 and 24: The frequency analysis of a time se
Page 25 and 26: That is, the cross-spectrum power i
Page 27 and 28: Demo). 11- Third Compute Coherence
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Page 31 and 32: frame” of consciousness (Thatcher
Page 33 and 34: Fig. 9 - Various degrees and types
Page 35 and 36: Fig. 10 - Illustrations of phase re
Page 37 and 38: when the 1 st derivative of the pha
Page 39 and 40: Fig. 13 shows an example of two 10
Page 41 and 42: Figure 15 - The x-axis are differen
Page 43 and 44: Fig - 16. Top is coherence (y-axis)
Page 45 and 46: Coherence inflation is defined as a
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Page 51 and 52: Example of conventional digital EEG
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Fig. 22 - Illustration of phase str
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measure of the area under the curve
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Fig. 24 - Top is filtered EEG at 25
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Fig. 26 - Top is the spontaneous EE
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27- How to compute Auto-channel Cro
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30- Cross-Channel Cross-Frequency C
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degrees or radians and is “normal
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exhibits statistically significant
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www.appliedneuroscience.com 37 - Co
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elationships as defined by the phas
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Fig. 33 - Example of phase differen
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Fig. 35 - Example of 4 different fr
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Fig. 37 - Example of a measure of c
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Fig. 39- Illustration of how cross-
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Fig. 41- Example of cross-frequency
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Gray, C.M., Konig, P., Engle, A.K.
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localization and functional connect
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35 - 48, 2007. Thatcher, R.W., Bive
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Cannon, R., Lubar, J., Sokhadze, E.
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Lubar, J., Congedo, M. and Askew, J
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Yoo, S. S. et al. Increasing cortic
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F and likewise, F iω0t iϕ t [ y e
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6- Phase Delay or phase difference
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