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

More documents

Recommendations

Info

The time series can be divided into two general categories: 1- the statistics of short-time biological data and other short-time interval events such as economic, sociological, etc. and 2- long time span events such as astronomical, meterological, geologic and geophysical data . . . . . . . – to be continued 44- Appendix – B Instantaneous Coherence and Phase Difference Complex demodulation was used to compute instantaneous coherence and phasedifferences (Granger and Hatanaka, 1964; Otnes and Enochson, 1972; Bloomfield, 2000). This method first multiples a time series by the complex function of a sine and cosine at a particular frequency followed by a low pass filter which removes all but very low frequencies and transforms the time series into instantaneous amplitude and phase and an “instantaneous” spectrum (Bloomfield, 2000). We place quotations around the term “instantaneous” to emphasize that there is always a trade-off between time resolution and frequency resolution. The broader the band width the higher the time resolution but the lower the frequency resolution and vice versa (Bloomfield, 2000). For example, if we multiply a time series {x t , t = 1, . . . , n} by sine ω 0 t and cos ω 0 t and then apply a low pass filter F, we have Z′ t = F( xt sinω 0t) Z′ ′= F( x cosω 0t) t 2 [ ] 1 / 2 2 and ( Z ) + ( Z′ ) t 2 ′ t t is an estimate of the “instantaneous” amplitude of the Z′ −1 t frequency ω 0 at time t and tan is an estimate of the “instantaneous” phase at ′′ Z t time t. The instantaneous cross-spectrum is computed when there are two time series {y t , t = 1, . . . , n} and {y’ t , t = 1, . . . , n} and if F [ ] is a filter passing only frequencies near zero, then, as above R t 2 2 2 iω [ ] [ ] [ ] 2 0t y sinω t + F y cosω t = F y e = F is the estimate of the amplitude of frequency ω 0 at time t and t 0 [ y ] ⎞ t sinω 0t [ y cosω t] ⎟⎟ t −1⎛ F ϕ t = tan ⎜⎜ is an estimate of the phase of frequency ω 0 at ⎝ F t 0 ⎠ time t and since 0 t
F and likewise, F iω0t iϕ t [ y e ] = R e t iω 0t iϕ t [ y′ e ] = R′ e t the instantaneous cross-spectrum is t t i t [ y e ] F[ y e ] ω 0 − i ω ′ t 0 Vt = F t t , = R R′ e t t [ ϕ −ϕ′ ] i t t and the instantaneous coherence is R V t ≡ 2 2 t R′t 1 however coherence is computed as the average of the sine and cosine functions over an interval of time or C ( t, ω ) = V 2 t t R R′ 2 t ϕ ϕ′ The instantaneous phase-difference is t − t which is also the arctan of the imaginary part of V t divided by the real part (or the quadspectrum divided by the cospectrum). Computation of the First Derivatives of the Time Series of Coherence and Phase Angles The first derivative of the time series of phase-differences between all pair wise combinations of two channels was computed in order to detect advancements and reset of phase-differences. The Savitzgy-Golay procedure was used to compute the first derivatives of the time series using a window length of 3 time points and the polynomial degree of 2 (Press et al, 1994). The units of the 1 st derivative are in degrees/point which is normalized to degrees/second and degrees/millisecond in the case of EEG. The second derivative was computed using a window length of 5 and a polynomial degree of 3 and is in units of degrees/second 2 or degrees/millisecond 2 in the case of EEG. For simplicity, in NeuroGuide the units of the first derivative of a phase time series is degrees per centiseconds (i.e., degrees/cs 2 = degrees/10 msec. 2 ). 45- Appendix – C
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 and 14:
coefficient is a valid and importan
Page 15 and 16:
Eq. 1- r = ∑ N ∑ N ( X ( X −
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
Page 29 and 30:
100 90 80 70 60 50 40 30 20 10 0 0
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
Page 47 and 48:
time over which the average is comp
Page 49 and 50:
Nonetheless, “instantaneous” ph
Page 51 and 52:
Example of conventional digital EEG
Page 53 and 54:
score; 2- Simplification by providi
Page 55 and 56: the brain. The integration of QEEG
Page 57 and 58: Multiple frequencies and multiple m
Page 59 and 60: Example of one of the progress char
Page 61 and 62: advantage of imaging of non-cortica
Page 63 and 64: where n is the number of observatio
Page 65 and 66: Fig. 22 - Illustration of phase str
Page 67 and 68: measure of the area under the curve
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
Page 77 and 78: degrees or radians and is “normal
Page 79 and 80: exhibits statistically significant
Page 81 and 82: www.appliedneuroscience.com 37 - Co
Page 83 and 84: elationships as defined by the phas
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.
Page 97 and 98: localization and functional connect
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: Yoo, S. S. et al. Increasing cortic
Page 109 and 110: 6- Phase Delay or phase difference
show all

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

Create successful ePaper yourself

Delete template?

Save as template?