Transcript Synchrony measures for newborn EEG analysis

Synchrony measures for newborn
EEG analysis
Amir Omidvarnia
3 June 2011
Outline
 The concept of cointegration
 Johansen test
 Bivariate phase synchronization
 Multivariate phase synchronization
 Surrogate data method for phase synchronization
 Results
 White noise
 Combination of white noise and random walk
 Asymm./Asynch. data
The concept of cointegration
 Suppose two drunkards are wandering aimlessly, while they
don’t know each other.
 Their movement can be considered as two independent
random walks.
 Mathematically, ‘random walk’ refers to a trajectory that
consists of taking successive random steps.
The concept of cointegration
Random walk
The concept of cointegration
Bivariate cointegration: A drunk and her dog [2]
 Now, imagine a drunk walking with her dog.
 Each of the two trajectories is still a random walk by itself.
But, the distance between two paths is fairly predictable, as
the location of the one can roughly tell us the location of the
other one.
The concept of cointegration
Bivariate cointegration: A drunk and her dog
The concept of cointegration
Bivariate cointegration
 A long-run equilibrium relationship between the drunk
and her dog causes a stationary distance between their random
walks.
 The co-movement between two random walks is called a
bivariate cointegrating relationship.
The concept of cointegration
Bivariate cointegration
x1(t), x2(t): Random walk
-d < αx1(t) +x2(t) < d
x  x1(t) ( / α)x2(t) ~ white noise
[α ] : cointegrating vector
The concept of cointegration
multivariate cointegration:
A sheep herd and Shepherd dogs [6]
 Imagine a herd of sheep wandering aimlessly in the field
(multiple random walks).
 Consider that a herding dog guards the flock by running
around and returning the sheep that have gone too far back to
the herd.
The concept of cointegration
multivariate cointegration:
A sheep herd and Shepherd dogs
 The dog makes a co-movement among the sheep. Thus, we can say
that there is a cointegrating relationship (rank of one) within the
sheep trajectories.
 It is obvious that two dogs (rank of two) are able to restrict the
movements within the flock more than one dog.
The higher cointegration rank,
the more restricted movements
Johansen test
 A test for estimation of the cointegrating vectors and co-
movements within a K-channel signal.
 Two types of hypothesis testing are applied on a mathematical
feature extracted from the signal:
Trace test
 H0: There are r cointegrating relations
 H1: There are K cointegrating relations
Maximum eigenvalue test
 H0: There are r cointegrating relations
 H1: There are r+1 cointegrating relations.
Johansen test
 Johansen test gives all linear combinations between the dimensions
which represent a stationary random process.
 Let x(t) be K-channel signal (t = 1,...,T). A cointegrating
relationship among the channels (out of r, r<K) can be
represented as follows:
C1x1(t)+ C2x2(t) +....+ CKxK(t) = zi(t)
i = 1,...,r
zi(t) : stationary white noise
r: number of cointegrating relationships
Phase synchronization
Phase synchronization
 Let x(t) = rx(t)*exp(i*φx(t)).
 Let y(t) = ry(t)*exp(i*φy(t)).
 x(t) and y(t) are phase-locked of order (m,n) if:
|m*φx(t)-n*φy(t)|<const.
 m and n are assumed to be integers (m/n should be
rational).
Is integer relationship between phases
necessary?
 Consider two signals x(t) and y(t) start from t=0
simultaneously.
 If m/n is rational, it means that there is a least common
multiple (LCM) between m and n at which x(t) and y(t) will
reach at the same time point.
 If m/n is irrational, x(t) and y(t) never reach to a same point
over time. So, they cannot be synchronized theoretically.
Is integer relationship between phases
necessary?
 An example of a two-dimensional state space for two multiple-
frequency signals.
ref: Wikipedia
 The trajectory is a closed curve which implies that two signals
have started from a single point and finished at the same point.
 The total number of turns around the middle point is always
integer (so-called winding number of rotation number).
Is integer relationship between phases
necessary?
x(t) = sin(2*t)
y(t) = sin(3*t)
x(t) = sin(2*t)
y(t) = sin(2.1*t)
Bivariate phase synchronization
measures
 Mean Phase Coherence (MPC) [3]
a. Analytic signals of two real signals x1(t) and x2(t) are
computed using Hilbert transform.
b. Phase traces of the analytic signals are extracted
(φ1(t), φ2(t) ).
MPC = sqrt( (cos(φ1(t)-φ2(t)))2 + (sin(φ1(t)-φ2(t)))2 )
Bivariate phase synchronization
measures
 Phase Locking Value (PLV) [5]
a. Each signal of x1(t) and x2(t) is filtered by a FIR band pass
filter around a central frequency (f±2 Hz).
b. Both signals are convolved by a complex Gabor wavelet.
c. The amplitude of the phase difference of the convolved signals
is extracted (θ(t)=φ1(t)-φ2(t) ).
Multivariate phase synchronization [6]
 Let φX(t) be the multivariate phase signal extracted from the
K-channel X(t)=(X1(t),..., XK(t)), t=1,...,T.
 Let r be the number of cointegrating relationships within the
phase dimensions (r ≤ K).
 Using cointegration analysis, r linear relationships can be
obtained between the phase dimensions as follows:
α1φX1(t)+ α2φX2(t)+....+ αKφXK(t)=zi(t)
i=1,...,r, r ≤ K
zi(t)=N(μ,σ): Gaussian white noise
Multivariate phase synchronization
 We can consider a phase synchronization of rank r within the
dimensions of the multichannel signal X(t).
 Despite the classical definition of phase synchronization, coefficients of
the phase signals (αi) are not necessarily integer.
r=0  Complete asynchrony
r=K  Complete synchrony
The higher cointegration rank,
the stronger long-run relationship
within phase signals
Surrogate data
(only for bivariate measures)
 For each segment, the first electrode is kept unchanged and the
second electrode is shuffled.
x1'(n)  x1(n)
x2'(n)x2(shuffle(n))
 The phase synchronization measure is then extracted between
x1'(n) and a certain number of x2'(n) to build the null distribution.
 The measure extracted from x1(n) and x2(n) is compared with the
null distribution at a certain confidence level.
Preliminary results
Used parameters
 Fs: 256 Hz
 Number of channels: 8
 Channels used for the EEG data: (Fp2-F4), (F4-C4),(C4-P4),
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(P4-O2), (Fp1-F3), (F3-C3), (C3-P3), (P3-O1)
Signal length: 2 minutes (30720 samples)
Window length: 5 seconds
Overlap: 2.5 seconds
number of surrogates (only for MPC and PLV): 50
Confidence level: 99%
Multichannel white noise
Complete synchrony
Two-channel signal
8-channel signal
Combination of random walk and
white noise
Partial synchrony
Two-channel signal
8-channel signal
Asynchrony EEG data (from Sampsa)
First 30 seconds
Asynchrony EEG data (from Sampsa)
Second 30 seconds
Asynchrony EEG data (from Sampsa)
Third 30 seconds
Asynchrony EEG data (from Sampsa)
Fourth 30 seconds
Conclusion
 Cointegration rank is always above 1. This observation may
reflect the long-run equilibrium relationships between all
channels (e.g., SAT’s and artifact).
 The phase distance between channels need to be limited in a
certain band. Therefore, the parameters of zi(t) should be
controlled.
 Bivariate measures fluctuate much more than the
multivariate one. Therefore, their interpretation over time
doesn't seem such straightforward.
Questions
 Can the cointegrating relationships within the EEG channels
(or their phase signals) be related to mutual brain source
activities?
 Does the cointegration-based phase synchrony have any
meaning from medical point of view?
 Is it necessary to only investigate the phase difference of the
integer multiples of the phase signals? What are the probable
drawbacks of extending phase synchrony measures to noninteger coefficients?
Useful references
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[1]
S. Vanhatalo, and J. M. Palva, “Phase brings a new phase to the exploration of the elusive
neonatal EEG,” Clinical Neurophysiology, vol. 122, no. 4, pp. 645-647, 2011.

[2]
M. P. Murray, “A Drunk and Her Dog: An Illustration of Cointegration and Error
Correction,” The American Statistician, vol. 48, no. 1, pp. 37-39, 1994.
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[3]
F. Mormann, K. Lehnertz, P. David et al., “Mean phase coherence as a measure for phase
synchronization and its application to the EEG of epilepsy patients,” Physica D: Nonlinear
Phenomena, vol. 144, no. 3-4, pp. 358-369, 2000.
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[4]
M. Korürek, and A. Özkaya, “A new method to estimate short-run and long-run
interaction mechanisms in interictal state,” Digital Signal Processing, vol. 20, no. 2, pp. 347-358,
2010.
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[5]
J.-P. Lachaux, E. Rodriguez, J. Martinerie et al., “Measuring phase synchrony in brain
signals,” Human Brain Mapping, vol. 8, no. 4, pp. 194-208, 1999.
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[6]
W. Chaovalitwongse, P. M. Pardalos, P. Xanthopoulos et al., "Analysis of Multichannel
EEG Recordings Based on Generalized Phase Synchronization and Cointegrated VAR,"
Computational Neuroscience, Springer Optimization and Its Applications, pp. 317-339: Springer
New York, 2010.
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[7]
http://www.r-bloggers.com/introduction-to-cointegration-and-pairs-trading/