Amir Omidvarnia 3 June 2011 Synchrony measures for newborn EEG analysis

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 x 1 (t), x 2 (t): Random walk -d < α x 1 (t) +  x 2 (t) < d  x  x 1 (t)  (  / α )x 2 (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 higher cointegration rank, the more restricted movementsthe 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 H 0 : There are r cointegrating relations H 1 : There are K cointegrating relations  Maximum eigenvalue test H 0 : There are r cointegrating relations H 1 : 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: C 1 x 1 (t)+ C 2 x 2 (t) C K x K (t) = z i (t) i = 1,...,r z i (t) : stationary white noise r: number of cointegrating relationships

Phase synchronization

Let x(t) = r x (t) * exp(i * φ x (t)). Let y(t) = r y (t) * exp(i * φ y (t)). x(t) and y(t) are phase-locked of order (m,n) if: |m * φ x (t)-n * φ y (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. 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). ref: Wikipedia

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 x 1 (t) and x 2 (t) are computed using Hilbert transform. b. Phase traces of the analytic signals are extracted ( φ 1 (t), φ 2 (t) ). MPC = sqrt( (cos( φ 1 (t)- φ 2 (t))) 2 + (sin( φ 1 (t)- φ 2 (t))) 2 )

Bivariate phase synchronization measures Phase Locking Value (PLV) [5] a. Each signal of x 1 (t) and x 2 (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)= φ 1 (t)- φ 2 (t) ).

Multivariate phase synchronization [6] Let φ X (t) be the multivariate phase signal extracted from the K-channel X(t)=(X 1 (t),..., X K (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: α 1 φ X 1 (t)+ α 2 φ X 2 (t) α K φ X K (t)=z i (t) i=1,...,r, r ≤ K z i (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 ( α i ) are not necessarily integer. r=0  Complete asynchrony r=K  Complete synchrony The higher cointegration rank,The higher cointegration rank, the stronger long-run relationshipthe stronger long-run relationship within phase signalswithin phase signals

Surrogate data (only for bivariate measures) For each segment, the first electrode is kept unchanged and the second electrode is shuffled. x 1 '(n)  x 1 (n) x 2 '(n)  x 2 (shuffle(n)) The phase synchronization measure is then extracted between x 1 '(n) and a certain number of x 2 '(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), (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 Two-channel signal8-channel signal Complete synchrony

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 z i (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 non- integer coefficients?

Useful references [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 , [2]M. P. Murray, “A Drunk and Her Dog: An Illustration of Cointegration and Error Correction,” The American Statistician, vol. 48, no. 1, pp , [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 , [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 , [5]J.-P. Lachaux, E. Rodriguez, J. Martinerie et al., “Measuring phase synchrony in brain signals,” Human Brain Mapping, vol. 8, no. 4, pp , [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 : Springer New York, [7]