Testing for significance of phase synchronisation dynamics in the EEG

Abstract

A number of tests exist to check for statistical significance of phase synchronisation within the Electroencephalogram (EEG); however, the majority suffer from a lack of generality and applicability. They may also fail to account for temporal dynamics in the phase synchronisation, regarding synchronisation as a constant state instead of a dynamical process. Therefore, a novel test is developed for identifying the statistical significance of phase synchronisation based upon a combination of work characterising temporal dynamics of multivariate time-series and Markov modelling. We show how this method is better able to assess the significance of phase synchronisation than a range of commonly used significance tests. We also show how the method may be applied to identify and classify significantly different phase synchronisation dynamics in both univariate and multivariate datasets.

Appendices

Appendix A: Semi-Markov models

To use a Semi-markov model two problems must be solved. The first of these is to identify, for a given set of trials, what the optimal associated prior, ε, transition Γ, and duration D probabilities are. The second problem is how to identify, for a given trial and two or more candidate models, which model is most likely to have generated that trial.

Because there is no hidden layer in the model the first problem may be estimated directly. The number of presentations of each label in each trial, the number of transitions from each label to each other label, the durations of each label and the number of times the first label in a given trial is equal to a particular value may be counted. Transition probabilities and prior probabilities may then be estimated via a maximum likelihood framework. Sojourn time probabilities may be estimated against a distribution. In this study the truncated normal distribution is chosen based upon the observed distribution of state durations in the datasets investigated in this study.

To solve the second problem the forward algorithm for HMMs is adapted to accommodate state sojourn time probabilities. Formally, the forward variable is defined as

$$ \alpha_t(q,l) = P(y_{t-l+1:t} | q,l) \sum\limits_{q'}\sum\limits_{l'} P(q,l|q',l')\alpha_{t-l}(q',l'), $$

(40)

where P(y _{t − l + 1:t} | q,l) denotes the probability of continuing to produce observations y for l steps given that the current state is q and the current length of duration in that state is l.

As with a HMM the forward variable may be solved by induction. Note that, as with HMMs, initial parameters are again drawn from uniform distributions.

• 1. Initialise the variable:

  $$ \alpha_0^*(j) = \pi_{(j)}. $$

  (41)

• 2. Induction:

  $$ \alpha_{t}(j) = \sum\limits_{d} P(y_{t-d+1:t} | j,d) P(d|j) \alpha_{t-d}^*(j), $$

  (42)
  $$ \alpha_t^*(j) = \sum\limits_i \alpha_t(i) a_{i,j}. $$

  (43)

• 3. Terminate:

  $$ P(O|\lambda) = \sum\limits_{i = 1}^N \alpha_T(i). $$

  (44)

The termination step to the forward variable calculation provides the solution to the second problem, the probability of a set of observations given a particular model. If a range of candidate models λ _i , i = 1,...,M are available, each trained on EEG recorded under a different condition, then the forward algorithm may be used to calculate P(O|λ _i ) for all M models. The most likely model is the one that yields the highest likelihood of the data sequence (assuming the priors for each model are equally likely). The data sequence is then classified as being recorded under the same conditions on which the model was trained.

Appendix B: Algorithm

The algorithm applied to model the dynamics of phase synchronisation by combining GPS with Markov modelling is summarised here. All development is done in Matlab version 7.11.

• 1. For each EEG channel calculate the instantaneous phase according to Eq. (3).

• 2. Reference the instantaneous phase to the phase on a reference channel as per Eq. (5).

• 3. At each sample point put the references phases on each channel into a phase vector (see Eq. (6)).

• 4. Calculate the instantaneous instability index via Eq. (7) and one of the Eqs. (8), (9), (10), or (11). Note, Eq. (11) is shown, in Section 5.1, to be prefereable.

• 5. Segment the instantaneous instability index into global phase synchronisation periods at points where the $50^{\mbox{th}}$ percentile is crossed according to Eqs. (12) and (13).

• 6. Cluster the GPS segments via K-means clustering with K = 6.

• 7. Train either an HMM on the GPS sequence or an SMM (as detailed in Appendix A).