Testing for significance of phase synchronisation dynamics in the EEG

Journal of Computational Neuroscience manuscript No. (will be inserted by the editor)

Testing for significance of phase synchronisation dynamics in the EEG Ian Daly · Catherine M. Sweeney-Reed · Slawomir J. Nasuto

Received: date / Accepted: date

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 also may 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. Keywords Phase synchronisation · statistical significance testing · EEG · HMM · SMM

I. Daly Institute for Knowledge Discovery, Laboratory of BrainComputer Interfaces, Graz University of Technology, Inffeldgasse 13/4, 8010 Graz, Austria Tel.: +43 316 873 30711 E-mail: [email protected] C. M. Sweeney-Reed Memory and Consciousness Research Group, University Clinic for Neurology and Stereotactic Neurosurgery, Medical Faculty, Otto von Guericke University, Leipziger Strasse 44, 39120 Magdeburg, Germany S.J. Nasuto Centre for Integrative Neuroscience and Neurodynamics, University of Reading, Reading, Berkshire, UK

1 Introduction Long range phase relationships in the electroencephalogram (EEG) may provide information about functional relationships between distinct cortical regions in the brain. In particular, long range phase synchronisation between two or more different cortical regions may indicate direct communication, a shared input or a common pathway of information flow between those regions [Lachaux et al., 1999, Alba et al., 2007]. Distributed cortical regions may thus form phase synchronisation networks during cognitive processing tasks such as movement generation and memory retention, which are reflective of the underlying functional relationships, both neighbouring and at distance to one another [Alba et al., 2007, Sauseng and Klimesch, 2008, Jensen and Tesche, 2002, Tallon-Baudry et al., 1996]. Synchrony is defined statistically, as a constant phase relationship between two signals over time, and we introduce here a novel approach to its assssment. We begin, however, by clarifying what we mean by phase synchrony. The difference between long and short range phase synchronisation is in the length of the scales - the former typically would be observed over distances of > 1−2cm and the latter over distances below 1-2cm [Menon et al., 1996]. Thus typically, short range synchronisation is usually only possible to note via local field potentials (LFPs) or via electrocorticography (ECoG) due to the spatial distribution of the EEG electrodes. However, long and short range synchronisation do not differ in terms of the index that defines them. Hence, this work is equally applicable to short range synchronisation, although only long range synchronisation is considered and tested here. Long and short range synchronisation are not to be confused with the event-related (de)synchronisation

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(ERD/S) which is a change in the amplitude of oscillations, most commonly seen in the alpha or beta band, and reflected in a change in spectral power [Pfurtscheller, 1997]. ERD/S is associated with the time series recorded during a number of cognitive processes including, for example, motor planning and execution and may be related to changes in synchrony of underlying populations of neurons [Pfurtscheller and Lopes da Silva, 1999]. Long range phase synchronisation, on the other hand, is a relative shift of phase in oscillations such that two or more time series have a stable phase difference over a period of time [Mizuhara et al., 2004]. This does not imply that either time series would experience a shift in spectral power. Hence, while the ERD/S may relate to synchronisation of neuronal populations, long range phase synchronisation does not imply ERD/S. To clarify, in this work we consider only long range phase synchronisation, however, for the sake of readability we abbreviate this to just phase synchronisation throughout. When identifying periods of phase synchronisation in the EEG it is important to determine whether the observed period of synchronisation relates to a process of physiological interest or whether the observed synchronisation occurred simply by chance. In other words, it is important to identify whether the phase synchronisation is statistically significant. Consider the case of two time series generated by independent, zero mean, unit variance Gaussian processes. When considered over sufficiently long time periods the probability that the two Gaussian processes exhibit short term synchronisation is increasingly likely. Thus, it would be desirable to discard such short term synchronisation as it is likely that it arose by chance from independent processes. On the other hand, two or more signals recorded from the brain may exhibit synchronisation for a short period of time followed by desynchronisation as their corresponding neural generators briefly communicate. It is clearly desirable to identify such short term synchronisation as a significant diversion from on-going activity, particularly in cases where such short term synchrony is part of an on-going pattern of synchronisation and desynchronisation which is stable over trials. The question, when considering phase synchrony, is how to ensure that observed periods of synchronisation may be accepted as significant and therefore likely to relate to a physiologically meaningful process. Indeed, very short periods of synchrony may be physiologically meaningful due to their part in a larger dynamic pattern of synchrony and desynchrony over time [Rubinov and Sporns, 2009, Stam, 2005]. This question is particularly challenging given that the EEG is a non-linear

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process exhibiting large degrees of nonstationarity and temporal, as well as inter-subject, variability. Consider the case where a particular period of synchronisation is observed in the EEG in response to, for example, the presentation of a particular stimulus. If one repeats this stimulus presentation and fails to observe the same period of synchronisation, that does not, necessarily, mean the synchronisation was not significant. The period of synchronisation may, for example, be observed slightly later in the signal (temporal jitter) or may be observed between different EEG channels (spatial jitter) [Engel et al., 1992]. Commonly used approaches to identify statistically significant periods of phase synchronisation in the EEG are to use tests based upon estimates of parametric distributions or bootstrapping methods. Parametric tests estimate the significance of a period of phase synchronisation against an estimate of the parameters of some well-chosen probability distribution [Allefeld and Kurths, 2004]. They therefore require some assumptions to be made about the underlying distribution of phases which may not be accurate. Bootstrapping, by contrast, generates replications of the phases of each signal under the null hypothesis that the phases are unrelated to one another. This may be done, for example, by randomly shuffling the order of the phases in the time series. The distribution of synchronisation under this null hypothesis is then estimated from the bootstrap replications and the phase synchronisation may be identified as significant if it falls below the α significance level. Such an approach therefore does not make any assumptions about the particular probability distribution of the phases. However, bootstrapping significance tests are not without their problems. Bootstrapping is based upon the assumption that samples are statistically independent of one another. The EEG time series exhibits large amounts of temporal, spatial and spectral dependencies which may, therefore, invalidate the traditional bootstrapping approach. This may be avoided by generating bootstraps by moving block re-sorting [Vogel and Shallcross, 1996]. In place of swapping individual sample points, blocks of several consecutive samples are resorted. However such an approach gives rise to further questions as to how one chooses the block size. Further to this, multiple sample points in the EEG time-series necessitate a multiple comparisons correction. The Bonferroni correction is the most commonly used method in such cases [Genovese and Wasserman, 2002]. The significance level is adjusted by the number of comparisons. Thus if 100 comparisons are made, 1 for each of 100 sample points, the significance level is adjusted by 1/100.

Testing for significance of phase synchronisation dynamics in the EEG

However, as with bootstrapping, Bonferroni correction is most accurate for independent tests (sample points) and may thus be overly conservative for EEG which has large degrees of dependency. In other words, if two consecutive sample points both fail to be rejected as insignificant via bootstrapping it is more likely that they are actually significant then if two non-consecutive sample points fail to be rejected. An alternative approach proposed in [Singh and Phillips, 2010] is to use hierarchical significance testing. The significance testing is broken down into a time-frequency hierarchy and hypothesis testing proceeds downwards. If there is a failure to reject the null hypothesis at a particular level then that node and its child nodes are pruned. Thus, this approach allows us to account for dependencies in the temporal, spectral or spatial dimensions. However, this approach requires for the EEG to be broken down into time-frequency groups in a welldefined manner driven by knowledge of the likely dependencies within the data. Otherwise, if no a priori knowledge exists about the dependencies, a search of different deconstruction methods must be performed. The temporal dynamics of phase synchrony in the EEG have been noted in a number of studies to be an important aspect in cognitive processing [Ito et al., 2007, Daly et al., 2011]. When considering phase synchronisation from such a view point the specific timing of periods of synchrony - subject to large amounts of variability - is, perhaps, less important than the underlying dynamical processes. Tests for the statistical significance of the temporal dynamics of phase synchronisation may therefore be designed to be robust to large amounts of variability while reliably identifying whether the dynamics of the synchronisation process are significant. Therefore, a novel test is introduced to identify the statistical significance of phase synchronisation dynamics in the EEG. Markov modelling of global patterns of phase synchronisation is used to model the temporal dynamics of synchronisation patterns across the cortex to determine if those patterns are significant or not. The approach taken models the dynamics of phase synchronisation as a probabilistic process. Hence, consecutive periods of synchronisation are more likely to be modelled and determined to be significant. Further to this, the use of global phase vectors [Ito et al., 2007] in conjunction with Markov modelling allows for the test to be robust to the effects of temporal and spatial drift in the locations of significant periods of phase synchronisation. To the best of our knowledge such a combination is proposed for the first time in the present study. Un-

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til now the global phase synchronisation method has been used to study phase stability in the resting brain [Ito et al., 2007] and Markov models have been applied to model temporal dynamics in a wide range of applications. The combination of the approaches allows the modelling of more complex datasets containing sequences of patterns of phase synchronisation periods. The method is first contrasted with traditional tests used to identify statistically significant synchronisation, which are shown to be inadequate when dealing with signals containing synchronisation which appears or disappears as a dynamical process. The method is evaluated against a range of current tests of statistical significance of phase synchronisation with datasets of increasing complexity and realism in approximating phase synchronisation patterns in EEG signals. Finally, the tests are evaluated against each other for their ability to identify different patterns of synchronisation in real EEG data. The method is first introduced in section 2, section 3 then introduces the tests for phase synchronisation significance against which it will be evaluated. Section 4 describes the datasets the methods will be evaluated on and section 5 presents the results of the comparisons of the tests on a number of datasets. Finally, section 6 discusses the results of the evaluation in the context of application to characterising and differentiating phase synchronisation dynamics in the EEG.

2 Method The dynamics of phase synchronisation patterns are characterised by combining the global phase synchronisation (GPS) approach [Ito et al., 2007] with Markov modelling methods [Rabiner and Juang, 1986]. GPS attempts to partition a multivariate phase sequence into periods of global phase stability. Markov models are then used to model the temporal dynamics of these partitioned periods of phase stability. Hence, the resulting method, which as far as we are aware is proposed for the first time in the present study, provides a novel way to test for particular temporal dynamics in multivariate phase sequences. Phase synchronisation is first defined before our method for identifying and classifying significantly different patterns of synchronisation dynamics is introduced.

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2.1 Phase synchronisation

2.2 Significance of phase dynamics

Phase of a narrow band signal may be defined (from [Allefeld and Kurths, 2004]) using an analytic representation of the signal

The aim of the following method is to characterise and classify the global temporal dynamics of phase synchronisation. The method may be applied to any number of signals from bivariate to multivariate trial analysis applied to EEG recorded across the entire cortex. First, - based upon an expansion of work in [Ito et al., 2007] - the phase values from a multivariate time series are used to define a relative phase vector by taking their phase relative to the phase on a reference channel. Note that this is not to be confused with scalp potential referencing which is performed as a pre-processing operation prior to any characterisation of the phase synchronisation. Formally,

x(t) = Aej(ωt+θ)

(1)

where A is the amplitude of signal x at time t, ω is the frequency, θ the phase, and the Hilbert transform, defined as

x ˆ(t) =

1¯ P π

∫

∞ −∞

x(τ ) dτ, t−τ

(2)

where P¯ is the Cauchy principal value operator and τ the convolution offset. Instantaneous phase at time t may then be calculated from the signal and its Hilbert transform as ( θ(t) = arctan

) x ˆ(t) . x(t)

(3)

Alternatively, Wavelets may be used to calculate the instantaneous phase by extracting the coefficient of the Wavelet transforms at the target frequency (as in [Mallat, 2009]). However, both the Wavelet and Hilbert transform based definitions of phase produce statistically insignificantly different results. In other words the two definitions are fundamentally equivalent when applied to neural signals [Le Van Quyen, 2001]. Therefore, the Hilbert transform is arbitrarily chosen for use in the remainder of this work. Phase synchronisation may be defined in the same way for either the Hilbert or Wavelet based definitions of phase. A pair of signals are said to be phase synchronised if their phase difference is stable over time. The phase difference is defined (from [Tass et al., 1998]) as Θn,m (t) = nθ1 (t) − mθ2 (t),

(4)

where θ1 (t), θ2 (t) define the instantaneous phases of signals 1 and 2, n and m are integers. ‘const’ defines a threshold below which the relative phase has to remain. Note, the terms n and m are both set to 1 throughout this work and are therefore omitted from future instances of this term. Thus a pair of signals are said to be phase synchronised if the difference between their phases at a particular pair of frequencies does not exceed a threshold value.

Φi (t) = θi (t) − θR (t),

(5)

where θi (t) defines the phase on channel i at time t and θR (t) defines the phase on a reference channel R at time t. To avoid bias in the choice of reference channel the mean of the difference between each channel and a small selection of reference channels may be taken. The number of channels in the multivariate set may range from 2, where one of the two channels is used as the reference, to the entire set of channels positioned over the cortex, and any subset in between, to focus on the phase synchronisation dynamics recorded over specific cortical regions. A relative phase pattern vector is then defined as Υ (t) = (Φ1 (t), ..., ΦN (t)),

(6)

where N is the number of channels on which relative phase Φi is calculated. The phase pattern vector characterises the phase synchronisation across the multivariate time series at a given moment in time. Thus, its temporal evolution is informative about the temporal dynamics of phase synchronisation across the cortex in the EEG. The patterns of phase synchronisation are segmented into regions of stable synchronisation. This is done via the Instantaneous Instability Index (I) of the relative phase pattern vectors, which is defined as v uN u∑ I(t) = t di (t)2

(7)

i=1

with di (t) = π − |π − |[Φi (t) − Φi (t − 1)]mod2π||

(8)


and represents the distance between Υ (t − 1) and Υ (t) on channel i. In [Ito et al., 2007] this method is chosen from a selection of methods for characterising phase dynamics in [Mardia and Jupp, 2000]. Other definitions of the term di (t) may be substituted here to characterise how the phase dynamics across the scalp change over time including the circular mean difference

di (t) =

N 1 ∑ {π − |(π − |Φh (t) − Φj (t)|)|}, N2

(9)

h=1

or di (t) = σ(t− δ¯ ):(t+ δ¯ ) , 2

(10)

2

where σ(t− δ¯ ):(t+ δ¯ ) is the standard deviation of the 2 2 phase differences between the two signals within the ¯ ¯ time window (t − 2δ ) : (t + 2δ ) where t are the time points in the signals and δ¯ is the length of the sliding time window used. An analogous method for calculating the circular mean difference may also be used

di (t) =

N 1 ∑ {1 − cos(Φi (t) − Φh (t))}. N2

(11)

h=1

These definitions provide alternative measures of, either the instantaneous distribution of phases across the scalp (9), (10), (11), or the difference between consecutive phase pattern vectors (8). An investigation into which of these methods is best suited to the partitioning of the EEG into periods of phase synchronisation is carried out as a part of this work. A period of phase synchronisation may be defined as a period for which I falls below a certain percentile; for example the 50th percentile - as used in [Ito et al., 2007] - may be used. A Global Phase Synchronisation (GPS) pattern vector is then defined across each of the periods of synchronisation. Formally, g pg = (Ξ1g , ..., ΞN ),

(12)

where

Ξig

−1

= tan

∑ g sin Φi (t) ∑ t∈l , t∈lg cos Φi (t)

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A period of synchronisation is denoted to be a period of consecutive samples lasting for 3 cycles for which I falls below the 50th percentile. The length of this period depends on the frequency at which synchronisation is sought, with twind = nocy/ωtarg , where nocy denotes the number of cycles, ωtarg the target frequency and twind the length (in seconds) of the period of consecutive samples [Lachaux et al., 2000]. We thereby aim to avoid spurious transient synchronisation that arises from the interaction of noisy signals while still including relatively short periods of synchronisation. The entire series of phase pattern vectors pg is then clustered and labelled via a K-means clustering approach to produce a labelled GPS time-series, sg , i.e. each GPS episode is given a label according to the Kmeans clustering approach. K cluster centroids are generated and GPS time-series points are grouped with their nearest centroid (as measured via Euclidian distance). In this work K = 6 based upon the choice made in [Ito et al., 2007]. (Different values of K were also tested and found to give broadly similar results.) Centroids are recalculated as the mean position of all the GPS time-series values grouped with them and GPS points are re-assigned. This process is iterated until all GPS points are clustered. GPS time-series points are assigned arbitrary labels according into which cluster they are grouped. This allows the identification and differentiation of different patterns of phase synchronisation dynamics within the time series. The temporal dynamics of phase synchronisation patterns (the labelled GPS time-series) is characterised by a Markov model which attempts to capture the temporal dynamics of the process by assuming an underlying stochastic system modelled by a series of state transitions. 2.3 Hidden Markov Model The Hidden Markov Model (HMM) is presented first. Each state in an HMM may generate observations according to its own probability distributions [Jelinek et al., 1975, Rabiner, 1989, Ani et al., 2004, Uguz et al., 2007]. Formally, an HMM may be defined by its prior (ϵ), state transition probability matrix (Γ ) and observation probabilities (B). These may be collated together as

(13)

where lg denotes the g th GPS episode, with 1 ≤ g ≤ M and M is equal to the number of GPS episodes. Thus, the vector pg gives the average phase pattern during a single episode of GPS.

λHMM = (Γ, B, ϵ)

(14)

where Γ = γι,κ = P (qκ at t + 1|qι at t); ι, κ = 1, 2, ..., S

(15)

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defines a state transition matrix, γι,κ denoting the probability of moving from state qι to state qκ , 1 ≤ γι,κ ≤ 0 and N ∑

γι,κ = 1, ∀ι ∈ [1, ..., S],

(16)

κ=1

with S denoting the total number of states in the model. We assume each state can generate observables - in the case of this work these would comprise the values taken by the labelled GPS time-series - and that this is done according to some probability distribution. The probability of generating a given observable from a specific state is obtained by B = bκ (k) where bι (k) = P (vk at t|qι at t)

(17)

and where vk is the generated GPS sequence label (generated via k-means clustering of GPS sequences), given that the current state is qι at time t. The initial state of the model at time t = 1 is defined by the prior state transition probabilities ϵι = P (qι at t = 1)

(18)

where ϵι defines the probability of being in state qι at time t = 1. HMMs may be used to model and classify the temporal dynamics of phase pattern vectors. Initial parameters are drawn from uniform distributions. Further details of how this may be done may be found in [Rabiner, 1989]. In this work the number of states in the HMM is set to 5 based upon experimentations with subsets of the data. The HMM toolbox provided by Murphy (1998) is used in this work Murphy [1998]. This is chosen due to its low computational cost and ready availability.

2.4 Semi-Markov Model An alternative approach investigated in this work is the Semi-Markov Model (SMM), an extension of the Markov chain. This is defined as a series of state transitions which are determined by a transition probability matrix and a set of prior probabilities defining the initial states of the model. SMMs add an additional state sojourn time probability determining the length of time the system is likely to remain in a single state. Figure 1 illustrates the HMM and SMM models. Both contain two states q1 and q2 which have associated conditional transition probabilities. For each state

in the HMM there are a number of possible observations - label values at each time point in the GPS time-series - each with probabilities of being observed given the state. Note that there is a self-transition probability for each state. In the case of the SMM each state may only generate one GPS label (hence there is no ”hidden” part to the model) and the self-transition probability is replaced with a duration probability. Formally an SMM is defined by λSMM = (ϵ, Γ, D)

(19)

where ϵ and Γ are defined as with the HMM and D defines the sojourn time probabilities Dι (∆) where ∆ is the number of steps for which the model remains in state ι. SMMs may be used to model processes for which the sojourn times are not subject to geometric probability distribution, for which there is an underlying stochastic transition process between states and for which each state generates only one observed value. They are therefore potentially better suited than HMMs to the modelling of the discrete labelled phase vectors produced by the GPS method. This is because the GPS sequences are not observed to remain in a state according to a geometric decay probability but rather until some period of synchronisation finishes, a length of time determined by complex interactions of underlying physiological phenomena. There are a couple of key problems to be solved before SMMs may be successfully applied to model and classify GPS patterns. Firstly, for a given set of trials containing sequences, of defined lengths, of labelled phase dynamics what are the associated prior, ϵ, transition Γ , and duration D probabilities that are most likely to have generated the observed GPS sequences. Secondly, for a given single GPS trial - containing a sequence of GPS episodes - and two or more candidate SMMs which model is most likely to have generated the observed GPS sequence. These are discussed in detail in Appendix A. 2.5 HMM/SMM voting To apply HMMs or SMMs to classify trials (GPS sequences of defined length) a voting based classification scheme is used. This allows multiple HMMs/SMMs trained on individual trials in a training set to inform the classification process for individual trials in a test set. Comparisons between this voting scheme and the use of single HMM/SMM classification shows the voting schemes to increase the classification accuracy. The voting scheme operates as follows

Testing for significance of phase synchronisation dynamics in the EEG 1 : 0.5 2 : 0.4 3 : 0.1

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1 : 0.2 2 : 0.8 3 : 0.0

1

2

0.4 0.2 0.6

q1

0.5

0.2

0.6

q2

q1

0.4

q2

0.4

0.6

D1

0.1

D2

Fig. 1 Schematic diagrams of a Hidden Markov Model (HMM) (left) and a Semi-Markov Model (SMM) (right). In the case of the HMM the states (q1 and q2 ) are able to generate observable values according to the observation probabilities (listed in the boxes above the states). In the case of the SMM each state is only able to generate one observable value. Transitions between the states proceed according to defined probabilities, examples of which are provided in the links between states in both Figures. In the case of SMMs each state has a sojourn time probability, the probability of remaining in that state for a given period of time, which is illustrated below the states (D1 and D2 ).

1. Apply the method, outlined in section 2, to generate GPS sequences over multiple trials recorded under the different conditions we wish to separate based upon their phase dynamics. 2. For each trial in each condition in the training set a single HMM or SMM is trained such that P (Oa,c |λa,c ) (the probability of observing trial a under condition c given HMM λa,c ), is maximised for that trial, where a is the trial number and c is the condition under which the EEG is recorded. Note, λ is used here to refer to the general case of a Markov model and may be used to refer to either an HMM or an SMM. 3. For each trial To in the test set under each condition calculate the likelihood of each HMM/SMM for each condition having generated that trial, ¯ ], P (To |λa,c ), ∀a ∈ [1, ..., N ∀c ∈ [1, ..., ν] ¯ is the number of models trained for a sinwhere N gle condition and ν is the number of conditions for which models have been trained. 4. The condition which gets the most votes (for which the likelihood of that condition generating the trial in the test set is the highest for the majority of the models) is taken to be the condition under which the trial was recorded. For example for trial T1 calculate Pa,1 = P (T1 |λa,1 ) and Pa,2 = P (T1 |λa,2 ) for a = 1, 2, ..., N , then N comparisons are made between likelihoods Pa,1 and Pa,2 . The number of times Pa,1 > Pa,2 is counted for all a ∈ [1, ..., N ]. Simultaneously the number of times Pa,1 < Pa,2 is counted. The majority vote indicates the most likely condition under which the trial T1 was recorded.

3 Significance tests The proposed method is compared to the following stateof-the-art significance tests. Note, the significance tests are applied to pairs of trials corresponding to potentially different conditions. Thus, the task of the significance tests is to determine if the phase synchronisation between the signals in one trial is significantly different from the synchronisation between the signals in the other trial. 3.1 Parametric tests Parametric tests of statistical significance are introduced in [Allefeld and Kurths, 2004] and make use of the circular statistical properties of the analytic signals in each trial to determine the significance of the synchronisation. Synchronisation is assumed to be a state and the trials must contain bivariate signals only. First, it is necessary to define some terms from circular statistics 1 ∑ C¯ = cos Θ(t) M˙

(20)

1 ∑ S¯ = sin Θ(t) M˙

(21)

¯= R

√ C¯ 2 + S¯2

S¯ θ¯ = arctan ¯ C

(22)

(23)

where Θ(t) defines the phase difference between a pair of signals of length M˙ in a single trial at sample t

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and the quantities C¯ and S¯ describe the components of the first empirical moment of the distribution of phase ¯ and θ¯ then describe the polar representations values. R of those moments. These quantities are used in many of the subsequent tests. The test statistic is defined as √ T¯ =

M˙ √ ¯ 1 ))− ( 2 arctan(R 2

√

M˙ √ ¯ 2 )) (24) ( 2 arctan(R 2

¯ 1 defines the quantity R ¯ calculated on the where R ¯2 phase difference of the pair of signals in trial 1 and R ¯ the quantity R calculated on the phase difference of the pair of signals in trial 2. The test statistic T¯ may then be compared to the desired percentile of the von Mises distribution - defined as

with s2Q¯ ∼ =

∑ 1 ¯ − Q) ¯ 2 (cos(Θ(t) − θ) M˙ (M˙ − 1)

(27)

¯ defined as and Q ∑ ¯ ¯= 1 Q cos(Θ(t) − θ), M˙

(28)

where Θ(t) is the difference in phase between the signal pairs at sample t and θ¯ is the mean phase difference between signal pairs over the length M of the trial.

3.2 Bootstrap tests 3.2.1 Bootstrapped T-test

1 eΩ cos(Θ−µ) 2πI0 (k)

where Io denotes the Bessel function of order o, µ the mean direction of the distribution and k > 0 the concentration. Note, when Ω = 0 the distribution is uniform and as k increases the distribution becomes more concentrated around µ and resembles a normal distribution. The moments are χ0 = I0 (Ω)/I0 Ωeipµ and χ1 = I1 (Ω)/I0 (Ω) denoting the mean direction and concentration (spread of the data points), respectively. Note, the test statistic may also be tested against the desired percentile of the wrapped normal distribution [Allefeld et al., 2007]. The choice of the usage of the von Mises distribution is arbitrary, however comparisons with the wrapped normal distribution reveal no significant differences.

A bootstrap test consists of the generation of replications of the statistic(s) of interest and the subsequent estimation of the distribution of these statistics from the replications [Efron et al., 1993]. To generate replications of the data the original phase difference values Θ(t) are drawn at random with replacement to create sets of ’bootstrap replications’. Formally, rt (t = 1, ..., M˙ ) are uniformly distributed and independent integers ranging from 1, ..., M˙ . Θ(r) is then a bootstrap replication of the original phase differences Θ. Statistics calculated on this sample are ¯ as defined in Eq. (28). The bootstrap replications of R, ¯ original statistic R may then be compared to the dis¯ ∗ which may tribution of its bootstrap replications R be assumed to be distributed according to either the wrapped normal or the von Mises distribution.

3.1.1 Parametric T-test

3.2.2 Bootstrap H0 test

A parametric test for statistical significance of the difference in synchronisation between trials may be formed based upon the t-statistic [Sheskin, 2004]. The difference in the mean synchrony of the trials is divided by the estimate of the standard deviation. This may then be compared to the Student t-distribution with f = 2(M − 1) degrees of freedom (M is defined, as above, as the number of samples in the identical length trials). Formally the t-statistic may be defined as

The bootstrap H0 approach [Theiler et al., 1992] first combines the phase differences of the two trials Θ1 and Θ2 as follows

pM (Θ; µ, Ω) =

¯1 − Q ¯2 Q t¯ = √ 2 s2Q,1 ¯ + sQ,2 ¯

(25)

(26)

{ θ0 =

Θ1 (t), Θ2 (t − M˙ ),

for t ≤ M˙ for t > M˙ ,

(29)

where t = 1, ..., 2M˙ . Bootstrap H0 replications are generated by re-sampling from this combined sequence with a sample ¯1 − R ¯ 2 |, size of M˙ . Replications of the test statistic, |R calculated from trials 1 and 2, are then calculated from this combined sample. The number of bootstrap replications (N ) generated is set to equal


9

where Hk is the conditional entropy defined as Nb =

200 , α

(30)

where α denotes the significance level, i.e. 4,000 bootstraps are generated for a significance level of 5%. The top 5th percentile, the significance level, is then used as the evaluation threshold of the test statistic ¯1 − R ¯ 2 |; trial pairs whose statistic |R ¯1 − R ¯ 2 | is greater |R than the threshold are categorised as having a significant phase synchronisation.

Hk = −

K ∑

P (vw |vk ) log P (vk |vw ),

(32)

w=1

with P (vk |vw ) giving the probability of the occurrence of symbol vk immediately after symbol vw and H defining the entropy. Note, vk and vw denote GPS state labels generated by the GPS algorithm. Formally

3.2.3 Bootstrap permutation test Under the previous bootstrapping method the distribution of the original samples is not preserved [Theiler et al., 1992]. Permutation based bootstrapping aims to solve this problem by generating new samples from the random exchange of samples between Θ1 (t) and Θ2 (t) for all sample points t. Formally rυ is a random permutation of the inte¯ 1 are then calculated gers 1, ..., 2M˙ . Replications of R ˙ ¯2 from Θ0 (rυ ) with υ = 1, ..., M and replications of R ˙ are calculated from Θ0 (rυ ) with υ = (M + 1), ..., 2M˙ . Calculations then follow on in the same way as for the bootstrap H0 test.

H=−

K ∑

P (vk ) log P (vk ),

(33)

k=1

where P (vk ) gives the probability of occurrence of a symbol vk ; (1 ≤ k ≤ K) from a set of K symbols. Instantaneous entropy may be used to characterise the uncertainty of the phase pattern dynamics around a specific time point t. It is defined as

Hins (t) = −

K ∑

pvk (t) log pvk (t),

(34)

k=1

3.2.4 Phase scattered surrogate data Again bootstrap replications of the signal pairs are created for this test. Each pair of surrogate signals has its phases scattered via random draw and replacement of the Fourier coefficients after application of the Fourier transform [Theiler et al., 1992]. ¯1 −R ¯ 2 | is then calculated on the The tests statistic |R original signal pair and on each of the pairs of signals generated by the surrogate data generation method. Hence, the significance of the phase synchronisation of the original pair of signals is evaluated against the distribution of phase scattered surrogates. Examples of this test may be seen in [Sweeney-Reed et al., 2005] and [Sweeney-Reed and Nasuto, 2007].

3.3 Other tests 3.3.1 Information theoretic tests of significance The entropy reduction rate may be used to evaluate the stepwise predictability of a sequence of symbols generated by the GPS method and is defined as

hred =

H−

∑K k=1

H

P (vk )Hk

,

(31)

where ∑

[

] δv ¯

g ks

τu

¯

D lu ∈ t− D 2 ,t+ 2

pvk (t) =

∑

[ lu ∈

] τu

,

(35)

¯ ¯ D t− D 2 ,t+ 2

where τ u gives the duration of the interval lu , sg ¯ the width denotes the labelled GPS time-series, and D of the time window in which to calculate probability. A surrogate data approach is taken to assess the statistical significance of the phase pattern vectors identified and measured by the test statistics hred or Hins . Surrogate datasets are generated by randomising phases in To . To preserve cross-correlations between To components identical random sequences are used for the generation of every component. Multiple surrogate datasets are generated and the test statistics are calculated for each. The value of the original test statistics against their estimated distributions from surrogate replications are then informative about the statistical significance of the temporal dynamics of phase synchronisation as characterised by the method.

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3.3.2 Phase synchronisation index For this test of significance of synchronisation of the two signals from a given trial the phase difference at time ti = i∆t is assumed to be modelled by an incremental process [Schelter et al., 2007],

x˙ A, ¯B ¯ = −ψA, ¯B ¯ yA, ¯B ¯ − zA, ¯B ¯ + Σ(xB, ¯ A ¯ − xA, ¯B ¯ ), y˙ A, ¯B ¯ = ψA, ¯B ¯ xA, ¯B ¯ + 0.15yA, ¯B ¯, z˙A, ¯B ¯ = 0.2 + zA, ¯B ¯ (xA, ¯B ¯ − 10),

(37)

where Θ(t) denotes the phase difference between a signal pair in one trial at time t, hence ∆Θ(t) is the change in phase difference from one time increment to the next. The evolution of Θ is modelled by a drift diffusion process. The distribution of the evolution of Θ given random diffusion is modelled by a χ2 distribution. The observed values of changes in Θ over time are then compared against this distribution and the statistical significance of the phase synchronisation is hence calculated. Full details of the operation of this method can be found in [Schelter et al., 2007].

where the ψA, ¯B ¯ = 1 ± ∆ψ and Σ governs the cou¯ pling strength between Rössler oscillators A¯ and B, and may be adjusted to produce systems which are either phase synchronised or unsynchronised [Arch et al., 1996]. To produce phase synchronised systems the parameters in this investigation are set to Σ = 0.15 and ∆ψ = 0.06 while to produce unsynchronised systems the coupling strength is adjusted to Σ = 0.05. Two pairs of signals are created with the Rössler oscillators; one with synchronisation and one without. Multiple trials of each of these signal pairs are created with different starting conditions of the oscillator to ensure different amplitudes across trials. These are organised into training and testing datasets such that each dataset contains 50 trial pairs where the synchronisation is similar and 50 pairs of trials where the synchronisation is markedly different.

4 Data

4.2 Simulated EEG

The introduced method is evaluated against a number of state-of-the-art tests of statistical significance of phase synchronisation (outlined in section 3). The evaluations are performed on three data types: Rössler oscillators, synthetic EEG and real EEG. The datasets are each split into two subsets, training and testing. Each of these subsets contains 50 pairs of trials with the same patterns of synchronisation dynamics and 50 pairs of trials with different patterns of synchronisation dynamics. The task of the significance tests is to identify statistically significant differences where they exist. Evaluation is performed using the following datasets

Simulated EEG with induced phase synchronisation dynamics is built in a multistage process. EEG is first simulated by a Neural Mass Model (NMM) [Friston and David, 2003]. It is then band-pass filtered into lowfrequency, target-frequency and high-frequency bands, where the target-frequency band is the band in which the desired phase synchronisation dynamics are to be located. A 10 Hz target frequency is used in this study as this is in the centre of the alpha frequency band (an important frequency band in the EEG). Copies are then taken of each of these frequency bands. The original signals are referred to as signal set A and the copies as signal set B. The phases are scattered at random on all three bands in signal set B. Finally, the phases within target time periods in the target frequency band are copied back from signal set B into signal set A at the target band, producing periods of phase synchronisation between the two signals within a particular set of target time ranges and at a particular target frequency. This method has been successfully used to generate synthetic signals containing periods of synchronisation in [Sweeney-Reed et al., 2005, Sweeney-Reed and Nasuto, 2007]. The neural mass model for simulating the EEG accepts a gaussian noise signal as input to represent the summation of neural activity from neighbouring regions. This is added to the activity of the excitatory spiny stellate cells, one of the three neural populations modelled:

Θ(t) = Θ(t − 1) + ∆Θ(t),

(36)

1. Rössler oscillators. 2. Simulated EEG datasets with periods of phase synchronisation introduced at specific time-frequency locations. 3. Synthetic multivariate EEG. 4. Real EEG recorded during a finger tapping task.

4.1 Rössler oscillators Rössler oscillators are an example of a coupled chaotic oscillator whose parameters may be adjusted to produce signals which are phase synchronised or not. Formally they are defined as,


(x1 ) spiny stellate cells, (x2 ) excitatory pyramidal cells and (x3 ) inhibitory interneurons, whose interactions, encapsulated in Eq. (38), reflect the couplings between the main neuron populations constituting local microcircuitry and giving rise to the EEG, x˙1 Λ˙1 x˙2 Λ˙2

2 e = H τe (G + S1 (Λ2 )) − τe x1 − = x1 2 1 i = H τi S2 (Λ3 ) − τi x2 − τ 2 Λ2

1 τe2 Λ1

i

= x2 e x˙3 = H τe S3 (Λ1 − Λ3 ) − Λ˙3 = x3

2 τe x3

−

(38)

1 τe2 Λ3

where G is the gaussian input to the excitatory spiny stellate neurons, He,i and τe,i are parameters that tune the mean inhibition and excitation responses of each population and S1,2,3 are sigmoid functions which simulate the electro-potential response over time of each neural population. They are defined as follows

Sη (v) =

c1η e0 1 + exp (r(v0 − c2k v))

(39)

where c1,2 η , r, e0 and v0 are parameters to tune the response of the sigmoid function. Parameter values are set to defaults described in [Friston and David, 2003] which allow generation of synthetic clean EEG. After generation of the simulated EEG signal the above method may be used to construct signal pairs with phase synchronisation dynamics at specific target frequencies. Two classes of synchronisation dynamics are constructed. Each class of phase dynamics is constructed from a Semi-markov model (SMM) containing two states. State A˙ determines when the two signals are to be synchronised and state B˙ determines when the signals are to be desynchronised. Thus, for example, if the SMM remains in state A˙ for 2 s before changing to state B˙ then the signal pair is synchronised for 2 s before becoming desynchronised. The two SMMs used in the construction of the two classes of phase synchrony dynamics have the parameters 1. SMM 1: σ = [0.5, 0.5], Γ = [0.5, 0.5; 0.5, 0.5], D = [50, 50] 2. SMM 2: σ = [0.5, 0.5], Γ = [0.5, 0.5; 0.5, 0.5], D = [100, 100] thus, each SMM has equal transition probability matrices however SMM 2 has state durations twice that of SMM 1. As the priors are also equal the only significant difference between trials generated via SMM

11

1 and trials generated by SMM 2 will be the temporal dynamics. Trials generated by SMM 2 will have longer dwell times in both the synchronised and desynchronised states. However, the average amount of time spent synchronised and desynchronised should also be the same between the classes. Thus, the only way to differentiate the classes is via their phase synchronisation dynamics. Multiple trials (100 related to each model) of each pattern are created to form a training set and a test set, with 50 trial of each class in the training and 50 in the testing set.

4.3 Multivariate synthetic EEG The combination of GPS and Markov modelling may also be used to model multivariate EEG datasets which contain many different pairs of synchronised channels. EEG trials containing synchronisation between different electrodes are generated. The method described in section 4.2 for introducing periods of phase synchronisation into a bivariate EEG time series at specified time-frequency locations is extended to allow introduction of phase synchronisation periods into multivariate EEG datasets. Synchronisation is introduced at different time points into eight channel synthetic EEG datasets to produce two types of dataset, each of which contains inter-channel synchrony. Both datasets are 10 s in length and generated at a sampling rate of 128 Hz. Dataset 1 is generated by applying SMM 1 to synchronise every channel pair while dataset 2 is generated by applying SMM 2 to synchronise every channel pair. All synchronisation is introduced at a target frequency of 10 Hz. To illustrate this an example of how the synchronisation pattern may be constructed is illustrated in Figure 2. The eight channel pairs are illustrated in the Figure. Synchrony is introduced between pairs of channels at specific times creating a detailed pattern of interchannel connectivity which changes as a function of time. For example in Figure 2 at time t = 1 channels 1 and 2, 1 and 4, 1 and 5 etc. are synchronised. Two hundred trials are generated, 100 of which contain one group of synchronisation patterns (labeled group 1; G1) and 100 of which contain a different group of synchronisation patterns (labeled group 2; G2). Meta data is maintained to allow for subsequent evaluation of the classification results. The datasets are split into training and verification sets with the training set containing 50 trials of G1 and 50 trials of G2.

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Fig. 2 Connectivity patterns generated between 8 channels in a set of signals. At each time point a grid indicates which channel pairs are synchronised at the indicated time point. Note, the length of time a pair of channels are synchronised for is determiend by the SMM.

4.4 Real EEG Real EEG data are recorded during a self-paced finger tapping task from brain areas that are known to exhibit synchronous activity during movements [SweeneyReed and Nasuto, 2009]. The supplementary motor area (SMA) and the contralateral primary motor cortex (M1) are thought to play roles in movement initiation and generation, respectively [Gross et al., 2005]. Additionally, the SMA exhibits both pre-movement and movement changes, with ERD prior to, and during, movement followed by ERS after cessation of movement [Pfurtscheller and Lopes da Silva, 1999]. Furthermore, functional connectivity between these areas has been confirmed during motor activity (see review by [Gross et al., 2005]). Therefore, confirmation of the method’s ability to test for phase synchronisation dynamics between these areas during a finger tapping test is a suitable test of the applicability of the methods to real EEG. EEG was recorded, using an Electrical Geodescics Inc. system, from two right handed subjects during selfpaced finger tapping with left and right hands. Periods of finger tapping were visually cued, preceded by a fixation cross, and subjects were free to choose whether to tap with the left or right index finger, while trying to balance whether left followed left or right and the other way round. Data were sampled at 500 Hz and band-pass filtered between 0.5 to 100 Hz with a 4th order, zero phase, digital Butterworth filter. The signals were first spatially filtered using the spline Laplacian transformation [Nunez and Srinivasan, 2006], then the data corresponding to electrode locations FCz, C3, and C1 were segmented around 3 s before to 3 s after each tap. Artifacts were identified and removed via visual inspection. A total of 82 trials were retained from participant A and 61 from participant B with signals extracted from the Laplacian-transformed data at FCz, C3 and C1. An additional filtering step was then applied, with a 4th order, zero phase, digital Butterworth

filter, to focus on the frequency band of interest in the range 8 - 12 Hz. This frequency range was chosen as it contains the peak frequencies for both subjects, which lie at 10 Hz and 11 Hz for subjects 1 and 2 respectively. It is known that patterns of synchrony may be observed between electrodes positioned over the M1. Furthermore, differing patterns of synchrony may be observed between electrodes positioned over M1 and the SMA [Sweeney-Reed and Nasuto, 2009]. Therefore, to investigate whether these different patterns also exhibit different temporal dynamics EEG is extracted from the Laplacian-transformed data at C1 and C3, both positioned over M1, and C1 and FCz, positioned over M1 and the SMA. These two sets of the Laplaciantransformed data pairs are hypothesised to exhibit different synchronisation dynamics and are hence split into two classes and into a training and verification set, with balanced numbers of trials per class in each. For further details on the recording of the dataset the reader is referred to the original paper on this dataset [Sweeney-Reed and Nasuto, 2009]. 5 Results 5.1 Calculation of Instantaneous Instability Index Some choices need to be made before the GPS method may be used to characterise the temporal dynamics of phase synchronisation. Calculation of the Instantaneous Instability Index involves the calculation of phase relative to the reference channels via the term di (t) in Eq. (7). This may be calculated via four different methods - Method A (Eq. (8)), Method B (Eq. (9)), Method C (Eq. (10)) and Method D (Eq. (11)). These equations are each evaluated for their ability to clearly identify periods of phase synchrony introduced into a pair of signals with superimposed gaussian noise. To establish which of these equations is most suitable for calculation of the Instantaneous Instability Index synthetic datasets are constructed with periods of phase synchronisation introduced at random time points. Two signals, each of duration 10 s, are constructed at a sampling rate of 128 Hz. Five sine waves, of 10 Hz frequency and duration 1 s, are introduced at identical times into each of the signals. Gaussian noise is introduced into the remaining periods of the signal, thus the constructed signal pairs contain 5, 1 s periods during which phase synchronisation is expected; distributed at random time points. Examples of the constructed signals are shown in Figure 3. The Instantaneous Instability Index is calculated from the phase synchronisation dynamics of these signal pairs using the different equations listed in section


Fig. 3 Synthetic signal pairs constructed for evaluation of calculation of the Instantaneous Instability Indexes via each method for di (t) described in section 2. A pair of signals is constructed with 5 s of synchronisation at 10 Hz at random time periods each of length 1 s.

13

Calculation method

Average area under ROC curve

Method A

0.878 (*)

Method B

0.661 (*)

Method C

0.894 (*)

Method D (Eq. (11))

0.912 (*)

Table 1 Average area under the ROC curve achieved by the different methods for calculating the Instantaneous Instability Index as outlined in section 2. Statistical significance (P < 0.05) is denoted by an asterisk (*) evaluated against the null hypothesis of random Instantaneous Index values.

1 0.9 0.8 0.7 Sensitivity

2. Examples of Instantaneous Instability Indices calculated on these signal pairs are shown in Figure 4. As seen in the Figure each of the methods performs differently at characterising the moment to moment changes in phase synchronisation. Thresholding the Instantaneous Instability Index partitions the signals into episodes containing differing levels of synchrony. In [Ito et al., 2007] thresholding is based upon the 50th percentile, however this is a relatively arbitrary choice. We investigate how the differentiation of synchronised vs. unsynchronised portions of the data is affected by the choice of threshold; ranging from the 0th percentile to the 100th percentile. Receiver Operating Characteristic (ROC) curves are constructed for each Instantaneous Instability Index calculation method and averaged over 100 different reshuffles of the periods of phase synchronisation within the signal. These curves give a measure of the ratio of sensitivity to specificity. The Area Under the ROC Curve (AUC) is indicative of the accuracy of each of the methods invariant to the particular choice of threshold. The AUCs each method achieves on average over 100 different phase synchronisation patterns are listed in Table 1 and the average ROC curves are shown in Figure 5. The accuracy may also be calculated for each threshold value allowing an identification of the best threshold to use for each method. Accuracies achieved with each method as a function of threshold are shown in Figure 6. Thresholds are ranged from the 0th to the 100th percentile (0.0 to 1.0). Note that above the 50th percentile no statistically significant results are achieved, hence the plot only illustrates thresholds from 0 to 0.5. The equation that achieves both the largest area under the ROC curve and the highest accuracy is Eq. (11). Note that the best threshold values are generally dataset specific and the optimal threshold changes as

0.6 0.5 0.4 0.3

Method A Method B Method C Method D

0.2 0.1 0

0

0.1

0.2 0.3 1 − Specificity

0.4

0.5

Fig. 5 ROC curves calculated from accuracies achieved by each of the methods for calculating the Instantaneous Instability Index averaged over 100 different signal pairs each with different patterns of synchronisation. Method A depicts the results of using Eq. (8), method B Eq. (9), method C Eq. (10) and method D Eq. (11).

additive gaussian noise is added to the signal in time intervals where phase synchronisation is expected. This is illustrated in Figure 7 which shows how the optimal threshold level to achieve the highest accuracy changes as the variance of the gaussian noise added to phase synchronised portions of the data is increased from 0 to 1. This highlights the importance of careful inspection of the amounts of noise present in a signal set before setting the threshold value for differentiation of Instantaneous Instability Index values. Note that the optimal threshold to accurately separate synchronised vs. unsynchronised portions of synthetic data increases as a function of the variance of the introduced gaussian noise. However, the increase in optimal threshold is not large and, as can be seen from Figure 7, a choice of threshold of approximately

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Ian Daly et al. 2.5 0.1 2 0.05

III

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(d) Method D Fig. 4 Instantaneous Instability Indexes (I) calculated with each method described in section 2 on signals synchronised from 1-2 s at 12 Hz. 4(a) - I calculated via method A (Eq. (8)), 4(b) - I calculated via method B (Eq. (9)), 4(c) - I calculated via method C (Eq. (10)) and 4(d) - I calculated via method D (Eq. (11)).

0.1 (the 10th percentile) is likely to produce very accurate separations of synchronised vs. non-synchronised portions of signal at all tested levels of noise when Eq. (11) is used to calculate the Instantaneous Instability Index. Therefore Eq. (11) is used for the remainder of this work in place of the original Eq. (8) proposed by [Ito et al., 2007].

5.2 Comparison with significance tests

For data-sets generated by the Rössler oscillators the task of the phase synchronisation significance tests is to differentiate between a trial containing bivariate signals which are synchronised and a trial which is unsynchronised. The dataset is organised such that the first 50 trials contain signal pairs, each of which contain the same pattern of synchronisation and the second 50 contain signal pairs, each of which contain different patterns of synchronisation.


15 Significance test

Accuracy

P

1 0.95

Method A Method B Method C Method D

0.9

Accuracy

0.85

Parametric test

0.890

0.000

Parametric T-test

0.950

0.000

Bootstrap T-test

0.920

0.000

Bootstrap H0 test

0.905

0.000

Bootstrap permutation test

0.990

0.000

Phase scattered surrogate data

1.000

0.000

Phase synchronisation index

0.905

0.000

Entropy reduction rate

0.485

0.618

Instantaneous entropy

0.545

0.184

GPS + HMM voting

0.660

0.001

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1.000

0.000

0.8 0.75 0.7 0.65 0.6 0.55 0.5

0

0.1

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Threshold

Fig. 6 Accuracies achieved by each of the methods for calculating the Instantaneous Instability Index listed in section 2 as a function of the threshold applied to each method. Accuracies are averaged over 100 different signal pairs each with different patterns of synchronisation. Method A depicts the results of using Eq. (8), method B Eq. (9), method C Eq. (10) and method D Eq. (11).

Table 2 Accuracies achieved by the statistical significance tests for differences in phase synchronisation applied to signals generated by R¨ ossler oscillators. Statistically significant accuracies (p < 0.05) are assessed against the null hypothesis of equal chance of classification of each condition. The final column indicates the significance level of the accuracy achieved with each test.

there is a significant difference between two different periods of phase synchronisation. The results of each test applied to the Rössler data are presented in Table 2. The statistical significance of the accuracy of each of the tests is evaluated against the null hypothesis of p = 0.5 chance correct classification in all the datasets.

Fig. 7 Mean threshold to achieve the maximum accuracy in separating synchronised vs. non-synchronised portions of data with overlaid gaussian noise at increasing amplitudes. The mean threshold is calculated over 100 randomly generated trials and the dotted lines depict ±1 standard deviation from the mean. The Instantaneous Instability Index is calculated via Eq. (11) which is shown to be most accurate.

Applying each of the tests to identify differences in the synchronisation between trials allows the calculation of classification accuracy, a measure of how accurate the test is at differentiating statistically significantly different patterns of synchronisation between trials. Thus, the success of each of the tests can be evaluated based upon how accurately it evaluates whether

The most accurate tests for identifying trials with similar phase synchronisation are the surrogate data based test with phase scattering and the SMM based test with a voting scheme. It is important to note that a number of other tests also produce very high accuracies which are not statistically significantly different from these tests (p > 0.05) as calculated against the null hypothesis of random two class classification (probability of correct classification (p = 0.5)). Specifically the parametric t-test, bootstrap t-test and bootstrap permutation test are all able to produce accurate classifications that are not statistically significantly different from the results achieved with the surrogate data and SMM tests. When the tests are applied to differentiate between patterns of phase synchronisation introduced into the EEG datasets simulated by the method described in

16

Ian Daly et al. Significance test

Accuracy

P

Significance test

Accuracy achieved

Parametric test

0.53

0.2421

GPS + HMM voting

0.93 (*)

Parametric T-test

0.49

0.5398

GPS + SMM voting

0.86 (*)

Bootstrap T-test

0.58

0.0666

Bootstrap H0 test

0.53

0.2421


0.54

0.1841


0.72

0.0000


0.43

0.9033


0.49

0.5398


0.51

0.3822

GPS + HMM voting

0.81

GPS + SMM voting

0.92

Table 4 Accuracies achieved by classifying the multivariate EEG dataset via GPS coupled with Markov modelling methods. Statistically significant accuracies (p < 0.05) are assessed against the null hypothesis of equal chance of classification of each condition and are indicated by an asterisk (*).

Subject 1 Acc.

P

Acc.

P

Parametric test

0.531

0.291

0.508

0.399

0.0000

Parametric T-test

0.524

0.291

0.533

0.304

0.0000

Bootstrap T-test

0.439

0.888

0.402

0.938

Bootstrap H0 test

0.512

0.370

0.549

0.221

Bootstrap permutation

0.537

0.219

0.541

0.221

Scattered surrogate

0.524

0.291

0.615

0.036

Synchronisation index

0.616

0.018

0.516

0.399


0.536

0.219

0.639

0.019


0.634

0.009

0.639

0.019

GPS + HMM voting

0.817

0.000

0.750

0.000

GPS + SMM voting

0.698

0.001

0.703

0.000

Table 3 Accuracies achieved by the statistical significance tests for differences in phase synchronisation applied to signals generated by the simulated EEG method. Statistically significant accuracies (p < 0.05) are assessed against the null hypothesis of equal chance of classification of each condition. The final column indicates the significance level of the accuracy achieved with each test.

section 4.2 the accuracies shown in table 3 are produced. The majority of the tests perform poorly when differentiating patterns of synchronisation in the simulated EEG, with most of the results not statistically significant (p < 0.05). Thus, while they are suited to identifying phase synchronisation differences when treating synchronisation as a state, they are not able to differentiate the temporal dynamics of synchronisation processes. In contrast, the methods which produce the highest accuracies are those based upon Markov processes; Hidden Markov Models and Semi-Markov Models. These techniques aim to characterise the temporal dynamics of a process and it is interesting to note that they are therefore able to characterise the different patterns of dynamics present in the GPS patterns with much higher levels of reliability when compared against the other tests applied to this dataset. The GPS method coupled with either HMM or SMM is applied to classify the multivariate synthetic EEG datasets. The results are present in table 4. High levels of accuracy are achieved by this method suggesting that the coupling of GPS with Markov modelling is also able to classify multivariate datasets well.

Significance test

Subject 2

Table 5 Accuracies achieved by the statistical significance tests for differences in phase synchronisation applied to real EEG signals recorded during a finger tapping exercise from both subjects. P-values denoting the significance against the null hypothesis of equal chance of classification of each class are listed.

Finally, when applying the significance tests to differentiate phase synchronisation dynamics in real EEG only a few of the tests are statistically significantly reliable. The results are listed in table 5. As can be seen the SMM and HMM based tests of statistical significance produce the highest levels of reliability when differentiating different patterns of phase synchronisation in real EEG. Many of the tests are not statistically significantly accurate when compared to the null hypotheses of random (p = 0.5) chance of correct determination of statistical significance. Although the phase surrogate, entropy reduction rate, synchronisation index, and instantaneous entropy tests also pro-


duce highly significant accuracies (p < 0.05), for one or both of the subjects, the SMM and HMM based tests are the most accurate. This suggests Markov model based tests are best suited to differentiating patterns of synchronisation in real EEG signals. It is also interesting to ask what Markov models are most likely to have generated the GPS labels corresponding to the finger tapping task recorded between each channel pair. Table 6 lists the Markov models generated corresponding to each channel pair. Note, the priors and observation probability matrices are very similar for both channel pairs while the transition matrices differ. This highlights the different temporal dynamics related to each channel pair which are identified via the use of Markov models.

6 Discussion and Conclusion Phase synchronisation may be detected in random signal pairs for short periods of time as the instantaneous phases of the two signals match purely by chance. It is therefore important to evaluate the statistical significance of periods of detected phase synchronisation in signal sets before anything physiologically meaningful can be said about their relationship. This work extends methods outlined in [Ito et al., 2007] for characterising and identifying significantly different phase dynamics in a multivariate time-series via the alternative calculation of di (tn ) (Eq. (11)) and the use of HMMs and SMMs. HMMs and SMMs are used to model the patterns of phase synchronisation dynamics and hence differentiate different patterns of synchronisation into cognitively relevant classes. The GPS method allows periods of global phase stability to be identified and clustered according to their similarities. The resulting time series of GPS labels allows the identification of particular trends and patterns in the progression of periods of global phase stability in the EEG. The key novel contribution of this work is to use Markov modelling approaches (HMM and SMM) to model the temporal dynamics of these GPS sequences. This allows the method to be used to describe more complex datasets in which periods of phase synchronisation wax and wane over time as a stochastic process. We show how the combination of the extended GPS method and Markov modelling may be used to differentiate patterns of phase synchronisation in bivariate and multivariate time-series. Comparisons between our approach and traditional approaches to testing for statistical significance of phase synchronisation - including state-of-the-art approaches used in [Ito et al., 2007]

17

and [Schelter et al., 2007] - reveal our approach to be the most accurate method for both bi- and multivariate synchronisation dynamics. There are currently no other tests for phase synchrony which are explicitly designed to be able to account for the temporal dynamics in the patterns of synchrony. Indeed, the other tests described here are designed to identify whether a given pair of signals are completely synchronised or not (i.e. they treat synchronisation as a state which is either on or off). They are very successful at identifying such synchrony, as shown by the results in table 2. However, one of the key findings of this paper is that these tests are not able to accurately identify situations when the phase is modulated dynamically as an on-going process (as highlighted in table 3). It is possible to analyse a single dataset and identify periods when the synchronisation is significant and periods when it is not. Indeed, this could be done with any of the current state-of-the-art tests presented. However, the tests would have to be applied in a sliding window, the length of which is determined by the number of cycles and the frequency at which one is looking for synchrony. Indeed the use of multiple trials, some of which are synchronised and some of which are not, is similar in approach to this (if one thinks of each sliding window position as a trial for example). However, when applying the tests in a sliding window edge effects are likely to play a decisive role in the accuracy with which synchronisation periods may be identified. The key point is that to do this each of the comparative tests (Parametric, bootstrap, surrogate etc), has to be modified. Furthermore these tests still do not model the dynamics of the phase transitions; they just detect when changes from synchronised to desynchronised states occur. The novel tests introduced by this paper are able to account for dynamics in the phase transitions and do so without any of the technical issues associated with the sliding window approach. The GPS method may cluster the data into GPS episodes according to how instantaneous phase differences are distributed across the scalp. A number of such measures are compared in section 5.1 and a measure of instantaneous phase distribution is found to be the best overall measure for accurately clustering GPS episodes. This could be because such a measure of the instantaneous distribution of phase differences across the scalp exhibits larger moment to moment transitions when the phase vector changes than the other measures. Thus, kmeans clustering of this metric is more likely to arrive at the correct partitions. The GPS method measures the instantaneous phase synchronisation between two or more EEG signals. The

18

Ian Daly et al. (a) M1-M1, Prior 0.185

0.000

0.286

0.000

(b) M1-SMA, Prior 0.530

0.221

(c) M1-M1, Transitions 0.182 0.004 0.384 0.016 0.459

0.014 0.399 0.017 0.663 0.019

0.470 0.012 0.044 0.005 0.161

0.019 0.563 0.008 0.309 0.014

0.012 0.527 0.000 0.000 0.000 0.460

0.037 0.000 0.963 0.000 0.000 0.000

0.009 0.541 0.000 0.000 0.000 0.449

0.215

0.000

0.564

(d) M1-SMA, Transitions 0.314 0.021 0.548 0.006 0.346

0.031 0.282 0.021 0.025 0.259

(e) M1-M1, Observations 0.004 0.000 0.000 0.599 0.397 0.000

0.000

0.006 0.047 0.025 0.024 0.116

0.060 0.098 0.417 0.484 0.002

0.055 0.049 0.535 0.456 0.016

0.848 0.524 0.002 0.012 0.607

(f) M1-SMA, Observations 1.000 0.000 0.000 0.000 0.000 0.000

0.002 0.000 0.000 0.551 0.448 0.000

0.001 0.575 0.000 0.000 0.000 0.424

0.001 0.000 0.999 0.000 0.000 0.000

0.001 0.580 0.000 0.000 0.000 0.419

1.000 0.000 0.000 0.000 0.000 0.000

Table 6 Hidden markov models trained on real EEG from subject 2. Sub-tables 5(a) and 5(b) list the prior probabilities for classes 1 (M1-M1 connectivity) and 2 (M1-SMA connectivity), sub-tables 5(c) and 5(d) list the transition probabilities, and sub-tables 5(e) and 5(f) list the observation probabilities.

GPS time series is then clustered via k-means clustering into different synchronisation periods. The choice of a particular value of k has to be carefully considered. The brain may be seen as moving through a series of interlinked states wherein each state is characterised by a short period of quasi-stable activity. Such a state is referred to as a micro-state [Lehmann et al., 1998] and determines which cortical regions are phase synchronised with one another and what the temporal dynamics of that synchronisation are. Thus, while two channels may be characterised as being in only one of two states (synchronised or unsynchronised) the underlying dynamics may be derived from a large number of microstates. GPS segments are based upon thresholding of the value of I below the 50th percentile. Thus, if one only considers synchronised vs. unsynchronised states the GPS segments define the lengths of periods of synchronised activity and Markov modelling may be used to model the dynamics. In such cases it is not necessary to use the K-means clustering step (or rather one may set k = 1). However, if one wishes to consider different synchronisation patterns in different GPS segments one may increase the value of k. In the case of our tests with synthetic data k = 1 and k = 2 both prove (as stated in section 2.2) to be good choices. However, k = 6 is used in this work as it is most consistent with the observations by [Ito et al., 2007] that EEG phase patterns may be best decomposed into 6 microstates. This is the case for all datasets considered in this paper. When time series continue to be synchronised over the recording period, as illustrated by our Rössler

oscillators data, the majority of the tests produce statistically significant accuracies (p < 0.05) with the SMM based test producing the highest accuracy by only a narrow margin, which is not statistically significantly different from the other tests. The good performance of our proposed tests on the Rössler oscillator generated datasets may initially seem surprising, given that our proposed tests are designed to deal with datasets for which there are different underlying dynamical patterns in the transitions between synchronised and unsynchronised states. However, the Rössler generated datasets may also contain underlying dynamics in which the level of synchronisation shifts over time between perfect synchronisation and a lesser degree of synchronisation that is non-the-less still synchronised. Hence, the clustering of the I measure into k = 6 states may allow differentiation of these dynamics via Markov modelling. By way of comparison, when synchronisation dynamics include large fluctuations in the dataset leading to periods of no synchronisation, as is the case with the simulated and real EEG datasets, the majority of the tests perform quite poorly. With synchronisation patterns in simulated EEG only differentiable at a statistically significant level via scattered surrogates, HMMs, and SMMs. With different patterns of synchronisation in finger tapping EEG a number of the tests produce statistically significant levels of differentiation. The most accurate tests are, again, SMM and HMM based tests. However, these tests are only a little better than the other tests.


It is interesting to note that two of the tests specifically intended for identifying significance in GPS patterns - instantaneous entropy and entropy reduction rate [Ito et al., 2007] - do not perform better than HMM/SMM based tests. These tests were originally conceived to identify whether there is a significant level of predictability in the phase pattern dynamics. The results suggest that for identifying significant differences in the dynamics of phase synchronisation different approaches may be suitable for different datasets. For finger tapping EEG data Markov models produce the highest accuracies, however these accuracies are only a little higher than some of the other tests. The simulated EEG dataset results suggest that the best approach may be to use the combined methods developed in this work. That is, first symbolising the phase pattern dynamics via the GPS method and then applying either an HMM or an SMM to test for statistical significance of differences in the dynamics of synchronisation. Furthermore, we note that the combination of GPS and Markov modelling is also able to differentiate multivariate phase synchrony patterns. Thus, datasets which contain multiple phase synchrony patterns and transistions between them may be differentated via the application of GPS coupled with either HMMs or SMMs. EEG is usually multivariate and contains, from moment to moment, transistions between a large number of different synchrony patterns involving differing numbers of channels. Thus, the success at differentiating these types of dataset based upon the dynamics of their global phase synchrony transitions is encouraging and suggests the method is highly applicable to real-world, multivariate EEG. It may be argued that when training the Markov model it is not possible to guarantee that the EEG contains synchrony. Indeed the approach is based upon the assumption that there is synchrony and that the dynamics of that synchrony are stable (i.e. it is phase locked over multiple presentations of some stimuli or event). It is therefore worth considering the case when the EEG contains no stable phase synchrony dynamics. In such instances the GPS label sequences will contain different dynamics for each trial. Markov models trained on these trials will therefore not be able to converge to a stable solution and will be unable to identify subsequent recordings of EEG corresponding to the same stimuli. In the general case it may be desirable to train one Markov model on surrogate data generated from randomly re-shuffling EEG. This may then be compared to Markov models trained on EEG in which one hopes to find stable patterns of synchrony. If the subsequent

19 Significance test

Complexity x Iterations

Parametric test

O(1) x 1

Parametric T-test

O(1) x 1

Bootstrap T-test

O(n) x 4,000

Bootstrap H0 test

O(n) x 4,000


O(n) x 4,000


O(n) x 4,000


O(n) x 4,000


O(n2 ) x 4,000


O(n2 ) x 4,000

GPS + HMM voting

O(n2 ) x n

GPS + SMM voting

O(n3 ) x n

Table 7 Computational complexities of the different tests of statistical significance of phase synchronisation patterns. The number of iterations necessary to calculate to the 95th percentile is also listed to indicate the approximate runtime more accurately. Note that n denotes the trial length in the dataset.

Markov models can be significantly differentiated from the surrogate model then their corresponding EEG sequences may be said to contain stable phase dynamics. It is necessary to consider the issue of multiple comparisons when discussing tests of statistical significance applied to phase synchronisation within the EEG. The EEG contains information at multiple time, frequency and spatial locations at which synchronisation may occur. If the significance of synchronisation at each of these locations is evaluated, then we can expect that a percentage of those evaluations will be reported as significant, with the percentage related to the alpha level of each of the significance tests [Singh and Phillips, 2010]. Approaches taken to correct for the multiple comparisons issue traditionally have been to use either Bonferroni correction - where the alpha level is adjusted by 1/n, where n is the number of comparisons made - or the false discovery rate (FDR) correction - where the probability of a single false positive is controlled [Chumbley and Friston, 2009]. More recently, hierarchical approaches to significance checking have been introduced to deal with the temporal, spectral and spatial dependencies in the EEG [Singh and Phillips, 2010]. However, the approach taken in this study circumvents the multiple comparisons problem by combining

20

prior knowledge with modelling sequences of GPS with Markov models. Firstly, the frequency band in which the phase synchronisation will be sought is selected a priori. This is often possible as existing studies indicate the frequency band of interest pertaining to the cognitive phenomenon in question. Then, a single phase vector is constructed encompassing information from all spatial locations. Hence the problem is, at this stage, reduced to assessing significance of a sequence of multiple temporal multivariate sample points. Secondly, given such data, the Markov modelling approach amounts to estimating the entire joint probability distribution of such patterns of phase synchronisation. Thus, in effect the method used in this study only performs a single comparison, which is the temporal pattern of synchronisation changes consistent with synchronisation fluctuations typical for one or other class. In practical terms, it is important to bear in mind that each of the tests takes different amounts of time to run. Table 7 lists the computational complexities of each of the methods and provides a rough indication of the approximate runtimes. Our approach of combining either an HMM or an SMM and the extended method from [Ito et al., 2007] can clearly be seen to be one of the slowest in terms of computational time, although the entropy reduction and instantaneous entropy are generally slower for small trial length (n) due to the number of iterations required. Thus, its use is a matter of judgement dependent upon the available knowledge of the phase synchronisation processes in the signals under study. If no prior knowledge is available about the expected synchronisation patterns the method developed here is the most suitable test for differentiating phase synchronisation patterns and states in bi- and multi-variate time series. Acknowledgements The authors would like to thank Dr Peter beim Graben for his many helpful comments and input throughout this work and Dr Brendan Z. Allison for proofreading.

Ian Daly et al.

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, π, transition A, 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

αt (q, l) = P (yt−l+1:t |q, l)

∑∑ q′

P (q, l|q ′ , l′ )αt−l (q ′ , l′ )

l′

(40) where P (yt−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 an 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: α∗ 0 (j) = π(j) ,

(41)

2. Induction: αt (j) =

∑

P (yt−d+1:t |j, d)P (d|j)α∗ t−d (j)

d

α∗ t (j) =

∑

αt (i)ai,j

(42) (43)

i

3. Terminate:

P (O|λ) =

N ∑

αT (i)

(44)

i=1

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 λi , 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|λi ) 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.


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 equation (3). 2. Reference the instantaneous phase to the phase on a reference channel as per equation (5). 3. At each sample point put the references phases on each channel into a phase vector (see equation (6)). 4. Calculate the instantaneous instability index via equation (7) and one of the equations, (8), (9), (10), or (11). Note, equation (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 50th percentile is crossed according to equations (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).

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Testing for significance of phase synchronisation dynamics in the EEG

Testing for significance of phase synchronisation dynamics in the EEG

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