Functional connectivity

Functional connectivity between two time series (e.g. signals recorded by two EEG electrodes) quantifies statistical similarities between those time series. Most measures of FC are examples of bivariate measures, which take a pair of time series x and y and quantifies coactivity of this pair of time series. Using bivariate measures for each pair of time series in a multivariate EEG, a functional network can be constructed where the nodes of the network correspond to a time series (e.g. an EEG electrode) and edge (i, j) corresponds to the strength of the func- tional connectivity between nodes i and j. Alternatively, some measures are mul- tivariate (accounting for all signals simultaneously) or global (giving information on synchrony in the whole network without information on edges) (Dauwels et al.,

2010). For brevity, a small number of relevant bivariant measures will be dis- cussed here, but excellent reviews of functional connectivity measures are given byDavid et al.(2004) andDauwels et al. (2010).

A simple measure of functional connectivity between time series x and y is the correlation coefficient, rx,y = 1 N N X k=1 (x(k) − ¯x) σx (y(k) − ¯y) σy , (1.22)

where N is the number of time points, ¯x denotes the mean of x, and σx denotes the standard deviation of x (and similar for y). Correlation is a linear measure quantifying the degree to which fluctuations of the signals about the mean are linearly related.

When working in sensor space (i.e. the analysed time series are those directly recorded by an EEG electrode), volume conduction must be considered. The ac- tivity of a point on the brain will conduct to multiple electrodes on the scalp, and this mixing is assumed to be linear and approximately instantaneous (Nunez and Srinivasan, 2006). Therefore volume conduction can result in spurious corre- lations in sensor space due to common sources. Source space reconstruction (Michel et al.,2004) can be used to account for the effects of volume conduction, but since the inverse problem is highly ill-posed these methods are often regular- ized and hence spurious instantaneous linear correlations may instead arise due to source leakage effects (Pascual-Marqui et al.,2011). Instead, approaches are often taken to study time lagged measures of functional connectivity.

One such measure is the maximum of the cross-correlation function (the cor- relation between x and y, shifted by lag τ , as a function of τ ) rejecting peaks

Quantifying neural dynamics

at zero lag (Schmidt et al., 2014). Taking the Fourier transform of the cross- correlation function results in the coherence between signals, which can be viewed as a frequency domain measure of linear correlation (Nunez and Srinivasan,

2006). Whilst it is not possible to explicitly remove zero lag correlations when using coherence,Nolte et al.(2004) showed that the imaginary part of coherence tends to zero as the phase difference tends to zero and hence cannot give rise to spurious connections due to linear mixing. However, the imaginary part of coher- ence will also underestimate functional connectivity for small but non-zero phase lags, so Pascual-Marqui et al. (2011) presented a lagged coherence measure normalized by the real part of coherence which reduces this underestimation of FC for small lags (Pascual-Marqui et al.,2011).

Coherence as a measure of synchrony mixes phase and amplitude effects (Lachaux et al., 1999). To test for phase binding without amplitude effects at a given frequency within a narrow frequency band,Lachaux et al.(1999) presented the phase locking factor (PLF),

cxy = 1 T     T X k=1 ei∆φxy(k)     , (1.23)

where T is the number of sampling points and ∆φxy(k)is the difference between the instantaneous phases of signals x and y at sampling point k (e.g. calculated by the Hilbert transform). As with the cross-correlation function, zero lag connec- tions can be manually removed. Alternatively, measures such as the phase lag index (PLI) (Stam et al.,2007b) which averages over the sign of the phase differ- ence at each time point are small if the lag fluctuates around zero, making them insensitive to zero lag connections. However, it should be noted that PLI does not quantify the degree of phase locking in the same manner as PLF; instead, PLI quantifies consistency in which time series is leading/lagged. The weighted PLI is a measuring which combines PLI and imaginary coherence to reduce sample- size bias, reduce sensitivity to uncorrelated noise, and increased power to detect phase synchronization (Vinck et al.,2011).

Following calculation of a functional network, it is important to test whether edges in the network are significant or an effect of finite window size in the data. A method to do so is to generate surrogate data which preserves the univariate properties of the data (e.g. amplitude distribution, power spectrum), but elimi- nate multivariate dependencies (Schreiber and Schmitz, 1996; Sun et al.,2012;

Lancaster et al., 2018). By calculating a large number of surrogate data sets and computing the functional connectivity between these surrogates, a null dis- tribution of edges can be generated to compare the empirical edges against and test for significance (Rummel et al.,2011;Sun et al.,2012;Schmidt et al.,2014). In this thesis, namelychapters 3 and chapter 4, the iterative amplitude adjusted

Fourier transform (IAAFT) method (Schreiber and Schmitz,1996) is used to gen- erate surrogates for testing significance of edges in a functional network (Rummel et al.,2011; Schmidt et al., 2014). IAAFT surrogates preserve the distribution of amplitudes and the power spectrum of the data, whilst removing any phase rela- tions in the multivariate data (Schreiber and Schmitz,1996).

Additionally, model based measures can give directed information, known as effective connectivity. Multivariate autoregressive (MVAR) models assume that the dynamics of a node are a stochastic process modulated by a linear mixture of its own history and the history of all other nodes in the networks, meaning it is an example of a multivariate measure. There exist a large number of ways to construct directed functional connectivity matrices from the coefficients of an MVAR model in both the frequency and time domain, reviewed byDauwels et al.

(2010). Models based in biophysics can also be used to estimate effective con- nectivity; for example, dynamic causal modelling involves placing neural mass models at each node and using Bayesian methods to infer the effective connec- tivity between regions (Friston et al.,2003). Dynamic causal modelling is not the only such technique, a range of parameter optimization methods are available to estimate the effective connectivity of a model based on functional data (Freestone et al.,2014;Shan et al.,2017;Dermitas et al.,2017).

Different functional (or effective) connectivity measures can give very different results for the same data (David et al., 2004; Dauwels et al., 2010; Jalili, 2016;

Hassan et al.,2017), so it is important an appropriate measure be chosen for the data. Many other factors can also affect the resulting functional network structure, including epoch length (Fraschini et al.,2016), network size (Joudaki et al.,2012), and choice of whether to binarize the network (e.g. by thresholding the network) or keeping weighted edges (Jalili,2016).