The EEG data in chapters 3-6 were analysed in the frequency domain using a suite of programs written by D. Halliday and J. Ogden (Division of Neuroscience and Biomedical Systems, Glasgow). The data in chapter 7 were analysed using a suite of programs written by A. Pogosyan (Sobell Department of Motor Neuroscience and Movement Disorders, London). Both analysis programs were based on methods outlined in Halliday et al. (1995). Subsequent comparisons of spectral power and coherence estimates were performed in Microsoft Excel.
The continuously varying waveforms that constitute EEG signals lead to analysis based on continuous time series. To characterise linear interactions between time series, frequency domain measures were derived using spectral estimation based on the fast Fourier transform (Halliday et al., 1995). Spectral methods are particularly suited to the study of systems displaying rhythmic behaviour (Conway et al., 1995). Several assumptions v/ere made about the data for this technique to be used. Firstly, the surface EEG signals were assumed to be realisations of time series, in other words the
data comprised an ordered sequence of values of a variable at equally spaced time intervals. Secondly, the time series data were assumed to be stationary. Stationarity is defined as a quality of a process in which the statistical parameters (mean, standard deviation) of that process do not change with time (Challis and Kitney, 1990). Thirdly, the data were assumed to meet the requirements for a mixing condition. The mixing condition assumes that sampled values, widely spaced in time, are statistically independent (Halliday et al., 1995). This allows confidence limits to be constructed for parameter estimates and thus hypotheses can be tested.
Local stationarity of the data was achieved by dividing the complete data record into non-overlapping disjoint segments prior to spectral analysis. The segment length used must be short enough to avoid non-stationary sections of data but long enough to obtain the desired level of frequency resolution. The frequency resolution of spectra calculated with the Fourier transform increases as the segment length used increases (see section 2.6.4).
2.6.1 Fourier transform
Mathematically, any signal in a given time interval can be decomposed into a sum of mutually orthogonal sinusoidal waves of different frequencies, amplitudes and phases. The Fourier Transform is a method of uncovering the periodic structure of EEG signals. It is a complex function of frequency used to obtain the frequency spectrum of a given signal. It is thus used to calculate the frequency, amplitude and phase of each sine wave making up a signal.
Spectral estimates were calculated by dividing each time series (total record length R) into L segments (each of length T), where R = LT. A finite Fourier transform was then performed on each segment. For a time series x(t), the Fourier transform of the segment at frequency X is defined as:
( / - I ) T / = ( / - ! )T
where i = . This transform can be thought of as a decomposition of the sampled waveform into constituent frequency components, which should highlight any distinct periodic components in the data.
The individual data processes are characterised by estimates of the power spectrum, or auto-spectrum, of each process. The pairwise relationships between the processes are characterised in the frequency domain by estimates of coherence.
2.6.2 Power spectra
The power spectrum, fxx(V> for the time series x(t) is given by the equation:
where the overbar — represents a complex conjugate. Power spectra then undergo a logarithmic transformation (log 10) to stabilise the variance that is then independent of the original value. Since the original power spectral values can be < 1, logarithmic transformation can result in negative values.
2.6.3 Coherence
The calculation of coherence is a method used to find out whether the same frequency components of two simultaneously recorded signals have similar phase shifts from trial to trial. The stability of the phase shift observed at certain frequencies may indicate that the corresponding rhythms in two signals are of mutual origin or are interacting with each other.
Coherence was used to assess the linear dependency between two time series, x(t)
and y(t), at a given frequency (k). Coherence is written as the function and defined by the equation:
J x x {Z )fy y {Z ]
Coherence functions provide a bounded (values lie between 0 and 1), unitless and normative measure of association, where 0 indicates independence of the two signals and 1 indicates a perfect linear relationship, whereby phase differences and amplitude ratios remain constant between the signals (Rosenberg et al., 1989). Coherence functions thus provide estimates of the strength of coupling between two EEG signals. As for the power spectra, a Fisher transform was applied to normalise the underlying distribution of correlation coefficients and stabilise the variances of the distributions
(Rosenberg et al., 1989; Farmer et al., 1993). The Fisher transform takes the hyperbolic inverse tangent (tanh'^) of the modulus of the coherency function, where coherency is given by the square root of the coherence.
2.6.4 Frequency resolution and segment length used
The segment length defines the minimum frequency that can be resolved and thus the spectral resolution, according to the equation:
Frequency resolution = sampling rate / segment length
The segment length in the Fourier transform in chapters 3-6 was 1024 data points and the sampling rate was 1 kHz. Thus the frequency resolution was 0.98 Hz. In chapter 7, data were sampled at 500 Hz with a segment length of 512 data points, again giving a frequency resolution of 0.98 Hz.
2.6.5 Statistical analysis and interpretation
Spectral power and coherence estimates in any given experiment were obtained under identical conditions for each subject. Prior to analysis, all spectral power estimated underwent logarithmic transformation and all coherence estimates underwent a Fisher transformation. Analysis of variance (ANOVA) was then used to compare data for two or more conditions across multiple subjects (chapters 3-6) and between multiple groups (chapter 7).
To reduce the data, log power and transformed coherence estimates for each subject and for each condition were averaged across set frequency bands (the bands used were similar in each experiment). In order to decrease the effect of inter-subject and inter-electrode pair variation in coherence data, task-related log power and task-related transformed coherence were calculated by subtracting the raw values at rest from those during the active state. Coherence was always interpreted in conjunction with power primarily to ensure that changes in coherence were not due to modulations in non-linearly related frequency components (Florian et al., 1998).
Analysis was performed using SPSS (version 10). For parametric data, a repeated measures general linear model analysis was performed. For non-parametric data, a Friedman ANOVA was used. In all parametric ANOVAs used, a Greenhouse-Geisser correction for sphericity was incorporated where necessary. ANOVA results were considered significant for p < 0.05. Post-hoc analysis was performed on significant parametric data sets using the Student’s paired t-test and on significant non-parametric
data sets using the Wilcoxon 2-sample test. Bonferroni corrections for multiple comparisons were applied where necessary.