 [page 16↓] 
Frequency analysis provides a few parameters whose knowledge is crucial to adequately interpret the results. This chapter will describe the methodological and analytical techniques that are in common to the experiments presented in Chapters 3–6. Specific methodological aspects of each experimental set up will be detailed at the start of each chapter. For all experiments corticomuscular and intermuscular frequency analysis was performed offline using a program written by J. Ogden and D. Halliday (Division of Neuroscience and Biomathematical Systems, University of Glasgow, UK) based on methods outlined by Halliday and colleagues (1995).
The very basis of all measurements in the frequency domain is the divison of the signal into discrete spectra. Spectra are usually determined using the fast Fourier transform (FFT), which was used in all the following experiments. A schematic summary of the different frequency analysis techniques is shown in Figure 2.1. In the FFT appraoch data are divided into serial, usually nonoverlapping windows, transformed and then averaged. The basic tradeoff to be considered in the FFT approach is between frequency resolution and spectral variance. As the size of the windows decrease, the variance goes down, but the spectral resolution becomes poorer. Spectra derived from a FFT approach are defined pointwise, and the frequency difference between two adjacent points is given by the sampling rate divided by the FFT window size (in samples).
Alternatively, spectra could be determined using MAR models. The latter have the desirable property of representing the characteristics of a signal with just a few coefficients, which can then be used to calculate the relevant spectra. Because of this property, MAR models are often useful for modelling short data sets. In addition, MAR spectra are continuous functions of frequency, and thus avoid the spectral resolution problems encountered with the FFT approach. In practice, however, the calculation of true confidence limits is
 [page 17↓] 
Fig. 2.1.: Schematic overview of the different methodological approaches to signal analysis in the frequency domain. Note that FFT based models can only be applied with signals assumed to be stationary whereas wavelet analysis and autoregressive models can additionally analyse nonstationary signals. For details see text  

 [page 18↓] 
problematic and the approximate limits that can be calculated are generally wider than their FFT counterparts. (Cassidy and Brown, 2002). Also, the computation time for FFT methods is much faster than for MAR modelling. The MAR representation can also be embedded into more complex nonstationary models, which are often necessary in the analysis of signals whose statistical properties change substantially over time.
Finally, coherence estimation can also be achieved using wavelet analysis. The major advantage of this technique is that, different to FFT based analysis, the data has not to be stationary and that it can detect short, significant episodes of coherence (Lachaux et al., 2001). Whichever technique is used autospectra and crossspectra may be derived, and from these coherence and phase are determined. For a general introduction to coherence see Challis and Kitney (1991), and for a more detailed discussion of the measures derived from frequency analysis to Rosenberg et al. (1989) and Halliday et al. (1995) for FFT approaches, Cassidy and Brown (2002) for MAR approaches and Lachaux et al. (2001) for wavelet analysis.
The main parameters deriving from the division of signals into spectra are as follows:
The coherence between signals a and b at frequency λ is an extension of Pearson's correlation coefficient and is defined as the absolute square of the crossspectrum normalised by the autospectra:
In this equation, faa , fbb and fab give the values of the auto and crossspectra as a function of frequency λ and are assumed to be realisations of stationary zero mean time series. Coherence is a measure of the linear association between 2 signals. It is a bounded measure taking values from 0 to 1 where 0 indicates that there is no linear association (that is signal [page 19↓] a is of no use in linearly predicting signal b) and 1 indicates a perfect linear association between the two. Here, coherence was considered to be significant if it exceeded the 95% confidence level.
Because coherence ranges between 0 and 1, its variance must be stabilised by transformation before statistical comparison for scientific purposes in larger studies. In practice this makes relatively small difference to small coherences, but is important with coherences of more than 0.6. The variance of the coherence is usually normalised by transforming the square root of the coherence (a complex valued function termed coherency) at each frequency using the Fisher transform:
This results in values of constant variance for each record given by 1/2L where L is the number of segment lengths used to calculate the coherence (Rosenberg et al., 1989), which can then lead to coherences greater than 1.
Phase, φ _{ab} ( λ ), is expressed mathematically as the argument of the crossspectra:
It comprises two factors, the constant time lag given by the slope of the phase spectrum, when linear, and a constant phase shift, which is reflected in the intercept and is due to differences in the shapes of the signals (Mima and Hallett, 1999a). To calculate the temporal delaybetween the two signals the following equation is used (where the phase is in radians):
 [page 20↓] 
The phase estimate from a single point is ambiguous (Gotman, 1983). Measuring phase relationships that are linear over a band of frequencies reduces this ambiguity. Under these circumstances the temporal delay between the signals can be calculated from the gradient of the line. A negative gradient indicates that the input/reference signal leads.
The cumulant density, equivalent to the crosscorrelation between signals, is calculated from the inverse Fourier transform of the crossspectrum. When the input/reference signal is EMG this cumulant density estimate resembles a backaveraged EEG record.
The recording of scalp EEG is not always easy, for example in children, and in movement disorders, in particular, the signal can be marred by muscle artefact. Thus it is fortunate that the same drive that leads to coherence between cortex and muscle also leads to coherence between the EMG signals of agonist muscles coactivated in the same task (Kilner et al., 1999). EMGEMG coherence analysis can be performed using single or multimotor unit intramuscular needle recordings or surface EMG. Studies of single units tend to be less informative (smaller signal to noise ratio in coherence spectra) than multiunit needle or surface recordings (Christakos, 1997). Surface EMG is more practical but may be limited by volume conduction between muscles. The latter can be ruled out if there is a constant phase lag between the two EMG signals in the range of significant coherence. Thus it is generally best to chose muscle pairs that are separated (such as forearm extensors and intrinsic hand muscles), where one would expect physiological coupling to involve a phase difference. Alternatively, volume conduction can be limited by appropriate levelling of both signals and analysing the coherence between the resulting point processes. The [page 21↓]principle that intermuscular coherence may give comparable information about descending cortical drives as corticomuscular coupling has been validated in cortical myoclonus (Brown et al., 1999).
Nevertheless, it should be remembered that oscillatory presynaptic drives to spinal motoneurons other than those of cortical origin will also be reflected in the synchronisation of motor unit discharge, where these contribute to muscle activity. Thus EMGEMG coherence may afford an additional insight into subcortical motor drives.
This section considers some specific problems of recording and interpretation relevant to the investigation of corticomuscular coupling.
The first problem is the signal itself and the question of how closely it matches the activity to be modelled. For example, the skull and scalp act as a low pass filter so that scalp EEG may not reflect cortical activities at higher frequencies which are otherwise evident in electrocorticographic or MEG recordings. Another factor is the focality of the cortical area sampled by scalp EEG. This can be increased by Laplacian derivations such as the current source density and Hjorth transformation (Hjorth, 1975; Horth 1980). The latter also tend to give higher EEGEMG coherence estimates, whereas common average references and balanced noncephalic references may give misleading results because of possible EMG contamination (Mima and Hallett, 1999b). In addition, it is necessary to sample the signal at a rate that is greater than twice the lowpass filter setting so as to avoid aliasing and the identification of spurious spectral elements.
Additionally, filter settings deserve specific consideration. In all experiments EMG was [page 22↓]band pass filtered between 53 and 1000 Hz. The highpass filter was chosen to limit contamination by movement artifact (see Fig. 2.2.), which otherwise would have lead to greatly inflated coherence estimates.
Fig. 2.2.: Example of data processing. (A) Raw EMG from 1DI highpass filtered at 0.53 Hz and recorded during selfpaced movement at ~5 Hz. Note prominent movement artefact between EMG bursts. (B) Simultaneously recorded raw EMG highpass filtered at 53 Hz. Movement artefact is much reduced. (C) EMG as in (B) but fullwave rectified. (D) Product of levelling signal in (C) to give a point process. (E) Power spectra corresponding to EMG in (A) and (B). Power between the two differs by a factor of ~100 (note logarithmic scale), although qualitatively the autospectra are similar. The difference in power is most marked at the tremor frequency of 5 Hz and is largely due to the presence of movement artefact with a highpass filter of 0.53 Hz. (F) Power spectrum of rectified highpass filtered EMG from (C). Rectification increases power and emphasises the tremor peak at 5 Hz. (G) Spectra of point processes derived from levelling rectified EMG filtered at 0.53 Hz and 53 Hz. Power spectra are almost identical, confirming that high pass filtering at 53 Hz does not diminish information about interspike intervals in the multiunit EMG record. It is the spike timing information that is important in determining the coherence between different EMG signals. Levelling, however, diminishes the effects of lowlevel signals such as movement artefact or volume conduction.  

 [page 23↓] 
An most important point is that as coherence is a measure of linear dependence between two signals in the frequency domain, any artefact common between channels leads to high coherence values over the relevant frequency band. This is most commonly evident in the case of mains artefact, but any volume conduction of signals between electrodes or crosstalk within leads or amplifiers will also lead to inflated coherences. Such artefacts occur with zero phase delay, and are reasonably obvious in paradigms in which biologically related signals would be expected to demonstrate phase differences, such as when investigating the coupling between EEG and EMG or EMG and tremor.
Two confounding factors must be remembered when the temporal delay between two signals is calculated from the phase. First, low pass filters, such as the skull and scalp, may introduce phase shifts that may underestimate real conduction delays (Lopez da Silva, 1989). Second, it is possible that more than one coherent activity may overlap in the same frequency band, in which case the phase estimate will be a mixture of the different phases. This may help explain why the temporal differences calculated between EEG or MEG and EMG are often shorter than those predicted from transcranial stimulation of the motor cortex (Brown et al., 1998c; Mima et al., 1998b; Salenius et al., 1997a), as both efferent and afferent corticomuscular coupling may occur in overlapping frequency bands (Mima et al., 2001a). Coexisting bidirectional oscillatory flows between neural networks can be separated through application of the directed transfer function (Kaminski and Blinowska, 1991), although so far there has been only one report of the use of this in the motor sphere (Mima et al., 2001a).
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