Distinguishing low frequency oscillations within the 1/f spectral behaviour of electromagnetic brain signals
© Demanuele et al; licensee BioMed Central Ltd. 2007
Received: 16 August 2007
Accepted: 10 December 2007
Published: 10 December 2007
It has been acknowledged that the frequency spectrum of measured electromagnetic (EM) brain signals shows a decrease in power with increasing frequency. This spectral behaviour may lead to difficulty in distinguishing event-related peaks from ongoing brain activity in the electro- and magnetoencephalographic (EEG and MEG) signal spectra. This can become an issue especially in the analysis of low frequency oscillations (LFOs) – below 0.5 Hz – which are currently being observed in signal recordings linked with specific pathologies such as epileptic seizures or attention deficit hyperactivity disorder (ADHD), in sleep studies, etc.
In this work we propose a simple method that can be used to compensate for this 1/f trend hence achieving spectral normalisation. This method involves filtering the raw measured EM signal through a differentiator prior to further data analysis.
Applying the proposed method to various exemplary datasets including very low frequency EEG recordings, epileptic seizure recordings, MEG data and Evoked Response data showed that this compensating procedure provides a flat spectral base onto which event related peaks can be clearly observed.
Findings suggest that the proposed filter is a useful tool for the analysis of physiological data especially in revealing very low frequency peaks which may otherwise be obscured by the 1/f spectral activity inherent in EEG/MEG recordings.
The significance of slow waves in literature
Whereas conventional, clinical, EEG recordings consider brain activity within a frequency range of 0.5–50 Hz, several studies have shown that physiological and pathological EEG activity ranges from 0 Hz to several hundreds of Hz. Hence, the emergence of the recent term, full band EEG, which considers both infraslow and ultrafast EEG activity . Ultrafast activity is likely to be apparent in the electrophysiology of neurocognition and motor initiation . Infraslow (ISO) EEG oscillations have been recorded in the EEG of preterm neonates (0.01 to 0.1 Hz) and during epileptic seizure activity .
Several studies have concentrated on these ISOs. Some recent work demonstrates large amplitude oscillations in the human cortex ranging from 0.02 to 0.2 Hz during sleep [3, 4]. Here, each slow wave is accompanied also by an increase in EEG synchronisation between different scalp areas. These ISO fluctuations in EEG synchrony are characterised by the stable dynamical structure of sleep . The work by P. Achermann et al. in  illustrates the presence of distinct slow-wave components below 1 Hz in human sleep and the work by I.G. Campbell et al. in  suggests that the physiological and molecular mechanisms of these very low frequency EEG activity differ from those of higher frequency bands such as the delta band (1–4 Hz). Supra-second oscillations have been recorded also in the thalamus and basal ganglia of rats, in the cerebral cortex of cats , and in the monkey visual cortex .
Moreover, the study for the classification of neuronal oscillations in the mammalian cortex in various frequency bands suggests that there are spontaneous coherent low frequency neuronal oscillations within a neuro-anatomically robust default network of brain activity . This has been reinforced by further work by Buzsaki et al. in  and M. Steriade et al. in  where studies have been carried out attempting to link neuronal activity to behaviour.
All this suggests that the basic rhythms of the EEG (δ, α, β, γ) represent only a part of the measured brain activity. Very low frequency oscillations are a salient feature of the EEG and can give access to new insight into brain function.
Issues in recording and analysing very low frequency EEG activity
Infraslow signal recording requires genuine DC-coupled amplifiers with high input impedance, high DC stability and a wide dynamic range. Sufficiently stable electrodes and adequate gels should be used to provide a faithful EEG recording at such low frequencies . DC drift which is superimposed on any meaningful event-related slow activity can be an issue unless amplifiers are reset every three minutes  to ensure that the signal is kept in the optimal range of the amplifier throughout the recording.
where S x (f) is the power spectral density, f is the frequency and γ is some spectral parameter which is usually close to 1 but can lie in the range 0 <γ < 2 , and could be greater than 2 in the presence of noise sources.
This 1/f γ spectral behaviour was first reported in 1925 in an electric current passing through a vacuum tube . It appears also in economic and communication systems, in electronic transistors and diodes, in the annual amount of rainfall and in the rate of traffic flow . Biological data such as the potential measured across nerves and physiological systems such as the cardiovascular and respiratory mammalian systems also exhibit this kind of behaviour .
The Origin of the 1/f γ spectral behaviour
The inverse relation of the power density of the EEG with frequency in the mammalian cortex is a result of the physical structure of neuronal networks and the limited speed of neuronal communication arising from axon conduction and synaptic delays . A large cluster of neurons, each generating a unit activity, forms a functional network which is held together by the neurons' synchronisation that ensures activity control . Such synchronised behaviour seems to attract further neurons and causes the oscillation amplitude to increase. Moreover, the period of oscillation is determined by the size of this neuronal cluster that constitutes a given cycle . Thus, large neuronal areas are associated with slow, high amplitude oscillations whereas a small, localised concentration of neurons gives rise to higher frequency, low amplitude signals . This explains why most of the power of the EEG signals is concentrated in the low frequency spectrum.
MEG recorded data, which is sometimes preferred over EEG recordings due to its high spatial resolution and the extremely high temporal resolution, also exhibits this 1/f γ behaviour inherent in its power spectrum. This is to be expected since these two systems share the same underlying model – MEG measures the minute magnetic field generated by the electrical activity of neurons. This electrical activity corresponds to that detected by EEG electrodes.
In order to be able to analyse any spectral activity superimposed on this 1/f trend the EEG/MEG power spectrum can be normalised by removing this trend . Makinen et al. in  employ a technique called Partition-Referenced Moment and use it with wavelet transforms to obtain a level base spectrum. This is used for examining ongoing oscillations and auditory event-related brain processes recorded by MEG. Apart from being an involved approach, the proposed method is based in the frequency domain thus destroying the phase information of the raw signal.
In this paper we describe two methods that can be used to achieve spectral normalisation, i.e. removal of the intrinsic 1/f γ to provide a flat spectral base onto which event-related brain activity is superimposed. The first method is based in the frequency domain – its main aim being to investigate the spectral characteristics of electrophysiological signals and to provide us with a basis for normalisation. The second method is a time domain approach where spectral normalisation is achieved by filtering the raw signals prior to further data analysis. This is a simple and effective method which conserves the phase information of the input signal. For these reasons, we use it on real electrophysiological data, namely epileptic seizures, very low frequency EEG recordings, MEG recordings and Evoked Responses to illustrate its function.
Methods for spectral normalisation
A. Normalisation in the frequency domain
Normalisation of the spectrum can be achieved in the frequency domain by dividing any EEG spectrum by an established background 1/f γ spectrum. This concept was tested on the multi-channel EEG recordings of two participants. 32 EEG channels were used and these included ear references at Tp9 and Tp10, two electrooculogram (EOG) channels that monitored the linked vertical and horizontal eye movement, and one electrocardiogram (ECG) channel. Electrode placement was in accordance to the 10–20 system and an electrode cap was used. Impedance levels were set at less than 5 kΩ.
No filters were switched on during the recordings such that DC activity could be captured and DC-stable sintered electrodes were used. The data was sampled at 250 Hz and was digitally stored in a 12 bit ADC. Before analysis the data was decimated by a factor of 25 – this was necessary to ensure an adequate number of samples for the analysis of very low frequencies. Detrending was then carried out to remove the mean shift (over 5 or 10 minutes) in each recording.
In these recordings, every participant followed a ten minutes driving task – where the participant was meant to trail a plain winding track on screen by pressing the arrow buttons on the keyboard. This was followed by five minutes, eyes-closed resting condition during which the participant was seated on a reclining chair. A ten minute arrows task followed, during which the participant was asked to press a button whenever the arrow appearing on screen pointed left or right (according to the instruction given). The participant's recording was concluded by a five minutes eyes-open resting period, again seated on a reclining chair. For every participant these segments of data were analysed separately. The expectation is that under this type of mental load which requires the participants' attention there is predominant low frequency activity around 0.1 Hz. More information behind this can be found in .
B. Time domain spectral normalisation approach
where A/f γ is the EEG spectrum with the intrinsic 1/f γ characteristics, Bf γ is the inverse filter contribution and AB is the result of their interaction, implying that the output is a normalised spectrum. The 1/f γ curve can be modelled as a finite impulse response (FIR) model such that its inverse will be an infinite impulse response (IIR) model. However the problem is the lack of control on the FIR coefficients since these are already predetermined by the shape of the normalisation curve. Thus if the resultant FIR model is not minimum phase the IIR model stability becomes a major issue. Moreover we require the filter to be a linear phase filter to avoid phase distortion of the input EEG signal – and an IIR filter will not meet this requirement.
Approximating the inverse filter by a differentiator
Figure 1 shows 1/f γ curves superimposed on the power spectral density of a typical EEG channel, with γ varying from 1 to 3. It is clear that the 1/f curve follows closely the EEG spectral trend across the entire frequency band. This was verified for a number of participants under different task and rest conditions. Therefore the normalisation curve can be approximated to be a 1/f curve, i.e. setting γ = 1, and the inverse filter can be obtained by applying a differentiator. This could be done using the cfirpm function in MATLAB, which provides a set of filter coefficients that simulate a linear phase differentiator.
where x(t) is the input signal, y(t) is the filtered output and Δt = T s = 1/f s , f s being the sampling frequency. Thus the MA filter coefficients can be set as 1/Ts and -1/Ts.
This analysis shows that performing normalisation in the time domain by using a differentiator is a very viable approach. This is because, although the frequency domain method produces a normalised spectrum without assuming that the underlying bias is a strict 1/f curve, (since the normalisation curve is modelled directly from the dataset), the phase information of the EEG signal is lost. Consequently, it is not possible to reconstruct the time-domain series from the normalised spectrum. Moroever the resultant time series cannot be used to extract its phase interactions with other normalised data. For these reasons, the differentiator is used here to filter different datasets in order to demonstrate its function as a tool for spectral normalisation. The results are illustrated in the following section.
Applying the differentiator to various datasets
The differentiator was applied to two sinusoidal signals of frequencies 0.1 Hz and 0.5 Hz respectively, superimposed on normal background EEG. The signal to noise ratio (SNR) of the higher frequency signal (SNR2) was kept fixed at 15 dB whereas that of the lower frequency signal (SNR1) was varied from 0 to 47 dB. Each SNR was measured by calculating the ratio of power of the sine wave to that of the background EEG.
Epileptic seizure data
Focal epileptic seizure data recorded using twenty-five electrodes placed on the scalp according to the International 10–20 electrode placement system with reference at FCz, was used as input to the differentiator. The three minute long data recording was sampled at 200 Hz and digitally stored at 12 bit resolution. The recording included pre-ictal, ictal and post-ictal activity. The seizure was focused at the left-temporal lobe (around T3).
When considering seizure data the relationship between SNR before and after filtering shows the same linear trend like that obtained for synthetic data. Here, the SNR was loosely computed by finding the ratio of power of the three minute EEG data incorporating seizure activity to the power of ongoing background EEG activity of the same duration recorded for the same subject. This procedure was carried out on the data before and after passing it through the differentiator.
EEG data with LFO activity
Thus the resultant spectrogram obtained by normalising in the frequency domain and that obtained by normalising in the time domain are very similar – they both show a peak at 0.1 Hz (or 0.2 Hz) and a flat spectrogram when no extra low frequency activity is expected. One can note similar activity around 0.1 Hz in Figure 3(b) (normalisation in the frequency domain) and Figure 14(d) (normalisation in the time domain) for the same data set.
Another useful application of the differentiator is in the analysis of MEG data. MEG, the recording of the magnetic activity of the brain is measured in a whole-head system. A CTF Systems 151 channel MEG system was used to record over 20 minutes of ongoing activity in a normal, healthy volunteer. The examples given here demonstrate that passing MEG data through this type of filter achieves normalisation of its spectra.
Evoked response data
Another relevant set of EEG signals are those involving evoked responses. The data set used here contains P300 evoked potentials. One minute's worth of EEG data sampled at 240 Hz was used and the analysis was focused on electrodes C3, Cz and C4 where the P300 response was expected. The participant was presented with a six by six matrix of characters. The task was to focus attention on characters in a word that was prescribed by the investigator (i.e. one character at a time). The data contained 35 epochs of 1.5 second duration each. The P300 stimulus appeared at 0.5 seconds in every epoch so the P300 response was expected to occur at 0.8 seconds within the epoch.
In this paper the importance of spectral normalisation has been emphasised as a way of revealing event-related peaks which may otherwise be obscured by the intrinsic 1/f spectral activity in EM brain signal recordings. Because of this 1/f trend EM brain signals have spectra with high power at low frequencies. Normalisation renders a flat base spectrum when no extra low frequency activity is present and reveals distinct peaks related to specific cognitive tasks or mental conditions. This is particularly important for the analysis of very low frequency oscillations (0.05 – 0.5 Hz) apparent in EEG and/or MEG signals. Frequency domain normalisation destroys the phase information of the input signal and excludes the possibility of signal reconstruction after spectral whitening. In this work we propose a time domain approach which employs a differentiator to cancel the 1/f trend. This is a simple solution which does not require the selection of various parameters. Moreover, the filter leaves the signal phase intact due to its linear phase characteristics. Here we have shown its application in a broad range of physiological signals including epileptic seizure data, EEG data with very low frequency characteristics, MEG data and evoked response recordings. In each case, the spectral normalisation helped to highlight peaks of interest across the spectra.
Spectral normalisation is achieved in all of the mentioned datasets. This implies that:
The underlying background electrophysiological activity does indeed follow a 1/f trend
The approximation to model the inverse of the normalisation curve by a differentiator is suitable.
Future work involves the use of this filter for the analysis of data that has very low frequency activity that is of interest. This technique can also be used for comparison of power between the very low frequency and the conventional or higher frequency bands during different cognitive processes. Its linear phase response makes it possible to investigate the phase synchronisation between pairs of filtered data at those frequency bands where prominent peaks appear in the normalised spectra.
Funding from the Rayleigh Scholarship (Institute of Sound and Vibration Research) and the School of Phsychology at the University of Southampton is gratefully acknowledged.
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