In a general way, the EEG source estimation must be regarded as an inverse problem. By way of a scalp-montage, surface electrical fields are recorded continuously. These fields are mainly generated by the cortical pyramidal cells that can be considered as sources of current, and called elementary dipoles. The pyramidal cells are situated in the grey matter of the brain. The EEG source estimation problem consists of calculating the position and the orientation (three coordinates) of these elementary dipoles from the distribution of scalp potentials.

A large number of mathematical approaches can be used for computation, but every method needs explicit or implicit assumptions about the physical properties and geometrical configuration of the pyramidal cells in the brain. For example, a specific stimulus (auditory, visual, etc.) usually causes a significant change of the scalp electrical field that reflects the activation of pyramidal cells restricted to a small area in the grey matter (auditory cortex, etc.). So, the source estimation of these evoked potentials often assume that the electrical field is only generated by a single or a few dipoles. The difference between the electrical field approximation on the basis of this assumption and the actual experimental electrical field is attributed to unknown sources or recording noise.

With regard to a study during a long period of time (several minutes), thousands or more pyramidal cells contribute to the electrical fields and the mathematical approach must take into account a wide distribution of dipoles. In this point of view, the LORETA algorithm developed by Pascual-Marqui uses 2394 dipoles evenly distributed in the grey matter. For an elementary dipole $\overline{j}$ at a given position, we compute the contribution of its electrical field, including its three moments $\left(j_x,j_y,j_z\right)$ at each electrode of the array. The head is modelled by homogenous conductive sphere (brain) surrounded by inner (skull) and outer (scalp) shells of differing conductivity (three sphere model). Then, an exact mathematical expression 〚15, 16〛 gives the lumped electrical influence on each electrode of dipoles located inside the first sphere (brain). Consequently, a linear formalism between the 2394 dipoles and the 21-electrode network (Fp1, Fpz, Fp2, F7, F3, Fz, F4, F8, T3, C3, Cz, C4, T4, T5, P3, Pz, P4, T6, O1,Oz, O2 in the standard 10–20 system, see 〚17〛) can be established:

As a standard head model reference, LORETA makes use of the three-shell spherical head model registered to the Talairach human brain atlas 〚19〛, available as a digitised MRI from the Brain Imaging Centre, Montreal Neurological Institute. Topological adjustments of spherical model and realistic head geometry use EEG electrode coordinates reported by Towle et al. 〚20〛.

In few words, LORETA method computes an estimation of the electrical activity distribution in the grey matter (standard anatomical head model) from the EEG cartographic recording (electrode array) of a subject.

This estimation is computed at each sample point and provides an instantaneous representation of cerebral activity. According to this short theoretical presentation, LORETA method seems very appropriate with evoked potential (transient EEG reactions) studies to monitor information processing. More, evoked potentials are usually characterised by a high signal to noise ratio that leads to very favourable conditions for LORETA method application. Nevertheless, we are going to show that, with take of precautions, LORETA is a very interesting tool to study long-term and spontaneous EEG activity too.

For sleep analysis, the all-night cerebral activity is most of time described as a series of sleep stages (deep sleep, light sleep, awake, paradoxical sleep, etc.). Each sleep stage presumably corresponds to a fundamental behavioural state that can be classified by the spectral composition of the EEG signals. For example, a decrease of vigilance (awareness) is associated with a rise of low frequency components of the EEG signal. In a simplified way, the EEG signal during sleep, may be split up in four frequency bands: δ 0.5–3.5 Hz (low frequencies), θ 4.0–7.5 Hz and α 8.0–12.5 Hz (medium frequencies), β 13.0–32.0 Hz (high frequencies). By now many classical investigations have confirmed these frequency bands during sleep that have led researchers to build automated detection device to classify sleep 〚21〛. Moreover, on the one hand, the quantification by spectral power has been validated as a method for characterising pathologies and pharmacological effects. On the other hand, the combination between localisation methods and frequency analysis of the sleep EEG signals remains an unexplored domain.

Our methodological approach consists in selecting a period of time (several minutes) inside each continuous sleep stage. During this period, the EEG signal of each electrode (21 signal recordings sampled at 128 Hz) is processed with a pass band filter (a moving finite-duration window (2 s) and a fast Fourier transform). Next, we compute the LORETA distribution at each sample of this period, and finally a time average gives us estimation of the cerebral activity related to a frequency band. To make easier the comparison between different frequency bands, this average estimation has been normalised on the mean dipole amplitude (whole volume). The operation is repeated for each one of the four frequency bands given above.

At first sight, it might be thought that this filtering process provides us with a localisation image related to each frequency band. As a matter of fact, so obtained images are nearly the same: an important density in the central positions of the head that scatter in more occipital areas. An argument may be suggested to explain the similarity of the image pattern: the signal to noise ratio is low in the present case. The average localisation for a quite long time, emphasize the global influence of the white noise. In the LORETA method, a weight, inversely proportional to the gain that depends on the depth, is applied to each point source of the mesh. This leads to preferential solutions at the centre of the head in case of white noise. So, the previously obtained images are mainly due to the white noise, and, to a large extent, correspond to this weight distribution.

As a consequence, the frequency approach in itself does not lead to a relevant imaging methodology. To solve this fundamental problem, we advocate a relative imaging methodology. In concrete terms, to point out the specificity of each frequential normalised average distribution, we compare these distributions with the normalised average distribution for the full frequency band (0.5–32.0 Hz). And the computation for each band of the relative density consists in the difference of each estimated dipole amplitude with regard to this full frequency distribution: (band–full)/full.

To sum up, this method gives an estimation of the regional relative cerebral activity associated with a frequency band during a determined EEG condition for each individual subject.

A lack of contribution for a particular frequency to the cerebral activity results in a relative difference that equals zero. So, comparisons to test null hypothesis (t-values), i.e., that this relative difference equals zero, have been carried out for each voxel. For each frequency band, the global three-dimensional distribution of the voxel-by-voxel t-values was visualised in a Talairach human brain atlas. The red and blue colour corresponds to positive and negative differences from the full spectrum, respectively. Comparisons between sleep stages are made in a qualitative way, and only obvious differences by visual inspection are described.

The recording study population was a group of 19 young male (mean age: 25.8 years, SD = 6.6) healthy volunteers. After one habituation night, the all-night sleep of each subject was recorded from a 28-electrode array: using a common linked-ear reference. The electronic device used for amplification and analogue filtering (type: Bessel order 2) was adjusted to 0.5–32 Hz as cut-off frequencies. The EEG signals were digitised at a sampling rate of 128 Hz. Afterwards, each all-night recording was analysed and, for each 30 s epoch, the sleep stage was visually identified, according to Rechtschaffen et Kales, by at least two confirmed scorers. Then, 21 EEG electrodes were selected and our frequency approach in source localisation was applied to periods of 20 consecutive epochs of paradoxical sleep (REM sleep), deep sleep (slow wave sleep) and light sleep (stage 2). For the display of the so-obtained three-dimensional distributions in the Talairach atlas, the LORETA–KEY software, developed by R.D. Pascual-Marqui, was used. Results are shown in the three orthogonal brain views, sliced at the level of the region of the maximum positive t-value.

For each subject, the epoch of 10 continuous minutes of paradoxical sleep was selected at the end of the night, during the fourth REM–non REM cycle. The results are shown in Fig. 1; it can be noticed that, in a general way (cf. Figs. 2 and 3), generators for the low frequency activity (δ-band) are localised in anterior regions, and higher frequency activity (θ-, α- and β-bands) in more posterior regions.

δ-Band relative activity is located predominantly in the superior frontal lobe and more pronounced in the motor cortex. For θ- and α-bands, an obvious asymmetry between right and left hemisphere can be observed. The relative source-activity of θ-band is important in both sides of the occipital part of the brain (visual cortex), but a relative positive contribution is observed in the right temporal lobe, whereas the relative activity is negative in the left temporal lobe. In the same way, the relative influence of the alpha band is far more important in the right temporal lobe.

With regard to the β-band activity, the distribution seems more central and the maximum is located in the region of posterior cingulate gyrus.

The three-dimensional distribution of relative activity for each frequency band during 10 min of continuous slow wave sleep (either stage 3 or stage 4) in the first REM–non-REM cycle is shown in Fig. 2. For the δ-band, we can notice that the superior frontal lobe is far less important, and the maximum of the relative activity is localised predominantly in the left anterior inferior temporal lobe. Concerning θ and α frequency bands, the relative activity is important in the right parietal and occipital lobes. The localisation of the cerebral activity for the β-band does not seem to be fundamentally different from that for the paradoxical sleep period.

The third period of sleep that we explored consists of 10 min of continuous stage 2 sleep during either the second or the third REM–non REM cycle. From a global point of view, the distribution of the cerebral activity, related to light sleep, occupies an intermediate position between deep sleep and paradoxical sleep. For the δ-band, the most abundant relative activity is found in the left inferior frontal lobe (Fig. 3). For the θ-activity, the distribution of positive and negative areas is not very different from that for paradoxical sleep. The maximum, however, is observed in the right parietal and occipital lobes (cf. deep sleep). Concerning α-band again, an asymmetry is present, with maximal cerebral activity in the right inferior posterior temporal lobe.