In this paper, we have used a combination of theoretical and experimental analyses to investigate the spectral structure of EEG and MEG signals. In the first part of the paper, we presented a theoretical investigation showing that if the extracellular medium is purely resistive, the equations of the frequency dependence of electric field and magnetic induction take a simple form, because the admittance tensor does not depend on spatial coordinates. Thus, the macroscopic magnetic induction does not depend on the electric field outside the neuronal sources, but only depends on currents inside neurons. In this case, the frequency scaling of the PSD should be the same for EEG and MEG signals. This conclusion is only valid in the linear regime, and for low frequencies.
An assumption behind this formalism is that the spatial and frequency dependence of the current density factorize (Eq. (19
)). We have shown in the Appendices
that this is equivalent to consider the different current sources as independent. Thus, the formalism will best apply to states where the activity of synapses is intense and of very low correlation. This is the case for desynchronized-EEG states or more generally “high-conductance states”, in which the activity of neurons is intense, of low correlation, and the neuronal membrane is dominated by synaptic conductances (Destexhe et al. 2003
). In such conditions, the dendrites are bombarded by intense synaptic inputs which are essentially uncorrelated, and one can consider the current sources as independent (Bédard et al. 2010
). In the present paper, we analyzed EEG and MEG recordings in such desynchronized states, where this formalism best applies.
Note that the above reasoning neglects the possible effect of abrupt variations of impedances between different media (e.g., between dura matter and cerebrospinal fluid). However, there is evidence that this may not be influential. First, our previous modeling work (Bédard et al. 2004
) showed that abrupt variations of impedance have a negligible effect on low frequencies, suggesting that even in the presence of such abrupt variations should not play a role at low frequencies. Second, in the frequency range considered here, the skull and the skin are very close to be resistive at low frequencies (Gabriel et al. 1996b
), so it is very unlikely that they play a role in the frequency scaling in EEG and MEG power spectra even at high frequencies.
In the second part of the paper, we have analyzed simultaneous EEG and MEG signals recorded in four healthy human subjects while awake and eyes open (with desynchronized EEG). Because of the large number of channels involved, we used an automatic procedure (B-splines analysis) to calculate the frequency scaling. As found in previous studies (Pritchard 1992
; Freeman et al. 2000
; Bédard et al. 2006a
), we confirm here that the EEG displays frequency scaling close to 1/f
at low frequencies.5
However, this 1/f
scaling was most typical of the midline channels, while temporal and frontal leads tended to scale with slightly larger exponents, up to 1/f2
(see Fig. (a)). The same pattern was observed in all four patients.
This approach differs from previous studies in two aspects. First, in contrast to prior studies (such as Novikov et al. 1997
; Linkenkaer-Hansen et al. 2001
), we calculated the frequency scaling of all the sensors and did not confine our analysis to a specific region. Second, unlike other investigators (such as Hwa and Ferree 2002a
), we did not limit our evaluations to either EEG or MEG alone, but rather analyzed the scaling of both type of signals simultaneously. Such a strategy enables us to provide an extended spatial analysis of the frequency scaling. It also provides a chance to compare the scaling properties of these signals in relation to their physical differences.
For the MEG recordings, the frequency scaling at low frequencies was significantly lower compared to the EEG (see Fig. ). This difference in frequency scaling was also accompanied by spatial variability patterns (see Fig. ) showing three distinct regions:(1) a frontal area where the exponents had their highest values in the case of MEG; (2) a central area where the values of exponents of EEG and MEG get closer to each other and (3) a parietotemporal horseshoe region showing the lowest exponents for MEG with bimodal characteristics (Fig. ). In some cases, the scaling of the uncorrected and corrected MEG signal was also close to 1/f
, as reported previously (Novikov et al. 1997
). In the frontal area (FR mask), the scaling exponent of the EEG was generally larger. At Vertex (VX mask), EEG and MEG had similar values and at the Parietotemporal region (PT mask), MEG showed a bimodal property with a much broader range of scaling exponent in comparison to EEG (see Fig. ). Note that to avoid the effect of spurious peaks, Novikov et al. used the spectrum of signal differences and argued for the existence of a local similarity regime in brain activity. This approach fundamentally changes the spectral characteristics of Magnetometers (which measure the absolute magnitude of the magnetic induction) into a measure that only for the neighboring sensors approximates the behavior of the gradiometers (which measures the gradient of the magnetic induction). So it is not clear how to relate their values to the ones obtained here.
To make sure that the differences of frequency scaling between EEG and MEG were not due to environmental or instrumental noise, we have used five different methods to remove the effect of noise. These methods are based on different assumptions about the nature and effect of the noise. A first possibility is to correct for the noise induced by the MEG sensors. It is known that the SQUID detectors used in MEG recordings are very sensitive to environmental noise and they can produce 1/f
noise (Hämäläinen et al. 1993
). Under this assumption, part of the scaling of the MEG could be due to “filtering” by the sensors themselves, which justifies a simple subtraction of scaling exponents to remove the effects of this filtering. Note that such empty-room recordings were not possible for the EEG, although the noise from the recording setup could be estimated (see Miller et al. 2009
for example). Because in some cases both MEG and emptyroom signals have similar scaling, a simple correction by subtracting the exponents would almost entirely abolish the frequency scaling while in other cases it may even revert the sign of the scaling exponent (see Fig. (h), Supplementary
However, if noise is not due to the sensors but is of additive uncorrelated nature, then another method for noise correction must be used. For this reason, we have used a second class of well-established methods consisting of spectral subtraction (Boll 1979
; Sim et al. 1998
). Using three of such methods (LMSS, NMSS and WF) changed the scaling exponent, without fundamentally changing its spatial pattern (Fig. (d)–(f)). The largest correction was obtained by non-linear methods which take into account the SNR information in the MEG signal. We also applied another class of method which uses the collective characteristics of all frequencies in noise correction (PLS). Similar to exponent subtraction, this method nearly abolished all the scaling of the MEG (Fig. (g)). In conclusion, although different methods for noise subtraction give rise to different predictions about frequency scaling, all of the used methods enhanced the difference between EEG and MEG scaling. Thus, we conclude that the difference of EEG and MEG scaling cannot be attributed to noise, but is significant, therefore reinforcing the conclusion that the medium must be non-resistive.
An alternative method to investigate this is the “Detrended Fluctuation Analysis” (DFA; see Kantelhardt et al. 2001
; Linkenkaer-Hansen et al. 2001
; Hwa and Ferree 2002a
). Like many nonlinear approaches, DFA results are very vulnerable to the selection of certain parameters. Different filters severely affect the scaling properties of the electromagnetic signals to different extents, and therefore the parameters estimated through the DFA analysis could be false or lead to distorted interpretations of real phenomena (Valencia et al. 2008
), and these effects are especially prominent for lower frequencies, which are precisely our focus of investigation here. One of the fields for which DFA can provide robust results is to analyze surrogate data with known characteristics. Although the use of DFA to evaluate the scaling exponents of EEG was vigorously criticized (Valencia et al. 2008
), a previous analysis (Hwa and Ferree 2002a
) reported two different regions, a central and a more frontal, which somehow correlate with the FR and VX regions identified in our analysis. Similarly, a study by Buiatti et al. (Buiatti et al. (2007
)) using DFA provided evidence for topographical differences in scaling exponents of EEG recordings. They report that scaling exponents were homogeneous over the posterior half of the scalp and became more pronounced toward the frontal areas. In contrast to Linkenkaer-Hansen et al. (2001
) (where envelope of alpha oscillations was used for DFA estimation), this study uses the raw signal in its DFA analysis and yields values closer to those reported here.
Both uncorrected signals and empty-room correction show that there is a fundamentally different frequency scaling between EEG and MEG signals, with near-1/f
scaling in EEG, while MEG shows a wider range at low frequencies. Although it is possible that non-neuronal sources affect the lower end (<1Hz) of the evaluated frequency domain (Voipio et al. 1989
), the solution to avoid these possible effects remain limited to invasive methods such as inserting the electrode into the scalp (Ferree et al. 2001
) or using intracranial EEG recordings (similar to Miller et al. 2009
). This approach would render wide range spatial recording as well as simultaneous invasive EEG and MEG recordings technically demanding or impractical. However, if technically feasible, such methods could provide a way to bypass non-neuronal effects at very low frequency. It could also provide a chance to evaluate the effects of spatial correlation on spectral structure at a multiscale level.
The power spectral structure we observe here is consistent with a scenario proposed previously (Bédard et al. 2006a
): the 1/f
structure of the EEG and LFP signals is essentially due to a frequency-filtering effect of the signal through extracellular space; this type of scaling can be explained by ionic diffusion and its associated Warburg impedance6
(see Bédard and Destexhe 2009
). It is also consistent with the matching of LFPs with multi-unit extracellular recordings, which can be reconciled only assuming a 1/f
filter (Bédard et al. 2006a
). Finally, it is also consistent with the recent evidence from the transfer function calculated from intracellular and LFP recordings, which also showed that the extracellular medium is well described by a Warburg impedance (Bédard et al. 2010
, , submitted to this issue). If this non-resistive aspect of extracellular media is confirmed, it may influence the results of models of source localization, which may need to be reformulated by including more realistic extracellular impedances.
In conclusion, the present theoretical and experimental analysis suggests that the scaling of EEG and MEG signals cannot be reconciled using a resistive extracellular medium. The 1/f
structure of EEG with smaller scaling exponents for MEG is consistent with non-resistive extracellular impedances, such as capacitive media or diffusion (Warburg) impedances. Including such impedances in the formalism is non trivial because these impedances are strongly frequency dependent. The Poisson integral (the solution of Poisson’s law
) would not apply anymore (see Bédard et al. 2004
; Bédard and Destexhe 2009
). Work is under way to generalize the formalism and include frequency-dependent impedances.
Finally, it is arguable that the scaling could also be influenced by the cancellation and the extent of spatial averaging of microscopic signals, which are different in EEG and MEG (for more details on cancellation see Ahlfors et al. 2010
; for details on spatial sensitivity profile see Cuffin and Cohen 1979
). Such a possible role of the complex geometrical arrangement of underlying current sources should be investigated by 3D models which could test specific assumptions about the geometry of the current sources and dipoles, and their possible effect on frequency scaling. Such a scenario constitutes another possible extension of the present study.