Comparative power spectral analysis of simultaneous electroencephalographic and magnetoencephalographic recordings in humans suggests non-resistive extracellular media

Abstract

The resistive or non-resistive nature of the extracellular space in the brain is still debated, and is an important issue for correctly modeling extracellular potentials. Here, we first show theoretically that if the medium is resistive, the frequency scaling should be the same for electroencephalogram (EEG) and magnetoencephalogram (MEG) signals at low frequencies (<10 Hz). To test this prediction, we analyzed the spectrum of simultaneous EEG and MEG measurements in four human subjects. The frequency scaling of EEG displays coherent variations across the brain, in general between 1/f and 1/f ². In a given region, although the variability of the frequency scaling exponent was higher for MEG compared to EEG, both signals consistently scale with a different exponent. In some cases, the scaling was similar, but only when the signal-to-noise ratio of the MEG was low. Several methods of noise correction for environmental and instrumental noise were tested, and they all increased the difference between EEG and MEG scaling. In conclusion, there is a significant difference in frequency scaling between EEG and MEG, which can be explained if the extracellular medium (including other layers such as dura matter and skull) is globally non-resistive.

Appendices

Appendix 1: Frequency dependence of electric field and magnetic induction

To compare the frequency dependence of magnetic induction and electric field, we evaluate them in a dendritic cable, expressed differentially. For a differential element of dendrite, in Fourier space, the current produced by a magnetic field (Ampère–Laplace law) is given by the following expression (see Appendix 2):

$$ \delta\bf{B}_f(\bf{r})= \frac{\mu_o}{4\pi} \bf{j}_f^p(\bf{r'}) \times \frac{\bf{r}- \bf{r'}}{\|\bf{r}-\bf{r'}\|^3} \delta v' $$

(20)

when the extracellular medium is resistive. Note that the source of magnetic induction is essentially given by the component of $\bf{j}_f^p$ along the axial direction ($j_f^{\,i}$) within each differential element of dendrite because the perpendicular (membrane) current does not participate to producing the magnetic induction if we assume a cylindrical symmetry.

For the electric potential, we have the following differential expression for a resistive medium (see Appendix 3):

$$ \delta\bf{V}_f(\bf{r})=\frac{1}{4\pi\gamma}\frac{ \delta I_f^{\perp}(\bf{r}')}{\| \bf{r}-\bf{r'}\|} =\frac{1}{4\pi\gamma}\frac{j_f^{\,m}(\bf{r}')}{\| \bf{r}-\bf{r'}\|}\delta S' $$

(21)

where $j_f^{\,m}$ is the transmembrane current per unit of surface.

If we consider the differential expressions for the magnetic induction (Eq. (20)) and electric potential (Eq. (21)), one can see that the frequency dependence of the ratio of their modulus is completely determined by the frequency dependence of the ratio of current density $j_f^{\,m}$ and $j_f^{\,i}$. In Appendix 4, we show that this ratio is quasi-independent of frequency for a resistive medium, for low frequencies (smaller that ∼10 Hz), and if the current sources are of very low correlation.

Thus, magnetic induction and electric potential can be very well approximated by:

$$\begin{array}{lll} V_{f}(\bf{r}) &=& N<V>= N<\sum\limits_{l=1}^{N} \delta V_{f}^{l}> \\ \bf{B}_f(\bf{r})&=& N<\bf{B}>= N<\sum\limits_{l=1}^{N} \delta \bf{B}_f^l> \end{array}$$

(22)

for sufficiently small differential dendritic elements (N/l large).

Because the functions of spatial and frequency are statistically independent, we can write the following expressions for the square modulus of the fields (see Eqs. (20) and (21)):

$$\begin{array}{lll} |V_{f}(\bf{r})|^2 &=& N^2|<\sum\limits_{l=1}^{N}V^l(\bf{r})G_l^m(f)>|^2 \\ &=& |V(\bf{r})|^2|G(f)|^2 \\ \|\bf{B}_f(\bf{r})\|^2 &=& N^2\|<\sum\limits_{i=1}^{N}\bf{B}^{\,i}(\bf{r})G_l^m(f)>\|^2 \\ &=& \|\bf{W}(\bf{r})\|^2|G(f)|^2 \end{array}$$

(23)

where$G(f)=<G_l^m(f)>$, $V^l(\bf{r})=<V^l(\bf{r})>$ and $\bf{W}(\bf{r})=<\bf{B}^l(\bf{r})>$. Thus, the scaling of the PSDs of the electric potential and magnetic induction must be the same for low frequencies (smaller than ~10 Hz) if the medium is resistive and when the current sources have very low correlation.

Appendix 2: Differential expression for the magnetic induction

According to Maxwell equations, the magnetic induction is given by:

$$ \bf{B}_f(\bf{r})= \frac{\mu_o}{4\pi}\iiint\limits_{\rm head} \frac{\nabla'\times\bf{j}_f^p(\bf{r'})}{\|\bf{r}-\bf{r'} \|} dv' $$

(24)

where dv′ = dx ^′1 dx ^′2 dx ^′3 and

$$ \nabla'\left(\frac{1}{\|\bf{r}-\bf{r'}\|}\right)=\frac{\bf{r}- \bf{r'}}{\|\bf{r}-\bf{r'}\|^3} $$

for a perfectly resistive medium.

We now show that this expression is equivalent to Ampere–Laplace law.

From the identity $\nabla' \times (g\bf{A})= g(\nabla' \times \bf{A})+\nabla' g\times\bf{A}$, where $\nabla' = \hat{e}_x\frac{\partial}{\partial x'}+\hat{e}_y\frac{\partial}{\partial y'}+\hat{e}_z\frac{\partial}{\partial z'}$, we can write:

$$\begin{array}{lll} \bf{B}_f(\bf{r})&=& \frac{\mu_o}{4\pi} \iiint\limits_{\rm head} \left[\nabla'\times \left(\frac{\bf{j}_f^p(\bf{r'})}{\|\bf{r}-\bf{r'}\|}\right) + \frac{\mu}{4\pi} \bf{j}_f^p(\bf{r'})\right. \\ &&\qquad\qquad\qquad\;\left.\times\nabla'\frac{1}{\|\bf{r}-\bf{r'}\|}\vphantom{\left(\frac{\bf{j}_f^p(\bf{r'})}{\|\bf{r}-\bf{r'}\|}\right)}\right] dv' \end{array}$$

(25)

Moreover, we also have the following identity

$$ \iiint\limits_{\rm head} \nabla' \times \left(\frac{\bf{j}_f^p(\bf{r'})}{\|\bf{r} - \bf{r'}\|}\right) dv' = -\iint\limits_{\partial {\rm head}} \frac{\bf{j}_f^p(\bf{r'})}{\|\bf{r} - \bf{r'}\|} \times \hat{n} dS' $$

(26)

where $\hat{n}$ is a unitary vector perpendicular to the integration surface and going outwards from that surface. Extending the volume integral outside the head, the surface integral is certainly zero because the current is zero outside of the head. It follows that:

$$ \bf{B}_f(\bf{r})= \frac{\mu_o}{4\pi}\iiint\limits_{\rm head} \bf{j}_f^p(\bf{r'})\times \frac{\bf{r}- \bf{r'}}{\|\bf{r}-\bf{r'}\|^3} dv' $$

(27)

where dv′ = dx ^′1 dx ^′2 dx ^′3 because

$$ \nabla'\left(\frac{1}{\|\bf{r}-\bf{r'}\|}\right)=\frac{\bf{r}- \bf{r'}}{\|\bf{r}-\bf{r'}\|^3} $$

Equation (27) is called the Ampère–Laplace law (see Eq. (13) in Hämäläinen et al. 1993). It is important to note that this expression for the magnetic induction is not valid when the medium is not resistive.

Finally, from the last expression, the magnetic induction for a differential element of dendrite can be written as:

$$ \delta\bf{B}_f(\bf{r})= \frac{\mu_o}{4\pi} \bf{j}_f^p(\bf{r'})\times \frac{\bf{r}- \bf{r'}}{\|\bf{r}-\bf{r'}\|^3} \delta v' $$

(28)

Appendix 3: Differential expression of the electric field and electric potential

In this appendix, we derive the differential expression for the electric field. Starting from Eq. (10), we obtain the solution for the electric potential:

$$ V_{f}(\bf{r})=-\frac{1}{4\pi\gamma_f}\iiint\limits_{\rm head} \frac{\nabla\cdot \bf{j}_f^p(\bf{r}')}{\Vert \bf{r}-\bf{r'}\Vert} dv' $$

(29)

It follows that the electric field produced by the ensemble of sources can be expressed as:

$$ \bf{E}_f(\bf{r})= - \nabla V_{f}(\bf{r}) = \frac{1}{4\pi\gamma_f}\iiint\limits_{\rm head} \nabla\cdot \bf{j}_f^p(\bf{r}') \cdot \frac{ \bf{r} - \bf{r'}}{\| \bf{r} - \bf{r'}\|^3} dv' $$

(30)

such that every differential element of dendrite produces the following electric field:

$$ \delta\bf{E}_f(\bf{r})= \frac{\nabla\cdot \bf{j}_f^p(\bf{r}')}{4\pi\gamma_f} \cdot \frac{ \bf{r}-\bf{r'}}{\| \bf{r}-\bf{r'}\|^3} \delta v' $$

(31)

The transmembrane current $\delta I_f^{\perp}$ obeys $\delta I_f^{\perp}=i\omega\rho_f(\bf{r}') \delta v'$ because we are in a quasi-stationary regime in a differential dendritic element. Taking into account the differential law of charge conservation $\nabla\cdot \bf{j}_f(\bf{r}')=-i\omega\rho_f(\bf{r}')$, we have:

$$ \delta\bf{E}_f(\bf{r})=\frac{\delta I_f^{\perp}(\bf{r}')}{4\pi\gamma_f}\frac{ \bf{r}-\bf{r'}}{\| \bf{r}-\bf{r'}\|^3} =\frac{j_f^{\,m}(\bf{r}')}{4\pi\gamma_f}\frac{ \bf{r}-\bf{r'}}{\| \bf{r}-\bf{r'}\|^3}\delta S' $$

(32)

where $j_f^{\,m}$ is the density of transmembrane current per unit surface and δS′ is the surface area of a differential dendritic element. This approximation is certainly valid for frequencies lower than 1,000 Hz because the Maxwell–Wagner time (see Bédard et al. 2006b) of the cytoplasm ($\tau_{mw}^{cyto}=\varepsilon/\sigma \sim 10^{-10}~{\rm s}$). is much smaller than the typical membrane time constant of a neuron (τ _m ∼5 − 20 ms).

Finally the contribution of a differential element of dendrite to the electric potential at position $\bf{r}$ is given by

$$ \delta\bf{V}_f(\bf{r})=\frac{1}{4\pi\gamma_f}\frac{ \delta I_f^{\perp}(\bf{r}')}{\| \bf{r}-\bf{r'}\|} =\frac{1}{4\pi\gamma_f}\frac{ j_f^{\,m}(\bf{r}')}{\| \bf{r}-\bf{r'}\|}\delta S' $$

(33)

We note that the expressions for the electric field and potential produced by each differential element of dendrite have the same frequency dependence because it is directly proportional to $\frac{j_f^{\,m}}{\gamma_f}$ for the two expressions. Also note that if the medium is resistive, then γ _f  = γ and the frequency dependence of the electric field and potential are solely determined by that of the transmembrane current $j_f^{\,m}$.

Appendix 4: Frequency dependence of the ratio ${ j_f^{\,i}(\bf{x})} / {j_f^{\,m}(\bf{x})}$

For each differential element of dendrite, we consider the standard cable model, in which the impedance of the medium is usually neglected (it is usually considered negligible compared to the membrane impedance). In this case, we have:

$$ \left\{ \begin{array}{ccc} j_f^{\,m} &=& \dfrac{V_{f}^{\,m}}{r_m}+i\omega c_m V_{f}^{\,m} \\ j_f^{\,i} &=& -\sigma\dfrac{\partial V_{f}^{\,m}}{\partial x} = -\dfrac{1}{r_i}\dfrac{\partial V_{f}^{\,m}}{\partial x} \end{array} \right .$$

(34)

where $V_{\!f}^{\,m}$, $j_f^{\,i}$, $j_f^{\,m}$, c _m , r _m et r _i are respectively the membrane potential, the current density in the axial direction, the transmembrane current density, the specific capacitance (F/m ²), the specific membrane resistance (Ω.m ²) and the cytoplasm resistivity (Ω.m).

It follows that

$$ \frac{ j_f^{\,i}(\bf{x})}{j_f^{\,m}(\bf{x})} =\frac{r_m}{r_i(1+i\omega\tau_m)} \cdot\frac{\partial}{\partial x}\big[ln(V_{f}^{\,m}(\bf{x})\big] $$

(35)

where τ _m  = r _m c _m .

Under in vivo-like conditions, the activity of neurons is intense and of very low correlation. This is the case for desynchronized-EEG states, such as awake eyes-open conditions, where the activity of neurons is characterized by very low levels of correlations. There is also evidence that in such conditions, neurons are in “high-conductance states” (Destexhe et al. 2003), in which the synaptic activity dominates the conductance of the membrane and primes over intrinsic currents. In such conditions, we can assume that the synaptic current sources are essentially uncorrelated and dominant, such that the deterministic link between current sources will be small and can be neglected (see Bédard et al. 2010). Further assuming that the electric properties of extracellular medium are homogeneous, then each differential element of dendrite can be considered as independent and the voltages V _m have similar power spectra.

In such conditions, we have:

$$ V_{f}^{\,m} (\bf{x}) = F^{\,m}(\bf{x})G^{\,m}(f) $$

(36)

Note that this expression implies that we have in general for each differential element of dendrite:

$$ \left\{ \begin{array}{ccc} j_f^{\,m} (\bf{x}) &=& F^{\,m}(\bf{x})\left(\dfrac{1+i\omega\tau_m}{r_m}\right)G^{\,m}(f) \\ j_f^{\,i} (\bf{x}) &=& -\dfrac{1}{r_i}\dfrac{\partial F^{\,m}(\bf{x})}{\partial x}G^{\,m}(f)= F^{\,i}(\bf{x})G^{\,m}(f) \end{array} \right . $$

(37)

according to Eq. (34).

It follows that

$$\begin{array}{lll} \frac{ j_f^{\,i}(\bf{x})}{ j_f^{\,m}(\bf{x})} &=&\frac{r_m}{r_i(1+i\omega\tau_m)} \cdot\frac{\partial}{\partial x}[ln(F(\bf{x}))] \\ &\approx& \frac{r_m}{r_i} \cdot\frac{\partial}{\partial x}[ln(F(\bf{x}))] \end{array}$$

(38)

Thus, for frequencies smaller than 1/(ωτ _m ) (about 10 to 30 Hz for τ _m of 5–20 ms), the ratio $\frac{ j_f^{\,i}(\bf{x})}{ j_f^{\,m}(\bf{x})}$ will be frequency independent, and for each differential element of dendrite, we have:

$$ \left\{ \begin{array}{ccc} j_f^{\,m} (\bf{x}) &=& F^{\,m}(\bf{x})G^{\,m}(f)\\ j_f^{\,i} (\bf{x}) &=& F^{\,i}(\bf{x})G^{\,m}(f) \end{array} \right. $$

(39)

for frequencies smaller than ∼10 Hz.