Generalized theory for current-source-density analysis in brain tissue

The current-source density (CSD) analysis is a widely used method in brain electrophysiology, but this method rests on a series of assumptions, namely that the surrounding extracellular medium is resistive and uniform, and in some versions of the theory, that the current sources are exclusively made by dipoles. Because of these assumptions, this standard model does not correctly describe the contributions of monopolar sources or of nonresistive aspects of the extracellular medium. We propose here a general framework to model electric fields and potentials resulting from current source densities, without relying on the above assumptions. We develop a mean-field formalism that is a generalization of the standard model and that can directly incorporate nonresistive (nonohmic) properties of the extracellular medium, such as ionic diffusion effects. This formalism recovers the classic results of the standard model such as the CSD analysis, but in addition, we provide expressions to generalize the CSD approach to situations with nonresistive media and arbitrarily complex multipolar configurations of current sources. We found that the power spectrum of the signal contains the signature of the nature of current sources and extracellular medium, which provides a direct way to estimate those properties from experimental data and, in particular, estimate the possible contribution of electric monopoles.

Figure 1

Attenuation profile of the extracellular potential as a function of distance. The profile of the potential with distance is shown for two positive values of $\langle \beta \rangle$: one value comparable to the diffusion coefficients of $k^{+},N{a}^{+},C{l}^{-}$ (left), and another value 100 times smaller (right). When $\langle \beta \rangle$ is comparable to the diffusion coefficients, the attenuation according to Yukawa potential is similar to Coulomb's potential and we have a Warburg impedance. For smaller values of $\langle \beta \rangle$, the Yukawa potential determines a significant additional low-pass filter. The attenuation law is very steep for frequencies larger than 100 Hz when ${\langle \beta \rangle |}_m={10}^{-11}{\mathrm{m}}^2/\mathrm{s}$. The different curves indicated are the Coulomb's attenuation law as $1/r$ (monopoles; solid line) and as $1/r^2$ (dipoles; dotted line; thick gray lines correspond to attenuation according to a Yukawa potential; thin lines correspond to Yukawa attenuation combined with a Warburg impedance, which is the most complete case taking into account ionic diffusion effects. In each case, different frequencies are compared (1, 10, 100, and 1000 Hz) and are shown by different colors and dashed lines, as indicated. The current source has a radius of 5 nm.

Figure 2

Scheme of an isopotential membrane compartment with ion channels. The blue circles indicate ion channels in the membrane. The dotted line indicates a Gauss surface delimiting the interior of the compartment. The values of electric permittivity $\varepsilon$ are equal to ${\varepsilon }_m$ in the membrane and ${\varepsilon }_c$ in the cytoplasm. The electric conductivity $\sigma$ is equal to ${\sigma }_c$ in the open channels and is assumed to be zero in the membrane.

Figure 3

Representation of $\kappa$ as a function of the exponent $b$. The values of $\kappa$ were calculated from expression $\frac{1}{\frac{-2}{\pi }{\int -}_0^{\infty }\frac{1}{x^{1-b}(x^2-1)}dx}$, where ${\int -}_0^{\infty }\frac{1}{x^{1-b}(x^2-1)}dx={\lim }_{x\to 1}{\int }_0^x[x^{-(1-b)}+x^{(1-b)}]\frac{1}{(x^2-1)}dx$.