Claude Bedard

Macroscopic models of local field potentials and the apparent 1/f noise in brain activity.

The power spectrum of local field potentials (LFPs) has been reported to scale as the inverse of the frequency, but the origin of this 1/f noise is at present unclear. Macroscopic measurements in cortical tissue demonstrated that electric conductivity (as well as permittivity) is frequency-dependent, while other measurements failed to evidence any dependence on frequency. In this article, we propose a model of the genesis of LFPs that accounts for the above data and contradictions. Starting from first principles (Maxwell equations), we introduce a macroscopic formalism in which macroscopic measurements are naturally incorporated, and also examine different physical causes for the frequency dependence. We suggest that ionic diffusion primes over electric field effects, and is responsible for the frequency dependence. This explains the contradictory observations, and also reproduces the 1/f power spectral structure of LFPs, as well as more complex frequency scaling. Finally, we suggest a measurement method to reveal the frequency dependence of current propagation in biological tissue, and which could be used to directly test the predictions of this formalism.

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.

Intracellular Impedance Measurements Reveal Non-ohmic Properties of the Extracellular Medium around Neurons

Determining the electrical properties of the extracellular space around neurons is important for understanding the genesis of extracellular potentials, as well as for localizing neuronal activity from extracellular recordings. However, the exact nature of these extracellular properties is still uncertain. Here, we introduce a method to measure the impedance of the tissue, one that preserves the intact cell-medium interface using whole-cell patch-clamp recordings in vivo and in vitro. We find that neural tissue has marked non-ohmic and frequency-filtering properties, which are not consistent with a resistive (ohmic) medium, as often assumed. The amplitude and phase profiles of the measured impedance are consistent with the contribution of ionic diffusion. We also show that the impact of such frequency-filtering properties is possibly important on the genesis of local field potentials, as well as on the cable properties of neurons. These results show non-ohmic properties of the extracellular medium around neurons, and suggest that source estimation methods, as well as the cable properties of neurons, which all assume ohmic extracellular medium, may need to be reevaluated.

A Modified Cable Formalism for Modeling Neuronal Membranes at High Frequencies

Intracellular recordings of cortical neurons in vivo display intense subthreshold membrane potential (V(m)) activity. The power spectral density of the V(m) displays a power-law structure at high frequencies (>50 Hz) with a slope of approximately -2.5. This type of frequency scaling cannot be accounted for by traditional models, as either single-compartment models or models based on reconstructed cell morphologies display a frequency scaling with a slope close to -4. This slope is due to the fact that the membrane resistance is short-circuited by the capacitance for high frequencies, a situation which may not be realistic. Here, we integrate nonideal capacitors in cable equations to reflect the fact that the capacitance cannot be charged instantaneously. We show that the resulting nonideal cable model can be solved analytically using Fourier transforms. Numerical simulations using a ball-and-stick model yield membrane potential activity with similar frequency scaling as in the experiments. We also discuss the consequences of using nonideal capacitors on other cellular properties such as the transmission of high frequencies, which is boosted in nonideal cables, or voltage attenuation in dendrites. These results suggest that cable equations based on nonideal capacitors should be used to capture the behavior of neuronal membranes at high frequencies.

Do neurons generate monopolar current sources

According to the “standard model,” electric potentials such as the local field potential (LFP) or the electroencephalogram (EEG) are generated by current dipoles made by cerebral cortex neurons arranged in parallel. In this issue of Journal of Neurophysiology , [Riera et al. (2012)][1] present

A framework to reconcile frequency scaling measurements, from intracellular recordings, local-field potentials, up to EEG and MEG signals

In this viewpoint article, we discuss the electric properties of the medium around neurons, which are important to correctly interpret extracellular potentials or electric field effects in neural tissue. We focus on how these electric properties shape the frequency scaling of brain signals at different scales, such as intracellular recordings, the local field potential (LFP), the electroencephalogram (EEG) or the magnetoencephalogram (MEG). These signals display frequency-scaling properties which are not consistent with resistive media. The medium appears to exert a frequency filtering scaling as 1/f, which is the typical frequency scaling of ionic diffusion. Such a scaling was also found recently by impedance measurements in physiological conditions. Ionic diffusion appears to be the only possible explanation to reconcile these measurements and the frequency-scaling properties found in different brain signals. However, other measurements suggest that the extracellular medium is essentially resistive. To resolve this discrepancy, we show new evidence that metal-electrode measurements can be perturbed by shunt currents going through the surface of the brain. Such a shunt may explain the contradictory measurements, and together with ionic diffusion, provides a framework where all observations can be reconciled. Finally, we propose a method to perform measurements avoiding shunting effects, thus enabling to test the predictions of this framework.

Kramers-Kronig Relations and the Properties of Conductivity and Permittivity in Heterogeneous Media

The macroscopic electric permittivity of a given medium may depend on frequency, but this frequency dependence cannot be arbitrary, its real and imaginary parts are related by the well-known Kramers-Kronig relations. Here, we show that an analogous paradigm applies to the macroscopic electric conductivity. If the causality principle is taken into account, there exists Kramers-Kronig relations for conductivity, which are mathematically equivalent to the Hilbert transform. These relations impose strong constraints that models of heterogeneous media should satisfy to have a physically plausible frequency dependence of the conductivity and permittivity. We illustrate these relations and constraints by a few examples of known physical media. These extended relations constitute important constraints to test the consistency of past and future experimental measurements of the electric properties of heterogeneous media.

Generalized Cable Models of Neurons and Dendrites

Cable theory was introduced for neurons by Wilfrid Rall more than half a century ago, and is widely used today for modeling the voltage and current flow in neuronal and dendritic structures. This theory was derived assuming that the extracellular medium is either inexistent, or modeled as a resistor. For modeling neurons in more realistic situations, where the extracellular medium has more complex electric properties, it is necessary to generalize Rall’s cable equations. We summarize here such generalized cable equations, and show that the nature of the surrounding extracellular medium can exert non-negligible influences on the cable properties of neurons.

Microscale impedance measurements suggest that ionic diffusion is implicated in generating extracellular potentials.

The genesis of the Local Field Potential (LFP) highly depends on the electric properties of the extracellular medium, but such properties are still subject to a controversy because of contradictory measurements. One possibility is that the use of metal electrodes as current sources in previous studies provides non-physiological results. We tested this possibility by performing impedance measurements in conditions as close as possible to physiological conditions. We generated single-cell LFPs by injecting subthreshold inputs in single neurons using patch-clamp recordings, combined with extracellular recordings with micropipettes. Various measurement configurations show that (1) the extracellular medium has strong low-pass filtering properties and cannot be accounted by a resistive medium; (2) the frequency scaling of the filtering, as well as its phase, show that the system seems intermediate between resistive and capacitive. The extracellular impedance was also measured from in vivo experiments in rats under anesthesia. In this case, recording with intracellular (whole-cell) electrodes, together with extracellular LFP, showed results consistent with the in vitro experiments. Finally, we developed a theoretical model based on Maxwell equations, which shows that all measurements can be explained if the extracellular medium is of diffusive type (Warburg impedance). This model predicts that the phase difference between intracellular and extracellular signals should provide a signature of the physical nature of the impedance, with 45 degrees phase difference for purely diffusive type. The experiments show that indeed, the phase is that of a RC soma in series with a diffusive impedance (between 0 and -45 degrees), therefore confirming the diffusive nature of the extracellular impedance. These findings have potentially important consequences for interpreting LFP measurements and source estimation such as CSD analysis.

Reply to Gratiy et al.

to the editor: In their commentary, Gratiy et al. (2013) suggest that according to Maxwell theory of electromagnetism, monopoles are impossible in neurons. However, to reach this conclusion, the authors make an approximation, which we argue below may not be valid in biological media. Maxwell theory