Generalized cable formalism to calculate the magnetic field of single neurons and neuronal populations

Neurons generate magnetic fields which can be recorded with macroscopic techniques such as magneto-encephalography. The theory that accounts for the genesis of neuronal magnetic fields involves dendritic cable structures in homogeneous resistive extracellular media. Here, we generalize this model by considering dendritic cables in extracellular media with arbitrarily complex electric properties. This method is based on a multi-scale mean-field theory where the neuron is considered in interaction with a "mean" extracellular medium (characterized by a specific impedance). We first show that, as expected, the generalized cable equation and the standard cable generate magnetic fields that mostly depend on the axial current in the cable, with a moderate contribution of extracellular currents. Less expected, we also show that the nature of the extracellular and intracellular media influence the axial current, and thus also influence neuronal magnetic fields. We illustrate these properties by numerical simulations and suggest experiments to test these findings.Comment: Physical Review E (in press); 24 pages, 16 figure

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.Comment: 17 pages, 2 figure

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 the present paper, we propose a model of the genesis of LFPs which 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 the present formalism

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 $1/\sqrt{f}$, 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.Comment: (in press

A modified cable formalism for modeling neuronal membranes at high frequencies

Intracellular recordings of cortical neurons in vivo display intense subthreshold membrane potential (Vm) activity. The power spectral density (PSD) of the Vm displays a power-law structure at high frequencies (>50 Hz) with a slope of about -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 non-ideal capacitors in cable equations to reflect the fact that the capacitance cannot be charged instantaneously. We show that the resulting "non-ideal" 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 non-ideal capacitors on other cellular properties such as the transmission of high frequencies, which is boosted in non-ideal cables, or voltage attenuation in dendrites. These results suggest that cable equations based on non-ideal capacitors should be used to capture the behavior of neuronal membranes at high frequencies.Comment: To appear in Biophysical Journal; Submitted on May 25, 2007; accepted on Sept 11th, 200

Comparative Evaluation of Child Behavior Checklist-Derived Scales in Children Clinically Referred for Emotional and Behavioral Dysregulation

Background: We recently developed the Child Behavior Checklist-Mania Scale (CBCL-MS), a novel and short instrument for the assessment of mania-like symptoms in children and adolescents derived from the CBCL item pool and have demonstrated its construct validity and temporal stability in a longitudinal general population sample. Objective: The aim of this study was to evaluate the construct validity of the 19-item CBCL-MS in a clinical sample and to compare its discriminatory ability to that of the 40-item CBCL-dysregulation profile (CBCL-DP) and the 34-item CBCL-Externalizing Scale. Methods: The study sample comprised 202 children, aged 7–12 years, diagnosed with DSM-defined attention deficit hyperactivity disorder (ADHD), conduct disorder (CD), oppositional defiant disorder (ODD), and mood and anxiety disorders based on the Diagnostic Interview Schedule for Children. The construct validity of the CBCL-MS was tested by means of a confirmatory factor analysis. Receiver operating characteristics (ROC) curves and logistic regression analyses adjusted for sex and age were used to assess the discriminatory ability relative to that of the CBCL-DP and the CBCLExternalizing Scale. Results: The CBCL-MS had excellent construct validity (comparative fit index = 0.97; Tucker–Lewis index = 0.96; root mean square error of approximation = 0.04). Despite similar overall performance across scales, the clinical range scores of the CBCL-DP and the CBCL-Externalizing Scale were associated with higher odds for ODD and CD, while the clinical range scores of the CBCL-MS were associated with higher odds for mood disorders. The concordance rate among the children who scored within the clinical range of each scale was over 90%. Conclusion: CBCL-MS has good construct validity in general population and clinical samples and is therefore suitable for both clinical practice and research