Models of local field potentials
Local field potentials (LFPs) describe the electrical potential recorded by an extracellular electrode. The genesis of LFPs is a complex problem because LFPs are due to the movement of charges in the highly-tortuous extracellular medium in which neurons are embedded. However, besides this complexity, extracellular recordings always show that the action potentials are recorded only for neurons immediately adjacent to the electrode, while slower events, such as synaptic potentials, seem to propagate and summate over large distances. The usual model of LFPs, which consists in a set of current sources embedded in a homogeneous extracellular medium, does not reproduce this fundamental property. In collaboration with Claude Bedard and Helmut Kröger (Laval University, Canada), we have developed theoretical models to understand the genesis of such frequency-dependent attenuation with distance . Starting from Maxwell equations, it can be shown that the extracellular potential can display frequency-dependent attenuation, but only if the extracellular conductivity is non-homogeneous. In this case, there is induction of non-homogeneous charge densities which may result in a low-pass filter. Because the extracellular space is a complex aggregate of processes of different conductivity, such as fluids (extracellular and intracellular) and membranes (dendrites, axons, myelin, glial cells, etc), its conductivity is necessarily highly non-homogeneous. Thus, the theory shows that these non-homogeneities are the primary cause for the frequency-dependent properties of LFPs , and in particular the induced electric fields in this non-homogeneous structure .
However, although this formalism provides a plausible explanation for the frequency-dependent properties of LFPs, it is difficult to apply in practice because one would need to explicitly take into account the complex three-dimensional structure of extracellular processes around neurons. To apply these ideas to standard neuron simulations, simplified models are needed. A first approach assumes that current sources are punctual and surrounded by a medium with spherically-symmetric conductivity/permittivity gradients around each source . This simple model can display low-pass filtering behavior, in which fast electrical events (such as Na+ -mediated action potentials) attenuate very steeply with distance, while slower (K+ -mediated) events propagate over larger distances in extracellular space, in qualitative agreement with experimental observations. This simple model can be used to obtain frequency-dependent extracellular field potentials without taking into account explicitly the complex folding of extracellular space (see details in ).
An explicit structure of extracellular space was considered in a second type of model . In this model, extracellular space was approximated by a high density of spherical cellular membranes packed around the source, and embedded in a conductive fluid. It is shown that these extracellular membranes, if considered as passive (glial cells), respond by polarization. Because of the finite velocity of ionic charge movement, this polarization will not be instantaneous. Consequently, the induced electric field will be frequency-dependent, and much reduced for high frequencies. This model suggests that (a) high frequencies are attenuated very steeply, and ignore the neighboring membranes around neuronal current sources; (b) low frequencies participate to successive polarizations of membranes, and are "transported" over much larger distances. Thus, we suggest that the physical origin of the frequency filtering properties of LFPs is the induced electric fields in passive cells surrounding neurons.
More recently, we investigated the physical basis of the 1/f frequency scaling of LFPs . Many complex systems display self-organized critical (SOC) states characterized by 1/f frequency scaling of power spectra. Global variables such as the electroencephalogram, are known to scale as 1/f, which could be the sign of SOC states in neuronal activity. We analyzed simultaneous recordings of global and neuronal activities, and confirmed the 1/f scaling of global variables (LFPs) for selected frequency bands. However, by analyzing neuronal activities, we did not find the typical power-law scaling of SOC states ("avalanche analysis"), which suggests that neuronal activity does not stem from critical states. The 1/f scaling of LFPs can be explained by a model which does not rely on critical states, but is rather due to a filtering process from extracellular space .
The latter theme was investigated in more detail in a recent paper . A macroscopic formalism was developed, based on Maxwell equations, in which macroscopic measurements of permittivity and conductivity are naturally incorporated. The study evidences that ionic diffusion must be taken into account to match the frequency dependence of electric parameters observed experimentally (in addition to electric field effects). The same mechanisms also reproduce the 1/f frequency filtering effect described above. Thus, this model suggests that plausible physical causes can account for the 1/f frequency dependence of local field potentials, without the need for critical states. The predictions of this model are testable experimentally, and are presently under investigation.
The nature of extracellular medium can also be inferred indirectly, by relating different signals recorded simultaneously in brain tissue. First, by relating simultaneous recordings of LFP and intracellular activity, it is possible, under some approximation, to estimate the impedance of the extracellular medium. This analysis was developed and tested in intracellular and LFP measurements from rat barrel cortex, and revealed that the impedance of the extracellular medium is close to a Warburg impedance (1/f filtering) , thus confirming the above analysis. Second, we also analyzed simultaneous electroencephalogram (EEG) and magnetoencephalogram (MEG) measurements in humans. We showed that if the tissue is resistive, then the power spectral structure of the two signal should have the same frequency scaling exponent. By analyzing simultaneous EEG and MEG recordings from 3 patients, we showed that the scaling exponent is indeed different from the two signals , which also supports the fact that the medium is not resistive. Further work is needed to predict the difference of scaling exponent if the medium is not resistive (work in progress).
The above considerations about the frequency filtering properties of extracellular media, their possible physical basis and their impact on the modeling of LFP signals, were summarized in a recent review chapter  and a Scholarpedia article . We also wrote a short review on the mean-field formulation of Maxwell equations .
The same approach was also applied to the current-source density (CSD) analysis. The CSD analaysis is based on the charge conservation law, and if the medium is non-resistive, this approach is not correct (the equations are invalid in this case). To address this caveat, we have developed a mean-field formalism which is a generalization of the standard model, and which can directly incorporate non-resistive (non-ohmic) properties of the extracellular medium, such as ionic diffusion effects. This formalism recovers the classic results of the standard model such as the classic CSD analysis, but in addition, provides expressions to generalize the CSD approach to situations with non-resistive 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 medium, which provides a direct way to estimate those properties from experimental data, and in particular, estimate the possible contribution of electric monopoles (see  for details).
The CSD profiles were also simulated from computational models of sleep oscillations . In this case, it was shown that models of CSD relying uniquely on resistive extracellular media, failed to reproduce the CSD profiles found experimentally. However, if frequency-filtering properties were considered, such as a non-homogeneous extracellular resistivity, the model could reproduce the CSD profiles seen experimentally. This study provides another evidence that the extracellular medium is non resistive and frequency dependent.
The presence of monopolar sources in the brain is puzzeling and was the subject of a commentary paper , which explored possible physical bases for such monopolar current sources in neurons. According to the "standard model'', electric potentials such as the LFP or the EEG, are generated by current dipoles made by cerebral cortex neurons arranged in parallel. However, there is recent evidence that this standard model may be insufficient to account for LFP and EEG signals and that monopolar components (not predicted by the standard model) are necessary to explain LFP and EEG signals. In this commentary paper, we briefly summarize the experimental evidences for monopolar sources, and then speculate on possible physical mechanisms to explain these surprising results. In another commentary, we clarified that electric monopoles are perfectly compatible with Maxwell equations of electromagnetism .
Recently, collaboration with Thierry Bal and Laurent Venance (Collège de France), we performed experiments to test the ohmic nature of the extracellular medium . For the first time, we were able to evaluate the extracellular impedance in "natural" conditions, with no metal electrode serving as current source, but instead, a neuron acts as current source, driven by an intracellular microelectrode. Such intracellular measurement of the impedance was performed both in vivo and in vitro, and showed that the impedance displays amplitude and phase variations in frequency, that are incompatible with a resistive medium. Instead, the measurements were compatible with the impedance of ionic diffusion.
These experiments have consequences for other signals. The non-ohmic properties of the extracellular medium may impact on cable properties , as well as on the genesis of LFPs and magnetic fields by neurons . The exact nature of the impedance predicts a specific frequency-scaling property of the LFP and magnetic fields, as we showed previously in human MEG recordings . Such properties should be investigated experimentally.
In a recent study, we characterized the spatiotenporal properties of beta and gamma frequency oscillations in the LFP of human and monkey . The oscillation dynamics exhibited propagating wave properties, in both wake and sleep states. Interestingly, the highest spatiotemporal coherence of the oscillations was found during slow-wave sleep, which suggests that sleep replays wake patterns with higher coherence, perhaps in relation to memory consolidation .
With Romain Brette, we have edited a book where the biophysical aspects of the genesis of different brain signals were reviewed by several specialists of the field .
 Bedard, C., Kröger, H. and Destexhe, A. Modeling extracellular field potentials and the frequency-filtering properties of extracellular space. Biophysical Journal 86: 1829-1842, 2004 (see abstract).
 Bedard, C., Kröger, H. and Destexhe, A. Model of low-pass filtering of local field potentials in brain tissue. Physical Review E 73: 051911, 2006 (see abstract).
 Bedard, C., Kröger, H. and Destexhe, A. Does the 1/f frequency-scaling of brain signals reflect self-organized critical states ? Physical Review Letters 97: 118102, 2006 (see abstract).
 Bedard, C. and Destexhe, A. Macroscopic models of local field potentials the apparent 1/f noise in brain activity. Biophysical Journal 96: 2589-2603, 2009 (see abstract).
 Bedard, C., Rodrigues, S., Roy, N., Contreras, D. and Destexhe, A. Evidence for frequency-dependent extracellular impedance from the transfer function between extracellular and intracellular potentials. J. Computational Neurosci. 29: 389-403, 2010 (see abstract).
 Dehghani, N, Bedard, C., Cash, S.S., Halgren, E. and Destexhe, A. Comparative power spectral analysis of simultaneous elecroencephalographic and magnetoencephalographic recordings in humans suggests non-resistive extracellular media. J. Computational Neurosci. 29: 405-421, 2010 (see abstract).
 Bedard, C. and Destexhe, A. Modeling local field potentials and their interaction with the extracellular medium. In: Handbook of Neural Activity Measurement, Edited by Brette R. and Destexhe A., Cambridge University Press, Cambridge, UK, pp. 136-191, 2012 (see abstract).
 Bedard, C. and Destexhe, A. A generalized theory for current-source density analysis in brain tissue. Physical Review E 84: 041909, 2011 (see abstract).
 Bazhenov, M., Lonjers, P., Skorheim, S., Bedard, C. and Destexhe, A. Non-homogeneous extracellular resistivity affects the current-source density profiles of Up/Down state oscillations. Phil. Trans. Roy. Soc. A 369: 3802-3819, 2011 (see abstract).
 Destexhe, A. and Bedard, C. Do neurons generate monopolar current sources? J. Neurophysiol. 108: 953-955, 2012 (see abstract).
 Bedard, C. and Destexhe, A. Electric monopoles are indeed compatible with Maxwell equations. J. Neurophysiol. 109: 1683, 2013 (see abstract)
 Destexhe, A. and Bedard, C. Local field potential. Scholarpedia 8: 10713, 2013. ( see article)
 Bedard, C. and Destexhe, A. Mean-field formulation of Maxwell equations to model electrically inhomogeneous and isotropic media. J. Electromagnetic Analysis and Applications 6: 296-302, 2014 (see abstract)
 Gomes, J-M., Bedard, C., Valtcheva, S., Nelson, M., Khokhlova, V., Pouget, P., Venance, L., Bal, T. and Destexhe, A. Intracellular impedance measurements reveal non-ohmic properties of the extracellular medium around neurons. Biophys J. 110: 234-246, 2016. see abstract)
 Bedard, C. and Destexhe, A. Generalized cable theory for neurons in complex and heterogeneous media. Physical Review E 88: 022709, 2013 (see abstract)
 Bedard, C. and Destexhe, A. Generalized cable formalism to calculate the magnetic field of single neurons and neuronal populations. Phys. Rev. E 90: 042723, 2014 (see abstract)
 Brette, R. and Destexhe, A. (Editors) Handbook of Neural Activity Measurement, Cambridge University Press, Cambridge, UK, 2012 (ISBN 978-0-521-51622-8).
 Le Van Quyen, M., Muller, L., Telenczuk, B., Cash, S.S., Halgren, E., Hatsopoulos, N.G., Dehghani, N. and Destexhe, A. High-frequency oscillations in human and monkey neocortex during the wake-sleep cycle. Proc. Natl. Acad. Sci. USA 113: 9363-9368, 2016 (see abstract)
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