Models of local field potentialsLocal 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 highlytortuous 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 frequencydependent attenuation with distance [1]. Starting from Maxwell equations, it can be shown that the extracellular potential can display frequencydependent attenuation if the extracellular conductivity is nonhomogeneous. In this case, there is induction of nonhomogeneous charge densities which may result in a lowpass 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 nonhomogeneous. Thus, the theory shows that these nonhomogeneities constitute a possible physical origin for the frequencydependent properties of LFPs [1], and in particular the induced electric fields in this nonhomogeneous structure [2]. However, although this formalism provides a plausible explanation for the frequencydependent properties of LFPs, it is difficult to apply in practice because one would need to explicitly take into account the complex threedimensional 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 sphericallysymmetric conductivity/permittivity gradients around each source [1]. This simple model can display lowpass 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 frequencydependent extracellular field potentials without taking into account explicitly the complex folding of extracellular space (see details in [1]). An explicit structure of extracellular space was considered in a second type of model [2]. 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 frequencydependent, 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 another physical origin of the frequency filtering properties of LFPs is the induced electric fields in passive cells surrounding neurons. We next investigated the physical basis of the 1/f frequency scaling of LFPs [3]. Many complex systems display selforganized 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 powerlaw 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 [3]. The latter theme was investigated in more detail in a subsequent paper [4]. 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) [5], 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 [6], 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 [7] and a Scholarpedia article [12]. We also wrote a short review on the meanfield formulation of Maxwell equations [13], as well as a recent review summarizing the different measurements and evidence for nonresistive media [20]. In the latter paper, we suggested a framework where all experiments are explained, and also a way to test it experimentally. This was also discussed in a recent commentary [23]. The same approach was also applied to the currentsource density (CSD) analysis. The CSD analaysis is based on the charge conservation law, and if the medium is nonresistive, this method is not correct (the equations are invalid in this case). To address this caveat, we have developed a meanfield formalism which is a generalization of the standard model, and which 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 classic CSD analysis, but in addition, provides 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 medium, which provides a direct way to estimate those properties from experimental data, and in particular, estimate the possible contribution of electric monopoles (see [8] for details). The CSD profiles were also simulated from computational models of sleep oscillations [9]. 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 frequencyfiltering properties were considered, such as a nonhomogeneous 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 [10], 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 [11]. Recently, collaboration with Thierry Bal and Laurent Venance (Collège de France), we performed experiments to test the ohmic nature of the extracellular medium [14]. 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 nonohmic properties of the extracellular medium may impact on cable properties [15], as well as on the genesis of LFPs and magnetic fields by neurons [16]. The exact nature of the impedance predicts a specific frequencyscaling property of the LFP and magnetic fields, as we showed previously in human MEG recordings [6]. 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 [18]. The oscillation dynamics exhibited propagating wave properties, in both wake and sleep states. Interestingly, the highest spatiotemporal coherence of the oscillations was found during slowwave sleep, which suggests that sleep replays wake patterns with higher coherence, perhaps in relation to memory consolidation [18]. 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 [17]. We also characterized the spatiotemporal properties of LFPs, in particular in the beta and gamma frequency range in human and monkey [18]. The oscillation dynamics exhibited propagating wave properties, in both wake and sleep states. Interestingly, the highest spatiotemporal coherence of the oscillations was found during slowwave sleep [18]. Last year, we published a study which we believe is important for the understanding of the LFP signal. Using human and monkey Utaharray recordings, we could relate the unit and LFP signals, taking advantage of the RSFS cells discrimination [19]. We discovered that most of the LFP signal is correlated with inhibitory cell activity, and thus seems generated mostly by IPSPs. This finding, if confirmed, may fundamentally change our interpretation of LFP signals. We are presently using compartmental modeling to understand what are the possible biophysical origins of this dominance of IPSPs. Recently, we investigated the genesis of LFPs from detailed morphologies, and in particular the role of the axon initial segment (IS) [21]. This study showed that the IS is determinant to the exact shape of the extracellular spike, and that changes in the channel distribution can change the shape of extracellular spikes, which has implications for longterm unit recordings. We also studied the conditions for measuring the electric and magnetic properties of biological media [22, 23]. We found extended relations for the conductivity [22], which constitute important constraints to test the consistency of past and future experimental measurements of the electric properties of heterogeneous media. We also discussed the coherence of recent measurements, and suggested new experiments in a commentary article [23]. Thus, these experimental and modeling studies show that the LFP signal is highly non trivial, it may contain significant contributions from the filtering through the extracellular medium, ionic diffusion, cell polarization or effects due to the inhomogeneities of conductivity. Most importantly, LFPs may reflect primarily the contribution of inhibitory postsynaptic currents, which may potentially have strong consequences on their interpretation. Further models and experiments are needed to fully characterize the LFP and how to decode this signal into the activity of neural populations. [1] Bedard, C., Kröger, H. and Destexhe, A. Modeling extracellular field potentials and the frequencyfiltering properties of extracellular space. Biophysical Journal 86: 18291842, 2004 (see abstract). [2] Bedard, C., Kröger, H. and Destexhe, A. Model of lowpass filtering of local field potentials in brain tissue. Physical Review E 73: 051911, 2006 (see abstract). [3] Bedard, C., Kröger, H. and Destexhe, A. Does the 1/f frequencyscaling of brain signals reflect selforganized critical states ? Physical Review Letters 97: 118102, 2006 (see abstract). [4] Bedard, C. and Destexhe, A. Macroscopic models of local field potentials the apparent 1/f noise in brain activity. Biophysical Journal 96: 25892603, 2009 (see abstract). [5] Bedard, C., Rodrigues, S., Roy, N., Contreras, D. and Destexhe, A. Evidence for frequencydependent extracellular impedance from the transfer function between extracellular and intracellular potentials. J. Computational Neurosci. 29: 389403, 2010 (see abstract). [6] 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 nonresistive extracellular media. J. Computational Neurosci. 29: 405421, 2010 (see abstract). [7] 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. 136191, 2012 (see abstract). [8] Bedard, C. and Destexhe, A. A generalized theory for currentsource density analysis in brain tissue. Physical Review E 84: 041909, 2011 (see abstract). [9] Bazhenov, M., Lonjers, P., Skorheim, S., Bedard, C. and Destexhe, A. Nonhomogeneous extracellular resistivity affects the currentsource density profiles of Up/Down state oscillations. Phil. Trans. Roy. Soc. A 369: 38023819, 2011 (see abstract). [10] Destexhe, A. and Bedard, C. Do neurons generate monopolar current sources? J. Neurophysiol. 108: 953955, 2012 (see abstract). [11] Bedard, C. and Destexhe, A. Electric monopoles are indeed compatible with Maxwell equations. J. Neurophysiol. 109: 1683, 2013 (see abstract) [12] Destexhe, A. and Bedard, C. Local field potential. Scholarpedia 8: 10713, 2013. ( see article) [13] Bedard, C. and Destexhe, A. Meanfield formulation of Maxwell equations to model electrically inhomogeneous and isotropic media. J. Electromagnetic Analysis and Applications 6: 296302, 2014 (see abstract) [14] Gomes, JM., Bedard, C., Valtcheva, S., Nelson, M., Khokhlova, V., Pouget, P., Venance, L., Bal, T. and Destexhe, A. Intracellular impedance measurements reveal nonohmic properties of the extracellular medium around neurons. Biophys J. 110: 234246, 2016. see abstract) [15] Bedard, C. and Destexhe, A. Generalized cable theory for neurons in complex and heterogeneous media. Physical Review E 88: 022709, 2013 (see abstract) [16] 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) [17] Brette, R. and Destexhe, A. (Editors) Handbook of Neural Activity Measurement, Cambridge University Press, Cambridge, UK, 2012 (ISBN 9780521516228). [18] Le Van Quyen, M., Muller, L., Telenczuk, B., Cash, S.S., Halgren, E., Hatsopoulos, N.G., Dehghani, N. and Destexhe, A. Highfrequency oscillations in human and monkey neocortex during the wakesleep cycle. Proc. Natl. Acad. Sci. USA 113: 93639368, 2016 (see abstract) [19] Telenczuk, B., Dehghani, N., Le Van Quyen, M., Cash, S., Halgren, E., Hatsopoulos, N.G. and Destexhe, A. Local field potentials primarily reflect inhibitory neuron activity in human and monkey cortex. Nature Scientific Reports 7: 40211, 2017 (see abstract) [20] Bedard, C., Gomes, JM., Bal, T. and Destexhe, A. A framework to reconcile frequency scaling measurements, from intracellular recordings, localfield potentials, up to EEG and MEG signals. J. Integrative Neurosci. 16: 318, 2017 (see abstract) [21] Telenczuk, M., Brette, R., Destexhe, A. and Telenczuk, B. Contribution of the axon initial segment to action potentials recorded extracellularly. eNeuro 5: 006818, 2018 (see abstract) [22] Bedard, C. and Destexhe, A. KramersKronig relations and the properties of conductivity and permittivity in heterogeneous media. J. Electromag. Analysis Applic. 10: 3451, 2018 (see abstract) [23] Bedard, C. and Destexhe, A. Is the extracellular impedance high and nonresistive in cerebral cortex? Biophys. J. 113: 16391642, 2017 (see abstract)
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