# Local Field Potential Interaction with the Extracellular Medium

**DOI:**https://doi.org/10.1007/978-1-4614-7320-6_720-1

## Keywords

Electric Parameter Ionic Diffusion Extracellular Medium Local Field Potential Macroscopic Model## Definition

The local field potential (LFP) is the electric potential in the extracellular space around neurons. The LFP is generated by electric currents and charges and is also interacting in several possible ways with the extracellular medium, such as capacitive interactions, polarization, or trough ionic diffusion. These types of interaction confer specific frequency dependence of the extracellular electric potential and thus are important for correctly interpreting the LFP, as well as estimating the underlying neuronal sources (inverse problem).

## Detailed Description

### Introduction

In contrast to the electroencephalogram (EEG) recorded at the surface of the scalp, the local field potential (LFP) is recorded by inserting microelectrodes into neural tissue, and it is recordable using a variety of electrode systems, such as metal or silicon electrodes or glass micropipettes. Early studies established that action potentials have a limited participation to EEGs or LFPs (Bremer 1938, 1949; Eccles 1951; Klee et al. 1965; Creutzfeldt et al. 1966a, b). The current view is that EEG and LFPs are generated by synchronized synaptic currents arising on cortical neurons, possibly through the formation of dipoles (Nunez 1981; Niedermeyer and Lopes da Silva 1998).

The fact that action potentials have little participation to the LFP indicates strong frequency filtering properties of cortical tissue. High frequencies (greater than ≈ 100 Hz), such as that produced by action potentials, are subject to a severe attenuation and therefore are visible only for electrodes immediately adjacent to the recorded cell. On the other hand, low-frequency events, such as synaptic potentials, attenuate less with distance. These events can therefore propagate over large distances in extracellular space and be recordable as far as on the surface of the scalp, where they can participate in the genesis of the EEG. This frequency-dependent behavior is also seen routinely in extracellular unit recordings: the amplitude of extracellularly recorded spikes is very sensitive to the position of the electrode, but slow events show much less sensitivity to electrode position. In contrast, action potentials are recorded only for the cell(s) immediately adjacent to the electrode and are therefore very sensitive to changes in electrode position. This property is fundamental because it allows resolving single units from extracellular recordings.

EEG or LFP measurements also display approximately 1/*f* frequency scaling in their power spectra at low frequencies (Bhattacharya and Petsche 2001; Bédard et al. 2006b; Novikov et al. 1997; Pritchard 1992; Dehghani et al. 2010). The origin of such 1/*f* “noise” is at present unclear. 1/*f* spectra can result from self-organized critical phenomena (Jensen 1998), suggesting that neuronal activity may be working according to such states (Beggs and Plenz 2003). Alternatively, the 1/*f* scaling may be due to filtering properties of the currents through extracellular media (Bédard et al. 2006b). This conclusion was reached by noting that the global activity reconstructed from multisite unit recording scales identically as the LFP if a “1/*f* filter” is assumed, without the need to assume self-organized critical states in neural activity.

We review here possible physical bases for such frequency filtering properties, from the interaction between electric events (such as membrane currents) and the extracellular space around neurons.

### The “Standard Model”

*resistive*or

*ohmic*, with no capacitive component. In this case, one considers a set of punctual current sources embedded in a homogeneous conductive medium of conductivity σ. The extracellular potential due to a single punctual current source can be deduced simply as follows. Starting from Ohm’s law \( \left(\overrightarrow{j}=\upsigma \overrightarrow{E}\right) \), combined with the law of current conservation \( \left(\overrightarrow{j}=\frac{i}{4\uppi {r}^2}\widehat{r}\right) \) in spherical symmetry around the source, and by integrating along a straight line from the source to a given point in extracellular space, the extracellular potential at some distance

*r*from the source

*i*is given by

*V*(

*∞*) = 0. This relation can also be deduced directly from Maxwell-Gauss law \( \left(\nabla \cdot \overrightarrow{D}=\uprho \right) \) within the quasistatic approximation in the electric limit (Montigny and Rousseaux 2006) a priori. (Note that this “electric limit” quasistatic approximation assumes that the magnetic field is negligible compared to the electric field \( \left(\mid \mid \overrightarrow{E}\mid \mid <\frac{1}{\sqrt{\mu \upvarepsilon}}\mid \mid \overrightarrow{B}\mid \mid \right) \). The other quasistatic approximation (“magnetic limit”) considers that the electromagnetic induction is not negligible.) In the case of a set of

*n*current sources

*i*

_{ j }, the superposition principle applies, and one can write

*r*

_{ j }is the distance from the source

*i*

_{ j }to the position

*r*in extracellular space. This expression captures many effects such as dipoles or multipolar configurations (if the distance is large compared to the size of the source). Equation 2 is widely used to model extracellular potentials, since early models (Rall and Shepherd 1968) to today’s models of extracellular activity (Protopapas et al. 1998; Destexhe 1998; Nunez and Srinivasan 2005; Gold et al. 2006; Pettersen and Einevoll 2011).

### Microscopic Model of Frequency Filtering

A first possible physical mechanism giving rise to frequency filtering properties is the fact that the extracellular space around neurons is not homogeneous but is subject to strong variations in conductivity and permittivity. In cerebral cortex, the different cellular processes are densely packed (Peters et al. 1991), and the extracellular space consists of a mixture of membranes of very low conductivity and extracellular highly conductive fluids. To formalize this situation, however, the equations above cannot be used, and one must restart from first principles, as done previously (Bédard et al. 2004). To do this, we will consider a “microscopic” model where the electric parameters are subject to spatial variations according to the type of medium (fluid, membranes).

^{ f }are, respectively, the electric displacement, the free-charge current density, and free-charge charge density in the medium surrounding the sources.

In this model, the electric parameters σ and ε take their “microscopic” values and are assumed to be independent of frequency. Indeed, the electric parameters of the extracellular fluid can be considered constant for frequencies lower than 1,000 Hz (Gabriel et al. 1996c). In this case, the Fourier transforms of the above equations are, respectively, \( {\overrightarrow{D}}_{\upomega}={\upvarepsilon}_{\upomega}{\overrightarrow{E}}_{\upomega}=\upvarepsilon {\overrightarrow{E}}_{\upomega} \) and \( {\overrightarrow{j}}_{\upomega}={\upsigma}_{\upomega}{\overrightarrow{E}}_{\upomega}=\upsigma {\overrightarrow{E}}_{\upomega}={\upsigma}_z{\overrightarrow{E}}_{\upomega} \) (where ω = 2π*f* where *f* is the frequency). It is important to note that σ_{ω} is a real number in this microscopic model, but this is not necessarily the case in a macroscopic model (as in next section).

This is the main equation to calculate the LFP in nonhomogeneous media, as derived previously (Bédard et al. 2004). It is general enough to calculate the propagation of the extracellular potential in extracellular media which can have a complex or inhomogeneous structure, as well as frequency-dependent electric parameters.

^{2}

*V*

_{ω}= 0) when the medium is homogeneous with respect to σ and ε. Thus, Eq. 6 is a generalization of Laplace equation for media where σ and ε are nonhomogeneous. Except for particular cases where the ratio ε/σ is independent of position, Eq. 6 shows that, in nonhomogeneous media, the extracellular potential will necessarily have a different power spectral structure compared to that of the current sources, because the extracellular potential must be solution of a differential equation with frequency-dependent coefficients (Note that in this case, the electric potential does not satisfy Laplace equation (∇

^{2}

*V*

_{ω}= 0) but obeys

This “microscopic” model was simulated with different spatial profiles of conductivity and permittivity around neurons (Bédard et al. 2004). It was found that according to the profile of σ and ε, one can have a high-pass or low-pass filter, but in general, when σ is high at short distances and decays with larger distances, a low-pass filter is observed. For example, with an exponentially decaying conductivity, a low-pass filter attenuates more strongly the fast frequencies compared to low frequencies. A simulation of the LFP generated showed that the extracellular waveform of an action potential changes as the distance to the source is increased (Bédard et al. 2004). Thus, this model accounts for basic features of frequency filtering.

*I*

_{ω}

^{ f }is the free-charge current (in Fourier space). Note that the membrane current is the sum of the free-charge current (ion channels) and the capacitive current. Thus, the free-charge current is

*I*

_{ m }is the membrane current of the source, σ(

*R*) is the electric conductivity, and ε(

*R*) is the electric permittivity of the membrane at the source. (This relation is obtained from \( {\overrightarrow{j}}_{\upomega}^f={\upsigma}_{\upomega}{\overrightarrow{E}}_{\upomega} \) and \( {\overrightarrow{j}}_{\upomega}^c=i{\upomega \upvarepsilon}_{\upomega}{\overrightarrow{E}}_{\upomega} \), where \( {\overrightarrow{j}}_{\upomega}^c \) is the capacitive current density). It follows that the PSD of the voltage varies as

*I*

^{2}

_{ m }(ω)/ω

^{2}for high frequencies, because in this case

*V*

_{ω}becomes proportional to

*I*

_{ m }(ω)/ω.

### Macroscopic Models of Extracellular Space

In principle, it is sufficient to solve Eq. 6 in the extracellular medium to simulate the LFP in inhomogeneous media. However, if the extracellular medium has a complex spatial structure, it may be tedious to assign the space dependence of the electric parameters σ and ε. One way to solve this problem is to consider a macroscopic or mean-field approach at a larger scale, in which the electric parameters will be considered as constant in space but are the result of a microscopic mixture of different media such as fluids and membranes.

Note that Maxwell equations are invariant under change of scale if electric parameters are renormalized appropriately (see details in Jackson 1999; Bédard and Destexhe 2011). This approach is further justified by the fact that the values measured experimentally are averaged values, which precision depends on the measurement technique. Ideally, one should have a formalism that can integrate those macroscopic measurements. Thus, we will use a macroscopic model, in which we take spatial averages of Eq. 6 and make a continuous approximation for the spatial variations of these average values (see details in Bédard and Destexhe 2009).

^{ M }and σ

^{ M }, by taking an average of the “microscopic” electric parameters over some volume

*V*:

We assume that *V* is of the order of 10^{3} μm^{3}, and is thus much smaller than the cortical volume, so that the mean values will be dependent of the position in cortex.

^{ M }and ε

^{ M }. The second term translates the fact that there is an inertia time for polarization (also called Maxwell-Wagner time), because the charges do not move instantaneously. This implies that the electric field will take a characteristic time to settle; this time is given by the Maxwell-Wagner time of the region in which the average is taken (see details in Planck 1932, and for application to biological membranes, see Bédard et al. 2006a).

_{ z }

^{ M }is the complex macroscopic conductivity.

*z*distinguishes from the real-valued permittivity ε

_{ω}

^{ M }. We can then write

*σ*

_{ z }

^{ M }= (

*σ*

_{ ω }

^{ M }+

*iωε*

_{ z }

^{ M }) and \( \left\langle {\overrightarrow{E}}_{\omega}\right\rangle =-\nabla \left\langle {V}_{\omega}\right\rangle, \) the expressions above (Eq. 10) can also be written in the form

We note that starting from the continuum model (Bédard et al. 2004), where only spatial variations were considered, and generalizing this model by including frequency-dependent electric parameters, gives the same mathematical form as the original model (compare with Eq. 6; see details in Bédard and Destexhe 2011). This form invariance will allow us to introduce different phenomena, such as surface polarization or ionic diffusion, by including an ad hoc frequency dependence in σ _{ω} ^{ M } and ε _{ω} ^{ M } . The physical causes of this macroscopic frequency dependence is that the cortical medium is microscopically non-neutral (although the cortical tissue is macroscopically neutral). Such a local non-neutrality was already postulated in a previous model of surface polarization (Bédard et al. 2006a). This situation cannot be accounted by Eq. 6 if σ _{ω} ^{ M } and ε _{ω} ^{ M } are frequency independent (in which case ρ_{ω} = 0 when ∇*V* _{ω} = 0). Thus, including the frequency dependence of these parameters enables the model to capture a much broader range of physical phenomena.

_{ω}

^{ M }and ε

_{ω}

^{ M }cannot take arbitrary values but are related to each other by the Kramers-Kronig relations (Foster et al. 1989; Kronig 1926; Landau and Lifshitz 1981):

### Conclusions

In this entry, we have overviewed three formalisms to model LFPs. The first formalism, mostly based on Coulomb’s law, expresses the potential as resulting from a set of current sources in a homogeneous resistive medium. This model is the one used by the majority of studies to simulate the LFP. It is justified by some measurements, suggesting that the extracellular medium is resistive (Logothetis et al. 2007). However, it does not account for other measurements, suggesting that the extracellular medium is nonohmic (Gabriel et al. 1996b; Bédard et al. 2010; Dehghani et al. 2010). There is also at present no convincing simulation of the spatial aspect of LFPs and why the spikes from only a few neurons are visible in extracellular recordings.

A second formalism, which we called “microscopic,” considers the specific case of extracellular media that are nonhomogeneous, in which the electric parameters σ and ε are strongly dependent on position. In this case, one must return to Maxwell equations to derive the correct equation to calculate the extracellular potential. The reason is that in such media, there will be charge accumulation, and the local free-charge conservation law \( \left(\nabla \cdot {\overrightarrow{j}}^f=0\right) \) does not apply (one must take into account the displacement current \( \nabla .\left(\nabla \times \overrightarrow{H}\right)=\nabla .\left({\overrightarrow{j}}^f+\frac{\partial \overrightarrow{D}}{\partial t}\right)=\nabla .{\overrightarrow{j}}^f+\frac{\partial \rho }{\partial t} \)) (see Bedard and Destexhe 2004). This formalism was derived previously and was shown to simulate the low-pass filtering properties of extracellular space (Bédard et al. 2004). However, this “microscopic” formalism cannot integrate “macroscopic” measurements of σ and ε (Gabriel et al. 1996b), nor the fact that the frequency filter deduced from experimental measurements is of 1/*f* type (Bédard et al. 2006b, 2010).

A third type of formalism, which we called “macroscopic,” describes the tissue at a larger coarse graining, in which the electric parameters will be considered as constant in space but are the result of a mixture of different media such as fluids and membranes (Bédard and Destexhe 2009). This macroscopic formalism can directly integrate the measurements of σ and e and thus can be used to simulate a wide variety of physical mechanisms. For example, it can be used to simulate the polarization phenomena that the electric field induces on neuronal membranes; it can also be used to integrate the effect of ionic diffusion.

This approach was possible because of the scale invariance of the mathematical structure of Maxwell equations. This property allows one to renormalize the electromagnetic parameters such that even if the microscopic parameters do not depend on frequency, the macroscopic parameters can be frequency dependent in some media (see details in Bédard and Destexhe 2011). This is a consequence of the fact that the Maxwell-Wagner time is non-negligible in such media. Note that the standard model postulates that there is no such frequency dependence, which amounts to neglect the Maxwell-Wagner time and consider that the charges move infinitely fast, which corresponds to the simplest possible model in electromagnetism theory.

Using such formalisms, it was shown that ionic diffusion gives macroscopic electric parameters σ _{ω} ^{ M } and ε _{ω} ^{ M } displaying a frequency dependence as \( 1/\sqrt{\upomega}, \) which gives a 1/*f* filter in power spectra. Such a 1/*f* filter is consistent with previous measurements, to relate LFPs with extracellularly recorded units (Bédard et al. 2006b), or LFPs with intracellular recordings (Bédard et al. 2010). It is also consistent with conductivity measurements showing resistive media (Logothetis et al. 2007), because these measurements were designed to specifically avoid ionic diffusion (see discussion in Logothetis et al. 2007). Thus, if ionic diffusion has dominant effects and if other effects such as inhomogeneities and polarization are negligible, then all measurements can be reconciled.

Thus, one can conclude that ionic diffusion is at present the only possible physical mechanism that can account for all measurements done to date. A macroscopic model of LFPs (such as Eq. 11) is the most appropriate formalism to model ionic diffusion and was shown to reproduce the frequency scaling properties of LFPs quantitatively (Bédard and Destexhe 2009).

Finally, it is important to note that capturing the correct frequency dependence of the electric parameters is very important in the so-called inverse problem of estimating neuronal sources from measurements of extracellular voltage, such as the current-source density method (Mitzdorf 1985; Nicholson and Freeman 1975). It is evident that such a frequency dependence, if present, will distort the extracellular electric potential, and inversely, starting from the electric potential, the estimation of the source will be different if the frequency dependence is neglected (as in the classic CSD analysis). This problem only starts to be evaluated now; for example, it was shown that generalizing the CSD analysis to frequency-dependent media may lead to large differences in the mathematical estimates of neuronal sources (Bédard and Destexhe 2011).

In conclusion, we have shown that the expressions to calculate the extracellular potential from membrane currents are vastly dependent on the nature of the extracellular medium. If phenomena like inhomogeneities, capacitive effects, polarization, or ionic diffusion are taken into account, the medium exerts a strong frequency filtering action on the propagation of the extracellular potentials. As a consequence, if such effects are confirmed, it is imperative to take them into account to correctly model the extracellular potentials. Similarly, estimating neuronal sources from extracellular measurements also requires to know precisely the nature of the frequency filtering properties of the medium. So far, measurements are contradictory (Gabriel et al. 1996a, b, c; Logothetis et al. 2007), presumably due to the fact that the currents used were nonphysiological. This issue should be settled by performing measurements under physiological conditions.

## References

- Bédard C, Destexhe A (2009) Macroscopic models of local field potentials and the apparent 1/f noise in brain activity. Biophys J 96:2589–2603PubMedCentralPubMedCrossRefGoogle Scholar
- Bédard C, Destexhe A (2011) A generalized theory for current-source density analysis in brain tissue. Phys Rev E 84:041909CrossRefGoogle Scholar
- Bedard C, Destexhe A (2012) Modeling local field potentials and their interaction with the extracellular medium. In: Brette R, Destexhe A (eds) Handbook of neural activity measurement. Cambridge University Press, Cambridge, pp 136–191CrossRefGoogle Scholar
- Bédard C, Kröger H, Destexhe A (2004) Modeling extracellular field potentials and the frequency-filtering properties of extracellular space. Biophys J 86:1829–1842PubMedCentralPubMedCrossRefGoogle Scholar
- Bédard C, Kröger H, Destexhe A (2006a) Model of low-pass filtering of local field potentials in brain tissue. Phys Rev E 73:051911CrossRefGoogle Scholar
- Bédard C, Kröger H, Destexhe A (2006b) Does the 1/f frequency scaling of brain signals reflect self-organized critical states? Phys Rev Lett 97:118102PubMedCrossRefGoogle Scholar
- Bédard C, Rodrigues S, Roy N, Contreras D, Destexhe A (2010) Evidence for frequency-dependent extracellular impedance from the transfer function between extracellular and intracellular potentials. J Comput Neurosci 29:389–403PubMedCrossRefGoogle Scholar
- Beggs J, Plenz D (2003) Neuronal avalanches in neocortical circuits. J Neurosci 23:11167–11177PubMedGoogle Scholar
- Bhattacharya J, Petsche H (2001) Universality in the brain while listening to music. Proc Biol Sci 268:2423–2433PubMedCentralPubMedCrossRefGoogle Scholar
- Bremer F (1938) L’activité électrique de l’écorce cérébrale. Actualités Scientifiques et Industrielles 658:3–46Google Scholar
- Bremer F (1949) Considérations sur l’origine et la nature des “ondes” cérébrales. Electroencephalogr Clin Neurophysiol 1:177–193PubMedCrossRefGoogle Scholar
- Creutzfeldt O, Watanabe S, Lux HD (1966a) Relation between EEG phenomena and potentials of single cortical cells. I. Evoked responses after thalamic and epicortical stimulation. Electroencephalogr Clin Neurophysiol 20:1–18PubMedCrossRefGoogle Scholar
- Creutzfeldt O, Watanabe S, Lux HD (1966b) Relation between EEG phenomena and potentials of single cortical cells. II. Spontaneous and convulsoid activity. Electroencephalogr Clin Neurophysiol 20:19–37PubMedCrossRefGoogle Scholar
- de Montigny M, Rousseaux G (2006) On the electrodynamics of moving bodies at low velocities. Eur J Phys 27:755–768CrossRefGoogle Scholar
- Dehghani Nima C, Bédard SS, Cash EH, Destexhe A (2010) Comparative power spectral analysis of simultaneous electroencephalographic and magnetoencephalographic recordings in humans suggests non-resistive extracellular media. J Computational Neurosci 29:405–421 (special issue “
*Modeling Extracellular Potentials*”)CrossRefGoogle Scholar - Destexhe A (1998) Spike-and-wave oscillations based on the properties of GABA
_{B}receptors. J Neurosci 18:9099–9111PubMedGoogle Scholar - Eccles JC (1951) Interpretation of action potentials evoked in the cerebral cortex. J Neurophysiol 3:449–464Google Scholar
- Foster KR, Schwan HP (1989) Dielectric properties of tissues and biological materials: a critical review. Crit Rev Biomed Eng 17:25–104PubMedGoogle Scholar
- Gabriel S, Lau RW, Gabriel C (1996a) The dielectric properties of biological tissues: I. Literature survey. Phys Med Biol 41:2231–2249PubMedCrossRefGoogle Scholar
- Gabriel S, Lau RW, Gabriel C (1996b) The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz. Phys Med Biol 41:2251–2269PubMedCrossRefGoogle Scholar
- Gabriel S, Lau RW, Gabriel C (1996c) The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum tissues. Phys Med Biol 41:2271–2293PubMedCrossRefGoogle Scholar
- Gold C, Henze DA, Koch C, Buzsaki G (2006) On the origin of the extracellular action potential waveform: a modeling study. J Neurophysiol 95:3113–3128PubMedCrossRefGoogle Scholar
- Jackson JD (1999) Classical electrodynamics, 3rd edn. Wiley, New YorkGoogle Scholar
- Klee MR, Offenloch K, Tigges J (1965) Cross-correlation analysis of electroencephalographic potentials and slow membrane transients. Science 147:519–521PubMedCrossRefGoogle Scholar
- Kronig RDL (1926) On the theory of dispersion of X-rays. J Opt Soc Am 12:547CrossRefGoogle Scholar
- Landau LD, Lifshitz EM (1981) Electrodynamics of continuous media. Pergamon Press, MoscowGoogle Scholar
- Logothetis NK, Kayser C, Oeltermann A (2007) In vivo measurement of cortical impedance spectrum in monkeys: implications for signal propagation. Neuron 55:809–823PubMedCrossRefGoogle Scholar
- Mitzdorf U (1985) Current source-density method and application in cat cerebral cortex: investigation of evoked potentials and EEG phenomena. Physiol Rev 65:37–100PubMedGoogle Scholar
- Nicholson C, Freeman JA (1975) Theory of current source-density analysis and determination of conductivity tensor for anuran cerebellum. J Neurophysiol 38(2):356–368PubMedGoogle Scholar
- Niedermeyer E, Lopes da Silva F (eds) (1998) Electroencephalography, 4th edn. Williams and Wilkins, BaltimoreGoogle Scholar
- Novikov E, Novikov A, Shannahoff-Khalsa D, Schwartz B, Wright J (1997) Scale-similar activity in the brain. Phys Rev E 56:R2387–R2389CrossRefGoogle Scholar
- Nunez PL (1981) Electric fields of the brain. The neurophysics of EEG. Oxford University Press, Oxford, UKGoogle Scholar
- Nunez PL, Srinivasan R (2005) Electric fields of the brain, 2nd edn. Oxford University Press, Oxford, UKGoogle Scholar
- Peters A, Palay SL, Webster HF (1991) The fine structure of the nervous system. Oxford University Press, Oxford, UKGoogle Scholar
- Pettersen KH, Einevoll GT (2011) Amplitude variability and extracellular low-pass filtering of neuronal spikes. Biophys J 94:784–802CrossRefGoogle Scholar
- Planck M (1932) Theory of electricity and magnetism. Macmillan, London, 80441143Google Scholar
- Protopapas AD, Vanier M, Bower J (1998) Simulating large-scale networks of neurons. In: Koch C, Segev I (eds) Methods in neuronal modeling, 2nd edn. MIT Press, Cambridge, MA, pp 461–498Google Scholar
- Rall W, Shepherd GM (1968) Theoretical reconstruction of field potentials and dendrodendritic synaptic interactions in olfactory bulb. J Neurophysiol 31:884–915PubMedGoogle Scholar

## Further Reading

- Koch C, Segev I (eds) (1998) Methods in neuronal modeling, 2nd edn. MIT Press, Cambridge, MAGoogle Scholar