View source for Local field potential

The <strong>local field potential</strong> (LFP) refers to the electric 
potential in the extracellular space around [[neurons]].  The LFP is a widely
available signal in many recording configurations, ranging from
single-electrode recordings to multi-electrode arrays.  In this
article, we review the main properties of the LFP as well as different
types of LFP models.

<!---------------------------- FIGURE ------------------------------>
[[Image:HumanLFP.jpg|thumb|400px|right|Local field potentials recorded in humans
  using an array of extracellular electrodes.  A. Array of 100 silicon
  electrodes (Utah array, BlackRock systems), with 400 $\mu$m
  inter-electrode distance, and which can be inserted into cerebral
  cortex.  B. Plan of implementation of surface electrodes (black
  dots).  Inset: magnified view of the medial temporal lobe, with four
  surface electrodes (green) and the Utah array (gray square).  C.
  Typical recordings obtained with surface electrodes (EcoG, green),
  local field potentials (LFP) and identified neurons (Units;
  red=inhibitory, blue=excitatory).  The 4 nearby surface EcoG are
  shown (green), while 4 randomly chosen LFPs were shown among all the
  electrodes available.  This example was recorded during slow-wave
  sleep, and one can see slow delta waves in the surface EcoG as well
  as in the LFPs, and also modulate the firing of units (modified from
  Peyrache et al., 2012).]]
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= Introduction = 

The Local Field Potential (LFP) is the electric potential recorded in
the extracellular space in brain tissue, typically using
micro-electrodes (metal, silicon or glass micropipettes).  LFPs differ
from the electroencephalogram (EEG), which is recorded at the surface
of the scalp, and with macro-electrodes.  It also differs from the 
electro-corticogram (EcoG), which are recorded from the surface of
the brain using large subdural electrodes, while LFPs are recorded
in depth, from within the cortical tissue (or other deep brain 
structures).

Besides their invasive
aspect, LFPs also sample relatively localized populations of neurons,
as these signals can be very different for electrodes separated by
1 mm (Destexhe et al., 1999) or by a few hundred microns (Katzner et
al., 2009).  In contrast, the EEG samples much larger populations of
neurons (Niedermeyer and Lopes da Silva, 1998).  The difference is due
to the fact that the EEG signals must propagate through various media,
such as cerebrospinal fluid, dura matter, cranium, muscle and skin,
and are therefore much more subject to filtering and diffusion
phenomena across these media.  However, even if recorded close to the
neuronal sources, LFP signals are also filtered, because the recording
electrode is separated from the sources by portions of cortical
tissue.  Besides these differences, EEG and LFP signals display the
same type of oscillations during wake and sleep states (Steriade,
2003).

Early studies demonstrated that action potentials have a limited
participation to the genesis of the EEG or LFPs.  The initial theory
about the genesis of the EEG and LFP oscillations, the "circus
movement theory", postulated that the frequency of oscillations was
due to travelling pulses along loops of connected neurons (Rothberger,
1931; Bishop, 1936).  Bremer (1938, 1949) was an opponent to this
theory and he was the first to propose that the EEG is not generated
by action potentials but rather by oscillations of the membrane
potential of neurons.  Eccles (1951) proposed that LFP and EEG
activities are generated by summated postsynaptic potentials arising
from the synchronized excitation of neurons. Intracellular recordings
from cortical neurons later demonstrated a close correspondence
between EEG/LFP activity and synaptic potentials (Klee et al., 1965;
Creutzfeldt et al., 1966a, 1966b).  The current view is that EEG and
LFPs are generated by synchronized synaptic currents arising on
cortical neurons, possibly through the formation of dipoles
(Niedermeyer and Lopes da Silva, 1998; Nunez and Srinivasan, 2005).

One of the possible explanation for the fact that action potentials
have a limited participation to EEG and LFP activities is that the
cortical tissue exerts strong frequency-filtering properties.  High
frequencies (greater than $\approx$ 100 Hz), such as that produced by
action potentials, are subject to steep attenuation, while
low-frequency events, such as synaptic potentials, attenuate less with
distance.  Consequently, the extracellular image of action potentials
is visible only for electrodes immediately adjacent to the recorded
cell, while synaptic events may propagate over large distances in
extracellular space and be recordable as far as on the surface of the
scalp, where they participate to the genesis of the EEG.  This
frequency-dependent behavior is a property routinely seen in
extracellular recordings of neuronal units: the amplitude of
extracellularly-recorded spikes is very sensitive to the position of
the electrode, presumably because action potentials are visible only
for the cell(s) immediately adjacent to the electrode.  By contrast,
slow events presumably originate from synaptic events in a large
number of neighboring neurons, and thus are more stable to small
changes of electrode position.  This property is fundamental to allow
one to resolve single units from extracellular recordings.

In this article, we review the main spatial and temporal (or spectral)
properties of LFPs, as well as different ways to model them.

= Local field potentials in human and animal recordings =

In this section, we review the main properties of LFPs in different
experimental preparations. 

== Spatial and temporal properties of LFPs ==

Figure 1A shows an example of one of the most popular
extracellular multi-electrode array systems, called the "Utah
array", developed at the University of Utah and by BlackRock
Microsystems.  This 100-electrode array was implemented in human
medial temporal cortex together with surface electrodes
(Fig. 1B).  The surface electrodes record the
electrocorticogram (EcoG in Fig. 1), which record the electric
potential at the surface of the cortex (green signals in
Fig. 1C).  By contrast, the LFP provides a recording of the
electric potential from within the brain tissue, and samples a more 
local population of neurons (black signals in Fig. 1C).  One can
clearly see slow delta waves that are visible both in the surface EcoG
and in depth LFP recordings (this example was recorded during slow-wave
sleep).  A number of discriminated units are also
shown, and were recorded using the same set of electrodes.

<!---------------------------- FIGURE ------------------------------>
[[Image:8EEGs2.jpg|thumb|400px|right|Local field potentials in cats during
  wake and sleep states.  Eight bipolar tungsten electrodes
  (inter-electrode distance of 1 mm) were inserted into the depth
  (1 mm) of areas 5-7 of cat parietal cortex (suprasylvian gyrus, area
  5-7; see top scheme for arrangement of electrodes).  Local field
  potentials (LFPs) are shown (left panels) together with a
  representation of the correlations as a function of distance
  (Spatial correlations; middle panels) and time (Temporal
  correlations; right panels).  A. When the animal was awake, LFPs
  were characterized by low-amplitude fast activities in the
  beta/gamma frequency range (15-75 Hz).  Correlations decayed steeply
  with distance and time.  B. During slow-wave sleep, the LFPs were
  dominated by large-amplitude slow-wave complexes recurring at a low
  frequency ($<$1 Hz; up to 4 Hz).  Correlations stayed high for large
  distances.  C.  During episodes of REM sleep, LFPs and correlations
  had similar characteristics as during wake periods (* indicates a
  PGO wave).  Modified from Destexhe et al., 1999.]]
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Figure 2 shows LFPs recorded by 8 sets of bipolar tungsten electrodes from
cat parietal cortex during wake and sleep states.  The LFP displays
the same essential features as classically seen from the EEG: during
wakefulness, the LFP was dominated by low-amplitude fast-frequency
"desynchronized" activity (Fig. 2A), while during slow-wave sleep, the 
LFP displayed high-amplitude slow waves (Fig. 2B).  During Rapid-Eye
Movement (REM) sleep (Fig. 2C), the LFP reverts to desynchronized 
activity with nearly identical characteristics as during wakefulness.  
Note that, contrary
to the EEG, the LFP signals are not contaminated by muscular (EMG)
activity, and therefore constitute ideal signals for detailed analyses,
even at high frequencies.  For instance, it was shown that the high
similarity between Wake and REM states extends to about everything
that can be measured electrophysiologically, such as the spectral
content, the correlation dynamics and the unit activity (Destexhe et
al., 1999).  On the other hand, slow-wave activity is clearly
different, as shown for example by their different pattern of
coherence as a function of distance compared to the fast activity of
wake and REM states.  This difference can be quantified by calculating
spatial correlations, and in particular their decay with distance.  As
expected from the LFP signals, the correlations remained high for
larger distances only during slow-wave sleep (Fig. 2, middle
panels).  In contrast, the temporal correlations displayed similar
profiles between different brain states (Fig. 2, right
panels).

== Relation between LFP and neuronal activity ==

The relationship between LFP and the firing of neurons depends on the
brain state, as illustrated in Fig. 3.  During
wakefulness, the low-amplitude fast-frequency LFP was associated with
sustained and very irregular firing activity in the units
(Fig. 3A,B, Wake).  There was no apparent relation
between units and LFP by visual inspection, although a statistical
analysis revealed that the depth-negative deflections were on average
related to an increase of firing activity in the units
(Fig. 3C, Wake; see details in Destexhe et al., 1999).
During slow-wave sleep, the ensemble activity was surprisingly similar
to wakefulness, although more bursty (Fig. 3A, SWS).
However, at closer scrutiny (Fig. 3B), it appears that
synchronous "silences" in all the units occur systematically and
simultaneously with the depth-positive part of the slow-wave
(Fig. 3B, SWS; blue shaded area).  This type of
synchronized silence is often referred as "Down-state", and was also
visible in the human recordings (see Fig. 1C).  This
modulation of firing by the slow wave is also visible when computing
the statistical relation between units and slow-waves
(Fig. 3C, SWS). The tight relation between slow waves
and Up and Down states in single neurons was firmly demonstrated using
intracellular recordings in natural slow-wave sleep in cats (Steriade
et al., 2001; reviewed in Steriade, 2003).  
The human and cat recordings display essentially the same features of
the relation between LFPs and neuronal units.


<!---------------------------- FIGURE ------------------------------>
[[Image:LFPspikes2.jpg|thumb|400px|right|Relation between LFP and unit
  activity in different brain states.  A. ''Left:'' During
  wakefulness, the low-amplitude fast-frequency LFP activity is
  associated with highly irregular firing activity of the 8
  multi-units recorded simultaneously (same experimental setup as in
  Fig. 2). ''Right:'' During slow-wave sleep (SWS), the
  activity is similar as wakefulness, except that synchronized
  "silences" of firing activity occur in all cells simultaneously,
  and in relation to the slow waves.  B. Same activity as in A at 20
  times higher temporal resolution.  The blue shaded area indicates
  the synchronized silence ("Down state") simultaneous in all cells,
  and occurring in parallel with slow waves.  C.  Wave-triggered
  averages of spiking activity.  During wakefulness, the LFP negative
  peaks were correlated with an increased firing activity in the
  units.  During SWS, the negative peak of the slow wave was
  correlated with a strong decrease of firing in the units (Down
  state), followed by a rebound a sustained activity (Up state).
  Modified from Destexhe et al., 1999.]]
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== Power spectra and frequency scaling of LFPs ==

One of the characteristics of EEG and LFP signals is that their
power-spectrum exhibits $1/f$ frequency scaling at low frequencies, as
shown by a number of studies (Bhattacharya, 2001; Bedard et al.,
2006a; Novikov et al., 1997; Pritchard, 1992; Dehghani et al., 2010a).
Figure 4 illustrates such scaling in the human EEG and in the
LFP recorded with tungsten electrodes in cat parietal cortex, both
when the subject was awake and attentive.  In these conditions, the
low-frequency end of the PSD scales as $1/f$ (red lines).  A more detailed study
showed that the EEG scaling exponent varies from 1 to 2 according to the
brain region considered (Dehghani et al., 2010a).

<!---------------------------- FIGURE ------------------------------>
[[Image:EEG-LFP.jpg|thumb|300px|right|$1/f$ scaling of EEG and LFP signals.
  The top traces show examples of human EEG recordings (left, vertex
  EEG) and LFP recording from cat parietal cortex (right) during awake
  and attentive states.  The corresponding power spectra (lower
  panels) display approximate $1/f$ scaling at low frequencies.  The
  straight lines (red) indicate a slope of -1 in this log-log
  representation.  The signals were not filtered, except for a notch
  filter at 60 Hz (*) for the EEG, and the data acquisition filters at
  high frequencies (not visible at this scale).]]
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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), but this subject is controverted
(Bedard et al., 2006a; Dehghani et al., 2012).  The morphology of the
neuron may also be responsible for filtering in the $1/f$ to $1/f^2$
range (Pettersen and Einevoll, 2008), but this scaling applies to high
frequencies and cannot explain the $1/f$ scaling at low frequencies.
Finally, the $1/f$ scaling may be due to filtering properties of the
currents through extracellular media (Bedard et al., 2006a).  This
conclusion was reached by noting that the global activity
reconstructed from multisite unit recordings scales identically as the
LFP if a "$1/f$ filter" is assumed, and without the need to assume
self-organized critical states in neural activity (Fig. 5).
However, the latter study made the point that $1/f$ filter may be
necessary to explain the experimental results, but no mechanism was
provided to explain such a $1/f$ filter.  We will see below that ionic
diffusion may explain these observations.

<!---------------------------- FIGURE ------------------------------>
[[Image:reconstructed-LFP.jpg|thumb|400px|right|Failure to reconstruct LFP power spectra
  from unit activity.  A. LFP recording in awake cat parietal cortex
  (same recording as in Fig. 4, left).  B. Power spectral
  density (PSD) of the LFP, showing two scaling regions, $1/f$ at low
  frequencies (red line, slope=-1), and $1/f^3$ at high frequencies
  (blue line; slope=-3).  C. Unit activity from the same experiment,
  recorded with a system of 8 tungsten electrodes.  D. Attempt to
  reconstruct the LFP signal from the unit activity.  The low
  frequency end of the PSD was constant (zero slope), while the
  high-frequency end scaled as $1/f^2$ (blue line, slope=-2).  An
  exponent of -1 is missing to reproduce the LFP scaling (modified
  from Bedard et al., 2006a).]]
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= Modeling LFPs =

The LFP is generated by electric currents and charges in brain cells,
including neurons and glial cells.  In particular, all ionic currents
in neurons can potentially contribute to the LFP.  The main
contribution to LFPs is believed to be the synaptic currents in
neurons, although intrinsic voltage-dependent currents and spikes can
also contribute.  Today's prevalent model is that EEG and LFPs are
generated by synchronized synaptic currents arising on cortical
pyramidal neurons, possibly through the formation of dipoles
(Niedermeyer and Lopes da Silva, 1998; Nunez and Srinivasan, 2005).
The LFP also interacts with the extracellular medium, trough several
mechanisms, such as capacitive effects, polarization, or ionic
diffusion.  These types of interaction confer specific frequency
dependence of the extracellular electric potential, and thus are
important for correctly interpreting the LFP.  In this section, we
review different models of LFPs, from simple to more complex models.

== The simplest model of LFP ==

The simplest model for LFP generation assumes that 
the LFP is generated by transmembrane currents and that neurons
are embedded in a perfectly resistive medium (also called Ohmic
medium, such as salted water, with no capacitive or other component).
This is equivalent to assume that the electric conductivity $\sigma$
and permittivity $\epsilon$ of the extracellular medium are constant
everywhere and do not depend on frequency.  In this case, one first
considers the electric potential generated by a punctual current
source.  Combining Ohm's law with the charge conservation law, and
considering a spherical symmetry, we obtain:
\begin{equation}
 V(r) = {1 \over 4 \pi \sigma} {I_0 \over |r-r_0|}   ,
\end{equation}
where $V(r)$ is the extracellular potential at a position $r$ in
extracellular space, $I_0$ is the current source, and $|r-r_0|$ is the
absolute distance between $r$ and the position of the current source
$r_0$.

According to this model, it is easy to verify from the Equation 
above that $V(r)$ is solution of Laplace equation:
\begin{equation}
 \nabla^2 V = 0   .
\end{equation}

Because the principle of linear superposition applies, the potential
resulting from a set of current sources is given by:
\begin{equation}
 V(r) = {1 \over 4 \pi \sigma} \ \sum_j {I_j \over |r-r_j|}   .
\end{equation}
This equation will be referred to as the "standard model".
This expression can be used to calculate the LFP resulting from
complex morphologies, or from networks of neurons.  This model 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, 2008).

This model is also called the "point current source model".  A
variant of this model, called the "line-source approximation" (Holt
and Koch, 1999), in which the membrane current source of a cylindric
compartment is evenly distributed along a line corresponding to the
axis of the cylinder, and which constitutes a more accurate
approximation for compartmental models.

== Example of LFPs generated by a simple model ==

To illustrate the standard LFP model, we consider a model of the
genesis of absence seizures in the thalamocortical system (Destexhe,
1998).  This type of generalized epileptic seizure generates typical
EEG patterns consisting of "Spikes" and "Waves", oscillating at a
frequency around 3 Hz.  Spike-and-wave patterns are seen in many other
types of epileptic seizures as well (Niedermeyer and Lopes da Silva,
1998).  To model such seizures, it is necessary to take into account
the complex intrinsic firing properties of cortical and thalamic
neurons, as well as the different types of receptors present in the
system.  In particular, this model showed that an increase of cortical
excitability can give rise to absence seizures with spike-and-wave
patterns (Destexhe, 1998; see details in the [[Spike-and-Wave Oscillations]] 
Scholarpedia article).  In this model, the
spike-and-wave patterns were generated using the standard model (see above),
using a linear arrangement of pyramidal (PY)
neurons (20 $\mu$m intercellular distance), where $I_j$ was the
transmembrane current of each pyramidal neuron, excluding the currents
responsible for action potentials (see details in Destexhe, 1998). The
simulation showed that spike-and-wave patterns can be reproduced by
this simple model, where synchronized postsynaptic potentials
(EPSP/IPSP sequences) generate the negative "spike", while slow
positive currents (also synchronized) generate the slow positive
"wave" (Fig. 6).  This illustrates that the standard model of
LFPs can generate the spike-and-wave patterns typically seen during
seizures.

<!---------------------------- FIGURE ------------------------------>
[[Image:SW2.jpg|thumb|300px|right|Modeling spike-and-wave LFPs from a
  thalamocortical network model of absence seizures.  A. Spike and wave
  oscillations in human EEG during an absence epileptic seizure.  B.
  Spike and wave oscillations in the LFP recorded using a tungsten
  electrode in the parietal cortex of an epileptic cat.  
  C. Computational model of spike and wave oscillations.  The model
  consists of a network of single-compartment Hodgkin-Huxley type
  model neurons, where synaptic interactions were modeled by
  conductance-based kinetic models of glutamate (AMPA, NMDA) and
  $\gamma$-amino-butyric acid (GABA$_A$ and GABA$_B$) synaptic
  receptors.  Reducing the fast (GABA$_A$-mediated) inhibition in
  cortex resulted in the emergence of hyper-synchronized oscillations
  at a frequency of around 3 Hz.  The LFP was calculated from
  monopolar sources (pyramidal neurons only) using all ionic currents
  except the fast $I_{Na}$/$I_K$ currents involved in spike
  generation.  The model LFP generated the typical spike-and-wave
  patterns seen experimentally during seizures.  When all cells fired
  in synchrony, the LFP generated a negative "spike", while the
  neuronal silences were associated with a slow positive "wave".
  The "spike" was generated by the synchronized EPSP/IPSP sequences,
  while the "wave" was generated by slow K$^+$ currents in pyramidal
  neurons (arrows).  PY = cortical pyramidal neurons, IN = cortical
  interneurons, RE = thalamic reticular neurons, TC = thalamic relay
  cells.  Modified from Destexhe, 1998; see details in the
  [[Spike-and-Wave Oscillations]] Scholarpedia article.]]
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== Frequency filtering properties of LFPs ==

As mentioned in the introduction, the fact that action potentials
have little participation to the LFP may be explained by
frequency-filtering properties of the extracellular medium.  If
the medium acts as a low-pass filter, it may attenuate more severely
the high frequencies (greater than $\approx$ 100 Hz), such as that
produced by action potentials, while the attenuation will be less
severe for lower frequencies such as synaptic events.  As a
consequence, an extracellular electrode at a given position will
"see" only the action potentials for cells immediately adjacent to
the electrode.  On the other hand, the low-frequencies of the LFP 
will be a compound signal of slower events (such as synaptic 
events) from a very large population of cells.

It is important to realize that the fact that spikes are only visible
for cells laying within a few microns from the electrode is the
property that allows one to resolve single units.  If there were no
such frequency-dependent behavior, extracellular recordings would
always be dominated by a signal where thousands of units contribute,
and it would be impossible to distinguish a given unit in such
multi-unit activity.  Thus, frequency filtering is a great opportunity
for the experimentalist.


== How to model frequency-filtering properties? ==

Several possible origins for the frequency-filtering properties of
LFPs have been proposed.  As mentioned above, it was proposed that the
morphology of the neuron is responsible for some type of filtering
(Pettersen and Einevoll, 2008; Linden et al., 2010).  In this case,
the return current, which participates to generating the LFP, is
filtered by the membrane and therefore gives rise to filtered LFPs.
However, it was also shown (Bedard et al., 2010) that this type of
filtering is only present for isolated inputs, and disappears for
''in vivo''--like conditions where the neuron is subject to synaptic
bombardment at many synapses simultaneously (see [[High-conductance State]] 
Scholarpedia article).  This suggests that, under ''in vivo'' 
conditions, the filtering due to morphology can be neglected.
Another possible source of filtering is the fact that action
potentials could be quadrupoles in dendritic and axonal structures
(Milstein and Koch, 2008), and therefore attenuate steeply
(quadrupoles attenuate as $1/r^3$ compared to $1/r^2$ with dipoles).
This contribution may also participate, although the postulated
quadripolar configuration for action potentials still needs to be
demonstrated experimentally.  Besides, none of these possibilities
explain the $1/f$ structure at low frequencies.

A third possible cause for frequency filtering, which was suggested 
earlier (Bedard et al., 2004), is due to the non-resistive or 
inhomogeneous properties of extracellular media.  We explore this 
possible cause in more detail below, with several types of 
extracellular filtering.  We will see that some of them can also
explain the $1/f$ power spectral structure at low frequencies.

A first type of extracellular frequency filtering is due to
electrical non-homogeneity.  The extracellular space is not a
homogeneous fluid (as assumed in the standard model), but is a compact
aggregate of fluids and membranes, as shown by morphological studies
(a nice review of the structure of extracellular space, or neuropil,
can be found in Peters et al., 1991).  In terms of electric
properties, because fluids are highly conductive and membranes are
excellent insulators, it means that the extracellular space has a
highly inhomogeneous electric structure.  To correctly model
extracellular space, one must take into account this complex mixture
of conductive and non-conductive media, as well as the fact that
membranes are capacitors, and therefore the extracellular medium, by
its structure, is not a resistor but also contains myriads of
capacitors.

How to model such complex and heterogeneous media ?  Unfortunately,
the equations given above for the standard model cannot be used,
because if $\sigma$ and $\epsilon$ depend on space, Laplace equation
is not valid anymore.  To obtain a consistent formalism, one must
restart the model from first principles, the Maxwell equations of
electromagnetism. Two approaches are possible, first to design a
"microscopic model", in which the microscopic variations of
conductivity (and permittivity) are taken into account explicitly, to
account for the mixture of fluids and membranes.  A second approach is
to derive a "macroscopic" (or mean-field) model, in which the
extracellular medium is seen as an average of the microscopic
properties.  Equivalently, one could say that the extracellular medium
is viewed with a spatial resolution which is larger than the
microscopic variations of conductivity.  These two approaches are
considered successively below.

== Model of LFP in inhomogeneous extracellular space ==

To design a correct microscopic model, one needs to start from
Maxwell equations.  This was done previously (Bedard et al., 2004),
and the main result is that, in a non-homogeneous medium, the 
extracellular potential obeys the following equation (written in
Fourier frequency space):
\begin{equation}
\nabla\cdot((\sigma+i\omega\epsilon)\nabla V_{\omega)} = 0   .
\end{equation}
This equation gives the spatial evolution of the $\omega$-frequency component
of the extracellular potential, $V_{\omega}$, as a function of the
electric conductivity $\sigma$ and permittivity $\epsilon$, which both
vary in space.  This equation is general enough to calculate the
propagation of the extracellular potential in extracellular media
which can have a complex or inhomogeneous structure.

It is important to note that the equation above reduces to Laplace
equation ($\nabla^2 V_{\omega}=0$) when the medium is homogeneous
with respect to $\sigma$ and $\epsilon$.  Thus, this equation is
a generalization of Laplace equation for media where $\sigma$ and
$\epsilon$ are non-homogeneous. 

This model of non-homogeneous extracellular space was simulated with
different spatial profiles of conductivity and permittivity around
neurons (Bedard et al., 2004).  It was found that according to the spatial
profile of $\sigma$ and $\epsilon$, one can have a high-pass or
low-pass filter, but in general when $\sigma$ is high at short
distance of the membrane source, and decays at larger distances
from the neuron, then a low-pass filter is observed.  For example, if 
the conductivity decays exponentially with distance from the 
neuron, a low-pass filter attenuates more strongly the fast frequencies 
compared to low frequencies.  A simulation of the LFP generated in such
conditions showed that the
extracellular waveform of an action potential changes as the distance
to the source is increased (Bedard et al., 2004; see Fig. 7).
Thus, this model accounts for basic features of frequency filtering.

Assuming a point current source with a spherical symmetry, one can
show that the extracellular potential at a distance $R$ from the 
source is given by
\begin{equation}
V_{\omega} = \frac{I_{\omega}^f}{4\pi\sigma(R)}
\int_R^{\infty} \frac{1}{r'^2}\cdot\frac{\sigma(R) +i\omega\varepsilon(R)}{\sigma(r') +i\omega\varepsilon(r')} dr' ~ ,
\end{equation}
where $I_{\omega}^f$ is the free-charge current (in Fourier space).
One can see from this equation that the $\omega$-frequency component
of the extracellular potential is given by a similar term as in the
standard model, $\frac{I_{\omega}^f}{4\pi\sigma(R)}$, but that this 
term is modulated by the complex expression in the integral.
One can also see from this expression that the result of the integral 
will be different for different frequencies $\omega$, but only if 
$\sigma$ and $\epsilon$ are dependent on position.  Therefore, the
inhomogeneity of $\sigma$ and $\epsilon$ will necessarily create a 
filtering.  Note that if $\sigma$ and $\epsilon$ are constant, the 
integral equals $1/R$, and one recovers the standard model.

== Example of a model of frequency-filtering LFPs ==

To illustrate the frequency-filtering model explained above, we
consider a simulation of the genesis of action potentials and how the
corresponding LFP is filtered in extracellular space.  We consider a
standard [[Hodgkin-Huxley model]] of a neuron, together with a model of
the extracellular space with spatially-variable conductivity and
permittivity.  The neuronal membrane is immediately adjacent to a
highly conductive extracellular fluid, while at further distances,
there is a higher and higher probability of finding low-conductivity
obstacles (membranes, vessels).  This situation was modelled as a
profile where the conductivity is high for short distances (to 
account for the presence of highly conductive fluid adjacent to the 
membrane), and decays exponentially with distance.  Simulating this 
model resulted in strong low-pass filtering properties, as 
illustrated in Fig. 7.


<!---------------------------- FIGURE ------------------------------>
[[Image:lowpass.jpg|thumb|400px|right|A model of the low-pass filtering
  properties of extracellular media.  A.  Membrane potential of a
  single-compartment model containing voltage-dependent Na$^+$ and
  K$^+$ conductances and a glutamatergic synaptic conductance.  The
  glutamatergic synapse was stimulated at t = 5 ms (arrow) and evoked
  an action potential. B. Total membrane current generated by this
  model.  Negative currents correspond to Na$^+$ and glutamatergic
  conductances (inward currents), while positive currents correspond
  to K$^+$ conductances (outward currents). C. Extracellular potential
  calculated at various distances from the source (5, 100, 500 and
  1000 $\mu$m). The frequency filtering properties can be seen by
  comparing the negative and positive deflections of the extracellular
  potential. The fast negative deflection almost disappeared at
  1000 $\mu$m (red) whereas the slow positive deflection was still
  present.  The inset in C (Overlay) shows the traces at 5 (blue) and
  1000 $\mu$m (red) overlayed (only the amplitude was scaled; modified
  from Bedard et al., 2004).]]
<!------------------------------------------------------------------>



== Mean-field model of LFPs ==

As mentioned above, another approach is to adopt a more macroscopic
point of view, and consider a mean-field model of the LFP.  A great
help for designing mean-field models is that Maxwell equations are
invariant under change of scale, if electric parameters $\sigma$ and
$\epsilon$ are renormalized appropriately (see details in Jackson,
1999, Bedard and Destexhe, 2011). A mean-field approach is also
justified by the fact that the values of $\sigma$ and $\epsilon$
measured experimentally are averaged values, which precision depends
on the measurement technique.  Ideally, one should have a formalism
that can integrate those macroscopic measurements, as well as the
resolution scale of the recording.  For all these reasons, it is
justified to design mean-field models of LFPs (see details in Bedard
and Destexhe, 2009,2011).

The first step to obtain a mean-field model of LFPs is to define
macroscopic electric parameters, $\epsilon^M$ and $\sigma^M$, by
taking an average of the "microscopic" electric parameters over some
volume ''Vol'', located at position $\vec{x}$:
$$
\epsilon_{\omega}^M(\vec{x}) = <\epsilon_{\omega}(\vec{x})>_{|_{Vol}}=f(\vec{x},\omega)
$$
and
$$
\sigma_{\omega}^M(\vec{x}) = <\sigma_{\omega}(\vec{x})>_{|_{Vol}}=g(\vec{x},\omega) ~ .
$$
The typical order of magnitude of the volume ''Vol'' considered here will be
around $10^3 \mu m^3$, which is much smaller than the typical volume
of brain tissue.  Consequently, these macroscopic electric parameters
will depend on the position in the brain.

It is important to note that the macroscopic electric parameters
$\sigma^M$ and $\epsilon^M$ are frequency-dependent.  This frequency
dependence arises from the renormalization process, and is necessary
for the macroscopic equations to be consistent [[#foot1|<sup>[1]</sup>]].
It can be shown that
different physical processes will result in different frequency
dependence of these parameters.  A perfectly resistive and homogeneous
medium will result in no frequency dependence of $\sigma^M$ and
$\epsilon^M$, while processes such as polarization, capacitive effects
and ionic diffusion, will all lead to specific frequency-dependent
profiles of $\sigma^M$ and $\epsilon^M$ . For example, for a diffusive
medium (a medium where ionic diffusion is taken into account), 
we have $\sigma^M +i \epsilon^M = k\sqrt{\omega}$ (where $k$ is
a complex number), for a polarizable medium we have $\sigma^M +i
\epsilon^M = k(1+i\omega\tau)$ (where $k$ is a real number; see
details in Bedard et al., 2006b; Bedard and Destexhe, 2009).
  
Defining similarly a macroscopic extracellular potential
$<V_{\omega}>_{|_{Vol}}$ by averaging $V_{\omega}$ over a volume
$Vol$, leads to the following mean-field equation for $V$: 
\begin{equation}
  \nabla\cdot((\sigma_{\omega}^M \ + \ i\omega\epsilon_{\omega}^M)\nabla
  <V_{\omega}>_{|_{Vol}}) =0 ~ .
\end{equation}
Solving this equation will give access to the macroscopic variations
of $V$.

It is interesting to note that this equation has the same mathematical
form compared to the microscopic model of the preceding sections.  This
form invariance is very useful, because one can introduce different
physical phenomena (inhomogeneity, polarization, diffusion...) in the
model, by simply introducing the corresponding frequency dependence of
the macroscopic electric parameters $\sigma_{\omega}^M$ and
$\epsilon_{\omega}^M$.  

It is also important to keep in mind that the frequency dependences of
the electrical parameters $\sigma_{\omega}^M$ and
$\epsilon_{\omega}^M$ cannot take arbitrary values, but are related to
each-other by the Kramers-Kronig relations (Foster and Schwan, 1989;
Kronig, 1926; Landau and Lifshitz, 1981).  These relations are of
course critical to relate the model to experiments, as shown
previously (Jackson 1999; Gabriel et al., 1996).

== Modeling the 1/f scaling of LFPs ==

As shown above, it was not possible to reconstruct the spectral
properties of LFPs, and in particular the $1/f$ frequency scaling at
low frequencies, from the unit activity using a resistive model (Fig. 5). 
A $1/f$-filter was missing to explain the scaling properties of LFPs
(Bedard et al., 2006a).

A plausible mechanism for this $1/f$ scaling was provided later based
on the participation of ionic diffusion (Bedard and Destexhe, 2009,
2011).  Using the macroscopic modeling approach outlined above, it was
shown that ionic diffusion can give rise to a $1/f$ frequency scaling
at low frequencies (Fig. 8).  This is equivalent to postulate that there
is a filter scaling as $1/\sqrt{\omega}$ (which will give $1/\omega$ 
in the power spectrum).  This $1/\sqrt{\omega}$ is well known in 
electrochemistry as the "diffusion impedance" or "Warburg impedance"
(Diart et al., 1999; Taylor and Gileadi, 1995).  Ionic diffusion appears,
therefore, as a plausible physical mechanism to explain the $1/f$
spectral structure of LFPs, while being also consistent with the unit
activity (Bedard and Destexhe, 2011) [[#foot2|<sup>[2]</sup>]].

<!---------------------------- FIGURE ------------------------------>
[[Image:diffusion-LFP.jpg|thumb|400px|right|Successful reconstruction of LFP power
  spectra from unit activity when ionic diffusion is taken into
  account.  A. Unit activity from awake cat (same as in
  Fig. 5C).  B. PSD of the reconstructed LFP, taking into
  account ionic diffusion.  The reconstructed LFP displays the same
  frequency scaling as the real LFP (compare with Fig. 5B;
  modified from Bedard and Destexhe, 2009).]]
<!------------------------------------------------------------------>


== Controversies about the sources of the LFP ==

To finish this review, it is interesting to mention a recent
controversy concerning the possible contribution of electric monopoles
in generating LFPs (Riera et al., 2012).  Using a set of multi-contact
electrodes inserted in the somatosensory cortex of the rat, these 
authors obtained recordings of neuronal activity in 3 dimensions, for 
both units and LFPs.  In addition, these experiments included a detailed
surface EEG recording of the rat brain.  With these simultaneous
recordings, the authors were able to establish first that the 
recordings do not obey instantaneously the electro-neutrality 
principle, suggesting the existence of monopolar sources.  Such 
monopolar components might be an alternative explanation for previous 
results showing that dipole estimates realized from EEG or MEG 
(magnetoencephalogram) measurements do not match (Dehghani et al., 
2010b).

In a second set of experiments, using a variant of the CSD analysis 
technique for data collected using linear probes (Nicholson and Freeman, 
1975), Riera et al. (2012) quantified the relative contribution of the 
different multipolar components (monopoles, dipoles, quadrupoles, ...) 
to the scalp EEG data. Surprisingly, while there were significant 
dipolar and multipolar components as expected, they found an unexpected 
strong monopolar component. These data support prior evidence that in 
some cases, LFPs attenuate in $1/r$, as if they were generated by
monopoles (Hunt et al., 2011).  Riera et al. (2012) seem to confirm
these preliminary observations.

In a commentary to this article (Destexhe and Bedard, 2012), several
possible physical mechanisms were proposed to account for electric
monopoles in neurons.  It was pointed out that the neuronal membrane
is very different from the electric circuit analogue of the membrane,
which is widely used for modeling neurons.  In particular, the
mobility of charges (ions) is several order of magnitude lower than
electrons, the charge carrier in an electronic circuit.  The "ionic
circuit", therefore, may not obey the well-known Kirchhoff's current
laws, because charges will take a significant time to start moving,
and an inward current will not be instantaneously balanced by an
outward current.  Therefore, transient charge accumulation may occur,
and will be seen transiently as electric monopoles.  This
interpretation has triggered some discussion, some authors arguing
that monopoles are not consistent with the usual formalism for
modeling neurons (Gratiy et al., 2013), while other authors pointed
out that electric monopoles are possible but they require to use
Maxwell equations without the usual approximations (Bedard and
Destexhe, 2013).

The consequences of electric monopoles are potentially very important.
Because monopolar sources predict a different attenuation ($1/r$)
compared to dipoles ($1/r^2$), it will evidently impact our
understanding of the genesis of LFP and EEG signals.  If a given
method to estimate neuronal activity from LFP or EEG assumes the wrong
distance dependence, then this method will provide wrong estimates.
It is tempting to relate this to the mismatch found by Dehghani et
al. (2010b) for dipole estimates from EEG and MEG.  If monopoles are
present, the source estimates from EEG will be greatly affected and
such estimation methods must be revised accordingly.

Future experimental work is necessary to confirm these observations of
strong monopolar components in neurons.  If confirmed, the presence of
electric monopoles in neurons will have devastating consequences, not
only to the interpretation of EEG/LFP or current source estimates, as
explained above, but it will also require theoreticians to refine the
current theories of electric phenomena in neurons.  It may be that we
have to profoundly revise our current models of neurons and population
of neurons in a near future.

= Conclusions =

The LFP is a very important variable because it is available in many
types of experimental recording configurations, ranging from glass
micropipettes to multi-electrode arrays.  Despite its widespread
availability, the LFP is under-used because its biophysical origins
remain incompletely understood.  In this contribution, we have reviewed
different properties of LFPs and different ways to model them.  We
reviewed simple models where the LFP is assumed to result from
punctual (or line) current sources in a homogeneous resistive medium,
as well as models taking into account more complex electrical
properties of the extracellular medium.

The simple LFP model, where neurons are embedded in a uniform and
resistive extracellular medium, is the one used in the vast majority
of studies, and is therefore today's "standard model".  This model
provides an acceptable simulation of the LFP in many cases (e.g., Rall
and Shepherd, 1968; Protopapas et al., 1998; Destexhe, 1998; Holt and
Koch, 1999; Nunez and Srinivasan, 2005; Gold et al., 2006; Milstein et
al., 2008; Pettersen and Einevoll, 2008; Linden et al., 2011).  The
procedures to implement this standard model are now available in
popular simulators such as NEURON or GENESIS.

The fact of using resistive extracellular media is justified by some
measurements (Logothetis et al., 2007), but it does not account for
other measurements suggesting that the extracellular medium is
non-ohmic (Gabriel et al., 1996; Bedard et al., 2010; Dehghani et al.,
2010a; Wagner et al., 2013).  But most importantly, the microstructure
of extracellular space shows that it is far from being uniform, and
rather consists of an aggregate of highly-conductive fluids with
low-conductive membranes.  There is therefore a serious possibility
that the medium is more complex than a simple resistor, although this
issue is still unclear today.

To account for such inhomogeneities, the standard model cannot be
used, because the underlying formalism is only valid in resistive
media.  One must restart from first principles (Maxwell equations) to
integrate electric parameters that depend on the spatial position in
extracellular space.  Such a formalism revealed that the
inhomogeneities of conductivity can give rise to strong filtering
properties of LFPs (Bedard et al., 2004). Such a model should be used
if one needs to account in detail for the attenuation of LFPs with
distance.  It was also shown that this model accounts for the
characteristics of the current-source density (CSD) analysis of the
slow oscillation in cerebral cortex (Bazhenov et al., 2011).  Despite
these important results, this model is too complex to apply in practice,
because it requires to know in detail the 3D arrangement of fluids and
membranes in the extracellular space around each neuron.

It must be noted that alternative mechanisms exist to explain the 
frequency filtering properties, in particular the very steep attenuation
of the extracellular LFP signal of spikes.  There can be a contribution
of the low-pass filtering of the membrane (the "morphology filtering"; 
see Pettersen and Einevoll, 2008).  This mechanism may explain a form of 
low-pass filtering, but seems incompatible with the $1/f$ scaling at 
low frequencies, and was also claimed to be negligible under ``in vivo''--like
conditions (Bedard et al., 2010).  There can also be a contribution due
to the fact that spikes may be quadrupoles (Milstein and Koch, 2008),
and therefore have a steeper attenuation.  Together with the filtering due 
to the extracellular medium, all these mechanisms contribute to filter the
high frequency components of the LFP.  The relative weight of these
different mechanisms should be quantified by future experiments.

Another modeling formalism, easier to implement in practice, describes 
the tissue at a larger coarse-graining, and in which the electric
parameters will be considered as an average over some volume of space,
and thus are the result of a mixture of different media such as fluids
and membranes.  This mean-field model of LFPs (Bedard and Destexhe,
2009, 2011) also presents the advantage that it can directly
incorporate macroscopic measurements of $\sigma$ and $\epsilon$ in
brain tissue (Gabriel et al., 1996).  This approach also can integrate
physical mechanisms such as ionic diffusion, which turns out to be the
one that explains various measurements including the $1/f$ scaling
properties of LFPs at low frequencies (Fig. 8).  This
mean-field approach can also be applied to generalize the CSD analysis
(which is only valid in resistive media) to extracellular media with
arbitrarily complex electrical properties (Bedard and Destexhe, 2011).
Such models can also integrate electric monopoles, and thus, we expect
that mean-field LFP models will play an increasingly important role in
the future.


= Footnotes =

#<div id="foot1">Intuitively, we can see that if a given medium is a low-pass filter because of inhomogeneities of $\sigma$ and $\epsilon$, then if we take the system at a more macroscopic level, it has to remain a low-pass filter.  The only way to keep this property is that $\sigma^M$ and $\epsilon^M$ are frequency dependent.</div>
#<div id="foot2">There is presently no other theory that can explain all the data in a coherent framework.</div>





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= Further Reading =

* Brette R. and A. Destexhe, editors. (2012) ''Handbook of Neural Activity Measurement'', Cambridge University Press, Cambridge, UK.

* Koch, C. and I. Segev, editors. (1998) ''Methods in Neuronal Modeling'' (2nd ed).  MIT Press, Cambridge MA.



= Internal links =

[[Electroencephalogram]]

[[Fast oscillations]]

[[General anaesthesia]]

[[High-conductance state]]

[[Neural fields]]

[[Spike-and-Wave oscillations]]

[[Thalamocortical oscillations]]

[[Up and Down states]]

[[Volume conduction]]