# A formalism for the long-distance magnetic field generated by populations of
neurons

Alexis O. García Rodríguez  Alain Destexhe  Paris-Saclay University, Institute
of Neuroscience, CNRS, Gif sur Yvette, France

###### Abstract

Brain activity can be measured using magnetic fields located at some distance
from the brain, a technique called magneto-encephalography (MEG). The origin
of such magnetic fields are the ionic currents involved in neuronal activity.
While these processes are well known at the microscopic scale, it is less
clear how large populations of neurons generate magnetic fields. Here, we
attempt to make such a link by deriving a mean-field theory of magnetic field
generation by populations of neurons. We use the concept of current dipoles to
build a theory that can be applied to populations of neurons. We also evaluate
the contributions of neuronal current sources such as the synaptic currents or
action potential currents. This theory should be useful to calculate the
magnetic field located at long distances from neuronal populations, such as in
MEG.

## Introduction

Magnetic field measurements constitute a very attractive method to measure
neuronal activity, in particular through magneto-encephalography (MEG), which
measures the magnetic field at some distance from the brain surface, as
reviewed previously [1, 2]. Contrary to the genesis of electric fields and
potentials, membrane currents do not contribute much to extracellular magnetic
fields, because they cancel due to the cylindrical shape of neuronal cables,
but that the main contribution to the extracellular magnetic field is the
axial current in neuronal dendrites [1]. It was shown previously that the
“microscopic” magnetic fields measured for muscle fibers using miniaturized
sensors, could be quantitatively matched by computational models taking into
account the axial currents [3].

Another difference between electric and magnetic fields is that the electric
potential around neurons crucially depends on the electric properties of the
extracellular medium, while magnetic fields are believed to have negligible
interactions with the extracellular medium [2]. Note that some studies have
found that the electric properties of extracellular media also affect magnetic
fields, because they influence the neuronal current sources that produce the
field [4]. We will assume here that there is no such dependence and will
consider neurons embedded in resistive (Ohmic) media, with no capacitive or
diffusive effects.

There exists well established theories to calculate the microscopic currents
generating extracellular electric potentials, such as the neuronal cable
theory [5]. Such theories can be formulated macroscopically, using a mean-
field formulation of Maxwell equations [6], yielding expressions for the
extracellular potential at scales larger than the variations of resistivity
and permittivity. Such theories can easily integrate the influence of multiple
neuronal current sources, and thus can be used to calculate the electric
potential resulting from populations of neurons.

However, how to account for “macroscopic” neuronal magnetic fields is less
clear. One of the reasons is that the vectorial nature of the magnetic field
will lead to partial cancellation of the fields generated by single neurons,
in a way that critically depends on their orientation. Taking advantage of the
fact that the principal neurons in cerebral cortex form electric dipoles,
which are arranged in parallel, we derive here a mean-field formulation to
account for macroscopic neuronal magnetic fields. We start from the notion of
current dipole, which we contrast with the notion of electric dipole, showing
that they constitute distinct concepts with their own definition of the dipole
moment. We next evaluate the contribution to different types of current
sources, such as the synaptic currents and action potential currents, to
generate the magnetic field located far away (centimenters) from neurons, such
as typically in MEG measurements.

##  I Current dipole

A current line segment can be seen as a current dipole. This is a combination
of a current sink and a current source of equal magnitude [7]. The current
enters at one end—a current sink—and leaves the segment at the other end—a
current source. Synaptic and action potential currents can be described as a
set of tiny current dipoles.

In the case of synaptic currents, we have a single current dipole at each
synapse. Here the current either flows into _or_ out of the cell. Transmitter
molecules in the synapse change the membrane permeability for specific ions.
In the excitatory (depolarization) case, $\displaystyle\mathrm{Na}^{+}$
channels open and current flows into the cell. For inhibitory
(hyperpolarization) synaptic currents, $\displaystyle\mathrm{K}^{+}$ or
$\displaystyle\mathrm{Cl}^{-}$ channels open and current flows out of the
cell.

Action potential currents, in turn, can be seen as two oppositely oriented
current dipoles forming a current quadrupole [1]. Here two currents flow in
opposite directions. $\displaystyle\mathrm{Na}^{+}$ ions go into the cell
causing the membrane to depolarize and $\displaystyle\mathrm{K}^{+}$ ions move
out of the cell causing repolarization. These currents produce a change in the
membrane potential that lasts about $\displaystyle
1\text{\,}\mathrm{m}\mathrm{s}$, a time scale $\displaystyle 10$ times smaller
than that of a postsynaptic potential lasting tens of
$\displaystyle\mathrm{m}\mathrm{s}$ [1].

The opposite currents creating an action potential produce magnetic fields
that tend to cancel each other. Hence a relatively small magnetic field,
compared to that of synaptic currents, is generated in that case. The brain
magnetic field 111More precisely, we mean here the magnitude of the magnetic
induction $\displaystyle{\bf\it B}$, called also magnetic flux density, which
is measured in MEG recordings. In a linear and isotropic medium with magnetic
permeability $\displaystyle\mu$ we have $\displaystyle{\bf\it H}={\bf\it
B}/\mu$, where $\displaystyle{\bf\it H}$ is the magnetic field is typically up
to $\displaystyle 5\text{\times}{10}^{-13}\text{\,}\mathrm{T}$ [1] which is
still very small compared to the earth magnetic field of about $\displaystyle
5\text{\times}{10}^{-5}\text{\,}\mathrm{T}$ [9]. Superconducting quantum
interference device (SQUID) detectors are employed to record the MEG signals.
These sensors work at liquid helium temperature (about $\displaystyle
4\text{\,}\mathrm{K}$=$\displaystyle-269\text{\,}\mathrm{\SIUnitSymbolCelsius}$)
and are placed some centimeters away from the scalp.

Our goal is to obtain the long-distance magnetic field produced by ionic
currents in small regions containing populations of neurons. The synaptic
currents in each region define an overall (time-dependent) current dipole. The
dimensions of each region must be small compared to the average distance from
the region to the measurement point. Considering this distance in the order of
centimeters, the region size can be thought in the order of millimeters. One
of those relatively small regions can then have a volume of about
$\displaystyle 1\text{\,}\mathrm{m}\mathrm{m}^{3}$. We can take the values of
the cortical surface, the average cortical thickness, and the total number of
neurons in the human cerebral cortex, given by about $\displaystyle
1.8\text{\times}{10}^{5}\text{\,}\mathrm{m}\mathrm{m}^{2}$, $\displaystyle
2.6\text{\,}\mathrm{m}\mathrm{m}$, and $\displaystyle 16$ billion neurons,
respectively [10]. We then find about $\displaystyle 4.7\text{\times}{10}^{5}$
regions of $\displaystyle 1\text{\,}\mathrm{m}\mathrm{m}^{3}$ in the human
cerebral cortex, each one with an average estimate of about $\displaystyle
3.4\text{\times}{10}^{4}$ neurons.

###  I.1 Long-distance magnetic field of a current line segment

The Biot–Savart law for a time-independent line current

${\bf\it B}({\bf\it r})=\frac{\mu}{4\pi}\,\mathcal{I}\int\differential{{\bf\it
l}^{\prime}}\crossproduct\frac{{\bf\it r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}^{3}},$ (1)

can be employed to estimate the magnetic induction $\displaystyle{\bf\it
B}_{seg}({\bf\it r})$ of a current line segment at a relatively long distance.
This leads to the concept of current dipole moment. The measurement point is
located at $\displaystyle{\bf\it r}$. We consider a linear and isotropic
medium with the same magnetic permeability $\displaystyle\mu$ throughout
space. $\displaystyle\mathcal{I}$ is the current and
$\displaystyle\differential{{\bf\it l}^{\prime}}$ is a vector element of
length in the current flow direction. $\displaystyle{\bf\it r}^{\prime}$
refers to a given point in the segment.

Given a long distance to the point of interest when compared to the size of
the current line segment, any point $\displaystyle{\bf\it r}_{0}^{\mathstrut}$
in it can be used to characterize its average position. We have that
$\displaystyle\vqty{{\bf\it r}^{\prime}-{\bf\it
r}_{0}^{\mathstrut}}\ll\vqty{{\bf\it r}-{\bf\it r}_{0}^{\mathstrut}}$ for
every point in the segment. The long-distance magnetic field can be estimated
by considering in Eq. (1) all $\displaystyle{\bf\it r}^{\prime}\approx{\bf\it
r}_{0}^{\mathstrut}$. We then have for the given segment

${\bf\it B}_{seg}({\bf\it
r})\approx\frac{\mu}{4\pi}\frac{{\bf\it\mathcal{Q}}_{seg}\crossproduct\hat{{\bf\it
R}}\pqty{{\bf\it r},{\bf\it r}_{0}^{\mathstrut}}}{\vqty{{\bf\it
R}\pqty{{\bf\it r},{\bf\it r}_{0}^{\mathstrut}}}^{2}},$ (2)

where

$\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it r}_{0}^{\mathstrut}}=\frac{{\bf\it
R}\pqty{{\bf\it r},{\bf\it r}_{0}^{\mathstrut}}}{\vqty{{\bf\it R}\pqty{{\bf\it
r},{\bf\it r}_{0}^{\mathstrut}}}},\quad{\bf\it R}\pqty{{\bf\it r},{\bf\it
r}_{0}^{\mathstrut}}={\bf\it r}-{\bf\it r}_{0}^{\mathstrut},$ (3)

and

${\bf\it\mathcal{Q}}_{seg}=\mathcal{I}\int\limits_{\mathcal{L}_{seg}}\differential{{\bf\it
l}^{\prime}}.$ (4)

Here $\displaystyle{\bf\it\mathcal{Q}}_{seg}$ is the current dipole moment of
the line segment $\displaystyle\mathcal{L}_{seg}$. Its magnitude is given by
the product of the current times the distance between the ends of the segment.
This is similar to an electric dipole moment which is given by the product of
the positive charge times the distance between the charges forming the dipole.

Eq. (2) shows a decrease with the square of the distance for the magnetic
field of a current dipole. This behavior can be expected for synaptic currents
due to their dipolar nature [1]. Below we reproduce that behavior and relate
the synaptic current dipole moment of a region to the currents driven by the
electric field.

Later we study the long-distance magnetic field for action potential currents.
It decreases faster—with the cube of the distance—than in the synaptic case.
This is characteristic for the quadrupolar case [1]. We derive such a decrease
by approximating to zero the current dipole moment in this case and then use
the _magnetic_ dipole moment for the action potential currents.

###  I.2 Current dipole moment for volume currents

The current dipole moment $\displaystyle{\bf\it\mathcal{Q}}_{reg}$ of a given
volume region $\displaystyle\mathcal{V}_{reg}$ can be expressed in terms of
the current density $\displaystyle{\bf\it\mathcal{J}}({\bf\it r}^{\prime})$ at
the different points $\displaystyle{\bf\it r}^{\prime}$ in the region.
Considering (4) we write

${\bf\it\mathcal{Q}}_{reg}=\int\limits_{\mathcal{V}_{reg}}\delta{\bf\it\mathcal{Q}},\quad\delta{\bf\it\mathcal{Q}}=\mathcal{I}^{\prime}\differential{{\bf\it
l}^{\prime}}=\mathcal{J}({\bf\it
r}^{\prime})\differential{s^{\prime}}\differential{{\bf\it l}^{\prime}},$ (5)

where $\displaystyle\differential{s^{\prime}}$ is a surface element
perpendicular to the current flow direction at point $\displaystyle{\bf\it
r}^{\prime}$. Then

$\delta{\bf\it\mathcal{Q}}=\mathcal{J}({\bf\it
r}^{\prime})\differential{s^{\prime}}\differential{{\bf\it
l}^{\prime}}={\bf\it\mathcal{J}}({\bf\it
r}^{\prime})\differential{s^{\prime}}\differential{l^{\prime}}={\bf\it\mathcal{J}}({\bf\it
r}^{\prime})\differential[3]{r^{\prime}},$ (6)

where $\displaystyle\differential[3]{r^{\prime}}$ is a volume element. The
current density $\displaystyle{\bf\it\mathcal{J}}({\bf\it r}^{\prime})$ is
therefore the density per unit volume, at point $\displaystyle{\bf\it
r}^{\prime}$, of the current dipole moment. We thus have

${\bf\it\mathcal{Q}}_{reg}=\int\limits_{\mathcal{V}_{reg}}{\bf\it\mathcal{J}}({\bf\it
r}^{\prime})\differential[3]{r^{\prime}}.$ (7)

##  II Biot–Savart law and the nonstatic case

###  II.1 The classic Biot-Savart law and its generalization

The Biot–Savart law for volume currents is

${\bf\it B}({\bf\it r})=\frac{\mu}{4\pi}\int{\bf\it J}({\bf\it
r}^{\prime})\crossproduct\frac{{\bf\it r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}^{3}}\differential[3]{r^{\prime}},$ (8)

where the integration is over all space and $\displaystyle{\bf\it J}$ is the
free current density which is assumed to be confined to a finite region. As
mentioned earlier, we consider a linear and isotropic medium with uniform
permeability $\displaystyle\mu$. Eq. (8) can be proved to be valid in the
static case where both $\displaystyle{\bf\it J}$ and the free charge density
$\displaystyle\rho$ are time-independent. It can be shown [9] that
$\displaystyle{\bf\it B}({\bf\it r})$ given by (8) satisfies the Ampère’s law
in differential form

$\curl{\bf\it B}({\bf\it r})=\mu{\bf\it J}({\bf\it r}),$ (9)

provided

$\divergence{\bf\it J}({\bf\it r})=0.$ (10)

This characterizes the magnetostatic case where $\displaystyle\rho$ is
independent of time $\displaystyle t$, i.e.,
$\displaystyle\partialderivative*{\rho}{t}=0$. The continuity equation
$\displaystyle\divergence{\bf\it J}+\partialderivative*{\rho}{t}=0$,
expressing charge conservation, becomes Eq. (10).

On the other hand, the brain tissue represents a nonstatic situation and the
use of the Biot–Savart law must be justified. Here the charge density, given
by the ion concentrations, depends on time at the different points in space.
As noted below, Eq. (8) still holds in the special nonstatic case where the
free charge density changes linearly with time [11]. The continuity equation
here implies that the free current density does not depend on time and the
same happens with $\displaystyle{\bf\it B}$ as shown by Eq. (8). However, MEG
measurements show time-varying magnetic signals necessarily corresponding to
time-varying currents in the brain. In this connection, a time-dependent
version of Eq. (8), given by

${\bf\it B}({\bf\it r},t)=\frac{\mu}{4\pi}\int{\bf\it J}({\bf\it
r}^{\prime},t)\crossproduct\frac{{\bf\it r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}^{3}}\differential[3]{r^{\prime}},$ (11)

could be proposed. Nevertheless, as noted also below, Eq. (11) is warranted in
the particular case where $\displaystyle{\bf\it J}({\bf\it r}^{\prime},t)$ is
linear in time [11]. Here $\displaystyle{\bf\it B}$, according to (11), is
also linear in time which neither corresponds to the situation in MEG where
oscillating magnetic signals are recorded.

Therefore the general nonstatic case, where $\displaystyle\rho$ and
$\displaystyle{\bf\it J}$ depend arbitrarily on time, must be considered.
Then, instead of (9), the Ampère–Maxwell equation

$\curl{\bf\it B}({\bf\it r},t)=\mu\bqty{{\bf\it J}({\bf\it
r},t)+\partialderivative{{\bf\it D}({\bf\it r},t)}{t}},$ (12)

must be satisfied. Here $\displaystyle{\bf\it D}$ is the vector of electric
displacement, such that

$\divergence{\bf\it D}({\bf\it r},t)=\rho({\bf\it r},t).$ (13)

The continuity equation

$\divergence{\bf\it J}({\bf\it r},t)+\partialderivative{\rho({\bf\it
r},t)}{t}=0,$ (14)

being a general statement of charge conservation, remains valid in this case.
It implies, using Eq. (13), that

$\divergence[{\bf\it J}({\bf\it r},t)+\partialderivative{{\bf\it D}({\bf\it
r},t)}{t}]=0,$ (15)

instead of Eq. (10). This can also be obtained directly from Eq. (12). The
terms inside the brackets form the generalized current density used previously
to include charge accumulation in cable equations [4, 6].

Aiming to obtain a time-dependent generalization of the Biot–Savart law, the
Maxwell’s equation $\displaystyle\divergence{\bf\it B}({\bf\it r},t)=0$ can be
considered. This allows us to introduce the vector potential
$\displaystyle{\bf\it A}({\bf\it r},t)$, so that

${\bf\it B}({\bf\it r},t)=\curl{\bf\it A}({\bf\it r},t).$ (16)

Substituting this into Eq. (12) results in a Biot–Savart-like expression
[[][.Thederivationgiveninthisreferenceforfreespacesimilarlyapplieswhenconsideringalinearandisotropicmediumwithuniformpermeability.]weber,
13], which reads

$\displaystyle{\bf\it B}({\bf\it r},t)=\frac{\mu}{4\pi}\int$
$\displaystyle\bqty{{\bf\it J}({\bf\it
r}^{\prime},t)+\partialderivative{{\bf\it D}({\bf\it r}^{\prime},t)}{t}}$ (17)
$\displaystyle\crossproduct\frac{{\bf\it r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}^{3}}\differential[3]{r^{\prime}}.$

This satisfies Eq. (12) by taking into account Eq. (15) and reduces to the
Biot–Savart law (8) in the static case.

From a practical point of view, however, Eq. (17) is not helpful as it turns
out to be self-referential [13, 11]. Eq. (17) requires the electric
displacement $\displaystyle{\bf\it D}$ to be known at all points of space.
Considering a linear and isotropic medium with permittivity
$\displaystyle\epsilon$, we have $\displaystyle{\bf\it D}=\epsilon{\bf\it E}$,
where $\displaystyle{\bf\it E}$ is the vector of electric field. The latter,
meanwhile, is determined by $\displaystyle{\bf\it B}$ through the Faraday’s
law of induction, $\displaystyle\curl{\bf\it E}({\bf\it
r},t)=-\partialderivative*{{\bf\it B}({\bf\it r},t)}{t}$. Therefore, in order
to evaluate $\displaystyle{\bf\it B}$ by employing Eq. (17), we must know
$\displaystyle{\bf\it B}$ itself at all points of space.

Eq. (17) cannot thus be actually used to obtain $\displaystyle{\bf\it B}$ for
time-dependent systems. It requires to know the displacement current density
$\displaystyle\partialderivative*{{\bf\it D}}{t}$ which has
$\displaystyle{\bf\it B}$ as its own source [13]. We employ below the proper
general solution given to this problem.

###  II.2 Jefimenko’s equation: The general nonstatic case

A suitable time-dependent generalization of the Biot–Savart law was derived by
Jefimenko [[][;2nded.(ElectretScientific, StarCity, 1989).]jefimenkobook, 13,
9], given by

$\displaystyle{\bf\it B}({\bf\it r},t)=\frac{\mu}{4\pi}\int$
$\displaystyle\bqty{{\bf\it J}({\bf\it
r}^{\prime},t_{ret})+\frac{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}}{v}\partialderivative{{\bf\it J}({\bf\it
r}^{\prime},t_{ret})}{t}}$ (18) $\displaystyle\crossproduct\frac{{\bf\it
r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}^{3}}\differential[3]{r^{\prime}}.$

An infinite, linear and isotropic medium with uniform permittivity
$\displaystyle\epsilon$ and permeability $\displaystyle\mu$ is assumed. Here

$t_{ret}=t-\vqty{{\bf\it r}-{\bf\it r}^{\prime}}/v,$ (19)

is the retarded time, and

$v=1/\sqrt{\epsilon\mu},$ (20)

is the velocity of light in the medium. Eq. (18) can be obtained from (16)
with

${\bf\it A}({\bf\it r},t)=\frac{\mu}{4\pi}\int\frac{{\bf\it J}({\bf\it
r}^{\prime},t_{ret})}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}}\differential[3]{r^{\prime}}.$ (21)

The current sources in the vector potential must thus be evaluated at the
retarded time (19). The status of the sources at this time determines
$\displaystyle{\bf\it B}$ at the later time $\displaystyle t$. It takes a time
$\displaystyle\vqty{{\bf\it r}-{\bf\it r}^{\prime}}/v$ for an electromagnetic
signal to travel from a source at point $\displaystyle{\bf\it r}^{\prime}$ to
the point $\displaystyle{\bf\it r}$ where $\displaystyle{\bf\it B}$ is
measured. In this sense, Eq. (18) is a causality-based general solution of
Maxwell’s equations.

Eq. (18) simplifies to the static form (8) of the Biot–Savart law in the case
of a time-independent current density. This does not require the free charge
density to be time-independent as well but, according to the continuity
equation (14), it may change linearly with time [11]. It can also be verified
that Eq. (18) takes the non-retarded form (11) when the current density is
linear in time [11], that is, when $\displaystyle{\bf\it J}({\bf\it
r}^{\prime},t)={\bf\it J}({\bf\it r}^{\prime},0)+t\mkern 1.0mu{\bf\it
C}({\bf\it r}^{\prime})$, where $\displaystyle{\bf\it C}({\bf\it r}^{\prime})$
is a time-independent function.

###  II.3 Magneto-quasistatic approximation

It is worth considering the Jefimenko’s equation (18) in the quasistatic case
where the current density $\displaystyle{\bf\it J}({\bf\it r}^{\prime},t)$, as
well as the corresponding charge density, changes relatively slowly with time
[11]. The characteristic time for this change to occur must be large compared
to the signal traveling time $\displaystyle t_{trav}({\bf\it r},{\bf\it
r}^{\prime})=\vqty{{\bf\it r}-{\bf\it r}^{\prime}}/v$. Considering (20) we
have

$t_{trav}({\bf\it r},{\bf\it r}^{\prime})=\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}\sqrt{\epsilon\mu}.$ (22)

Similarly to the static case, the current density $\displaystyle{\bf\it J}$ in
Eq. (18) is assumed to be confined to a finite region of space. The source
point $\displaystyle\bar{{\bf\it r}}^{\prime}$ which is farthest from the
field point $\displaystyle{\bf\it r}$ determines the largest traveling time
$\displaystyle t_{trav}({\bf\it r},\bar{{\bf\it r}}^{\prime})$.

In the quasistatic case it is useful to expand the current density
$\displaystyle{\bf\it J}({\bf\it r}^{\prime},t_{ret})={\bf\it J}\bqty{{\bf\it
r}^{\prime},t-t_{trav}({\bf\it r},{\bf\it r}^{\prime})}$ in Eq. (18) as a
Taylor’s series, so that

$\displaystyle{\bf\it J}({\bf\it r}^{\prime},t_{ret})$ $\displaystyle={\bf\it
J}({\bf\it r}^{\prime},t)-t_{trav}({\bf\it r},{\bf\it
r}^{\prime})\partialderivative{{\bf\it J}({\bf\it r}^{\prime},t)}{t}$ (23)
$\displaystyle\quad+\frac{1}{2}t_{trav}^{2}({\bf\it r},{\bf\it
r}^{\prime})\partialderivative[2]{{\bf\it J}({\bf\it
r}^{\prime},t)}{t}+\cdots.$

This leads to

$\displaystyle{\bf\it B}({\bf\it r},t)=\frac{\mu}{4\pi}\int$
$\displaystyle\left[{\bf\it J}({\bf\it
r}^{\prime},t)-\frac{1}{2}t_{trav}^{2}({\bf\it r},{\bf\it
r}^{\prime})\partialderivative[2]{{\bf\it J}({\bf\it
r}^{\prime},t)}{t}\right.$ (24)
$\displaystyle\left.\vphantom{\partialderivative[2]{{\bf\it J}({\bf\it
r}^{\prime},t)}{t}}+\cdots\right]\crossproduct\frac{{\bf\it r}-{\bf\it
r}^{\prime}}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}^{3}}\differential[3]{r^{\prime}}.$

The time dependency of the current density, and correspondingly that of
$\displaystyle{\bf\it B}$, can be described through the different frequency
components. A component of angular frequency $\displaystyle w$ corresponds to
a time oscillation with period $\displaystyle T(w)=2\pi/w$. Here it is helpful
to notice that the main contributions in neuromagnetism are from frequencies
below a typical value $\displaystyle f_{max}=$\displaystyle
100\text{\,}\mathrm{H}\mathrm{z}$$ [1]. We thus consider a maximum angular
frequency

$w_{max}=2\pi f_{max},$ (25)

and write

${\bf\it J}({\bf\it
r}^{\prime},t)\approx\int\limits_{0}^{w_{max}}e^{iwt}{\widetilde{{\bf\it
J}\mkern 6.0mu}\mkern-6.0mu}{}({\bf\it r}^{\prime},w)\differential w.$ (26)

Substituting this into Eq. (24) yields

${\bf\it B}({\bf\it
r},t)\approx\int\limits_{0}^{w_{max}}e^{iwt}{\widetilde{{\bf\it B}\mkern
4.0mu}\mkern-4.0mu}{}({\bf\it r},w)\differential w,$ (27)

where

$\displaystyle{\widetilde{{\bf\it B}\mkern 4.0mu}\mkern-4.0mu}{}({\bf\it
r},w)=\frac{\mu}{4\pi}\int$ $\displaystyle\left[{\widetilde{{\bf\it J}\mkern
6.0mu}\mkern-6.0mu}{}({\bf\it
r}^{\prime},w)+\frac{1}{2}w^{2}t_{trav}^{2}({\bf\it r},{\bf\it
r}^{\prime})\right.$ (28)
$\displaystyle\left.\vphantom{\frac{1}{2}}\times{\widetilde{{\bf\it J}\mkern
6.0mu}\mkern-6.0mu}{}({\bf\it r}^{\prime},w)+\cdots\right]$
$\displaystyle\crossproduct\frac{{\bf\it r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}^{3}}\differential[3]{r^{\prime}}.$

In the quasistatic case we must have at every source point
$\displaystyle{\bf\it r}^{\prime}$

$w^{2}t_{trav}^{2}({\bf\it r},{\bf\it r}^{\prime})\ll 1,\mbox{\quad for
all\quad}0<w<w_{max}.$ (29)

Since $\displaystyle wt_{trav}({\bf\it r},{\bf\it r}^{\prime})\leqslant
w_{max}t_{trav}({\bf\it r},\bar{{\bf\it r}}^{\prime})$, it is enough to
require that

$w_{max}^{2}t_{trav}^{2}({\bf\it r},\bar{{\bf\it r}}^{\prime})\ll 1.$ (30)

This describes the quasistatic case through the current frequency component
with the shortest oscillation period $\displaystyle 2\pi/w_{max}=1/f_{max}$.
This time must be large enough compared to the largest traveling time
$\displaystyle t_{trav}({\bf\it r},\bar{{\bf\it r}}^{\prime})$. In that case,
Eq. (28) can be replaced by

${\widetilde{{\bf\it B}\mkern 4.0mu}\mkern-4.0mu}{}({\bf\it
r},w)\approx\frac{\mu}{4\pi}\int{\widetilde{{\bf\it J}\mkern
6.0mu}\mkern-6.0mu}{}({\bf\it r}^{\prime},w)\crossproduct\frac{{\bf\it
r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}^{3}}\differential[3]{r^{\prime}},$ (31)

for all $\displaystyle 0<w<w_{max}$. Using (27) and (26), we then have

${\bf\it B}({\bf\it r},t)\approx\frac{\mu}{4\pi}\int{\bf\it J}({\bf\it
r}^{\prime},t)\crossproduct\frac{{\bf\it r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}^{3}}\differential[3]{r^{\prime}}.$ (32)

This is similar to the static form (8) of the Biot–Savart law. Expression (32)
is the quasistatic approximation of the general solution (18) of Maxwell’s
equations. It is valid provided (30) is satisfied.

We note that a similar expression to (30) is well known for the magneto-
quasistatic approximation of Maxwell’s equations
[[][;seealsosupplementarymaterial.ThediscussionhereisbasedonJ.R.Melcher,
\emph{ContinuumElectromechanics}(MITPress, Cambridge, 1981).]wagner, 16], as
well as their electro-quasistatic approximation [15, 16, 17, *gratiyadd]. In
these cases, however, instead of the traveling distance
$\displaystyle\vqty{{\bf\it r}-\bar{{\bf\it r}}^{\prime}}$ in Eq. (22), from
the most distant source point $\displaystyle\bar{{\bf\it r}}^{\prime}$ to the
field point $\displaystyle{\bf\it r}$, one typical or characteristic length
scale of the medium is employed. The system dimensions are assumed to be all
within a factor of two or so of each other [16]. The typical length is then
used to normalize [15] or approximate [16] all spatial derivatives in
Maxwell’s equations. In gray matter, the typical length is approximated by the
cortical thickness [15, 17]. The approaches in Refs. 15, 16, 17 are not based
on the general solution of Maxwell’s equations. This is given in the magnetic
case by Eq. (18). As shown above, this leads to expression (30) and therefore,
to the traveling distance $\displaystyle\vqty{{\bf\it r}-\bar{{\bf\it
r}}^{\prime}}$ as the appropriate length to define the quasistatic dynamics.

###  II.4 The quasistatic case of the brain

In the following we check whether the requirement (30) is fulfilled in the
specific case of the brain. According to Eqs. (25) and (22), we must have

$(2\pi f_{max})^{2}\vqty{{\bf\it r}-\bar{{\bf\it
r}}^{\prime}}^{2}\epsilon\mu\ll 1.$ (33)

As mentioned above, we typically have $\displaystyle f_{max}=$\displaystyle
100\text{\,}\mathrm{H}\mathrm{z}$$ [1]. We consider the distance
$\displaystyle\vqty{{\bf\it r}-\bar{{\bf\it r}}^{\prime}}$ mainly given by the
average size of the human brain of about $\displaystyle
15\text{\,}\mathrm{c}\mathrm{m}$. This is much larger than the cortical
thickness of the order of $\displaystyle 1\text{\,}\mathrm{m}\mathrm{m}$. The
distance of about few centimeters between sensors and the scalp is also
relatively small. We then take $\displaystyle\vqty{{\bf\it r}-\bar{{\bf\it
r}}^{\prime}}\approx$\displaystyle 0.1\text{\,}\mathrm{m}$$. With regard to
the permittivity, we have
$\displaystyle\epsilon\approx$\displaystyle{10}^{7}$\epsilon_{0}^{\mathstrut}$
for gray matter at frequencies below $\displaystyle
100\text{\,}\mathrm{H}\mathrm{z}$ [15, 19]. Here
$\displaystyle\epsilon_{0}^{\mathstrut}\approx$\displaystyle
8.85\text{\times}{10}^{-12}\text{\,}\mathrm{C}^{2}\mathrm{/}\mathrm{(}\mathrm{N}\,\mathrm{m}^{2}\mathrm{)}$$
is the permittivity of free space. A magnetic permeability close to that of
free space is typical for biological tissues [15]. We then assume
$\displaystyle\mu\approx\mu_{0}^{\mathstrut}\approx$\displaystyle
1.26\text{\times}{10}^{-6}\text{\,}\mathrm{N}\mathrm{/}\mathrm{A}^{2}$$, and
obtain 222It can be seen that using $\displaystyle f_{max}=$\displaystyle
10\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and a characteristic length of
$\displaystyle 1\text{\,}\mathrm{m}\mathrm{m}$, corresponding to the order of
the cortical thickness, yields also the value $\displaystyle
4\text{\times}{10}^{-7}$ for the electro-quasistatic approximation in Ref. 17

$(2\pi f_{max})^{2}\vqty{{\bf\it r}-\bar{{\bf\it
r}}^{\prime}}^{2}\epsilon\mu\approx$\displaystyle 4\text{\times}{10}^{-7}$\ll
1.$ (34)

The condition (30) is thus very well satisfied, indicating a quasistatic
dynamics. Therefore, we take (32) as a valid expression to calculate
$\displaystyle{\bf\it B}({\bf\it r},t)$.

In closing this section, we discuss about a condition which is different from
(30) and that has been proposed in Ref. 1 to justify the validity of
expression (32) in the quasistatic case. The analysis in Ref. 1 is based on
the Ampère–Maxwell equation, given here by Eq. (12). The frequency components
of the displacement term $\displaystyle\partialderivative*{{\bf\it D}({\bf\it
r},t)}{t}$ are compared to those of the ohmic part of the current density
$\displaystyle{\bf\it J}({\bf\it r},t)$. They are given, respectively, by
$\displaystyle iw\epsilon{\widetilde{{\bf\it E}\mkern
4.0mu}\mkern-4.0mu}{}({\bf\it r},w)$ and
$\displaystyle\sigma{\widetilde{{\bf\it E}\mkern 4.0mu}\mkern-4.0mu}{}({\bf\it
r},w)$, where $\displaystyle{\widetilde{{\bf\it E}\mkern
4.0mu}\mkern-4.0mu}{}({\bf\it r},w)$ is the frequency component of the
electric field, and $\displaystyle\sigma$ is the electrical conductivity of
the medium. The ratio between the absolute values of those frequency
components leads then to the condition

$w_{max}\tau_{MW}^{\mathstrut}\ll 1,$ (35)

where $\displaystyle\tau_{MW}^{\mathstrut}=\epsilon/\sigma$ is the
Maxwell–Wagner reaction time.

According to (35), the shortest oscillation period $\displaystyle
2\pi/w_{max}=1/f_{max}$ must be large enough compared to the Maxwell–Wagner
time. If that is the case, then the displacement term can be neglected and the
Ampère–Maxwell equation can be replaced by $\displaystyle\curl{\bf\it
B}({\bf\it r},t)\approx\mu{\bf\it J}({\bf\it r},t)$. This is similar to the
Ampère’s law (9) describing the static case and results in the expression
(32), as can be seen from Eq. (17).

The expressions (35) and (30) are clearly different from one another. In the
first place, one is linear while the other is quadratic and secondly, they
define the quasistatic dynamics in terms of essentially different times. These
are the Maxwell–Wagner reaction time and the traveling time which also differ
quantitatively. Using
$\displaystyle\epsilon\approx$\displaystyle{10}^{7}$\epsilon_{0}^{\mathstrut}$
as before [15, 19], and $\displaystyle\sigma\approx$\displaystyle
0.3\text{\,}\mathrm{\SIUnitSymbolOhm}^{-1}\,\mathrm{m}^{-1}$$ for gray matter
at frequencies below $\displaystyle 100\text{\,}\mathrm{H}\mathrm{z}$ [15, 1],
we have $\displaystyle\tau_{MW}^{\mathstrut}\approx$\displaystyle
3\text{\times}{10}^{-4}\text{\,}\mathrm{s}$$. Taking
$\displaystyle\vqty{{\bf\it r}-\bar{{\bf\it r}}^{\prime}}\approx$\displaystyle
0.1\text{\,}\mathrm{m}$$,
$\displaystyle\epsilon\approx$\displaystyle{10}^{7}$\epsilon_{0}^{\mathstrut}$,
and $\displaystyle\mu\approx\mu_{0}^{\mathstrut}$, we have $\displaystyle
t_{trav}({\bf\it r},\bar{{\bf\it r}}^{\prime})=\vqty{{\bf\it r}-\bar{{\bf\it
r}}^{\prime}}\sqrt{\epsilon\mu}\approx$\displaystyle
1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. The expression (35) thus requires a
relatively slower dynamics than in the case of expression (30).

Now we evaluate the fulfillment of the requirement (35). We use the typical
value $\displaystyle f_{max}=$\displaystyle 100\text{\,}\mathrm{H}\mathrm{z}$$
[1] and $\displaystyle\tau_{MW}^{\mathstrut}\approx$\displaystyle
3\text{\times}{10}^{-4}\text{\,}\mathrm{s}$$ which was obtained before. This
yields

$w_{max}\tau_{MW}^{\mathstrut}=2\pi
f_{max}\tau_{MW}^{\mathstrut}\approx$\displaystyle 0.2$.$ (36)

The condition (35) is thus minimally satisfied. We note that a permittivity
$\displaystyle\epsilon\approx$\displaystyle{10}^{5}$\epsilon_{0}^{\mathstrut}$
was considered in Ref. 1 giving $\displaystyle
w_{max}\tau_{MW}^{\mathstrut}\approx$\displaystyle 2\text{\times}{10}^{-3}$\ll
1$. However, according to Refs. 15 and 19,
$\displaystyle$\displaystyle{10}^{5}$\epsilon_{0}^{\mathstrut}$ is an
underestimated value of the permittivity for gray matter at frequencies below
$\displaystyle 100\text{\,}\mathrm{H}\mathrm{z}$.

The fact that the condition (35) is minimally fulfilled indicates that it is
not a very good approximation to neglect the displacement term in the
Ampère–Maxwell equation. That is the case according to typical values of
frequency, permittivity and conductivity in the brain. Neglecting the
displacement term is not a safe option and the general solution of Maxwell’s
equations, given above by Eq. (18), must be considered.

The expression (35), which corresponds to neglecting the displacement term in
the Ampère–Maxwell equation, is not a necessary condition to justify the
validity of expression (32) in the quasistatic case. As pointed out after Eq.
(17), the electric displacement $\displaystyle{\bf\it D}=\epsilon{\bf\it E}$
depends on $\displaystyle{\bf\it B}$ itself through the Faraday’s law of
induction, $\displaystyle\curl{\bf\it E}({\bf\it
r},t)=-\partialderivative*{{\bf\it B}({\bf\it r},t)}{t}$. Therefore the
displacement current density $\displaystyle\partialderivative*{{\bf\it D}}{t}$
must not be considered as a source of $\displaystyle{\bf\it B}$ in the
conventional sense [13]. The time-dependent generalization of the Biot–Savart
law, Eq. (18), does not contain indeed the displacement current density but
only $\displaystyle{\bf\it J}$. Eq. (18) introduces the signal traveling time
$\displaystyle\vqty{{\bf\it r}-{\bf\it r}^{\prime}}/v$ in terms of which the
quasistatic dynamics can be defined. Correspondingly, the Maxwell–Wagner time
is suitable for a quasistatic approximation of the Ampère–Maxwell equation but
not of the generalized Biot–Savart law (18).

##  III Magnetic field of the different currents

According to the previous section, we take (32) as a valid expression to
calculate $\displaystyle{\bf\it B}({\bf\it r},t)$. Considering the typical
value $\displaystyle\mu\approx\mu_{0}^{\mathstrut}$ [15], it becomes

${\bf\it B}({\bf\it r},t)\approx\frac{\mu_{0}^{\mathstrut}}{4\pi}\int{\bf\it
J}({\bf\it r}^{\prime},t)\crossproduct\frac{{\bf\it r}-{\bf\it
r}^{\prime}}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}^{3}}\differential[3]{r^{\prime}}.$ (37)

The current density $\displaystyle{\bf\it J}$ can be written as the sum of two
components [21]. First, the conduction current density

${\bf\it J}^{c}({\bf\it r},t)=\sigma({\bf\it r},t){\bf\it E}({\bf\it r},t),$
(38)

which corresponds to the current caused by the electric field
$\displaystyle{\bf\it E}$. This is produced by the charge carriers in the
conducting medium. In the electro-quasistatic approximation [17] we have
$\displaystyle\curl{\bf\it E}({\bf\it r},t)=-\partialderivative*{{\bf\it
B}({\bf\it r},t)}{t}\approx\mathbf{0}$. This allows to use the scalar
potential $\displaystyle\Phi({\bf\it r},t)$ so that $\displaystyle{\bf\it
E}({\bf\it r},t)\approx-\gradient\Phi({\bf\it r},t)$. Therefore we have

${\bf\it J}^{c}({\bf\it r},t)\approx-\sigma({\bf\it r},t)\gradient\Phi({\bf\it
r},t).$ (39)

Here the medium is described as a nonhomogeneous conductor with conductivity
$\displaystyle\sigma({\bf\it r},t)$ which is modeled on cellular scale. On the
same microscopic, micrometer scale, we have a second component of the current
density corresponding to the impressed currents [22] and represented by
$\displaystyle{\bf\it J}^{i}({\bf\it r},t)$. This stands for the current
density which is not produced by the electric field $\displaystyle{\bf\it E}$
but mainly, by gradients of concentration in diffusion processes and
electromotive chemical reactions in the cell pump mechanism. Then

${\bf\it J}({\bf\it r},t)={\bf\it J}^{c}({\bf\it r},t)+{\bf\it J}^{i}({\bf\it
r},t),$ (40)

with the conduction current density $\displaystyle{\bf\it J}^{c}({\bf\it
r},t)$ given by (39).

We assume the impressed current density mainly given by a current density
$\displaystyle{\bf\it J}^{s}({\bf\it r},t)$ corresponding to synaptic
currents, and a current density $\displaystyle{\bf\it J}^{a}({\bf\it r},t)$
corresponding to action potential currents. That is, we have

${\bf\it J}^{i}({\bf\it r},t)\approx{\bf\it J}^{s}({\bf\it r},t)+{\bf\it
J}^{a}({\bf\it r},t).$ (41)

Thus

${\bf\it J}({\bf\it r},t)\approx{\bf\it J}^{c}({\bf\it r},t)+{\bf\it
J}^{s}({\bf\it r},t)+{\bf\it J}^{a}({\bf\it r},t),$ (42)

and the expression (37) can be written in the form

${\bf\it B}({\bf\it r},t)\approx{\bf\it B}^{c}({\bf\it r},t)+{\bf\it
B}^{s}({\bf\it r},t)+{\bf\it B}^{a}({\bf\it r},t),$ (43)

where

${\bf\it B}^{c}({\bf\it r},t)=\frac{\mu_{0}^{\mathstrut}}{4\pi}\int{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\crossproduct\frac{{\bf\it r}-{\bf\it
r}^{\prime}}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}^{3}}\differential[3]{r^{\prime}},$ (44) ${\bf\it B}^{s}({\bf\it
r},t)=\frac{\mu_{0}^{\mathstrut}}{4\pi}\int{\bf\it J}^{s}({\bf\it
r}^{\prime},t)\crossproduct\frac{{\bf\it r}-{\bf\it r}^{\prime}}{\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}^{3}}\differential[3]{r^{\prime}},$ (45)

and

${\bf\it B}^{a}({\bf\it r},t)=\frac{\mu_{0}^{\mathstrut}}{4\pi}\int{\bf\it
J}^{a}({\bf\it r}^{\prime},t)\crossproduct\frac{{\bf\it r}-{\bf\it
r}^{\prime}}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}^{3}}\differential[3]{r^{\prime}},$ (46)

correspond, respectively, to the conduction, synaptic and action potential
currents.

We note that the case of conduction currents, given by expressions (44) and
(39), has been solved exactly [23]. Next we look at the synaptic and action
potential cases.

###  III.1 Long-distance magnetic field of synaptic currents

We estimate the synaptic contribution (45) by considering several relatively
small regions in the medium, such that

${\bf\it B}^{s}({\bf\it r},t)=\sum_{reg}{\bf\it B}_{reg}^{s}({\bf\it r},t),$
(47)

where

${\bf\it B}_{reg}^{s}({\bf\it
r},t)=\frac{\mu_{0}^{\mathstrut}}{4\pi}\int\limits_{\mathcal{V}_{reg}}{\bf\it
J}^{s}({\bf\it r}^{\prime},t)\crossproduct\frac{{\bf\it r}-{\bf\it
r}^{\prime}}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}^{3}}\differential[3]{r^{\prime}},$ (48)

corresponds to a region of volume $\displaystyle\mathcal{V}_{reg}$. The
dimensions of each region must be small compared to the average distance from
the region to the field point $\displaystyle{\bf\it r}$. Taking any point
$\displaystyle{\bf\it r}_{reg}^{\mathstrut}$ in the region, we must have

$\vqty{{\bf\it r}^{\prime}-{\bf\it r}_{reg}^{\mathstrut}}\ll\vqty{{\bf\it
r}-{\bf\it r}_{reg}^{\mathstrut}},$ (49)

for every region point $\displaystyle{\bf\it r}^{\prime}$.

Eq. (48) can be written in the form

${\bf\it B}_{reg}^{s}({\bf\it r},t)=\curl{\bf\it A}_{reg}^{s}({\bf\it r},t),$
(50)

with the vector potential

${\bf\it A}_{reg}^{s}({\bf\it
r},t)=\frac{\mu_{0}^{\mathstrut}}{4\pi}\int\limits_{\mathcal{V}_{reg}}\frac{{\bf\it
J}^{s}({\bf\it r}^{\prime},t)}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}}\differential[3]{r^{\prime}}.$ (51)

We expand the factor $\displaystyle 1/\vqty{{\bf\it r}-{\bf\it r}^{\prime}}$
around $\displaystyle{\bf\it r}^{\prime}={\bf\it r}_{reg}^{\mathstrut}$. Up to
the first order term, we have

$\displaystyle\frac{1}{\vqty{{\bf\it r}-{\bf\it r}^{\prime}}}$
$\displaystyle\approx\frac{1}{\vqty{{\bf\it r}-{\bf\it
r}_{reg}^{\mathstrut}}}$ (52)
$\displaystyle\quad+\evaluated{\gradient^{\prime}\frac{1}{\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}}}_{{\bf\it r}^{\prime}={\bf\it
r}_{reg}^{\mathstrut}}\dotproduct\pqty{{\bf\it r}^{\prime}-{\bf\it
r}_{reg}^{\mathstrut}},$

and therefore

$\displaystyle{\bf\it A}_{reg}^{s}({\bf\it r},t)$
$\displaystyle\approx\frac{\mu_{0}^{\mathstrut}}{4\pi}\frac{1}{\vqty{{\bf\it
r}-{\bf\it r}_{reg}^{\mathstrut}}}\int\limits_{\mathcal{V}_{reg}}{\bf\it
J}^{s}({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}$ (53)
$\displaystyle\quad+\frac{\mu_{0}^{\mathstrut}}{4\pi\vqty{{\bf\it r}-{\bf\it
r}_{reg}^{\mathstrut}}^{3}}\int\limits_{\mathcal{V}_{reg}}{\bf\it
J}^{s}({\bf\it r}^{\prime},t)$ $\displaystyle\qquad\times\bqty{\pqty{{\bf\it
r}-{\bf\it r}_{reg}^{\mathstrut}}\dotproduct\pqty{{\bf\it r}^{\prime}-{\bf\it
r}_{reg}^{\mathstrut}}}\differential[3]{r^{\prime}}.$

Similarly to Eq. (7), the quantity

${\bf\it Q}_{reg}^{s}(t)=\int\limits_{\mathcal{V}_{reg}}{\bf\it J}^{s}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$ (54)

in the first term of expression (53), defines the synaptic current dipole
moment. Assuming this is nonzero vector, we approximate the vector potential
by

${\bf\it A}_{reg}^{s}({\bf\it
r},t)\approx\frac{\mu_{0}^{\mathstrut}}{4\pi}\frac{{\bf\it
Q}_{reg}^{s}(t)}{\vqty{{\bf\it r}-{\bf\it r}_{reg}^{\mathstrut}}}.$ (55)

Using then Eq. (50), we have

${\bf\it B}_{reg}^{s}({\bf\it
r},t)\approx\frac{\mu_{0}^{\mathstrut}}{4\pi}\frac{{\bf\it
Q}_{reg}^{s}(t)\crossproduct\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}}{\vqty{{\bf\it R}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}}^{2}},$ (56)

where

$\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it r}_{reg}^{\mathstrut}}=\frac{{\bf\it
R}\pqty{{\bf\it r},{\bf\it r}_{reg}^{\mathstrut}}}{\vqty{{\bf\it
R}\pqty{{\bf\it r},{\bf\it r}_{reg}^{\mathstrut}}}},$ (57)

with

${\bf\it R}\pqty{{\bf\it r},{\bf\it r}_{reg}^{\mathstrut}}={\bf\it r}-{\bf\it
r}_{reg}^{\mathstrut}.$ (58)

Expression (56) shows a decrease with the square of the distance for synaptic
currents which is characteristic for the dipolar case [1].

###  III.2 Synaptic current dipole moment

We must find the synaptic current dipole moment, $\displaystyle{\bf\it
Q}_{reg}^{s}(t)$, defined by (54). Integrating the expression (42) of the
current density yields

${\bf\it Q}_{reg}(t)\approx{\bf\it Q}_{reg}^{c}(t)+{\bf\it
Q}_{reg}^{s}(t)+{\bf\it Q}_{reg}^{a}(t),$ (59)

where

${\bf\it Q}_{reg}(t)=\int\limits_{\mathcal{V}_{reg}}{\bf\it J}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$ (60)

is the total current dipole moment of the region. Correspondingly

${\bf\it Q}_{reg}^{c}(t)=\int\limits_{\mathcal{V}_{reg}}{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$ (61)

is the conduction current dipole moment, and

${\bf\it Q}_{reg}^{a}(t)=\int\limits_{\mathcal{V}_{reg}}{\bf\it J}^{a}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$ (62)

is the current dipole moment for action potential currents. We consider an
approximately cylindrical shape for the neuronal cables. Then we have opposite
transverse currents on opposite sides of the cables whose contributions, in
the current dipole moment (62) for action potential currents, approximately
cancel each other. We then take

${\bf\it Q}_{reg}^{a}(t)\approx\mathbf{0}.$ (63)

Thus, from expression (59) we have

${\bf\it Q}_{reg}^{s}(t)\approx{\bf\it Q}_{reg}(t)-{\bf\it Q}_{reg}^{c}(t).$
(64)

We assume quasistatic conditions such that, based on the continuity equation,
$\displaystyle\gradient^{\prime}\dotproduct{\bf\it J}({\bf\it
r}^{\prime},t)=-\partialderivative*{\rho({\bf\it r}^{\prime},t)}{t}\approx 0$
[21]. Then, provided the contribution of the impressed currents at the region
boundary is small compared to that of the conduction currents 333Specifically,
as can be seen from (93), this contribution of the impressed currents at the
surface $\displaystyle\mathcal{S}_{reg}$ bounding the region, is given by the
components of $\displaystyle\int\limits_{\mathcal{S}_{reg}}{\bf\it
r}^{\prime}\bqty{{\bf\it J}^{i}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}}$. This term
should be added to expression (65) and its equivalent (67) in a broader
approach, the total current dipole moment (60) is given by

${\bf\it Q}_{reg}(t)\approx\int\limits_{\mathcal{S}_{reg}}{\bf\it
r}^{\prime}\bqty{{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}},$ (65)

as shown in the Appendix. Here $\displaystyle\mathcal{S}_{reg}$ is the surface
bounding the region. Substituting (65) into (64) gives

${\bf\it Q}_{reg}^{s}(t)\approx\int\limits_{\mathcal{S}_{reg}}{\bf\it
r}^{\prime}\bqty{{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}}-{\bf\it
Q}_{reg}^{c}(t).$ (66)

Equivalently (see the Appendix), expression (65) can be written in the form

${\bf\it Q}_{reg}(t)\approx\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}+{\bf\it Q}_{reg}^{c}(t),$ (67)

which can also be obtained independently by integrating Eq. (40) of the
current density (see the Appendix). Importantly, (65) and (67) are independent
of the chosen coordinate origin if, and only if, the net impressed current
crossing the region boundary is approximately zero (see the Appendix). Using
(64) and (67) we obtain

${\bf\it Q}_{reg}^{s}(t)\approx\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}.$ (68)

We thus have two equivalent expressions, (66) and (68), of the synaptic
current dipole moment. We observe that they are given in terms of the
conduction current density. In the following we use its expression (39) to
write (66) and (68) in terms of the scalar potential.

####  III.2.1 First expression in terms of the scalar potential

Here we consider expression (66). By using (61) and (39), it takes the form

$\displaystyle{\bf\it Q}_{reg}^{s}(t)$
$\displaystyle\approx-\int\limits_{\mathcal{S}_{reg}}\sigma({\bf\it
r}^{\prime},t){\bf\it r}^{\prime}\bqty{\gradient^{\prime}\Phi({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}}$ (69)
$\displaystyle\quad+\int\limits_{\mathcal{V}_{reg}}\sigma({\bf\it
r}^{\prime},t)\gradient^{\prime}\Phi({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}.$

The second term on the right-hand side here can be transformed as follows
[21]. The nonhomogeneity in the conductivity can be taken into account by
visualizing the region as composed by $\displaystyle m$ homogeneous parts with
uniform conductivity $\displaystyle\sigma_{reg}^{(i)}(t)$, where
$\displaystyle i=1,2,\dots,m$. Representing the volume of the $\displaystyle
i$th part and its bounding surface by $\displaystyle\mathcal{V}_{reg}(i)$ and
$\displaystyle\mathcal{S}_{reg}(i)$, respectively, we have

$\displaystyle\int\limits_{\mathcal{V}_{reg}}\sigma({\bf\it
r}^{\prime},t)\gradient^{\prime}\Phi({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}$ (70)
$\displaystyle\enskip=\sum_{i=1}^{m}\sigma_{reg}^{(i)}(t)\int\limits_{\mathcal{S}_{reg}(i)}\Phi({\bf\it
r}^{\prime},t)\differential{{\bf\it S}^{\prime}_{i}},$

where the identity

$\int\limits_{\mathcal{V}_{reg}(i)}\gradient^{\prime}\Phi({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}=\int\limits_{\mathcal{S}_{reg}(i)}\Phi({\bf\it
r}^{\prime},t)\differential{{\bf\it S}^{\prime}_{i}},$ (71)

is utilized. Here $\displaystyle\differential{{\bf\it S}^{\prime}_{i}}$ is the
outward normal vector of surface element for the $\displaystyle i$th part.
Next, the surface $\displaystyle\mathcal{S}_{reg}(i)$ can be divided into an
internal component $\displaystyle\mathcal{S}_{reg}^{int}(i)$ which lies inside
the given region, and an external component
$\displaystyle\mathcal{S}_{reg}^{ext}(i)$ which makes part of the bounding
surface $\displaystyle\mathcal{S}_{reg}$ of the considered region. We then
write

$\displaystyle\int\limits_{\mathcal{S}_{reg}(i)}\Phi({\bf\it
r}^{\prime},t)\differential{{\bf\it S}^{\prime}_{i}}$
$\displaystyle=\int\limits_{\mathcal{S}_{reg}^{int}(i)}\Phi({\bf\it
r}^{\prime},t)\differential{{\bf\it S}^{\prime}_{i}}$ (72)
$\displaystyle\quad+\int\limits_{\mathcal{S}_{reg}^{ext}(i)}\Phi({\bf\it
r}^{\prime},t)\differential{{\bf\it S}^{\prime}_{i}}.$

Finally, representing by $\displaystyle\mathcal{S}_{reg}^{int}(i,j)$ the
common internal surface between two homogeneous parts $\displaystyle i$ and
$\displaystyle j$, the current dipole moment (69) is given by

$\displaystyle{\bf\it Q}_{reg}^{s}(t)$
$\displaystyle\approx-\int\limits_{\mathcal{S}_{reg}}\sigma({\bf\it
r}^{\prime},t){\bf\it r}^{\prime}\bqty{\gradient^{\prime}\Phi({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}}$ (73)
$\displaystyle\quad+\sum_{<ij>}\bqty{\sigma_{reg}^{(i)}(t)-\sigma_{reg}^{(j)}(t)}$
$\displaystyle\qquad\qquad\times\int\limits_{\mathcal{S}_{reg}^{int}(i,j)}\Phi({\bf\it
r}^{\prime},t)\differential{{\bf\it S}^{\prime}_{i,j}}$
$\displaystyle\quad+\int\limits_{\mathcal{S}_{reg}}\sigma({\bf\it
r}^{\prime},t)\Phi({\bf\it r}^{\prime},t)\differential{{\bf\it S}^{\prime}}.$

Here the sum is over all different pairs of adjacent homogeneous parts in the
given region. The normal vector of surface element
$\displaystyle\differential{{\bf\it S}^{\prime}_{i,j}}$ is directed from the
$\displaystyle i$th part to the $\displaystyle j$th part.

The above expression obtained for synaptic currents can be considered also
valid for impressed currents. By integrating (41) and using (63), we see that
the synaptic current dipole moment results to be approximately equal to the
impressed current dipole moment. We note that an expression for this is given
in Ref. 23. However, such expression does not contain the first term on the
right-hand side of (73). This term corresponds to the total current dipole
moment (65). In Ref. 23, an external medium with zero conductivity is assumed
implying that the normal component of the conduction current density vanishes
on both sides close to the region boundary. In contrast, we assume an external
medium with nonzero conductivity which corresponds to a non-isolated region.
As such, the last expression obtained above can be seen as a generalization of
the expression presented in Ref. 23.

####  III.2.2 Second expression in terms of the scalar potential

Here we consider expression (68). Using (39), we have

${\bf\it Q}_{reg}^{s}(t)\approx-\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\gradient^{\prime}\dotproduct\bqty{\sigma({\bf\it
r}^{\prime},t)\gradient^{\prime}\Phi({\bf\it
r}^{\prime},t)}\differential[3]{r^{\prime}}.$ (74)

Taking into account the nonhomogeneity in the conductivity as above, by
considering the region as composed by $\displaystyle m$ homogeneous parts with
uniform conductivity $\displaystyle\sigma_{reg}^{(i)}(t)$, where
$\displaystyle i=1,2,\dots,m$, and representing the volume of the
$\displaystyle i$th part by $\displaystyle\mathcal{V}_{reg}(i)$, we get

${\bf\it
Q}_{reg}^{s}(t)\approx-\sum_{i=1}^{m}\sigma_{reg}^{(i)}(t)\int\limits_{\mathcal{V}_{reg}(i)}{\bf\it
r}^{\prime}{\nabla^{\prime}}^{2}\Phi({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}.$ (75)

Here, different from (73), the synaptic current dipole moment is in terms of
the second spatial derivatives of the scalar potential inside each homogeneous
part.

The last expression can also be written in terms of the free charge density
$\displaystyle\rho$. In the electro-quasistatic approximation we have
$\displaystyle{\bf\it E}({\bf\it
r}^{\prime},t)\approx-\gradient^{\prime}\Phi({\bf\it r}^{\prime},t)$. This
leads to the Poisson’s expression

${\nabla^{\prime}}^{2}\Phi({\bf\it
r}^{\prime},t)\approx-\gradient^{\prime}\dotproduct{\bf\it E}({\bf\it
r}^{\prime},t).$ (76)

Describing the $\displaystyle i$th part by a linear and isotropic medium with
uniform permittivity $\displaystyle\epsilon_{reg}^{(i)}(t)$, we have
$\displaystyle{\bf\it D}({\bf\it r}^{\prime},t)=\epsilon_{reg}^{(i)}(t){\bf\it
E}({\bf\it r}^{\prime},t)$. Considering the Maxwell’s equation (13), we then
obtain

${\bf\it
Q}_{reg}^{s}(t)\approx\sum_{i=1}^{m}\frac{\sigma_{reg}^{(i)}(t)}{\epsilon_{reg}^{(i)}(t)}\int\limits_{\mathcal{V}_{reg}(i)}{\bf\it
r}^{\prime}\rho({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}.$ (77)

This can be written in a more compact manner by using

$\tau_{reg}^{(i)}(t)=\frac{\epsilon_{reg}^{(i)}(t)}{\sigma_{reg}^{(i)}(t)},$
(78)

which is the Maxwell–Wagner time for the $\displaystyle i$th part, and

${\bf\it p}_{reg}^{(i)}(t)=\int\limits_{\mathcal{V}_{reg}(i)}{\bf\it
r}^{\prime}\rho({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}},$ (79)

which is the electric dipole moment for the $\displaystyle i$th part. Then

${\bf\it Q}_{reg}^{s}(t)\approx\sum_{i=1}^{m}\frac{{\bf\it
p}_{reg}^{(i)}(t)}{\tau_{reg}^{(i)}(t)}.$ (80)

Thus, through the corresponding Maxwell–Wagner time, the synaptic current
dipole moment is related to the electric dipole moment of each homogeneous
part.

###  III.3 Long-distance magnetic field of action potential currents

Here we consider the case of action potential currents, given by Eq. (46).
Similarly to the synaptic case, we consider several relatively small regions
in the medium, such that

${\bf\it B}^{a}({\bf\it r},t)=\sum_{reg}{\bf\it B}_{reg}^{a}({\bf\it r},t),$
(81)

where

${\bf\it B}_{reg}^{a}({\bf\it
r},t)=\frac{\mu_{0}^{\mathstrut}}{4\pi}\int\limits_{\mathcal{V}_{reg}}{\bf\it
J}^{a}({\bf\it r}^{\prime},t)\crossproduct\frac{{\bf\it r}-{\bf\it
r}^{\prime}}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}^{3}}\differential[3]{r^{\prime}},$ (82)

corresponds to a region of volume $\displaystyle\mathcal{V}_{reg}$. As above,
the dimensions of each region must be small compared to the average distance
from the region to the field point $\displaystyle{\bf\it r}$. Taking any point
$\displaystyle{\bf\it r}_{reg}^{\mathstrut}$ in the region, we must have

$\vqty{{\bf\it r}^{\prime}-{\bf\it r}_{reg}^{\mathstrut}}\ll\vqty{{\bf\it
r}-{\bf\it r}_{reg}^{\mathstrut}},$ (83)

for every region point $\displaystyle{\bf\it r}^{\prime}$.

Eq. (82) can be written in the form

${\bf\it B}_{reg}^{a}({\bf\it r},t)=\curl{\bf\it A}_{reg}^{a}({\bf\it r},t),$
(84)

with the vector potential

${\bf\it A}_{reg}^{a}({\bf\it
r},t)=\frac{\mu_{0}^{\mathstrut}}{4\pi}\int\limits_{\mathcal{V}_{reg}}\frac{{\bf\it
J}^{a}({\bf\it r}^{\prime},t)}{\vqty{{\bf\it r}-{\bf\it
r}^{\prime}}}\differential[3]{r^{\prime}}.$ (85)

Employing the expansion (52) of the factor $\displaystyle 1/\vqty{{\bf\it
r}-{\bf\it r}^{\prime}}$ around $\displaystyle{\bf\it r}^{\prime}={\bf\it
r}_{reg}^{\mathstrut}$, and considering (63), we have

$\displaystyle{\bf\it A}_{reg}^{a}({\bf\it r},t)$
$\displaystyle\approx\frac{\mu_{0}^{\mathstrut}}{4\pi\vqty{{\bf\it r}-{\bf\it
r}_{reg}^{\mathstrut}}^{3}}$ (86)
$\displaystyle\quad\times\int\limits_{\mathcal{V}_{reg}}{\bf\it J}^{a}({\bf\it
r}^{\prime},t)\bqty{\pqty{{\bf\it r}-{\bf\it
r}_{reg}^{\mathstrut}}\dotproduct{\bf\it
r}^{\prime}}\differential[3]{r^{\prime}}.$

By using (84), we then obtain

$\displaystyle{\bf\it B}_{reg}^{a}({\bf\it r},t)$
$\displaystyle\approx-\frac{3\mu_{0}^{\mathstrut}}{4\pi}\frac{\hat{{\bf\it
R}}\pqty{{\bf\it r},{\bf\it r}_{reg}^{\mathstrut}}}{\vqty{{\bf\it
R}\pqty{{\bf\it r},{\bf\it r}_{reg}^{\mathstrut}}}^{3}}$ (87)
$\displaystyle\quad\crossproduct\int\limits_{\mathcal{V}_{reg}}{\bf\it
J}^{a}({\bf\it r}^{\prime},t)\bqty{\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}\dotproduct{\bf\it
r}^{\prime}}\differential[3]{r^{\prime}}$
$\displaystyle\quad+\frac{\mu_{0}^{\mathstrut}}{2\pi}\frac{{\bf\it
m}_{reg}^{a}(t)}{\vqty{{\bf\it R}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}}^{3}},$

where $\displaystyle\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}$ and $\displaystyle{\bf\it R}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}$ are given by (57) and (58), and

${\bf\it m}_{reg}^{a}(t)=\frac{1}{2}\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\crossproduct{\bf\it J}^{a}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$ (88)

is the magnetic dipole moment for the action potential currents. This, as well
as the integral in (87), are independent of the chosen coordinate origin,
which can be shown by using (63).

Expression (87) shows a decrease with the cube of the distance for action
potential currents. This is characteristic for the quadrupolar case [1]. Here
we have a faster decrease with the distance than in the quadratic synaptic
case given above by (56).

We note that expression (87) can be written in a familiar form for a magnetic
dipole, provided $\displaystyle\divergence{\bf\it J}^{a}\approx 0$ in the
given region, and $\displaystyle{\bf\it J}^{a}({\bf\it
r}^{\prime},t)=\mathbf{0}$ on its boundary [25, *wangsness]. In that case, the
integral in (87) can be expressed through the magnetic dipole moment (88), by

$\displaystyle\int\limits_{\mathcal{V}_{reg}}{\bf\it J}^{a}({\bf\it
r}^{\prime},t)\bqty{\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}\dotproduct{\bf\it
r}^{\prime}}\differential[3]{r^{\prime}}$ (89)
$\displaystyle\enskip\approx{\bf\it m}_{reg}^{a}(t)\crossproduct\hat{{\bf\it
R}}\pqty{{\bf\it r},{\bf\it r}_{reg}^{\mathstrut}},$

and (87) then takes the form

$\displaystyle{\bf\it B}_{reg}^{a}({\bf\it
r},t)\approx\frac{\mu_{0}^{\mathstrut}}{4\pi}$
$\displaystyle\left\\{3\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}\right.$ (90)
$\displaystyle\enskip\times\bqty{\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}\dotproduct{\bf\it m}_{reg}^{a}(t)}$
$\displaystyle\left.\vphantom{\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}}-{\bf\it m}_{reg}^{a}(t)\right\\}\frac{1}{\vqty{{\bf\it
R}\pqty{{\bf\it r},{\bf\it r}_{reg}^{\mathstrut}}}^{3}}.$

Additionally, using (89), the vector potential (86) reads

${\bf\it A}_{reg}^{a}({\bf\it
r},t)\approx\frac{\mu_{0}^{\mathstrut}}{4\pi}{\bf\it
m}_{reg}^{a}(t)\crossproduct\frac{\hat{{\bf\it R}}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}}{\vqty{{\bf\it R}\pqty{{\bf\it r},{\bf\it
r}_{reg}^{\mathstrut}}}^{2}},$ (91)

which, through (84), can also be utilized to obtain (90).

##  IV Discussion

Here we have developed a method for calculating the long-distance magnetic
field generated by neuronal populations. Since MEG measurements show magnetic
signals that change over time, we first considered the time-dependent
generalization of the Biot–Savart law given by Eq. (18). As the main
contributions in neuromagnetism come from relatively low frequencies that are
below a typical value $\displaystyle f_{max}=$\displaystyle
100\text{\,}\mathrm{H}\mathrm{z}$$ [1], we then followed the approach of
Griffiths and Heald [11] to approximate Eq. (18) in the quasistatic case. This
method differs from previous ones [1, 15, 16, 17] where the treatment does not
rely on the general solution of Maxwell’s equations, but on the quasistatic
approximation of these equations themselves.

The time dependency of the current density and that of the magnetic field can
be described via the different frequency components. We found that in the
quasistatic case, the general nonstatic solution (18) can be approximated by
expression (32) provided $\displaystyle w_{max}^{2}t_{trav}^{2}({\bf\it
r},\bar{{\bf\it r}}^{\prime})\ll 1$. This is expression (30) and describes the
quasistatic case through the current frequency component with the shortest
oscillation period $\displaystyle 2\pi/w_{max}=1/f_{max}$. This time must be
large enough compared to the largest traveling time $\displaystyle
t_{trav}({\bf\it r},\bar{{\bf\it r}}^{\prime})$. The latter is the time needed
for an electromagnetic signal to travel from the most distant source point
$\displaystyle\bar{{\bf\it r}}^{\prime}$ to the measurement point
$\displaystyle{\bf\it r}$.

We note that a similar expression to (30) is well known for the magneto-
quasistatic approximation of Maxwell’s equations [15, 16], as well as their
electro-quasistatic approximation [15, 16, 17]. In these cases, however,
instead of the traveling distance $\displaystyle\vqty{{\bf\it r}-\bar{{\bf\it
r}}^{\prime}}$, one typical or characteristic length scale of the medium is
employed. The system dimensions are assumed to be all within a factor of two
or so of each other [16]. The typical length of gray matter is estimated by
the cortical thickness [15, 17]. On the other hand, Eq. (18) leads to
expression (30) and therefore, to the traveling distance
$\displaystyle\vqty{{\bf\it r}-\bar{{\bf\it r}}^{\prime}}$ as the appropriate
length to define the quasistatic dynamics.

A clearly different expression from (30) can be found in Ref. 1 where the
quasistatic approximation of the Ampère–Maxwell equation was examined. It is
given by $\displaystyle w_{max}\tau_{MW}^{\mathstrut}\ll 1$, where
$\displaystyle\tau_{MW}^{\mathstrut}=\epsilon/\sigma$ is the Maxwell–Wagner
reaction time. That is expression (35) which we discussed in greater detail
earlier. Here $\displaystyle\epsilon$ and $\displaystyle\sigma$ are the
permittivity and the electrical conductivity of the medium, respectively. We
note that a permittivity
$\displaystyle\epsilon\approx$\displaystyle{10}^{5}$\epsilon_{0}^{\mathstrut}$,
where $\displaystyle\epsilon_{0}^{\mathstrut}$ is the permittivity of free
space, was considered in Ref. 1. This gives $\displaystyle
w_{max}\tau_{MW}^{\mathstrut}\approx$\displaystyle 2\text{\times}{10}^{-3}$\ll
1$. Here, however, based on Refs. 15 and 19, the permittivity for gray matter
at frequencies below $\displaystyle 100\text{\,}\mathrm{H}\mathrm{z}$ has been
underestimated by two orders of magnitude. Therefore, the condition
$\displaystyle w_{max}\tau_{MW}^{\mathstrut}\ll 1$ is actually fulfilled in a
minimal manner.

The quasistatic approximation (32) of the Biot–Savart law was justified with
(30) being well satisfied in the form $\displaystyle(2\pi
f_{max})^{2}\vqty{{\bf\it r}-\bar{{\bf\it
r}}^{\prime}}^{2}\epsilon\mu\approx$\displaystyle 4\text{\times}{10}^{-7}$\ll
1$. Here we estimated $\displaystyle\vqty{{\bf\it r}-\bar{{\bf\it
r}}^{\prime}}\approx$\displaystyle 0.1\text{\,}\mathrm{m}$$, mainly given by
the average size of the human brain. Additionally, we took the permittivity
$\displaystyle\epsilon\approx$\displaystyle{10}^{7}$\epsilon_{0}^{\mathstrut}$
for gray matter at frequencies below $\displaystyle
100\text{\,}\mathrm{H}\mathrm{z}$ [15, 19]. Finally, typical for biological
tissues [15], the permeability $\displaystyle\mu$ was approximated by that of
free space, $\displaystyle\mu_{0}^{\mathstrut}$.

We then focused on the magnetic field of the various currents. Following
previous work [21, 22], the current density was split into conduction and
impressed current densities. The conduction current is caused by the electric
field and its density is specified in the electro-quasistatic approximation by
$\displaystyle{\bf\it J}^{c}({\bf\it r}^{\prime},t)\approx-\sigma({\bf\it
r}^{\prime},t)\gradient^{\prime}\Phi({\bf\it r}^{\prime},t)$, where
$\displaystyle\sigma({\bf\it r}^{\prime},t)$ is the nonhomogeneous
conductivity of the medium and $\displaystyle\Phi({\bf\it r}^{\prime},t)$ is
the scalar potential. We assumed the impressed current density mostly given by
a synaptic current density $\displaystyle{\bf\it J}^{s}({\bf\it
r}^{\prime},t)$ and an action potential current density $\displaystyle{\bf\it
J}^{a}({\bf\it r}^{\prime},t)$. The magnetic field case of conduction currents
has been worked out exactly [23]. The long-distance magnetic field was studied
in the synaptic and action potential cases.

Each population of neurons was described as a relatively small region such
that its dimensions are small compared to the average distance from the region
to the field point $\displaystyle{\bf\it r}$. Taking any point
$\displaystyle{\bf\it r}_{reg}^{\mathstrut}$ in the region, we must have
$\displaystyle\vqty{{\bf\it r}^{\prime}-{\bf\it
r}_{reg}^{\mathstrut}}\ll\vqty{{\bf\it r}-{\bf\it r}_{reg}^{\mathstrut}}$, for
every region point $\displaystyle{\bf\it r}^{\prime}$. Considering the
distance to the measurement point in the order of centimeters, each region
volume $\displaystyle\mathcal{V}_{reg}$ can be thought in the order of
$\displaystyle 1\text{\,}\mathrm{m}\mathrm{m}^{3}$. Using known values of the
cortical surface, the average cortical thickness, and the total number of
neurons in the human cerebral cortex [10], we can find a quantity of
approximately $\displaystyle 4.7\text{\times}{10}^{5}$ regions of
$\displaystyle 1\text{\,}\mathrm{m}\mathrm{m}^{3}$ in the human cerebral
cortex, each one with an average estimate of about $\displaystyle
3.4\text{\times}{10}^{4}$ neurons. The contributions from the different
regions must be added together to obtain the magnetic field at a certain
point. This is indicated through Eqs. (47) and (81) for synaptic currents and
action potential currents, respectively.

The magnetic induction produced by synaptic currents at a relatively distant
point $\displaystyle{\bf\it r}$, for a region located at $\displaystyle{\bf\it
r}_{reg}^{\mathstrut}$, can be approximated by expression (56). Here
$\displaystyle{\bf\it Q}_{reg}^{s}(t)$ is the (time-dependent) synaptic
current dipole moment defined by Eq. (54). This was related to the conduction
current density leading to two equivalent expressions of $\displaystyle{\bf\it
Q}_{reg}^{s}(t)$ in terms of the scalar potential.

In conclusion, we have derived here the formalism to relate the magnetic field
at large distances (centimeters) from populations of neurons and estimated the
contributions from various current sources in neurons. The next step in this
approach is to apply this population-level description to mean-field models of
neuronal populations, in order to link these models with the genesis of the
MEG signal.

*

##  Appendix A Total current dipole moment

Here we derive expressions (65) and (67) for the total current dipole moment.
It is also checked how each of these expressions can yield the same result
regardless of the choice of the coordinate origin. The following are two key
assumptions. Firstly, we consider quasistatic conditions such that the
continuity equation gives

$\gradient^{\prime}\dotproduct{\bf\it J}({\bf\it
r}^{\prime},t)=-\partialderivative{\rho({\bf\it r}^{\prime},t)}{t}\approx 0.$
(92)

Secondly, we assume a small contribution of the impressed currents at the
surface $\displaystyle\mathcal{S}_{reg}$ bounding the given region, compared
to that of the conduction currents. We take

$\vqty{\,\int\limits_{\mathcal{S}_{reg}}x^{\prime}{\bf\it J}^{i}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it
S}^{\prime}}}\ll\vqty{\,\int\limits_{\mathcal{S}_{reg}}x^{\prime}{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}},$
(93)

and this similarly applies to the remaining components of
$\displaystyle\int\limits_{\mathcal{S}_{reg}}{\bf\it r}^{\prime}\bqty{{\bf\it
J}^{i}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}}$.

###  A.1 Expression (65)

We consider the $\displaystyle x$ component of the total current dipole moment
(60), given by

$Q_{reg}^{x}(t)=\int\limits_{\mathcal{V}_{reg}}J_{x}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$ (94)

and write $\displaystyle J_{x}({\bf\it r}^{\prime},t)$ in the form

$\displaystyle J_{x}({\bf\it r}^{\prime},t)={\bf\it J}({\bf\it
r}^{\prime},t)\dotproduct\gradient^{\prime}x^{\prime}$
$\displaystyle=\gradient^{\prime}\dotproduct[x^{\prime}{\bf\it J}({\bf\it
r}^{\prime},t)]$ (95)
$\displaystyle\quad-x^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}({\bf\it
r}^{\prime},t).$

By using (92), this simplifies to

$J_{x}({\bf\it
r}^{\prime},t)\approx\gradient^{\prime}\dotproduct[x^{\prime}{\bf\it
J}({\bf\it r}^{\prime},t)].$ (96)

Then

$\displaystyle Q_{reg}^{x}(t)$
$\displaystyle\approx\int\limits_{\mathcal{V}_{reg}}\gradient^{\prime}\dotproduct[x^{\prime}{\bf\it
J}({\bf\it r}^{\prime},t)]\differential[3]{r^{\prime}}$ (97)
$\displaystyle=\int\limits_{\mathcal{S}_{reg}}x^{\prime}{\bf\it J}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}},$

where $\displaystyle\mathcal{S}_{reg}$ is the surface bounding the region,
according to the divergence theorem. Furthermore, from Eq. (40),
$\displaystyle{\bf\it J}({\bf\it r}^{\prime},t)={\bf\it J}^{c}({\bf\it
r}^{\prime},t)+{\bf\it J}^{i}({\bf\it r}^{\prime},t)$. Taking then into
account (93), we write

$Q_{reg}^{x}(t)\approx\int\limits_{\mathcal{S}_{reg}}x^{\prime}{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}.$
(98)

Considering the $\displaystyle y$ and $\displaystyle z$ components of the
total current dipole moment, we similarly obtain

$Q_{reg}^{y}(t)\approx\int\limits_{\mathcal{S}_{reg}}y^{\prime}{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}},$
(99)

and

$Q_{reg}^{z}(t)\approx\int\limits_{\mathcal{S}_{reg}}z^{\prime}{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}.$
(100)

Therefore, we get

${\bf\it Q}_{reg}(t)\approx\int\limits_{\mathcal{S}_{reg}}{\bf\it
r}^{\prime}\bqty{{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}},$ (101)

that is, expression (65).

###  A.2 Expression (67)

####  A.2.1 Derivation from expression (65)

Considering the $\displaystyle x$ component in (65) (or (101)), given by (98),
we have

$Q_{reg}^{x}(t)\approx\int\limits_{\mathcal{V}_{reg}}\gradient^{\prime}\dotproduct[x^{\prime}{\bf\it
J}^{c}({\bf\it r}^{\prime},t)]\differential[3]{r^{\prime}},$ (102)

using the divergence theorem. Similarly to (95), we get

$\gradient^{\prime}\dotproduct[x^{\prime}{\bf\it J}^{c}({\bf\it
r}^{\prime},t)]=x^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)+J_{x}^{c}({\bf\it r}^{\prime},t).$ (103)

Then

$\displaystyle Q_{reg}^{x}(t)$
$\displaystyle\approx\int\limits_{\mathcal{V}_{reg}}x^{\prime}\gradient^{\prime}\dotproduct{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}$ (104)
$\displaystyle\quad+\int\limits_{\mathcal{V}_{reg}}J_{x}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}.$

The second term on the right-hand side here is the $\displaystyle x$ component
of the conduction current dipole moment, $\displaystyle{\bf\it
Q}_{reg}^{c}(t)$, defined by (61). Expressions of the same form as that of
(104) apply to the $\displaystyle y$ and $\displaystyle z$ components.
Therefore

${\bf\it Q}_{reg}(t)\approx\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}+{\bf\it Q}_{reg}^{c}(t),$ (105)

that is, we obtain expression (67).

####  A.2.2 Independent derivation

Expression (67) can also be derived starting from Eq. (40) of the current
density. Integrating it, we have

${\bf\it Q}_{reg}(t)={\bf\it Q}_{reg}^{c}(t)+{\bf\it Q}_{reg}^{i}(t),$ (106)

where

${\bf\it Q}_{reg}^{i}(t)=\int\limits_{\mathcal{V}_{reg}}{\bf\it J}^{i}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$ (107)

is the impressed current dipole moment, with $\displaystyle x$ component

$\displaystyle\int\limits_{\mathcal{V}_{reg}}J_{x}^{i}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}$
$\displaystyle=\int\limits_{\mathcal{V}_{reg}}{\bf\it J}^{i}({\bf\it
r}^{\prime},t)\dotproduct\gradient^{\prime}x^{\prime}\differential[3]{r^{\prime}}$
(108)
$\displaystyle=\int\limits_{\mathcal{V}_{reg}}\gradient^{\prime}\dotproduct[x^{\prime}{\bf\it
J}^{i}({\bf\it r}^{\prime},t)]\differential[3]{r^{\prime}}$
$\displaystyle\quad-\int\limits_{\mathcal{V}_{reg}}x^{\prime}\gradient^{\prime}\dotproduct{\bf\it
J}^{i}({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}.$

Then

$\displaystyle\int\limits_{\mathcal{V}_{reg}}J_{x}^{i}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}$
$\displaystyle=\int\limits_{\mathcal{S}_{reg}}x^{\prime}{\bf\it J}^{i}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}$ (109)
$\displaystyle\quad-\int\limits_{\mathcal{V}_{reg}}x^{\prime}\gradient^{\prime}\dotproduct{\bf\it
J}^{i}({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}.$

Considering again (40), we have

$\gradient^{\prime}\dotproduct{\bf\it J}({\bf\it
r}^{\prime},t)=\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)+\gradient^{\prime}\dotproduct{\bf\it J}^{i}({\bf\it
r}^{\prime},t),$ (110)

which through (92), implies

$\gradient^{\prime}\dotproduct{\bf\it J}^{i}({\bf\it
r}^{\prime},t)\approx-\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t).$ (111)

Thus

$\displaystyle\int\limits_{\mathcal{V}_{reg}}J_{x}^{i}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}$
$\displaystyle\approx\int\limits_{\mathcal{S}_{reg}}x^{\prime}{\bf\it
J}^{i}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}$
(112)
$\displaystyle\quad+\int\limits_{\mathcal{V}_{reg}}x^{\prime}\gradient^{\prime}\dotproduct{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}.$

Here we can substitute

$\displaystyle\int\limits_{\mathcal{V}_{reg}}x^{\prime}\gradient^{\prime}\dotproduct{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}$
$\displaystyle=\int\limits_{\mathcal{S}_{reg}}x^{\prime}{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}$ (113)
$\displaystyle\quad-\int\limits_{\mathcal{V}_{reg}}J_{x}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$

which is similar to (109). Using then (93), we get

$\int\limits_{\mathcal{V}_{reg}}J_{x}^{i}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}\approx\int\limits_{\mathcal{V}_{reg}}x^{\prime}\gradient^{\prime}\dotproduct{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}.$ (114)

Expressions of the same form hold for the $\displaystyle y$ and $\displaystyle
z$ components of the impressed current dipole moment. Therefore

${\bf\it Q}_{reg}^{i}(t)\approx\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}.$ (115)

Eq. (106) then gives

${\bf\it Q}_{reg}(t)\approx\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}+{\bf\it Q}_{reg}^{c}(t),$ (116)

thus reproducing expression (67).

###  A.3 Independence on the coordinate origin

The total current dipole moment

${\bf\it Q}_{reg}(t)=\int\limits_{\mathcal{V}_{reg}}{\bf\it J}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}},$ (117)

is independent of the chosen origin of the coordinate system. The current
density value at a given point clearly does not depend on the position of the
coordinate origin.

The independence of $\displaystyle{\bf\it Q}_{reg}(t)$ on the coordinate
origin can be explicitly shown as follows. Let a new origin be set at position
$\displaystyle{\bf\it\mathcal{R}}$ relative to the old origin. A given point
located at $\displaystyle{\bf\it r}^{\prime}$ in relation to the old origin is
now located at $\displaystyle{\bf\it r}^{\prime\prime}$ in relation to the new
origin, with $\displaystyle{\bf\it r}^{\prime\prime}={\bf\it
r}^{\prime}-{\bf\it\mathcal{R}}$. Correspondingly, the current density is
described by $\displaystyle{\bf\it\mathcal{J}}({\bf\it r}^{\prime\prime},t)$,
with $\displaystyle{\bf\it\mathcal{J}}({\bf\it r}^{\prime\prime},t)={\bf\it
J}({\bf\it\mathcal{R}}+{\bf\it r}^{\prime\prime},t)={\bf\it J}({\bf\it
r}^{\prime},t)$. The total current dipole moment when using the new origin is

$\displaystyle{\bf\it\mathcal{Q}}_{reg}(t)$
$\displaystyle=\int\limits_{\mathcal{V}_{reg}}{\bf\it\mathcal{J}}({\bf\it
r}^{\prime\prime},t)\differential[3]{r^{\prime\prime}}$ (118)
$\displaystyle=\int\limits_{\mathcal{V}_{reg}}{\bf\it
J}({\bf\it\mathcal{R}}+{\bf\it
r}^{\prime\prime},t)\differential[3]{r^{\prime\prime}},$

so that

${\bf\it\mathcal{Q}}_{reg}(t)=\int\limits_{\mathcal{V}_{reg}}{\bf\it
J}({\bf\it r}^{\prime},t)\differential[3]{r^{\prime}}={\bf\it Q}_{reg}(t).$
(119)

The mentioned independence on the coordinate origin must be verified for
expressions (65) and (67) of the total current dipole moment. Below, when
using a new origin as above, the conduction current density is described by

${\bf\it\mathcal{J}}^{c}({\bf\it r}^{\prime\prime},t)={\bf\it
J}^{c}({\bf\it\mathcal{R}}+{\bf\it r}^{\prime\prime},t)={\bf\it J}^{c}({\bf\it
r}^{\prime},t).$ (120)

####  A.3.1 First case

Regarding expression (65) (or (101)), we have

$\displaystyle{\bf\it\mathcal{Q}}_{reg}(t)$
$\displaystyle\approx\int\limits_{\mathcal{S}_{reg}}{\bf\it
r}^{\prime\prime}\bqty{{\bf\it\mathcal{J}}^{c}({\bf\it
r}^{\prime\prime},t)\dotproduct\differential{{\bf\it S}^{\prime\prime}}}$
(121) $\displaystyle=\int\limits_{\mathcal{S}_{reg}}\pqty{{\bf\it
r}^{\prime}-{\bf\it\mathcal{R}}}\bqty{{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}},$

so that

$\displaystyle{\bf\it\mathcal{Q}}_{reg}(t)$
$\displaystyle\approx\int\limits_{\mathcal{S}_{reg}}{\bf\it
r}^{\prime}\bqty{{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}}$ (122)
$\displaystyle\quad-{\bf\it\mathcal{R}}\int\limits_{\mathcal{S}_{reg}}{\bf\it
J}^{c}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}.$

Concerning the second term on the right-hand side here, by integrating (111)
we get

$\int\limits_{\mathcal{S}_{reg}}{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it
S}^{\prime}}\approx-\int\limits_{\mathcal{S}_{reg}}{\bf\it J}^{i}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}.$ (123)

Hence, for expression (65) to be independent of the coordinate origin, it is
enough, and necessary, that the net impressed current crossing the region
boundary is approximately zero. Namely, if, and only if, we have

$\int\limits_{\mathcal{S}_{reg}}{\bf\it J}^{i}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}\approx 0,$ (124)

then

$\int\limits_{\mathcal{S}_{reg}}{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}\approx 0,$ (125)

and consequently

${\bf\it\mathcal{Q}}_{reg}(t)\approx\int\limits_{\mathcal{S}_{reg}}{\bf\it
r}^{\prime}\bqty{{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}},$ (126)

which gives the same result as expression (65).

####  A.3.2 Second case

As for expression (67) (or (116)), when we employ a new origin it reads

$\displaystyle{\bf\it\mathcal{Q}}_{reg}(t)$
$\displaystyle\approx\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime\prime}\gradient^{\prime\prime}\dotproduct{\bf\it\mathcal{J}}^{c}({\bf\it
r}^{\prime\prime},t)\differential[3]{r^{\prime\prime}}$ (127)
$\displaystyle\quad+\int\limits_{\mathcal{V}_{reg}}{\bf\it\mathcal{J}}^{c}({\bf\it
r}^{\prime\prime},t)\differential[3]{r^{\prime\prime}},$

which, using the divergence theorem and (123), leads to

$\displaystyle{\bf\it\mathcal{Q}}_{reg}(t)$
$\displaystyle\approx\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}+{\bf\it Q}_{reg}^{c}(t)$ (128)
$\displaystyle\quad+{\bf\it\mathcal{R}}\int\limits_{\mathcal{S}_{reg}}{\bf\it
J}^{i}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}.$

Thus, similar to the case of expression (65), if, and only if, (124) is met,
we have

${\bf\it\mathcal{Q}}_{reg}(t)\approx\int\limits_{\mathcal{V}_{reg}}{\bf\it
r}^{\prime}\gradient^{\prime}\dotproduct{\bf\it J}^{c}({\bf\it
r}^{\prime},t)\differential[3]{r^{\prime}}+{\bf\it Q}_{reg}^{c}(t),$ (129)

which produces the same result as expression (67).

###### Acknowledgements.

Research funded by the CNRS, the European Community (H2020-785907), the ANR
(PARADOX) and the ICODE excellence network.

## References

  * Hämäläinen  _et al._ [1993] M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa, Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain, Rev. Mod. Phys. 65, 413 (1993).
  * Gulrajani [1998] R. M. Gulrajani,  _Bioelectricity and Biomagnetism_ (Wiley, New York, 1998).
  * Barbieri  _et al._ [2016] F. Barbieri, V. Trauchessec, L. Caruso, J. Trejo-Rosillo, B. Telenczuk, E. Paul, T. Bal, A. Destexhe, C. Fermon, M. Pannetier-Lecoeur, and G. Ouanounou, Local recording of biological magnetic fields using Giant Magneto Resistance-based micro-probes, Sci. Rep. 6, 39330 (2016).
  * Bedard and Destexhe [2014] C. Bedardand A. Destexhe, Generalized cable formalism to calculate the magnetic field of single neurons and neuronal populations, Phys. Rev. E 90, 042723 (2014).
  * Rall [1995] W. Rall,  _The Theoretical Foundation of Dendritic Function (Edited by Segev I, Rinzel J and Shepherd GM)_ (MIT Press, Cambridge, MA, 1995).
  * Bédard and Destexhe [2013] C. Bédardand A. Destexhe, Generalized cable theory for neurons in complex and heterogeneous media, Phys Rev E Stat Nonlin Soft Matter Phys 88, 022709 (2013).
  * Hobbie and Roth [2015] R. K. Hobbieand B. J. Roth,  _Intermediate Physics for Medicine and Biology_ (Springer, Cham, 2015).
  * Note [1] More precisely, we mean here the magnitude of the magnetic induction $\displaystyle{\bf\it B}$, called also magnetic flux density, which is measured in MEG recordings. In a linear and isotropic medium with magnetic permeability $\displaystyle\mu$ we have $\displaystyle{\bf\it H}={\bf\it B}/\mu$, where $\displaystyle{\bf\it H}$ is the magnetic field.
  * Griffiths [2013] D. J. Griffiths,  _Introduction to Electrodynamics_ (Pearson, Boston, 2013).
  * Van Essen  _et al._ [2018] D. C. Van Essen, C. J. Donahue, and M. F. Glasser, Development and evolution of cerebral and cerebellar cortex, Brain Behav. Evol. 91, 158 (2018).
  * Griffiths and Heald [1991] D. J. Griffithsand M. A. Heald, Time-dependent generalizations of the Biot–Savart and Coulomb laws, Am. J. Phys. 59, 111 (1991).
  * Weber and Macomb [1989] T. A. Weberand D. J. Macomb, On the equivalence of the laws of Biot–Savart and Ampere, Am. J. Phys. 57, 57 (1989).
  * Jefimenko [1990] O. D. Jefimenko, Comment on “On the equivalence of the laws of Biot–Savart and Ampere,” by T. A. Weber and D. J. Macomb [Am. J. Phys. $\displaystyle\mathbf{57}$, 57–59 (1989)], Am. J. Phys. 58, 505 (1990).
  * Jefimenko [1966] O. D. Jefimenko,  _Electricity and Magnetism_ (Appleton-Century-Crofts, New York, 1966).
  * Wagner  _et al._ [2014] T. Wagner, U. Eden, J. Rushmore, C. J. Russo, L. Dipietro, F. Fregni, S. Simon, S. Rotman, N. B. Pitskel, C. Ramos-Estebanez, A. Pascual-Leone, A. J. Grodzinsky, M. Zahn, and A. Valero-Cabré, Impact of brain tissue filtering on neurostimulation fields: A modeling study, NeuroImage 85, 1048 (2014).
  * Haus and Melcher [1989] H. A. Hausand J. R. Melcher,  _Electromagnetic Fields and Energy_ (Prentice Hall, Englewood Cliffs, 1989).
  * Gratiy  _et al._ [2017] S. L. Gratiy, G. Halnes, D. Denman, M. J. Hawrylycz, C. Koch, G. T. Einevoll, and C. A. Anastassiou, From Maxwell’s equations to the theory of current-source density analysis, Eur. J. Neurosci. 45, 1013 (2017).
  * [18] here the approach is based on Ref. 16.
  * Gabriel  _et al._ [1996] S. Gabriel, R. W. Lau, and C. Gabriel, The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz, Phys. Med. Biol. 41, 2251 (1996).
  * Note [2] It can be seen that using $\displaystyle f_{max}=$\displaystyle 10\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and a characteristic length of $\displaystyle 1\text{\,}\mathrm{m}\mathrm{m}$, corresponding to the order of the cortical thickness, yields also the value $\displaystyle 4\text{\times}{10}^{-7}$ for the electro-quasistatic approximation in Ref. 17.
  * Geselowitz [1967] D. B. Geselowitz, On bioelectric potentials in an inhomogeneous volume conductor, Biophys. J. 7, 1 (1967).
  * Ilmoniemi [1994] R. J. Ilmoniemi, Magnetic source imaging, in  _Biological Effects of Electric and Magnetic Fields_ , Vol. 2, edited by D. O. Carpenterand S. Ayrapetyan (Academic Press, San Diego, 1994).
  * Geselowitz [1970] D. B. Geselowitz, On the magnetic field generated outside an inhomogeneous volume conductor by internal current sources, IEEE Trans. Magn. 6, 346 (1970).
  * Note [3] Specifically, as can be seen from (93), this contribution of the impressed currents at the surface $\displaystyle\mathcal{S}_{reg}$ bounding the region, is given by the components of $\displaystyle\intop\ilimits@\limits_{\mathcal{S}_{reg}}{\bf\it r}^{\prime}\bqty{{\bf\it J}^{i}({\bf\it r}^{\prime},t)\dotproduct\differential{{\bf\it S}^{\prime}}}$. This term should be added to expression (65) and its equivalent (67) in a broader approach.
  * Jackson [1999] J. D. Jackson,  _Classical Electrodynamics_ (Wiley, New York, 1999).
  * Wangsness [1986] R. K. Wangsness,  _Electromagnetic Fields_ (Wiley, New York, 1986).