This document may require the windows character set to display mathematical symbols correctly. These are some notes put together to provide a cursory understanding of scalp current density. I do not have a tertiary education in mathematical physics. The notes are generated while exploring this issue and I do not guarantee the accuracy of the information. I have had some help and would like to aknowledge some patient guidance, comments and corrections from Dr. Jim Bashford (University of Tasmania) and Dr. Mike Teubner (University of Adelaide).
Perrin, Bertrand and Pernier (1987) report that scalp current density (SCD) can be calculated from a surface Laplacian of scalp potential values, multiplied by minus scalp tissue conductivity:
SCD = 1 * scalp conductivity * Laplacian of scalp potential
Mathematically, this is given by:
Ñ • J = σ Ñ^{2}φ
where Ñ ('del') is a partial differential vector operator (explained below), J is a current density vector (Ampere/m^{2} for a surface, Ampere/m^{3} for a volume), Ñ • J is a scalar product, called the divergence of current density, σ is conductivity (Siemens/m; assumed constant here and estimated at 0.33 Siemens/m for human scalp), and Ñ^{2}φ is the Laplacian of potential (φ volts). The units of measurement for the equation above are:
Potential
Laplacian
Conductivity
SCD
Volts ( V)
Volts/square meter ( V/m^2)
Siemens/meter (S/m)
Amperes/square meter ( A/m^2)
The term 'scalp current density' actually refers to the divergence of the current density in the scalp. It is the rate of change of current flowing into and through the scalp. To understand this quantity it is helpful to illustrate the physical system as follows (Katznelson, 1981):
This represents the volume conduction of current, generated in the brain, through the skull and into the scalp. Note, however, that this figure may be slightly misleading, because current approaching the scalp surface may not simply translate from a normal into a tangential orientation. Rather, as in optics, the current will flow along the path of least resistance, which can include reflection back into the volume at an angle dependent on the approach angle to the surface (Teubner, 2001, personal communication).
The following discussion explores the principles of electromagnetism that define Ñ • J. The physical meaning of this quantity is discussed in terms of current sources and sinks. To elaborate the meaning of this quantity, it is first necessary to cover some fundamentals of electric charge and electric fields and the vector mathematics of these properties. Vector calculus is beyond the scope of these notes and a greater appeciation of the discussion will be gained from a short revision of vector calculus. However, these notes attempt to integrate some relevant concepts of physics into a convenient discussion. The latter appendix of internet resources provides some links to excellent visual models and interactive applets to facilitate learning the principles of physics.
To say that a particle is charged is to assert that it has properties of attraction or repulsion of other charged particles. This is in contrast to an inert particle, which exhibits no properties of attraction or repulsion of other particles. Two fundamental charged particles are the electron, with negative charge, and the proton, with positive charge. One electron has a charge of 1.6019E19 coulombs, where one coulomb is approximately 6E18 electrons  note that one coulomb is much larger than an electron charge. Coulomb was a French scientist who's experiments around 1785 AD demonstrated the electric force of charged particles.
A positively charged particle generates repulsion forces against positive charges. In contrast, a negatively charged particle generates attraction forces for positive charges. Like charges repel, opposite charges attract.
These forces comprise a vector field, with a magnitude proportional to distance and a direction radial from a charged particle. Coulomb's law is given as: the force along a line between two static charged particles in free space, Q and Q_{2} (point charges), is directly proportional to the product of their charge (Q x Q_{2}) and inversely proportional to the squared distance between them, R (provided that R >> diameter of Q_{n}); mathematically:
F = k (Q * Q_{2}) / R^{2}  Coulomb's law (eq. C.i) 

where k is a proportionality constant. In SI units, F is in newtons (N), Q_{n} is in coulombs (C), R is in meters (m), and k = 1 / 4 π ε_{0}, where ε_{0} is the permittivity of free space (8.854E12 farads[F]/m), or k = 9E09 m/F. Given any two points in free space, the vector form of (eq. i) is given by:
F = [ k (Q * Q_{2}) / R^{2} ] a_{R}  Coulomb's law (eq. C.ii) 

where a_{R} is the unit vector in the direction of R (taking into account the charge signs of Q & Q_{2}). For example, if Q and Q_{2} lie on the xaxis, the unit vector is <1,0,0> in 3D Cartesian coordinates.
For more than two charged particles, the force on one particle, Q, from all others is obtained by the summation of the separate forces  this is the principle of superposition. For example, given three point charges in free space (Q, Q_{2}, Q_{3}), the force on Q becomes:
F = [ k (Q * Q_{2}) / R_{2}^{2} ] a_{R2} + [ k (Q * Q_{3}) / R_{3}^{2} ] a_{R3}  Coulomb's law (eq. C.iii) 

or more generally,
_{N}  
F = kQ  Σ  ( Q_{n} / R_{n}^{2} ) a_{Rn}  Coulomb's law (eq. C.iv) 

^{n = 2} 
In electrostatic fields, charged particles have no velocity and each particle contributes a force according to its charge and distance. In electrodynamic fields, charged particles are in motion, creating electric field disturbances that are transmitted at the speed of light. That is not to say that charged particles move at the speed of light, only that any change in their electric field is transmitted at the speed of light. It is important to note that Coulomb's law and the principle of superposition are given for electrostatic fields. It is also important to note that Coulomb's law applies to point charges in free space (a vacuum), not charges in matter. To simplify electrostatic fields, the next section considers electric field strength as a function of unit charge.
Electric field strength or electric field intensity is a measure of electric field force per unit charge, given by:
E = lim_{Q → 0} F / Q  Electric Fields (eq. E.i) 

E is a vector in the direction of F, measured in newtons/coulomb (N/C, ie, force per unit charge). For example, the electric field intensity generated by a point charge, Q, at a distance, R, is derived from Coulomb's law, given as:
E = ( k Q / R^{2} ) a_{R}  Electric Fields (eq. E.ii) 

or more generally,
_{N}  
E = k  Σ  ( Q_{n} / R_{n}^{2} ) a_{Rn}  Electric Fields (eq. E.iii) 

^{n = 1} 
Furthermore, E can be determined for any uniform charge distribution along a line, across a surface, or through a volume. Here, a unit charge is the charge density (ρ), obtained from the differential of charge with respect to space (line ρ_{L} dL [C/m], surface ρ_{S} dS [C/m^{2}], volume ρ_{V} dv [C/m^{3}]). From the principle of superposition, E is the sum or integral of all the electric forces from the charges comprising the distribution. From Coulomb's law, replacing a point charge, Q_{n}, with the charge density, ρ, E can be given by:
E = k ∫ ( ρ_{L} • dL / R^{2} ) a_{R}  Line (eq. E.iv) 

E = k ∫ ( ρ_{S} • dS / R^{2} ) a_{R}  Surface (eq. E.v) 

E = k ∫ ( ρ_{V} • dv / R^{2} ) a_{R}  Volume (eq. E.vi) 

Given that EEG is often measured at the scalp surface, it is useful to derive the sum of electric field strength for a surface. Electric field strength may be determined in any direction. However, for a surface, it is determined by only the electric field strength normal to the surface. The total electric field strength normal to a surface is the electric flux.
Gauss' law defines electric flux for inert, closed surfaces (a closed surface defines a volume, eg, a sphere). The electric field strength of an inert closed surface, S, is related to the distance from and magnitude of a charged region, given by:
Q ε_{0} ∫ E • dS  Gauss' Law (eq. G.i) 

where Q is the net charge within S, and E • dS is the integral of electric field strength normal to S. To illustrate, let a surface, S, surround a point charge, q, then:
The electric potential is defined as the work done in moving a charged particle from point A to another point B in an electric field. Electric potential is measured in volts, where 1 volt = 1 joule/coulomb. Given a zero potential at some infinite point B from point A, the electric potential at A is given by V_{A} = W/Q, where W is the work done by an external agent in moving the particle from infinity to A. The work done is independent of the path of motion from point A to point B, as all work done can be only in a radial direction from Q. By convention, electric potential is a magnitude only (it is not a vector, it has no direction) and it is defined in relation to positive charge. A positive electric potential measures the work done in moving a positive charge toward another positive charge (pushing the charge against a repulsion force) and a negative electric potential measures the work done in moving a positive charge away from a positive charge (releasing the charge into a repulsion force).
Electric potential is equal to minus the electric field strength. Over a surface (S), the relationship between electric potential (voltage, φ ) and electric field strength (E) is given by:
φ =  ∫ E • dS  Electric Potential (eq. P.i) 

Solving for E, given the inverse relationship between integrals and derivatives, we have:
E =  (dφ / dS)  Electric Potential (eq. P.ii) 

Given measures of φ from scalp electrodes in 3D Cartesian coordinates, φ = φ(x,y,z), we can calculate the x, y, z components of E from the partial derivatives (∂) of φ(x), φ(y), φ(z):
E_{x} =  (∂φ / ∂x)  
E_{y} =  (∂φ / ∂y)  
E_{z} =  (∂φ / ∂z)  Electric Potential (eq. P.iii) 

These equations give the first partial derivates of a scalar field with respect to the x,y,z axes, which are known as the directional derivatives, and they measure the rate of change or gradient in the direction of x,y,z. Thus, in terms of the Cartesian unit vectors i <1,0,0>, j <0,1,0>, k <0,0,1>, the electric field strength is given by:
E = E_{x} i + E_{y} j + E_{z} k =  ( (∂φ/∂x) i + (∂φ/∂y) j + (∂φ/∂z) k)  Electric Potential (eq. P.iv) 

In compact vector notation, we have:
E =  Ñφ =  grad(φ)  Electric Potential (eq. P.v) 

where the vector partial differential operator, Ñ ('del') is:
Ñ = (∂/∂x) i + (∂/∂y) j + (∂/∂z) k
Ñ f = grad f = (∂f /∂x) i + (∂f /∂y) j + (∂f /∂z) k
The sum of the directional derivatives given by (1.1.5.iii) is called the gradient vector, which is directed along the line for which the derivative of the field is maximal; in other words, along the line of maximum increase in a scalar field. Figure 1 provides some visual guides to this quantity. For example, if you were to walk up a hill along the path of steepest ascent, you would be walking in the direction of grad(H), the vector gradient of the height (H) of the hill. For an electric potential field, the gradient of any point is directed in the line of greatest potential difference (the path AB for which the greatest work is done moving a charge from A to B). The gradient of the electric potential has an opposite direction to the electric field, it points away from negative charge (sinks) and toward positive charge (sources). Given an electric field, the gradient of the electric potential in this field has the same magnitude as the electric field, only opposite direction. The gradient of a scalar field is said to be conservative because the work done in displacing an object from point A to B in the field depends only on A and B and not on the path of the displacement. Systems with friction, electrical resistance, or air resistance, are not conservative.
(a)
(b)
(c)
(d)
Figure 1. Gradient and divergence of a potential field. Graph (a) is a bipolar scalar potential field in x,y. Graph (b) is the 3D gradient vector field of the potential field. Graph (c) is the 2D projection of (b). Note that gradient vectors are directed toward positive potential, normal to the isocontours of potential. Graph (d) is the Laplacian of the potential field. Note the inverted polarity of the Laplacian, also called the divergence of the gradient, which measures the expansion or contraction of the gradient field. A positive Laplacian indicates expansion of the gradient, whereas a negative Laplacian indicates contraction of the gradient. [This figure is best viewed in color. The matlab commands to create these figures are given in Appendix B.]
The divergence of the electric field strength over an inert closed surface is zero, which satisfies Laplace's equation, giving:
Ñ • E = Ñ • Ñφ = Ñ^{2}φ = 0
The scalp potential, φ = φ(x,y,z), is a scalar field and the Laplacian of this field is given by:
Ñ^{2}φ = ∂^{2}φ / ∂x^{2} + ∂^{2}φ / ∂y^{2} + ∂^{2}φ / ∂z^{2}
To simplify and clarify these partial derivates, we can write the above in vector notation:
div( E ) =  div( grad(φ) ) = 0
where div( f ) is a scalar result of the vector product, Ñ • grad( f ), and grad( f ) is a vector resulting from multiplication of the vector operator, Ñ, by a scalar function, f. The divergence of a vector field (div f) is a scalar field representing the source intensity of the field. Put another way, the divergence measures the rate of expansion or contraction per unit volume (see Figure 1d).
Electric current is generated by electric charge in motion. For a conductive material, a conduction current is generated by the drift motion of electrons. Current (i) is the derivative, with respect to time, for a quantity of charge crossing a surface. Current is a macroscopic property of conductors; it is a scalar quantity similar to volume, length, or mass. Current is measured in Amperes (C/s)  the electric charge (coulombs, C) moving through an area per unit time (s):
i = dQ / dt across S (eq. .i)
When the rate of current across a surface varies over the surface, we can define current with respect to the current density (J), given by:
i = ∫ J • dS across S (eq. .iii)
From this relationship it can be seen that current density is a vector, the differential of current; that is, current density is the rate of current flow through a unit surface or volume. In other words, if you take a small area, dS, at a given place in a material (eg, human scalp), the current density is the amount of charge flowing across that area in a unit of time (C/s or Amperes) divided by the area (m^{2}). Current density is a microscopic property of conductors, it is a vector oriented in the direction that a positive point charge would move at a given point in the conductor. Given a unit cross section (a surface or volume) of a conductor, the current density is the rate of current flow in the normal direction through the unit surface or volume. Current density is measured in Ampere/m^{2} for a surface and Ampere/m^{3} for a volume.
For a physical medium (not a vacuum), the current density depends on both the electric field strength in the medium and the conductivity (σ) of the medium. This is similar to a gravity field, where an object gravitating (falling) toward the earth will do so faster through air than water, as the resistance (friction) against the object is less in air than water. Conductivity is inversely proportional to resistivity and the conductance of a medium is a factor of both its conductivity and volume (ie, conductivity is conductance per unit volume).
Conductivity depends on the amount of free electrons in a material and temperature. Materials with small conductivity (<<1) are called dielectrics or insulators (eg, distilled water 10^{4} S/m, glass 10^{12} S/m) and materials with high conductivity (>>1) are called metals (eg, silver 6.1x10^{7} S/m, copper 5.8x10^{7} S/m, gold 4.1x10^{7} S/m, aluminum 3.5x10^{7} S/m, zinc 1.7x10^{7} S/m), materials in between these extremes are called semiconductors (eg, biological tissues, see below). Some materials at temperatures approaching absolute zero (0^{o}K) have near infinite conductivity; these materials are called superconductors (eg, lead and aluminum). Magnetoencephalography (MEG) employs superconductors to measure magnetic fields of the brain in a device called a SQUID (superconducting, quantum inferference device).
Ohm's law describes the relationship between electric field strength (E), conduction current density (J) and conductivity (σ):
J = σ E in Sv (eq. .iii)
The divergence of this vector field provides a scalar measure of the current source intensity, given by:
div( J ) = div( σ E ) in Sv (eq. .iv)
For an isotropic linear volume, conductivity is constant and it can be factored out, to give:
div( J ) = σ div( E ) in Sv (eq. .v)
Given the electric potential, we can substitute E = grad(φ) to derive the divergence of current density, as follows:
div( J ) =  σ div( grad(φ) ) in Sv (eq. .vi)
In this form, we can see that the divergence of current density is proportional to minus the Laplacian of electric potential. Thus, the divergence of current density through a surface volume (Sv) can be calculated from σÑ^{2}φ (see Figure 2). For the human scalp, conductivity (σ) is estimated at 0.33 siemens/m and Ñ^{2}φ is the Laplacian of scalp electric potential (sampled by EEG). That is, multiplying minus conductivity by a Laplacian of scalp potential gives a scalar field, div(J). This is a mathematical representation or definition of 'scalp current density' (Perrin, Bertrand & Pernier, 1987; Oostendorp & van Oosterom, 1996).
The divergence of any vector field is a scalar field representing the source intensity of the field. Put another way, the divergence measures the rate of expansion or contraction per unit volume (see Figure 1d). For a surface, the divergence represents the rate of flow through a unit area of the surface. A positive divergence represents the rate of flow out of the surface (in the direction of the positive normal surface vector), sometimes called a source. A negative divergence represents the rate of flow into the surface (in the direction of the negative normal surface vector), sometimes called a sink.
(a) (b)
Figure 2. Scalp current density. Graph (a) is the Laplacian of a potential field (eg, volts/m^{2}). Graph (b) is the divergence of the current density (ampere/m^{2}), given here by minus the scalp conductivity (0.35 siemens/m) multiplied by the Laplacian of potential. Positive values indicate increasing current field density; negative values indicate decreasing current field density. [This figure is best viewed in color.]
Electrostatic boundary value problems are solved by application of Poisson's or Laplace's equation (both derived from Gauss's law). For a linear, heterogeneous material, the charge density at any point is given by Poisson's equation:
ρ = div ( ε grad( φ ) ) in S (eq. .i)
where ε is permittivity (Farads/m) φ is potential (Volts), and ρ is charge density (C/m^{2} for surface, C/m^{3} for a volume; Sadiku, 1995). For a homogeneous volume, ε is a constant, and we have:
div ( grad( φ ) ) = ρ / ε in S (eq. .ii)
Furthermore, for a charge free region, ρ = 0, giving Laplace's equation:
div ( grad( φ ) ) = Ñ^{2}φ = 0 in S (eq. .iii)
Conductivity, σ (S/m), is the differential of permittivity with respect to time. Current density (vector J, A/m^{2} for surface, A/m^{3} for a volume) is the product of velocity and charge density (J = ρv). Then, substituting conductivity and current density, Poisson's equation is:
Ñ • (  σ Ñφ ) = Ñ • J
This is an illustration of a small portion of the anatomy to be considered (Henry Gray [1825–1861]. Anatomy of the Human Body. 1918, Fig. 1196). Note that it is really more complex than anatomical models currently in use for source estimation. Most realistic anatomical models are based on computer tomography (CT) and magnetic resonance imaging (MRI) of the whole head, which may not give the kind of detail illustrated here.
The skin, or dermis, has considerable variation of tissue composition. The dermis is a stratified epithelium, consisting of several layers of epithelium that are more and more compressed nearer the surface. Inner layers of skin, the endodermis, contain stem cells that generate new outer layers of skin, the epidermis. The outer layers of skin are much thicker than the inner layers, serving as a protective layer.
"Of all the tissues of the body, the resistivity … for bone is the most variable. This is probably not too surprising, since bone is so varied in composition. For example, the skull consists of two dense, poorly conducting bony tables separated by a spongy region containing blood, a good conductor. Electrically the skull mimics a leaky capacitor rather than a resistor." (Geddes and Baker, 1967). The skull bones are composed of two tissue types, with different bone density. The dense bone is called compact tissue, while the fibrous reticular bone is called cancellous tissue. Both tissues are pourous, but the compact tissue has much smaller cavities than the cancellous tissue. The marrow is contained in the cancellous tissue, which has greater vasculature than the compact tissue. The bone is surrounded by a vascular tissue, called the periosteum. The cranium varies in thickness from a waifer thin transparency to several mm in depth. Thin sections are composed of largely compact tissue. Thicker sections are composed of two outer layers of compact tissue that enclose an inner layer of cacellous tissue.
The assumptions of boundary element models (BEM) require that a volume model of the skull is a closed volume or surface. The holes in the skull at the eyes, nose, ears and neck may not be modelled correctly. These holes provide areas of cerebral tissue where there is less resistance (greater conductivity) than other areas, so current flow will be greater in these areas than the current modelling estimates. Similarly, the scalp surface is assumed to be a closed surface. This surface is often modelled by cerebral MRI that is discontinuous at the neck. The neck area is often modelled as a flat outer surface of the scalp, so the conductivity of this region and the estimated current flow of this region is not modelled accurately.
The cerebrum has significant air pockets, which are often not modelled. Modelling the ventricles can make a difference to the source solution also.
In BEM, it is assumed that scalp, skull, and CSF are continuous charge free regions, containing no generating current sources/sinks. This means there is no divergence of current density (ie, Ñ • J = 0), there is a constant rate of flow or volume conduction for these tissues; these tissues serve purely as volume conductors for brain current generators (largely cortical piramidal cells). Current generators are assumed to be equipotential dipoles. Note that there is a substantial neurophysiology and computational literature on modelling single neuron current source and sink dynamics and neural networks (eg, Koch & Segev, 1998; Moore & Heinze, 2000).
One modelling assumption is that the skin is free of current generating sources and sinks. Is this true? Probably not.
When integrating EEG/MEG recordings with anatomical models, it is important
to locate the recording sensors on the anatomy. This is facilitated by
digitisation of the sensor positions in relation to anatomical landmarks. The
following figure illustrates the common landmarks used, namely the nasion, inion
and the preauricular points
(Henry Gray [1825–1861]. Anatomy of the Human Body. 1918, Fig. 1193).
Gray (1918) describes the preauricular points as follows:
When S is a human head, the following boundary condition applies at the scalp surface:
σ grad(φ) • n = 0 on S (eq. .iv)
where n is the surface normal vector. This states that the current vector in the direction normal to the scalp surface is zero. That is, electric current does not flow out of the head in a direction normal to the scalp surface. Just below the scalp surface, there is current in a direction normal to the surface, but the conservation of current requires that it is reflected back into the scalp surface.
Assuming the scalp is a Gaussian surface, with no inherent current sources or sinks, Laplace's equation applies and eq. 1.3.iv demonstrates that the divergence of the scalp current density is zero on the scalp surface. The current density vector field is given by:
J =  σ grad(φ) on S (eq. .v)
This can be separated into its tangential and radial components, given by:
J =  σ (∂φ/∂x i + ∂φ/∂y j + ∂φ/∂z k) on S (eq. .vi)
where ∂φ/∂x and ∂φ/∂y are the tangential derivates in x and y, and ∂φ/∂z is the radial or normal derivative in z. Furthermore, the divergence of the these components can be given by:
div( J ) =  σ ( ∂^{2}φ/∂x^{2} + ∂^{2}φ/∂y^{2} + ∂^{2}φ/∂z^{2} ) on S (eq. vii)
Now, note that Laplace's equation gives:
∂^{2}φ/∂x^{2} + ∂^{2}φ/∂y^{2} + ∂^{2}φ/∂z^{2} = 0 (eq. viii)
∂^{2}φ/∂x^{2} + ∂^{2}φ/∂y^{2} =  ∂^{2}φ/∂z^{2} (eq. ix)
This demonstrates that the x,y tangential components are equal to minus the normal component (or radial component for a sphere). The divergence of the current density on the scalp may be interpreted as the expansion or contraction of the tangential components of the current density on the scalp. An account of the components of scalp current density is provided in curvilinear coordinates below (Oostendorp & van Oosterom, 1996).
For practical purposes, the scalp potential (φ) is given by a limited measurement at finite electrode locations:
φ = V_{s}(x,y,z) on S_{1} (eq. .x)
where S_{1} is a subsurface of S covered by EEG electrodes. Interpolation of these measurements is often used to estimate the scalp potential. For interpolated points within the bounds of a dense electrode array, a linear or nearest neighbour method can provide accurate results, but often a cubic spline method is preferred.
Oostendorp and van Oosterom (1996) provide the following derivation of the surface Laplacian from the volume Laplacian in curvilinear coordinates. Given potential measures in curvilinear orthogonal coordinates (u_{1},u_{2},u_{3}), the volume Laplacian is given by:
Ñ^{2}φ = 1/h_{1}h_{2}h_{3} ( ∂/∂u_{1}(h_{2}h_{3}/h_{1} * ∂φ/∂u_{1}) + ∂/∂u_{2}(h_{1}h_{3}/h_{2} * ∂φ/∂u_{2}) + ∂/∂u_{3}(h_{1}h_{2}/h_{3} * ∂φ/∂u_{3}) )
where h_{1}=h_{1}(u_{1},u_{2},u_{3}), h_{2}=h_{2}(u_{1},u_{2},u_{3}), and h_{3}=h_{3}(u_{1},u_{2},u_{3}) are scaling factors of the curvilinear coordinates, providing distance units on the surface. This provides the divergence of the potential field (φ) in a volume.
On the outer boundary of the scalp surface, we can denote h_{3}∂u_{3} as ∂n, where n is the surface normal vector. On this boundary, lets assume h_{3}=1, so ∂u_{3}=∂n, and u_{1},u_{2} are orthogonal surface coordinates (tangents to the surface at any point on the surface). The surface Laplacian is then given by:
Ñ^{2}φ = 1/ h_{1}h_{2} ( ∂/∂u_{1}(h_{2}/h_{1} * ∂φ/∂u_{1}) + ∂/∂u_{2}(h_{1}/h_{2} * ∂φ/∂u_{2}) + ∂/∂n(h_{1}h_{2} * ∂φ/∂n) )
If this surface belongs to a source free region, Laplace's equation applies ( Ñ^{2}φ = 0) and we have:
Ñ^{2}φ =  1/ h_{1}h_{2} ∂/∂n( h_{1}h_{2} * ∂φ/∂n )
Ñ^{2}φ =  1/ h_{1}h_{2} ( ( ∂h_{1}/∂n * h_{2} + h_{1} * ∂h_{2}/∂n)∂φ/∂n + h_{1}h_{2} * ∂^{2}φ/∂n^{2} )
Where this surface bounds a source free region and there is no current leaving the region through the surface, for (u_{1},u_{2},n) approaching the surface from within the region, the normal component is zero (∂φ/∂n = 0), so:
0 = ( ∂h_{1}/∂n * h_{2} + h_{1} * ∂h_{2}/∂n )∂φ/∂n
Ñ^{2}φ =  1/h_{1}h_{2} ( 0 + h_{1}h_{2} * ∂^{2}φ/∂n^{2} )
Ñ^{2}φ =  ∂^{2}φ/∂n^{2}
This equation for the surface Laplacian applies whenever:
0 = ( ∂h_{1}/∂n * h_{2} + h_{1} * ∂h_{2}/∂n )∂φ/∂n
This occurs when the surface normal is zero (ie, ∂φ/∂n = 0) and when h_{1} or h_{2} are zero, such as when the surface is a plane.
Scalp current density is insensitive to deep current sources in the brain. The scalp current density is sensitive to superficial sources, with sensitivity falling off at appoximately r^{4}, r being the distance from a current source or sink and the scalp surface (Pernier et al., 1988; Oostendorp & van Oosterom, 1996). The implication of this is that superficial cortical sources will have greater impact on the scalp current density than deeper sources, some of which are cortical. For instance, Babiloni et al. (2000) report a study of motor potentials and an inverse solution that employs the scalp current density rather than the scalp potential. They argue that it provides greater sensitivity to the shallow current sources of the motor cortex (see also, Oostendorp & van Oosterom, 1996). On the other hand, use of the scalp current density in inverse source modelling may preclude accurate identification of deep cortical sources. For instance, the ventral and deeper sulcal cortical regions will contribute very little to the scalp current density. Oostendorp & van Oosterom (1996), for example, illustrate that the body potential has greater sensitivity to heart activity than the body Laplacian.
A cortical potential on the cortical surface (S_{c}) may be given by:
φ_{c} = V_{c}(x,y,z) on S_{c} (eq. .xi)
and φ_{c}(x,y,z) is recorded directly by cortical electrodes or estimated by an inverse solution of scalp recordings, given a model of the cortical surface, based on MRI data (using a boundary or finite element model of current volume conduction).
Can we determine whether a cortical potential indicates neuronal excitation or inhibition? Perhaps an answer to this problem requires an inverse solution that replaces equipotential dipoles with piramidal neuron models.
Geddes and Baker (1967) review resistivity measures from gross samples of various animal tissues (including work of Hans Berger). They conclude that biological tissues, composed of various cells, have considerable anisotropy, especially fibrous tissues such as nervous and skeletal muscle tissue. These measures employ various current frequencies for in vitro or in vivo (anethestised) organisms, where careful methods were employed to avoid electrode polarization and constant tissue temperature. A lot of the data cited for nervous tissues is from the Rabbit, published by Crile, Hosmer and Rowland (1922). In nervous tissue, where long figures are present, resistivity varies considerably from the transverse to the longitudinal orientation of the fibres (a ratio of about 5.7 to 9.4; Geddes & Baker, 1967). For instance, the average resistivity of whole brain is 580 Ω•cm, white matter is 682 Ω•cm, and grey matter is 284 Ω•cm. Note that the white matter is myelinated, which contains insulating lipids that have a high resistivity (especially in the transverse orientation). Given the larger proportion of white matter to grey matter, it is not surprising that the whole brain value is closer to the value for white matter than grey matter.
Recent attempts to measure cerebral tissue impedance, resistance, or conductivity employ current injection and inverse solution methods.
Nunez (1987) describes a method for measuring skull conductivity. This approach involves overdetermination of the inverse problem. Given a small region of cortical activity of known current, measurement of the tissue thickness for cortex, CSF, skull, and scalp, and measurement of electric potential from scalp electrodes near to the cortical region, it is possible to estimate the local skull resistance/conductivity.
Table 1. Resistivity of Tissues at Body Temperature (≈ 39 ^{o}C; Geddes & Baker, 1967; Nunez, 1981,p.80)
Tissue 
Resistivity (Ω•cm) 
Frequency 
Conductivity (S/m) 
Water Sea water Pure 
20 20,000,000 


White Matter (Rabbit) 
≈ 746  957 (avg) 
1 kc/s 
E05 
Gray Matter (Rabbit) 
208 ± 6 230 321 284 (avg) 
1 kc/s 5 kc/s 5 c/s 
3.50 E05 
Whole Brain 
570 (505  725) 

1.70 E05 
Cerebellum 
670 


Spinal Chord Longitudinal (Cat) Transverse (Cat) 
576 (386  863) 138  212 1,211 
1 kc/s 510 c/s 510 c/s 

CSF 
64.6 
130 kc/s (25 ^{o}C) 

Blood (40% red cells) 
≈ 165 (148176) 
1 kc/s 

Plasma 
66 


Bone 
≈ 1800 * 16,000 (? ^{o}C) 
1 Mc/s low freq. (ECG) 

Skull Wet Dry 
20,000 (avg., ? ^{o}C) 10,000,000,000,000 (? ^{o}C) 
low freq. low freq. 

Fat 
2180 (11005000) 
1 Mc/s 

Skeletal Muscle Transverse Longitudinal 
950 1600 300 


Eye (Bovine) Cortical Intermediate Central 
198 285 530 
300 kc/s 500 kc/s 500 kc/s 

Human Skin (with fat) Human Scalp 
289 230 ** 
1 Mc/s d.c. 

Note: conductivity (S/m) = 1 / (Resistivity (Ω•cm) * 100).
* see comments below. ** Assumes homogeneous current density distribution.
Waberski et al. (1998). Advanced…
skin = brain = 0.33 [S/m = 1/(Ohm • m)]
skull = 0.0042 [S/m = 1/(Ohm • m)]
The following table provides estimates of cerebral tissue conductivities (Source, Richard Greenblatt, EMSE training course notes, FBM symposium, May 2001, Melbourne, Australia).
Table .2. Estimated Conductivity of Cerebral Tissues (S/m)
Tissue 
Lowest 
Highest 
Baseline 
White Matter 
0.08 
1.18 
0.2 
Gray Matter 
0.16 
0.48 
0.33 
CSF 
1.0 
1.79 
1.79 
Skull 
0.004 
0.07 
0.0132 
Fat 
0.02 
0.07 
0.045 
Muscle 
0.043 
0.67 
0.35 
Eye 
0.5 
0.5 
0.5 
Skin 
0.35 
0.35 
0.35 
Pernier (2001, personal communication) uses a scalp conductivity of 0.45 Siemens/m, "In our lab we are using 0.45 S/m. This value was proposed by Rush and Driscoll (EEG electrode sensitivity. IEEE Trans. Biomed Eng; 1969, v16, 1522)." The Source Signal software, EMSE, and the Neuroscan software, Curry™, use the following conductivity estimates for BEM of closed surfaces (as of July, 2001):
SKIN 12 mm @ 0.3300 S/m
SKULL 10 mm @ 0.0042 S/m
BRAIN/CSF 08 mm @ 0.3300 S/m
Pernier (2001, personal communication) uses a scalp conductivity of 0.45 Siemens/m, "In our lab we are using 0.45 S/m. This value was proposed by Rush and Driscoll (EEG electrode sensitivity. IEEE Trans. Biomed Eng; 1969, v16, 1522). This is a mean value because the scalp conductivity varies from a point to another on the scalp." Note that scalp current density is given in milli Amperes per meter cubed (mA/m^3). For further informtion see:
http://www.bionic.fr/erpa/Erpa.html
http://www.lyon151.inserm.fr/unites/280_angl.html
Features include :
A scalar field in x,y is any function f(x,y). For example,
[x,y] = meshgrid(2:.2:2,1:.1:1);
z = x .* exp(x.^2  y.^2);
surfc(x,y,z), hold on;
view(20,20);
A vector field, such as the gradient of a scalar field, can be derived by the directional derivative of the scalar field. For example, in 3D:
[u,v,w] = surfnorm(x,y,z); % 3D surface normals
[gx,gy] = gradient(z,.2,.1);
quiver3(x,y,z,gx,gy,w)
Similarly, the 2D projection of the gradient is given by:
[gx,gy] = gradient(z,.2,.1);
figure; contour(x,y,z), hold on; quiver(x,y,gx,gy)
The Laplacian of scalar field, eg electric potential (voltage), gives a scalar field. For example:
lap = 4*del2(z); % Hjorth, nearest neighbour Laplacian
figure; surfc(x,y,lap); view(20,20); %plot Laplacian field
figure; surf(x,y,z,lap); view(20,20); %plot potential field, colored by Laplacian
Particle Physics:
http://ParticleAdventure.org
Interactive Physics Education, including Electromagnetic Principles:
http://www.Colorado.edu/physics/2000/cover.html
Founders of Electromagnetism
http://www.ee.umd.edu/~taylor/frame1.htm
International Journal of Bioelectromagnetism:
http://130.230.18.251/ijbem/volume3/number1/toc.htm
Electromagnetism Text Book (free)
http://www.plasma.uu.se/CED/Book/index.html
Leonid Zhukov's Physical equations of EEG/MEG:
http://waggle.gg.caltech.edu/~zhukov/research/eeg_meg/eeg_meg_physics/node1.html
Electromagnetism Online:
http://dmoz.org/Science/Physics/Electromagnetism/
Math & Physics with Maple:
http://www.mapleapps.com/powertools/education.html
http://www.maple4students.com/main.html
http://www.mapleapps.com/powertools/physics/Physics.html
DeMonte, Tim
Department of Electrical & Computer Engineering,
University of Toronto
25/01/00
Applications in Bioelectromagnetics: Magnetic Resonance and Current Density Imaging
Bioelectromagnetics is the study of electricity and magnetism and their interaction with living tissue. Research in this field continues to lead towards new medical applications such as hyperthermia, photodynamic therapy, microwave imaging and electrostimulation techniques (just to name a few).
Magnetic Resonance Imaging (MRI) is one of the most successful applications of bioelectromagnetics in recent times (brought into clinical use in the early 1980's). MRI is a noninvasive, highresolution imaging technique which can penetrate the entire human body by combining a static magnetic field, three magnetic gradient fields and a radio frequency (RF) magnetic field transceiver system.
Current Density Imaging (CDI) is a recent extension of MRI, developed at the University of Toronto, which is capable of spatially mapping electrical current pathways through tissue. CDI research is presently attempting to increase signaltonoise ratio (SNR), increase data acquisition speed, increase the amount of data acquired and develop new applications for the technique.
The original implementation of CDI injects low frequency (LF) electrical currents into the tissue. These electrical currents generate magnetic fields that are measurable using MRI techniques. Measurement of these magnetic fields then allows electrical current pathways to be computed and spatially mapped. This technique is limited by the requirement of at least two orthogonal orientations of the subject to measure a single component of current density. It is also limited by muscle twitching associated with the injection of LF electrical current. At present, LFCDI is limited to phantom and invitro studies and is being extended to a multislice sequence to increase the acquisition speed and the amount of data collected.
Radio frequency (RF) CDI is a rotating frame of reference version of LFCDI. It has two important advantages over LFCDI. First, a single component of current density can be measured without requiring subject rotation. Second, the high frequency current injected into tissue does not cause muscle twitching. Therefore, invivo measurements on human subjects are possible. At present, RFCDI is being extended to a multislice sequence to increase the acquisition speed and the amount of data collected. The frequency of the RF current is required to match the resonant frequency (Larmor frequency) of the MRI system. This requirement may limit the number of slices that can be acquired in RFCDI.