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Volume Conduction
C H A P T E R
6 Volume Conduction: Electric Fields in Electrolyte Solutions O U T L I N E Homogeneous Electric Field
154
The Monopole Field
155
The Dipole Field and Current Source Density Analysis
Multipole Fields and Current Source Density
157
References
159
156
Sources of electric current in water or in salt solutions give rise to stationary electric fields. The theory of electricity was developed in the 18th century with static forms of electricity. Charges in space, maintaining electrical potential differences, were kept from leaking away by dry air and by insulators like glass or ceramics. In a conducting medium, however, an electric current would flow and discharge any static source within a fraction of a second. Therefore, to generate an electric field in a fluid, we have to supply current continually (DC and/or AC). The mathematical descriptions of electric fields in water and in an insulating space are similar, but in a conducting medium, current sources replace the charges in a static field, and current densities replace field strengths. Three main forms of stationary electric field can be distinguished and can be used or encountered in the electrophysiological practice: 1. Homogeneous field 2. Monopole field 3. Dipole field These are illustrated in Fig. 6.1 and will be described briefly below.
Introduction to Electrophysiological Methods and Instrumentation, Second Edition https://doi.org/10.1016/B978-0-12-814210-3.00006-2
153
Copyright © 2019 Elsevier Ltd. All rights reserved.
154
6. VOLUME CONDUCTION: ELECTRIC FIELDS IN ELECTROLYTE SOLUTIONS
FIGURE 6.1
(A) Homogeneous, (B) monopole, and (C) dipole fields.
HOMOGENEOUS ELECTRIC FIELD A homogeneous field arises when an electric current is fed through two parallel electrodes that cover the short sides of an elongated tray. This configuration, sometimes called an electrolytic trough, is shown in Fig. 6.1A. It can be used to illustrate the quantities used to describe electric fields in conducting fluids. Let us assume that we feed a direct current (of strength I) through the trough from right to left, by making the right electrode positive with respect to the left one. Because of the constant cross-section of the tank and because the electrodes fit the short sides entirely, current flows in straight lines, parallel to the long sides of the trough. Since the properties of the field are the same throughout the tank, it is called a homogeneous field. The quantities that describe the electric field in the liquid are analogous to the quantities that describe current through an object, viz. an electrical resistance, but expressed per spatial unit. In analogy to resistance R, we have the resistivity r (rho), which is usually expressed in Ucm. In analogy with current, we have the current density J, expressed in A/cm2. Finally, the voltage gradient, grad(V) ¼ vV/vx expressed in V/cm, replaces the voltage. Resistivity, current density, and voltage gradient thus describe the properties of a unit cube of 1 cm side, which has R ¼ r, I ¼ J, and U ¼ vV/vx. Obviously, these quantities are related via an analogue of Ohm’s Law:
THE MONOPOLE FIELD
U ¼ IR4
155
vV ¼ Jr vx
To describe the potential, we need a reference value. In a practical circuit, one of the electrodes would be grounded, but theoretically, it is more attractive to take zero at the middle of the trough (note that otherwise the electrode polarization would introduce an error). Thus, the potential (U) can be determined as the voltage between a measuring electrode and a central reference electrode, l cm apart, and follows from: U ¼ l
vV vx
or
U ¼ Jrl
Lines connecting points of equal potential, or equipotential lines, are parallel to the electrodes that generate the field. The homogeneous electric field can be used to test and calibrate one’s measurement setup.
THE MONOPOLE FIELD The situation at an isolated electrode, or monopole, is completely different (Fig. 6.1B). A monopole field arises, theoretically, when the reference, or ground electrode, is situated at infinity (and is infinitely large). In practice, to approximate a monopole field sufficiently, the ground electrode has to be far away only with respect to the distances involved in the measurements. For example, the field around a point-shaped electrode in a 10 10 2 cm3 tray is sufficiently monopole-like over a radius of at least a few millimeters. The electrode itself must, theoretically again, be infinitely small, i.e., pointlike. If one uses the tip of a glass micropipette, the monopole situation is sufficiently approximated, even close to the tip. According to a general principle first established by Gauss, the current injected, I, spreads out in all directions and the current leaving a closed surface around the source must be identical to the current injected. Now, if the medium is isotropic, spherical symmetry may be ! assumed. The current density, J at a distance r from the tip of the electrode is then given by: ! J ¼
I br 4pr2
! The arrow over the J indicates that J is a vector, i.e., it has both a size and a direction, r is the radius of a sphere with surface 4pr2, and br is a unit vector pointing in the same direction ! as J (i.e., from the center of the sphere inward or outward depending on the sign of the current). If r is the resistivity of the medium, then according to Ohm’s law as we have seen above: ! E 1 vV ! J ¼ ¼ r r vr
156
6. VOLUME CONDUCTION: ELECTRIC FIELDS IN ELECTROLYTE SOLUTIONS
! where E is the electrical field and vV/vr is the potential gradient induced by the current. Note that V, the potential, is a scalar and not a vector. Combining the two equations gives: rI vV ¼ 2 4pr vr In order to obtain an equation in V rather than its gradient, we need to integrate: Z Z rI vV dr dr ¼ 2 4pr vr which simplifies to: VðrÞ ¼
rI 4p
Z
1 dr r2
and hence, VðrÞ ¼
rI 4pr
THE DIPOLE FIELD AND CURRENT SOURCE DENSITY ANALYSIS A dipole can be considered as two monopoles, the first of which carries a current þI. The other one “absorbs” the same current and hence carries a current eI. The electrodes are also known as source and sink, respectively. To compute the voltage gradient, current density, or potential at any point, the contributions of both poles simply sum, taking the respective distances ri into account. Therefore, current density, voltage gradient, and potential at any point p follow from: I I r I I þ gradðVÞ ¼ r$J ¼ þ JP ¼ P 4$p$r21 4$p$r22 4$p r21 r22 r I I VðpÞ ¼ þ 4$p jr1 j jr2 j
(6.1)
This leads to the pattern shown in Fig. 6.1C. Since in the middle between the poles the contributions of the two poles cancel, the line through the center of the dipole, perpendicular to the line connecting the poles, has zero potential. This midline is also shown in Fig. 6.1C. Close to the electrodes, the equipotential lines are almost circular (monopole-like) and too closely spaced to be distinguished in the figure. Multipole fields can be considered as a combination of poles, by extension of the dipole equations. Dipole fields arise in a number of situations, of both natural and technical origin.
MULTIPOLE FIELDS AND CURRENT SOURCE DENSITY
157
FIGURE 6.2 Iron filings showing magnetic field lines of a (hidden) bar magnet.
However, because of their complex form, dipole fields are less suited for testing and calibration purposes. Note that, properly speaking, current density and voltage gradient are vector quantities, having both a magnitude and a direction. For simplicity, the values given here are the maximum values that pertain to the mentioned directions. In directions perpendicular to the stated ones, both current density and voltage gradient are zero. To be more precise, they follow a cosine law: As an example, the current density at an angle of 45 degrees to the direction of the field lines, in the abovementioned homogeneous field, would be J cos(45 ), or about 0.71 J. The shape of the dipole field is best known from that of a bar magnet. In fact, the electric and magnetic dipole fields share a common mathematical form. In Fig. 6.2, the magnetic field lines of a hidden bar magnet are made visible by the layer of iron filings.
MULTIPOLE FIELDS AND CURRENT SOURCE DENSITY The situation where there are many sources and sinks is a generalization of Eq. (6.1). These current sources, Ik, located at ! rk , contribute to the potential at ! r by summing: Vð! rÞ ¼
X k
rI k ! ! 4p r rk
(6.2)
It is easy to calculate the field potential distribution in space with Eq. (6.2) if the distribution of current sources and sinks is known. This has been done as an exercise in Fig. 6.3A and B for one dimension. However, the problem is usually the inverse: given the field potential
158
6. VOLUME CONDUCTION: ELECTRIC FIELDS IN ELECTROLYTE SOLUTIONS
Simulation of field potentials and simplified current source density analysis in one dimension (x). Suppose six sources and sinks are distributed as in (A). The length of the arrows is proportional to the current. (B) The field potential (V) at each point along the x-axis can be calculated with Eq. (6.2). If the field potentials at an ensemble of points are known, the location of current sources and sinks can be estimated by taking the second derivative of V (D). The first derivative is shown in (C). Note that the second derivative of V is proportional to the original current source distribution in (A).
FIGURE 6.3
distribution where are the current sources and sinks located? That is because these indicate, for example, active brain regions and sources of EEG potentials (see Chapter 9). The methods of analysis that solve this problem are collectively called current source density (CSD) analysis. Unfortunately, CSD math is complicated and not very straightforward.1,2 We will therefore not attempt to develop it here but instead give an important result that can be obtained: 1 v2 V v2 V v2 V ! þ þ Cð r Þ ¼ r vx2 vy2 vz2 This is the Poisson equation, which states that if one takes the second partial derivative of the field potential with respect to the spatial coordinates, one obtains the 3current source density distribution, C ! r (see Fig. 6.3C and D). The dimension of C is A/m , hence current per volume, which is not the same as current per surface or current density (J). C(r/) can be thought of as the current emanating from a small volume. In the above equation, r is supposed to be uniform in all spatial directions. If r(x,y,z) depends on spatial coordinates, the equation becomes somewhat more intricate. However, CSD analysis is useful and often applied even in the absence of any knowledge of eventual resistivity variations in the tissue. In that case, C ! r cannot be deduced. For this reason, usually only the second derivative is taken and the result, expressed as V/m2.
REFERENCES
159
This description of the spatial distribution of electric currents in ion solutions ends our discussion of electrochemical processes in this and the two preceding chapters. More detailed information can be found in the excellent book on this subject by Bard and Faulkner.3
References 1. Einevoll GT, Kayser C, Logothetis NK, Panzeri S. Modelling and analysis of local field potentials for studying the function of cortical circuits. Nat Rev Neurosci 2013;14(11):770e85. 2. Nicholson C. Theoretical analysis of field potentials in anisotropic ensembles of neuronal elements. IEEE Trans Biomed Eng 1973;20(4):278e88. 3. Bard AJ, Faulkner LR. Electrochemical methods: fundamentals and applications. New York: Wiley; 2001.
6 Volume Conduction: Electric Fields in Electrolyte Solutions O U T L I N E Homogeneous Electric Field
154
The Monopole Field
155
The Dipole Field and Current Source Density Analysis
Multipole Fields and Current Source Density
157
References
159
156
Sources of electric current in water or in salt solutions give rise to stationary electric fields. The theory of electricity was developed in the 18th century with static forms of electricity. Charges in space, maintaining electrical potential differences, were kept from leaking away by dry air and by insulators like glass or ceramics. In a conducting medium, however, an electric current would flow and discharge any static source within a fraction of a second. Therefore, to generate an electric field in a fluid, we have to supply current continually (DC and/or AC). The mathematical descriptions of electric fields in water and in an insulating space are similar, but in a conducting medium, current sources replace the charges in a static field, and current densities replace field strengths. Three main forms of stationary electric field can be distinguished and can be used or encountered in the electrophysiological practice: 1. Homogeneous field 2. Monopole field 3. Dipole field These are illustrated in Fig. 6.1 and will be described briefly below.
Introduction to Electrophysiological Methods and Instrumentation, Second Edition https://doi.org/10.1016/B978-0-12-814210-3.00006-2
153
Copyright © 2019 Elsevier Ltd. All rights reserved.
154
6. VOLUME CONDUCTION: ELECTRIC FIELDS IN ELECTROLYTE SOLUTIONS
FIGURE 6.1
(A) Homogeneous, (B) monopole, and (C) dipole fields.
HOMOGENEOUS ELECTRIC FIELD A homogeneous field arises when an electric current is fed through two parallel electrodes that cover the short sides of an elongated tray. This configuration, sometimes called an electrolytic trough, is shown in Fig. 6.1A. It can be used to illustrate the quantities used to describe electric fields in conducting fluids. Let us assume that we feed a direct current (of strength I) through the trough from right to left, by making the right electrode positive with respect to the left one. Because of the constant cross-section of the tank and because the electrodes fit the short sides entirely, current flows in straight lines, parallel to the long sides of the trough. Since the properties of the field are the same throughout the tank, it is called a homogeneous field. The quantities that describe the electric field in the liquid are analogous to the quantities that describe current through an object, viz. an electrical resistance, but expressed per spatial unit. In analogy to resistance R, we have the resistivity r (rho), which is usually expressed in Ucm. In analogy with current, we have the current density J, expressed in A/cm2. Finally, the voltage gradient, grad(V) ¼ vV/vx expressed in V/cm, replaces the voltage. Resistivity, current density, and voltage gradient thus describe the properties of a unit cube of 1 cm side, which has R ¼ r, I ¼ J, and U ¼ vV/vx. Obviously, these quantities are related via an analogue of Ohm’s Law:
THE MONOPOLE FIELD
U ¼ IR4
155
vV ¼ Jr vx
To describe the potential, we need a reference value. In a practical circuit, one of the electrodes would be grounded, but theoretically, it is more attractive to take zero at the middle of the trough (note that otherwise the electrode polarization would introduce an error). Thus, the potential (U) can be determined as the voltage between a measuring electrode and a central reference electrode, l cm apart, and follows from: U ¼ l
vV vx
or
U ¼ Jrl
Lines connecting points of equal potential, or equipotential lines, are parallel to the electrodes that generate the field. The homogeneous electric field can be used to test and calibrate one’s measurement setup.
THE MONOPOLE FIELD The situation at an isolated electrode, or monopole, is completely different (Fig. 6.1B). A monopole field arises, theoretically, when the reference, or ground electrode, is situated at infinity (and is infinitely large). In practice, to approximate a monopole field sufficiently, the ground electrode has to be far away only with respect to the distances involved in the measurements. For example, the field around a point-shaped electrode in a 10 10 2 cm3 tray is sufficiently monopole-like over a radius of at least a few millimeters. The electrode itself must, theoretically again, be infinitely small, i.e., pointlike. If one uses the tip of a glass micropipette, the monopole situation is sufficiently approximated, even close to the tip. According to a general principle first established by Gauss, the current injected, I, spreads out in all directions and the current leaving a closed surface around the source must be identical to the current injected. Now, if the medium is isotropic, spherical symmetry may be ! assumed. The current density, J at a distance r from the tip of the electrode is then given by: ! J ¼
I br 4pr2
! The arrow over the J indicates that J is a vector, i.e., it has both a size and a direction, r is the radius of a sphere with surface 4pr2, and br is a unit vector pointing in the same direction ! as J (i.e., from the center of the sphere inward or outward depending on the sign of the current). If r is the resistivity of the medium, then according to Ohm’s law as we have seen above: ! E 1 vV ! J ¼ ¼ r r vr
156
6. VOLUME CONDUCTION: ELECTRIC FIELDS IN ELECTROLYTE SOLUTIONS
! where E is the electrical field and vV/vr is the potential gradient induced by the current. Note that V, the potential, is a scalar and not a vector. Combining the two equations gives: rI vV ¼ 2 4pr vr In order to obtain an equation in V rather than its gradient, we need to integrate: Z Z rI vV dr dr ¼ 2 4pr vr which simplifies to: VðrÞ ¼
rI 4p
Z
1 dr r2
and hence, VðrÞ ¼
rI 4pr
THE DIPOLE FIELD AND CURRENT SOURCE DENSITY ANALYSIS A dipole can be considered as two monopoles, the first of which carries a current þI. The other one “absorbs” the same current and hence carries a current eI. The electrodes are also known as source and sink, respectively. To compute the voltage gradient, current density, or potential at any point, the contributions of both poles simply sum, taking the respective distances ri into account. Therefore, current density, voltage gradient, and potential at any point p follow from: I I r I I þ gradðVÞ ¼ r$J ¼ þ JP ¼ P 4$p$r21 4$p$r22 4$p r21 r22 r I I VðpÞ ¼ þ 4$p jr1 j jr2 j
(6.1)
This leads to the pattern shown in Fig. 6.1C. Since in the middle between the poles the contributions of the two poles cancel, the line through the center of the dipole, perpendicular to the line connecting the poles, has zero potential. This midline is also shown in Fig. 6.1C. Close to the electrodes, the equipotential lines are almost circular (monopole-like) and too closely spaced to be distinguished in the figure. Multipole fields can be considered as a combination of poles, by extension of the dipole equations. Dipole fields arise in a number of situations, of both natural and technical origin.
MULTIPOLE FIELDS AND CURRENT SOURCE DENSITY
157
FIGURE 6.2 Iron filings showing magnetic field lines of a (hidden) bar magnet.
However, because of their complex form, dipole fields are less suited for testing and calibration purposes. Note that, properly speaking, current density and voltage gradient are vector quantities, having both a magnitude and a direction. For simplicity, the values given here are the maximum values that pertain to the mentioned directions. In directions perpendicular to the stated ones, both current density and voltage gradient are zero. To be more precise, they follow a cosine law: As an example, the current density at an angle of 45 degrees to the direction of the field lines, in the abovementioned homogeneous field, would be J cos(45 ), or about 0.71 J. The shape of the dipole field is best known from that of a bar magnet. In fact, the electric and magnetic dipole fields share a common mathematical form. In Fig. 6.2, the magnetic field lines of a hidden bar magnet are made visible by the layer of iron filings.
MULTIPOLE FIELDS AND CURRENT SOURCE DENSITY The situation where there are many sources and sinks is a generalization of Eq. (6.1). These current sources, Ik, located at ! rk , contribute to the potential at ! r by summing: Vð! rÞ ¼
X k
rI k ! ! 4p r rk
(6.2)
It is easy to calculate the field potential distribution in space with Eq. (6.2) if the distribution of current sources and sinks is known. This has been done as an exercise in Fig. 6.3A and B for one dimension. However, the problem is usually the inverse: given the field potential
158
6. VOLUME CONDUCTION: ELECTRIC FIELDS IN ELECTROLYTE SOLUTIONS
Simulation of field potentials and simplified current source density analysis in one dimension (x). Suppose six sources and sinks are distributed as in (A). The length of the arrows is proportional to the current. (B) The field potential (V) at each point along the x-axis can be calculated with Eq. (6.2). If the field potentials at an ensemble of points are known, the location of current sources and sinks can be estimated by taking the second derivative of V (D). The first derivative is shown in (C). Note that the second derivative of V is proportional to the original current source distribution in (A).
FIGURE 6.3
distribution where are the current sources and sinks located? That is because these indicate, for example, active brain regions and sources of EEG potentials (see Chapter 9). The methods of analysis that solve this problem are collectively called current source density (CSD) analysis. Unfortunately, CSD math is complicated and not very straightforward.1,2 We will therefore not attempt to develop it here but instead give an important result that can be obtained: 1 v2 V v2 V v2 V ! þ þ Cð r Þ ¼ r vx2 vy2 vz2 This is the Poisson equation, which states that if one takes the second partial derivative of the field potential with respect to the spatial coordinates, one obtains the 3current source density distribution, C ! r (see Fig. 6.3C and D). The dimension of C is A/m , hence current per volume, which is not the same as current per surface or current density (J). C(r/) can be thought of as the current emanating from a small volume. In the above equation, r is supposed to be uniform in all spatial directions. If r(x,y,z) depends on spatial coordinates, the equation becomes somewhat more intricate. However, CSD analysis is useful and often applied even in the absence of any knowledge of eventual resistivity variations in the tissue. In that case, C ! r cannot be deduced. For this reason, usually only the second derivative is taken and the result, expressed as V/m2.
REFERENCES
159
This description of the spatial distribution of electric currents in ion solutions ends our discussion of electrochemical processes in this and the two preceding chapters. More detailed information can be found in the excellent book on this subject by Bard and Faulkner.3
References 1. Einevoll GT, Kayser C, Logothetis NK, Panzeri S. Modelling and analysis of local field potentials for studying the function of cortical circuits. Nat Rev Neurosci 2013;14(11):770e85. 2. Nicholson C. Theoretical analysis of field potentials in anisotropic ensembles of neuronal elements. IEEE Trans Biomed Eng 1973;20(4):278e88. 3. Bard AJ, Faulkner LR. Electrochemical methods: fundamentals and applications. New York: Wiley; 2001.