Classification of the extracellular fields produced by activated neural structures
© Richerson et al; licensee BioMed Central Ltd. 2005
Received: 17 June 2005
Accepted: 07 September 2005
Published: 07 September 2005
Classifying the types of extracellular potentials recorded when neural structures are activated is an important component in understanding nerve pathophysiology. Varying definitions and approaches to understanding the factors that influence the potentials recorded during neural activity have made this issue complex.
In this article, many of the factors which influence the distribution of electric potential produced by a traveling action potential are discussed from a theoretical standpoint with illustrative simulations.
For an axon of arbitrary shape, it is shown that a quadrupolar potential is generated by action potentials traveling along a straight axon. However, a dipole moment is generated at any point where an axon bends or its diameter changes. Next, it is shown how asymmetric disturbances in the conductivity of the medium surrounding an axon produce dipolar potentials, even during propagation along a straight axon. Next, by studying the electric fields generated by a dipole source in an insulating cylinder, it is shown that in finite volume conductors, the extracellular potentials can be very different from those in infinite volume conductors. Finally, the effects of impulses propagating along axons with inhomogeneous cable properties are analyzed.
Because of the well-defined factors affecting extracellular potentials, the vague terms far-field and near-field potentials should be abandoned in favor of more accurate descriptions of the potentials.
Classical Properties of Near and Far Field Potentials
Relatively Independent of Recording Electrode Position
Strongly dependent on Position of Recording Electrode
Distribution on Skin
Negative or Positive
Monopolar or Bipolar
It should be noted that in many parts of the paper the terms charge, dipole, or quadrupole will be used although it is more proper to refer to point current sources, dipolar or quadrupolar current sources. This does not alter any of the fundamental conclusions in this paper.
Model derivations and simulations
A. Extracellular fields and axonal geometry
Understanding how the geometry of a generalized axon affects the extracellular fields produced when it is depolarized is particularly instructive. Previous work by Plonsey and Rosenfalck was particularly useful in defining extracellular fields for active fibers of finite and infinite length in infinite homogenous media [8–10]. However, these models used previously assume fibers have circular cross sections, are straight, and are located in uniform conducting media. Holt and Koch  have studied this problem from the viewpoint of the cable equations. The general solution we present here is based on the geometry of surfaces, a simplified version of which has been applied previously in analyzing magnetic stimulation of a bent neuron [11, 14]. Any surface, such as the surface of the generalized axon, can be characterized by a vector function of two variables . In this development, these two variables will be called s and θ. It will be instructive to think of s as the distance along the length of the axon and θ as the angular position around the axon. The extracellular field generated when a small region of a nerve along its long axis is depolarized is due to relatively localized movements of charges. Thus, the extracellular fields in an infinite volume conductor can be calculated using the multipole expansion, the first terms of which are the field generated by the net dipole moment and the net quadrupole moment of the source. Appendix A (see Additional file #1) demonstrates the calculation of the net dipole and quadrupole moments produced by depolarization of the nerve in the case in which the trans membrane potential V m (s, θ) = V m (s) is independent of θ. This leads to the following expression for the dipole moment per unit activated length:
and the expression for the magnitude of the dipole moment per unit length:
where a(s, θ) is the distance from the centroid at s to the point (s, θ) on the surface. This is the mean square radius of the axon at position s.
One implication of the above equations is that the dipole moment produced by each segment of the axon depends only on the curvature of the curve of centroids and the change in the mean square radius with distance. It does not depend on the detailed shape of the axon. It is also important to realize that the component of the dipole moment produced by changing axonal radius is directed along the tangent to the centroid curve X*(s) and the component produced by the curvature of the axon are oriented along the normal to this same curve.
The quadrupole moment tensor produced when a small axonal segment is depolarized can be evaluated in a similar way although with substantially more algebra (See appendix A (see Additional file #1)) :
where τ*(s) is the torsion  of and the and are integrals of the cube of the radius multiplied by sin(θ) and cos(θ) as noted in appendix A (see Additional file #1). This has a number of simple implications. First, whenever the axon cross section has inversion symmetry around the centroid (a(s, θ) = a(s, θ + π)), then and are both zero and:
B. Changes in local extracellular electrical environment
Once the effects of axon geometry on the recorded extracellular potentials have been explored, it is natural to explore the effects that changes in the extracellular electrical environment may have.
It has been suggested that the presence of inhomogeneities in the extracellular space can change the fundamental characteristics of the extracellular fields, producing "far-field" potentials (A discussion of the inverse problem of stimulating a nerve near a region with a localized change in conductivity can be found in Roth  and a discussion of the general effects of inhomogeneities in conductivity can be found in Geselowitz .). In order to understand these effects, it is helpful to begin by studying the fields generated by a small conducting sphere of radius a placed at position in an arbitrary electric potential . An approximate expression for the potential produced by the presence of the sphere far from the sphere is (see Appendix B (see Additional file #1)):
where, ε0 is the dielectric constant of the rest of the medium and δ ij is the Kronecker delta. A similar expression holds for the case of a conducting sphere with ε replaced by σ and ε0 replaced by σ0, the conductivity of the medium outside the sphere.
In contrast to the results obtained above, it is important to note that dipolar fields are NOT generated when a nerve impulse characterized by a quadrupolar source travels near a plane conducting boundary. The effect of the plane boundary on the extracellular fields can be simulated by image charges placed at the location that their optical images would have in the boundary plane. Thus, if the source charge density is quadrupolar, the image charges are quadrupolar.
C. Effects of finite volume conductors
Since practical recordings are not carried out in an infinite volume conductor, it is useful to consider a simple qualitative model demonstrating the effects that placing a charge in a finite volume conductor can have on the recorded extracellular fields (Appendix C (see Additional file #1)). In particular, the special case of charges placed inside a cylindrical volume conductor that is insulating except at its two ends is very instructive. Assuming that all charges are confined to a finite segment of the cylinder z0 - Δ <z <z0 + Δ, the potential outside the region in which the charges exist can be written:
It is instructive to consider the mean value of the potential along segments perpendicular to the long axis of the cylinder:
These integrals are not well defined in an infinite volume and so the results are restricted to finite cylindrical volume conductors. Integrating the Poisson equation over the radial and angular variables yields an expression for the mean potentials in the charge free regions and integrating the Poisson equation across the region containing charges produces the following expression for the averaged electrical potential:
In this equation, Q is the total charge contained in the region and dz is the component of the dipole moment of the charge along the long axis of the cylinder. a<and b<are constants chosen to satisfy the boundary conditions. This suggests that, far enough from the sources, the actual potential has a linear dependence on the axial coordinate. The form of the above equation suggests that if the source is dipolar there may be a step change in the potential across the region containing the charges that is proportional to the dipole moment. A quadrupolar field would be expected to be least influenced by the presence of the finite cylindrical volume conductor. These results are similar to those obtainable using the Green's function for the appropriate cylindrical boundary conditions .
The critical question is to define the regime in which these results apply. Some arguments relating to this question are presented in Appendix C (see Additional file #1). They suggest, for an infinitely long cylinder, that when the axial coordinate of the point of observation is many cylinder diameters from the charges (z >> a), the potentials develop the linear behavior discussed above. Similarly, when the point of observation is much less than a cylinder diameter away from the charge, it is expected that the potential will possess similarities to that seen in an infinite volume conductor.
The sensitivity of the recorded extracellular potentials to the shape, size and character of the volume conductor in which the sources are immersed is evident in these calculations. They are also very sensitive to the detailed structure of the boundary conditions imposed. These observations are consistent with the results obtained in a number of experiments performed by Jewett [6, 18] who recorded changes in extracellular potentials from isolated nerves as the impulse propagated into volume conductors of different sizes and shapes. These observations on the importance of finite volume effects are also consistent with the conclusions drawn from finite element models of Cunningham  who evaluated extracellular fields created in a 2-D finite element model of the hand, arm and torso. However, as the geometry of the system becomes more complex, the interpretation of results also becomes more complex. Mapping studies of the electric fields generated by artificial dipole and quadrupole sources by Dumitru and King [20, 21] reproduce the above theoretical and computational results and in particular demonstrate that dipole sources in cylinders are associated with large regions of constant potential away from the source. However, these authors state that quadrupolar sources do produce fields far from the source when inside a cylinder under certain circumstances. This may result from the fact that in these simulations discrete electrodes were used to generate the quadrupolar source.
Clinical studies [22–24] on the distribution of the potentials generated by median nerve stimulation in humans do demonstrate some potentials with a latency that is independent of electrode position that are broadly distributed over the skin. The spatial distribution of these potentials does not appear to be quadrupolar or bipolar in nature as they seem to decrease in amplitude very slowly with distance from the putative source. It is thus likely that boundary effects play a significant role in the generation of these potentials.
D. Changes in cable properties of an axon
The effects that a localized alteration in the cable properties of an axon has on the generated extracellular fields are also of interest . As long as the axon remains cable-like, straight and with a constant radius, equation (2) shows that only quadrupolar potentials will be generated even if cable properties of the axon change along its length. However, as in the case of the impulse approaching the plane boundary, the effective quadrupole moment will change near the point at which the cable properties change.
In order to illustrate these effects, a simple simulation was undertaken to find the transmembrane potential as a function of time for the situation in which a stimulator moving with a constant velocity injects a constant amount of current through the membrane. It is assumed that the membrane resistance quadruples at the point z = 100. The equations describing this simple model and the method of computing the extracellular fields is discussed in Appendix D (see Additional file #1).
F. Spectral analysis of extracellular potentials
As in the above section, it is clear that the frequency of the extracellular potentials produced as an action potential moves through a complex medium is an important factor in interpreting recordings. Thus, an understanding of the spectral content of the potentials recorded from a travelling action potential is important. This problem is discussed in detail in Appendix E (see Additional file #1) where it is shown that the power spectrum at angular frequency ω of the potential recorded a distance R from an axon along which an action potential propagates at a constant velocity v is given by:
where α is the quadrupole moment associated with the action potential, β is the spatial extent of the action potential and:
This argument demonstrates that applying a low frequency filter when recording electrodes are placed far from the source can remove the responses generated as the action potential traverses the axon. However, the frequency spectrum of the potentials generated when an impulse reaches any boundary must be similar to that of the "form factor" only since these potentials do not propagate down the axon (The "structure factor" component arises only out of propagation of the impulse as is evident since only this term contains the propagation velocity.). Thus, as in Table 2, if the distance between the recording electrodes and the axon is sufficiently great, a low frequency filter may eliminate the propagating potentials to a much greater degree than the potentials generated at interfaces. This may obscure the true nature of the generators of the recorded extracellular fields.
G. The effect of recording montage
H. Radiation fields
In this paper, a number of theoretical mechanisms underlying the generation and recording of extracellular potentials during propagation of a nerve impulse down a generalized neural structure were presented and illustrated with data from simulations. This approach provides a much greater degree of insight into the generation of extracellular fields than the use of simple analytic approaches, finite element simulations, or clinical recordings in isolation. In particular, although necessarily approximate and idealized, the theoretical calculations allow for predictions of the general categories of field types that can be encountered and estimates of their magnitude.
When a neural membrane is depolarized the change in transmembrane potential is associated with a dipole moment density oriented perpendicular to the surface of the membrane that is proportional to the change in potential. Because of the symmetry of the ideal cylindrical axon, the net dipole moment associated with the activation disappears and so the propagating nerve impulse is typically a quadrupolar field far from the axon. However, when the cylindrical symmetry is broken by bending the nerve, a net dipole moment along the normal to the axon may be generated. In addition, if there are changes in the axon radius along the length of the cylinder there will be dipole moments generated directed along the tangent to the axon.
The theoretical results and simulations discussed above are in agreement with the experimental results obtained by Dupree and Jewett  which showed a peak in the extracellular potential that is generated when frog sciatic nerve was bent. These recordings were however made with electrodes always within 200 mm of an axon placed in a finite volume conductor. Because of the relative proximity of the electrodes to the source the full transition from recording only traveling waves to recording only potentials as the impulse enters the bend is not seen in this experiment.
In addition, the above theoretical results are applicable to situations in which an axon terminates in a sealed end. This is equivalent to allowing the axon radius to change from its baseline value to zero at the end of the axon. As described by (2) a net dipole moment will appear at such points and a peak will be recorded in the extracellular potentials. Equation (2) does not suggest that a net dipole moment will appear in a cylindrical axon with constant radius that is cleanly cut (ie not sealed) at the end. Dumitru and Jewett  suggest that this should occur because, once the impulse reaches the end of the axon, there is no longer "neural tissue to support the leading dipole". They speculate that at this point only the fields generated by the trailing dipole appear. This is not the case, since the leading and trailing dipoles are not physical entities and are only used to represent the field generated by the depolarized membrane surface. In fact, as the impulse reaches the cut end, the equivalent leading dipole must remain at the end of the axon while the trailing dipole gradually reaches the end of the axon. Thus, the quadrupole moment of the impulse will decline linearly to zero as the impulse reaches the cut end of the nerve since the total area of depolarization will diminish gradually. Thus, no peak in extracellular potential should be recorded as the impulse reaches a pure "cut end". However, if there is any change in axon diameter at the end, a large potential may be generated.
As breaking the cylindrical symmetry of an axon by distorting the axon itself can produce net dipolar fields, so can placing an axon in an environment that is not cylindrically symmetric. In particular, placing a sphere of altered conductivity near the axon destroys cylindrical symmetry and results in a dipolar potential even when extracellular fields generated by the axon itself would be quadrupolar. Placing an infinite conducting plane perpendicular to the path of the traveling impulse, however, does not destroy the symmetry of the depolarized axonal segments and so, although the effective quadrupole moment of the propagating action potential changes as the impulse passes through this barrier, no dipolar potentials are generated. Similarly, when a propagating impulse reaches a point on the axon at which there is a sudden change in the cable properties, there is a change in the quadrupole moment but no dipole potential is generated.
The results obtained in this section should be compared to the experiments of Nakanishi  who placed nerves through multiple partitions and found that, with one electrode in each partitioned segment, potentials were generated that could be related to the passage of the impulse from one compartment into another. The amplitude of this potential was correlated with the impedance between the partitions, a measure of the size of the gap in the partition. This also suggested that the geometry of the partitions was critical to the development of the responses. This seems plausible and recordings with electrodes either far from the axon or in multiple locations to detect the angular dependence of the potential could be helpful in determining whether the true character of the responses are dipolar or quadrupolar.
The above simulations of the potentials produced as an impulse passes through a plane conducting barrier are different from those found by Stegeman [28, 29] who computed the potentials produced by a simulated impulse traveling in an insulating cylindrical shell filled with media of different conductivity. In this model, an action potential travels at constant velocity along the long axis (z axis) of the cylinder at its center and encounters a stepwise change in conductivity at the point z = 0. Their simulations suggested that, as the nerve impulse passes the point z = 0, it generates a stepwise constant shift in the electric potential at all points z > 0 although no or minimal change is seen for in the region of space z < 0. Although there are sudden changes in the potential in model described above as the impulse passes through the boundary between the regions of differing conductivity, all of the potentials do decline with distance from the source as would be expected from a quadrupolar source. This difference between Stegeman's model and the model of an action potential approaching a plane conducting barrier as discussed above relates to the unusual types of electric field that are generated in a finite cylindrical volume conductor. As discussed in section C, when sources are placed in an insulating cylinder, the recorded potential is similar in many ways to its value in an infinite volume conductor when the recording electrodes are very close to the source. However, neither dipolar or quadrupolar fields are recorded more than 2 or 3 cylinder radii from the source but the potential becomes a linear function of the axial coordinate. This behavior is specific to insulating cylindrical volume conductors. Although there are distortions of the potentials in a circular volume conductor and in a cylindrical volume conductor that is not perfectly insulating, many of the characteristics of the potentials recorded in an infinite volume conductor persist. Field distributions produced primarily as a result of boundary effects can be identified in actual recordings when the potential does not drop as or or when the expected angular behavior expected with a dipolar source, cos(θ), or a quadrupolar source, 3 cos2(θ)-1, are not evident.
The effect of recording montage on the recorded potentials was discussed. In any recordings of extracellular potentials in an infinite volume conductor, similar information is recorded from closely spaced electrodes and with a distant reference because the potentials drop off with distance in a predictable manner. However, the potentials generated by sources within a finite volume conductor with insulating boundary conditions become linear functions of distance far from the source and so bipolar recordings will not record changes in the location of the sources over time while monopolar or reference recordings will.
The spectral properties of the generated extracellular fields were also elucidated with the result that the frequency spectrum generated by a traveling impulse is cutoff at a frequency which is inversely proportional to the distance of the recording electrodes from the axon. Put differently, the frequency of potentials recorded far from a traveling quadrupolar action potential drops as the distance between the recording electrodes and the axon increases. On the other hand, the frequency spectrum generated when an impulse reaches a boundary is independent of the distance from the generator. In this case the frequency is cutoff at roughly the size of the nerve impulse divided by the conduction velocity. Understanding this frequency behavior is important in that it forms the scientific basis for choosing filters to optimally record potentials with different origins. For instance, if recording of traveling potentials far from the axon, is desired, it will be important to keep low frequency (high pass) filters as low as possible. However, if the goal of an experiment is to analyze the changes in the local environment that a nerve impulse encounters, it is best to set the low frequency filter high enough to reduce the amplitude of the traveling potentials .
Finally, in a discussion of the possibility that radiation fields are generated during the propagation of neural impulses or when an impulse encounters a localized change in conductivity, it was demonstrated that such fields are extremely tiny. The main reasons for this are the fact that the rate of change in charge distributions is slow and the radiated power increases as the fourth power of the frequency. The radiated power also depends on the fourth power of the size of the generator and neural structures are typically small.
Summary of Extracellular Field Types Far From An Axon
Change in Dipole Direction Moving Dipole Through Region of Altered Electrical Properties
cos(θ) -Free Particle. Complex Angular Dependence for Transition Radiation
Change in Axon Radius (Dipole Tangent To Axon)
Includes "Sealed End" Axons
Axon curvature (Dipole Normal To Axon)
Localized Changes in Extracellular Electrical Properties
Only When Cylindrical Symmetry is Broken
Impulse Travelling Along Uniform, Straight, Homogenous Axon in Isotropic Electrical Environment.
Changes in Cable Properties.
Localized Changes in Extracellular Electrical Properties
For Instance the Plane Conducting Boundary
Fields Generated Because of the Finite Volume In Which The Generator is Immersed
Depends Critically on Boundary Shape and Size
Stationary or Travelling
Potentials Linear Far From Source For A Cylindrical Volume Conductor
Danielle Perry and Mark Ingram were sponsored by the NSF REU grant to the department of Physics at Bucknell University. The authors also wish to thank Robert Marchitelli who also contributed to the calculations in this paper under the sponsorship of the NSF REU grant to the department of physics at Bucknell. Finally, we wish to thank Dr. Marty Ligare from the department of Physics at Bucknell University for his help in organizing this project and his ongoing assistance in all aspects of this project. Finally, we wish to thank one of the anonymous reviewers for comments that have significantly improved this manuscript.
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