Far-field potentials in circular volumes: the effect of different volume sizes and intercompartmental openings.

Preliminary investigations of circular volume conductors suggested that far-field potential magnitude declines progressively slower with increasing radial distance from a current source and follows a cosine function with angular displacement of the recording electrode from the electrical generator's axis. Using circular volumes of 6 differing radii, the mathematical relationship between angle, radii, and far-field potential amplitude is determined. Previous theoretical relationships of amplitude versus dipolar spacing, current, and distance from a dipole generator in a bounded volume conducting medium are verified for the near-field. Far-field potentials in circular volumes are found to become constant at radii greater than 75% of the bounded volume's radius. Additionally, an adjoining volume conductor acts simply as a passive fluid-filled electrode (wick electrode) to the circular volume containing the generator until the intercompartmental opening to the circular volume exceeds 20% of its circumference. This finding was clinically supported by recording similar P9 somatosensory-evoked far-field potentials generated caudal to the foramen magnum from various portions of the cranium, whose connections to the torso, foramen magnum, and neck, average 6.2% and 17.8%, respectively. Finally, 3 circular volume conductors were connected in series by channels less than 20% of the volume conductor's circumference. Both adjoining circular volumes were equipotential to the far-field potential present at the boundary of the first circular volume containing the dipole generator. This observation supports the clinical finding of far-field potential transmission through multiple human bodies in conductive contact.     

Far-field potentials in cylindrical and rectangular volume conductors

The occurrence of a transient dipole is one method of producing a far-field potential. This investigation qualitatively defines the characteristics of the near-field and far-field electrical potentials produced by a transient dipole in both cylindrical and rectangular volume conductors. Most body segments of electrophysiologic interest such as arms, legs, thorax, and neck are roughly cylindrical in shape. A centrally located dipole generator produces a nonzero equipotential region which is found to occur along the cylindrical wall at a distance from the dipole of approximately 1.4 times the cylinder's radius and 1.9 times the cylinder's radius for the center of the cylinder. This distance to the equipotential zone along the surface wall expands but remains less than 3.0 times the cylindrical radius when the dipole is eccentrically placed. The magnitude of the equipotential region resulting from an asymmetrically placed dipole remains identical to that when the dipole is centrally located. This behavior is found to be very similar in rectangular shallow conducting volumes that model a longitudinal slice of the cylinder, thus allowing a simple experimental model of the cylinder to be utilized. Amplitudes of the equipotential region are inversely proportional to the cylindrical or rectangular volume's cross-sectional area at the location of dipolar imbalance. This study predicts that referential electrode montages, when placed at 3.0 times the radius or greater from a dipolar axially aligned far-field generator in cylindrical homogeneous volume conductors, will record only equipotential far-field effects.     

Generators of the early and late median thenar premotor potentials

The generator sources of the median thenar premotor potentials (PMPs) have remained elusive despite debate in the literature. By studying the median nerve in the hand with a variety of bipolar and referential recording montages, we systematically examined the possible nearfield and far-field sources that may determine these potentials. The results suggest that the early PMP is a near-field potential recorded by G₁ and generated by the median nerve traversing the distal carpal tunnel. The late PMP represents a far-field potential generated by the median digital nerve fibers as they pass from the palm volume into the thumb volume. Characteristics of the late PMP are explained using the leading/trailing dipole (L/TD) model of far-field potential generation. The diagnostic utility of these PMPs is questionable, since they are recorded from “regions” along the nerve rather than from more clearly defined sites. © 1995 John Wiley & Sons, Inc.     

Far-field potentials

Far-field potentials are produced by neural generators located at a distance from the recording electrodes. These potentials were initially characterized incorrectly as being of positive polarity, widespread distribution, and constant latency; however, recent advances have clearly demonstrated that far-field potentials may be either positive or negative depending upon the location of the electrodes with respect to the orientation of the dipole generator. Additionally, peak latencies in the far-field can vary with alterations in body position and the spatial distribution of far-field potentials, while widespread, is not uniform. Recent studies of far-field potentials suggest how such waveforms are produced when the symmetry of an action potential, as recorded by distant electrodes, is broken by such factors as differing conductivities of volume conductor compartments, direction of action potential propagation, size differentials in adjoining body segments, or the termination of action potential propagation in excitable tissue. Human, animal, and computer experiments support the preceding generalizations. These new explanations are directly applicable to such far-field potentials as the short latency somatosensory-evoked potential. Furthermore, since far-field potentials can also occur in muscle tissue, one should expect that these generalizations will hold with respect to electromyographic potentials. © 1993 John Wiley & Sons, Inc.     

Far-field potentials in muscle

Far-field potentials have been predicted by computer simulations as well as demonstrated in both animals and humans with respect to the peripheral and central nervous systems. Computer simulations have also predicted far-field potentials originating at the termination of muscle tissue. This investigation demonstrates the occurrence of 2 far-field potentials in the human biceps muscle resulting from action potential termination at the musculotendonous junctions. A monophasic potential is produced at both the muscle's origin and insertion, and the polarity is entirely dependent upon the recording montage. Sequential stimulation of the biceps muscle at 2.5-cm increments resulted in the 2 far-field potentials and their respective latencies changing proportional to the distance between the stimulus site and the 2 musculotendonous junctions. Various stimulation and recording montages are used to investigate the properties of these far-field potentials. The leading/trailing dipole model is utilized to explain the production and polarity of far-field potentials generated by muscle tissue.     

Motor unit action potential components and physiologic duration.

Motor unit action potentials (MUAPs) recorded from the same motor unit at two distances along the biceps brachii muscle with monopolar needle electrodes at high amplifier gains (20 microV/division) and averaged 2000-3000 times reveal total potential durations of 39.6 +/- 4.6 ms. In addition, the terminal segment for each of these two MUAPs contained a late far-field potential with a mean duration of 23.8 +/- 4.1 ms. Computer simulations of MUAPs suggest that this long-duration positive far-field mirrors the true morphology of the intracellular action potential (IAP), which is monophasic positive, possessing a terminal repolarization phase approaching 30 ms. This investigation suggests that the MUAP's physiologic duration is directly proportional to the muscle fiber length and the IAP's duration, which becomes manifest as a positive far-field potential when the IAP encounters the musculotendinous junction and slowly dissipates. The leading/trailing dipole model is used to explain qualitatively this study's quantitative clinical and computer simulation findings.     

Far-field potentials generated by action potentials of isolated frog sciatic nerves in a spherical volume.

Previous results in cylindrical volumes have shown that action potentials generate far-field potentials when experimental conditions are such that quadrupolar components of the action potential are reduced to an equivalent dipole. We now show that the same conclusions are also reached within a spherical volume, again recording far-field potentials from isolated bullfrog nerves. A mathematical proof is given that shows that in a sphere, antipodal electrodes primarily detect far-field potentials from dipole generators and not quadrupole generators. A revised conception of the 'far-field' in evoked responses is discussed which equates far-field recordings with dipole detection.     

Far-field potentials due to action potentials traversing curved nerves, reaching cut nerve ends, and crossing boundaries between cylindrical volumes

A previously published computer simulation was tested in a biological preparation by recording action potentials from frog sciatic nerves within a volume conductor filled with Ringer's solution. Traveling in a straight line, nerve action potentials traversed a constricted cylinder before crossing into a larger, hemicylindrical volume. Recordings from widely spaced electrodes in the larger volume demonstrated a potential associated with the action potential crossing the boundary between the two volumes. Another potential was associated with the action potential reaching the nerve's cut end. These potentials did not diminish in amplitude with increasing distance from the source. In other recordings, a potential associated with a bend in the nerve was found which was dependent upon the angle of the bend. These results indicate that the simple model of a dipole in a bounded sphere in which potentials decrease as a function of distance from the generator does not explain all potentials that can be observed under conditions that approximate human and animal recordings.     

Physiologic basis of potentials recorded in electromyography

The extracellularly recorded configuration of a single muscle fiber discharge is generally appreciated to be triphasic with an initially positive deflection. However, careful attention to waveform appearance during the electrodiagnostic medicine examination reveals that both innervated and denervated muscle waveforms may display a pantheon of configurations. Further, despite the fact that innervated and denervated single muscle fiber discharges arise from distinctly different intracellular action potential (IAP) configurations, their extracellularly recorded waveforms can appear quite similar, leading to potential misidentification and, hence, the possibility of an erroneous diagnostic conclusion. The least appreciated, but nevertheless critical, aspect of explanations for muscle waveform configurations is the relationship between the muscle fiber and recording electrode. Additionally, it is important to appreciate both the near-field and far-field aspects of single fiber and compound muscle action potentials. In this review, the leading/trailing dipole model is used to explain muscle waveform configurations in both innervated and denervated tissues.     

Simulation of 'stationary' SAP and SEP phenomena by 2-dimensional potential field modelling

In order to model the distribution of potentials in the hand due to antidromic SAP propagation and in the body due to afferent conduction of the median nerve volley, 2-dimensional matrices of the appropriate shape were constructed, each containing a 'generator' consisting of up to 3 'source' and 3 'sink' points. The value of the field potential at other sites was calculated using a finite difference method. It was shown that the potential gradient is virtually zero in matrix zones which are separated from the region containing the generator by a constriction in the boundary of the conductor. Points on the far side of the constriction remain virtually equipotential, at a level determined by the potential at the junction. This is naturally influenced by the proximity of the generator, so that as the generator approaches the constriction a potential difference will develop between points on the far side, irrespective of their distance from the junction, and other remote parts of the matrix. In the context of human SAPs and SEPs, such factors may be of paramount importance in the generation of so-called 'stationary' or 'far-field' potentials. With additional postulates concerning the manner in which the SAP is attenuated by the termination of axons as it propagates through the hand, and the course taken by the median nerve volley between the arm and neck, it was possible to model the majority of stationary SAP phenomena described by Kimura et al. (1984), and also the distribution and latency of the P9 SEP component following median nerve stimulation.     

Dipole-tracing method applied to human brain potentials

A new computer-aided method was developed to estimate the location of an electric source generator (e.g. a current dipole) in the human brain. Brain activity such as somatosensory evoked potentials was recorded with 21 surface electrodes over the scalp. To solve the inverse problem, it was assumed that only one dipole is elicited at a given time, and that the head is embedded in an infinite and homogeneous conductor. The exact geometry of the human head was measured from 17 slices of CT-images of a real head to arrange a human head model. A dipole with a given moment and location is assumed in the head model. Potential distribution elicited by the dipole is compared with potential distribution which was the actual recorded one. The optimal dipole location was calculated, using the simplex method. Hence, the optimal dipole moment was obtained. The accuracy of estimation as an equivalent dipole was expressed in terms of dipolarity.     

The localization of equivalent dipoles of EEG sources by the application of electrical field theory.

A technique is described for finding the position, magnitude and orientation of the equivalent electrical dipole of EEG activity given the pattern of potential differences recorded at the scalp. The technique is based on an iterative computer program implementing equations describing the electrical field of a dipole in a spherical conductor. The computer program was tested in vitro against data obtained from an inert spherical conductor (a bowl containing physiological saline, fitted with recording electrodes and a movable dipole) and an anisotropic conductor (a similarly equipped human skill including a simulated scalp). In both practical models, the computer program accurately located the dipole, the mean differences between observed and computed loci being of the order of 1 cm at widely different locations. This accuracy was maintained in the anisotropic model even though potentials transmitted through the skill were attenuated by 80%. In vivo, the program successfully located the equivalent generator of blink artefacts within the remaining eye of a one-eyed subject and, in a normal subject, localized the dipole of corresponding potentials to a midline inter-ocular position. In further investigation of normal subjects, the distribution of amplitudes at latencies within Wave V of the visual evoked response confirmed the loci of equivalent generators within posterior cerebral regions. The technique was also applied to the alpha rhythm indicating a posterior locus compatible with the view that alpha rhythm is generated chiefly by posterior cerebral cortex. Factors affecting the accuracy of the technique and the limitations of one-dipole models are discussed.     

Far-field evoked potential components induced by a propagating generator: computational evidence

This study validates current hypotheses for the generation of so-called far-field or stationary somatosensory evoked potential (SEP) components. Changes in the volume conductor configuration and changes in the direction of nerve propagation are demonstrated to be capable of generating such components. Results are based on basic aspects of the theory of volume conduction. It is shown that in an essentially restricted volume conductor any disturbance of uniform nerve propagation in a homogeneous extracellular medium will lead to the generation of non-moving field components. A number of illustrative examples are presented in which intermingling of non-moving and propagating potential fields can be observed. These results can be helpful in unravelling complicated wave forms from the nervous system in which both types of potential field can be distinguished. It is shown that realistic changes within the volume conductor can lead to substantial far-field components. This type of volume conductor induced 'virtual generators' or 'secondary sources' is present in the peripheral nervous system and most probably also in the inhomogeneous structures of the brain.     

Far-field potentials recorded from action potentials and from a tripole in a hemicylindrical volume

There is growing evidence in support of the hypothesis that far-field potentials are recorded when action potentials encounter discontinuities in the surrounding volume. The present study found further support for this hypothesis using two methods of experimentation. The first method recorded potentials when the action potential from an isolated bullfrog sciatic nerve in a hemicylindrical volume (i) encountered a change in the shape of the surrounding volume, (ii) crossed a boundary between 2 volumes of differing resistivities, (iii) reached a bend in the nerve, or (iv) reached the functional end of the nerve. In the second method, potentials were recorded when an electrical tripole, constructed in a way to produce the electrical equivalent of an action potential, encountered the same discontinuities as well as when it was configured to simulate a curved nerve. These results are consistent with the hypothesis that dipole components of an action potential predominant in far-field recordings.     

Stationary waves recorded at the shoulder after median nerve stimulation

A new recording method with a reference electrode on the stimulated arm defined two discrete far-field potentials just before the propagated near-field nerve action potential was recorded at Erb's point. Both potentials were stationary waves with the same latency at all recording sites. The first potential had the same onset and peak latencies as the propagated wave at the axilla; it corresponded to the first component of the P9 far-field potential recorded with scalp to noncephalic reference montages. The latency of the second potential coincided with that of the propagated wave entering the neck and corresponded to the peak of the P9 potential. The occurrence of these potentials where there are significant changes in the morphology of the volume conductor suggests that the P9 far-field potential is due to a change in the conducting medium that surrounds the nerve.     

Stationary peaks from a moving source in far-field recording

In 20 radial nerves, referential recording of antidromic sensory potentials from the digits consisted of two stationary far-field peaks, PI-NI and PII-NII. When compared with a bipolar recording of a moving source, the onset of PI and PII was coincident with the sensory potential approaching the wrist and the base of the digit, respectively. Of the two stationary positive peaks, PI was identical in latency irrespective of the digits tested. In contrast, the latency of PII was smaller for the first digit than the others, reflecting different distances from the stimulus point to the base of the respective digit. The amplitude of PII was proportional to that of the propagating sensory nerve action potential recorded at the volume conductor junction. A bipolar recording registers a near-field potential over the sensory fibers along the length of the nerve. In contrast, a referential recording represents a mixture of the near- and far-field potentials with the latter frequently producing major alterations of the classical triphasic wave. The standing far-field peaks show a temporal relationship to the traveling volleys approaching the main borders of the volume conductor. At the moment the sensory impulse reaches the boundary, current density changes suddenly in the two adjacent conducting media, giving rise to an apparent standing potential. The same mechanism probably plays an important role for the generation of some of the short latency peaks in the scalp recorded SEPs.     

The biphasic morphology of voluntary and spontaneous single muscle fiber action potentials

The extracellular morphology of single muscle fiber action potentials (SMFAPs) is anticipated by volume conductor theory to be triphasic. Single muscle fiber action potentials recorded during single muscle fiber studies (500 Hz to 20 kHz) usually appear triphasic; however, when recorded with an open bandwidth (1 Hz to 20 kHz) they are found to be biphasic. Fibrillation potentials recorded with a single fiber electrode and open bandwidth have identical biphasic morphologies as volitional SMFAPs. Computer simulations suggest that the intracellular action potential models currently used to derive simulated extracellularly recorded SMFAPs must have the repolarization phase considerably prolonged to yield the clinically recorded potentials. This implies that either the models presently used require significant modification, or there is some distortion of the transmembrane source current induced in needle recording studies such that biphasic and not triphasic potentials are detected. © 1994 John Wiley & Sons, Inc.     

EEG source localization: a neural network approach

Functional activity in the brain is associated with the generation of currents and resultant voltages which may be observed on the scalp as the electroencephelogram. The current sources may be modeled as dipoles. The properties of the current dipole sources may be studied by solving either the forward or inverse problems. The forward problem utilizes a volume conductor model for the head, in which the potentials on the conductor surface are computed based on an assumed current dipole at an arbitrary location, orientation, and strength. In the inverse problem, on the other hand, a current dipole, or a group of dipoles, is identified based on the observed EEG. Both the forward and inverse problems are typically solved by numerical procedures, such as a boundary element method and an optimization algorithm. These approaches are highly time-consuming and unsuitable for the rapid evaluation of brain function. In this paper we present a different approach to these problems based on machine learning. We solve both problems using artificial neural networks which are trained off-line using back-propagation techniques to learn the complex source-potential relationships of head volume conduction. Once trained, these networks are able to generalize their knowledge to localize functional activity within the brain in a computationally efficient manner.     

N30 and the effect of explorative finger movements: a model of the contribution of the motor cortex to early somatosensory potentials.

OBJECTIVES: The source of the N30 potential in the median nerve somatosensory evoked potentials (SEP) has been previously attributed to a pre-central origin (motor cortex or the supplementary motor area, SMA) or a post-central located generator (somatosensory cortex). This attribution was made from results of lesion studies, the behavior of the potential under pathological conditions, and dipole source localization within spherical volume conductor models. METHODS: The present study applied dipole source localization and current density reconstruction within individual realistically shaped head models to median nerve SEPs obtained during explorative finger movements. RESULTS: The SEPs associated with movement of the stimulated hand showed a minor reduction of the N20 amplitude and a markedly reduced amplitude for the frontal N30 and parietal P27, exhibiting a residual frontal negativity around 25 ms. The brain-stem P14 remained unchanged. Mapping of the different SEPs (movement of the non-stimulated hand minus movement of the stimulated hand) showed a bipolar field pattern with a maximum around 30 ms post-stimulus. In eight out of ten normal subjects, both the N30 and the gN30 (subtraction data) sources resided within the pre-central gyrus, more medially than the post-centrally located N20. Two subjects, in contrast, showed rather post-centrally localized sources in this time range. A model of the cortical SEP sources is introduced, explaining the data with respect to previously described findings of dipole localization, and from lesion studies and the alterations seen in motor diseases. CONCLUSIONS: The results provide evidence for a pre-central N30 generator, predominantly tangentially oriented, located within the motor cortex, while no sources were detected elsewhere. It is suggested that the mechanisms underlying the 'gating' effect during explorative finger movements in the 30 ms time range predominantly arise in the motor cortex.     

Stationary negative potentials near the source vs. positive far-field potentials at a distance

We studied the field distribution of referentially recorded negative potentials after stimulation of the median nerve at the wrist in 15 normal subjects. When recorded from multiple sites along the lateral aspect of the arm with the reference electrode at the knee, 3 negative peaks, N3, N6 and N9, appeared at fixed latencies. Of these N3 and N6 were highest in amplitude at the distal insertion of the brachioradialis and the distal end of the deltoid, respectively, and N9, at the acromion. With stimulation of the finger, the negative peaks shifted in latency by about 3 msec, indicating an anatomically fixed generator source for each component. When compared to far-field potentials, N9 was of the same latency as scalp recorded P9, that extended to the arm contralateral to the side of stimulation and to the upper half of the trunk. In contrast, N6 extended to the scalp with P6 spreading to the lower half of the body. When two subjects were connected by the arm, stationary negative or positive peaks were transmissible from the stimulated to the non-stimulated subject. When the stimulated arm of the first subject was in contact with the second subject, N3, N6 and N9 were recorded in the latter. Only P9 was registered when the unstimulated arm was in contact with the second subject. We conclude that N3, N6 and N9 are stationary negative potentials generated at certain points along the nerve pathway, probably representing a negative counterfield for positive far-field peaks, P3, P6 and P9. These stationary potentials can spread widely in a volume conductor and can even be detected in a non-stimulated subject making a close contact to the generator source.