The leadinghrailing dipole model explains the production of far-field potentials as an asymmetry in the leading and trailing dipole moments of a propagating action potential detected by a referential montage. This investigation documents the production of far-field potentials produced by a pure dipole generator in a circular volume conductor. Multiple equipotential waveforms are recorded in an adjoining circular volume conductor attached to the one in which the dipole generator is located. This finding substantiates the “wick electrode” effect that explains the equipotential and instantaneous distribution of far-field potentials over relatively large distances in volume conductors. The present findings support a number of the leading/ trailing dipole model proposals which explain far-field potential generation. Key words: far-field potentials action potentials stationary potentials virtual dipoles MUSCLE & NERVE 15:lOl-105 1992

FAR=FIELD POTENTIALS I N CIRCULAR VOLUMES: EVIDENCE TO SUPPORT THE LEADINGITRAILING DIPOLE MODEL DANIEL DUMITRU, MD, and JOHN C. KING, MD

Far-field potentials were first recognized clinically in the recording of brainstem auditory evoked potentials, and subsequently observed in referential somatosensory evoked potential^.'*^*^*^ A number of theories postulated that far-field potentials arose directly from the traveling action potential, or various stationary neural generators. 12,14 Kimura and Yamada proposed that a far-field waveform was generated not solely by the action potential, but by an interaction between the action potential and some property of the encompassing volume conductor.9 Specifically, a propagating action potential could generate a far-field waveform if it encountered: (1) a geometric alteration in the volume conductor, (2) a change in the direction or neural branch point, or (3) regions of different conductivity. l o , ’ These proposals were critical in refocusing investigators’ attention from neural generators to the interaction between action po~

~~

~

From the Department of Rehabilitation, University of Texas Health Science Center at San Antonio, San Antonio, Texas. Acknowledgment: We thank Don L.Jewett MD, DPhil, for his support and advice. Address reprint requests to Daniel Durnitru, MD. Department of Rehabilitation, University of Texas Health Science Center at San Antonio, 7703 Floyd Curl Drive, San Antonio, TX 78284-7798. Accepted for publication February 14, 1991

CCC 0148-639X/92/010101-05 $04.00 0 1992 John Wiley & Sons, Inc.

Far-Field Potentials in Circular Volumes

tential current field distributions and the enveloping volume conductor. Computer simulations of various boundary conditions suggested that all three of the above proposed volume conductor/action potential interactions could result in monophasic far-field potent i a l ~A . ~significant ~ contribution to far-field theory occurred when Jewett and Deupree proposed the leadingltrailing dipole model.”’ This model qualitatively helped conceptualize how an asymmetric representation of the linear double dipole results in an unbalanced dipole moment detected at far-field recording sites as a far-field potential. Action potentials propagating along frog sciatic nerves encountered different volume conductor boundary conditions producing far-field potentials, and substantiated the results of computer simulations as well as demonstrating that a fourth or “cut-end’’ effect also generated a far-field potential. .7,6*8 This leadingitrailing dipole model relied upon the “wick electrode” effect to explain how a dipole moment imbalance at a volume conductor boundary projected a far-field potential essentially instantaneously and equipotentiall over large portions of the volume conductor.“,‘ As explained by Jewett et al.,37638 a wick electrode is a passive electrode with the capability of recording any potential to which it is in contact instantaneously throughout its volume. A familiar example is the

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micropipette electrode (filled with an electrolyte) that pierces a cell to record its intracellular potential. The solution within the micropipette, though being a volume conductor, acts as a wick electrode, transmitting the intracellular potential instantaneously from its tip to the other end. If the electrode length is increased, the acquisition time of the potential does not change. In a similar manner, various portions of the body can act as wick electrodes, and convey far-field potentials generated at volume conductor boundaries over large portions of the body instantaneously. The leading-trailing dipole model appears to be a rather promising conce.ptualization of farfield potential production. To date, animal experiments and computer simulations have taken advantage of either the quadrupole or tripole model to study the production of far-field waveforms. The present study was undertaken to specifically investigate whether a constant current dipole source under simple laboratory conditions could produce far-field potentials. A far-field waveform produced by a dipole generator would support the contention that when action potentials behave transiently as unbalanced dipole sources, far-field potentials are generated. Simple circular volumes are utilized to examine, in two dimensions, the more complex geometry of spheres with respect to dipole generator properties and far-field potentials. Additionally, the wick electrode phenomenon is investigated. MATERIALS AND METHODS

A Cadwell Excel1 (Cadwell Corp., Kennewick, WA) electrophysiologic instrument was utilized with high and low frequency settings of 20,000 Hz and 0.04 Hz, respectively, which permitted accurate reproduction of the square wave pulse delivered with an amplifier input impedance of 50 MR. A sweep speed of 1 ms/ div (sampling frequency of 64 kHz) with a sweeptrigger to stimulus delay of 4 ms and a sensitivity of 25,000 FV/div were employed. The dipole current source and sink were a pair of platinum subdermal electroencephalographic electrodes (Grass Corp. Quincy, MA) with a 3.0-cm separation centrally located within the circular volume conductor, connected to cathode and anode stimulator ports. The dipole generator utilized a 6-mA square wave constant current source with a pulse duration of 1.0 ms. The E-1 (active) and E-2 (reference) electrodes were similar subdermal electrodes to those used for the cathode/anode, and were completely submersed within the volume Instrumentation.

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conductor with an interelectrode impedance of 3 kn. The magnitude of the recorded dipole pulse was measured from the baseline to the square wave's negative peak. Ten trials of each experimental condition were averaged, which removed baseline noise. E-1 and E-2 electrode amplifier connections conform to the convention of a negative potential difference resulting in an upward deflection from the baseline. Two 55-cm diameter nonconducting circular containers, 1.5 cm in depth, were used to simulate simple volume conductors (Fig. 1). The containers were joined to each other at one region with a 3.0 mm (width) X 2 cm (length) rectangular connection. The two circular volumes were filled with a 0.9% saline solution to a depth of 1.25 cm and maintained at 23°C. A water-tight gate between the two volume conductors could be opened or closed (resistance greater than 10 MR) to permit or restrict electrical continuity, respectively, and was aligned with the cathode/anode axis of the dipole generator (Fig. 1).

Volume Conductor.

Single Volume. Initially, the water-tight gate between the two volumes was closed. The E-2 electrode was placed 27.5 cm radially away from the centrally located dipole constant current source perpendicular to a line midway between the cathode and anode, 270" clock-wise from the gate region (Fig. 1). A ground electrode was located at a radial distance of 27.5 cm from the volume's center and 250" clock-wise from the gate, and remained in this position for the entire experiment. The E-1 electrode was sequentially placed 2, 4, 8, 16, and 26 cm linearly from the cathode (E'-12-26/00, Fig. 1). At each location, 10 stimuli were delivered from the constant current dipole and averaged. The experiment was then repeated with the connection between the two circular volumes opened. A similar set of data were generated for the E-1 electrode located on a line 30" clockwise from the previous set of stimuli (E"12-,6/30", not shown in Fig. 1) as well as 45" (E"12-26/45"? Fig. 1) and 60" (Ef'-12-26/60",not shown in Fig. 1). Additionally, the E-1 electrode was placed 90" (E"-l2,.,/90") clock-wise from the gate area at 27.5 cm perpendicular to the anode/cathode axis aligned opposite the E-2 electrode (Fig. 1). Recording electrodes E-1 and E-2 were then placed 1 cm apart in a nonconducting plastic material and located 2 cm from the anode on a line between the cathode and water-tight gate. E-2 was

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E-le

FIGURE 1. The two circular volume conductors containing normal saline are pictured along with the various electrode montage arrays. The (+ -) signifies the dipole generator constant current source placed in the center of one circular volume. The ground (G) and reference (E-2) electrodes are also shown. The vertical dotted line between the two volumes signifies the variable position, water-tight gate.

positioned further from the cathode than E-1. Ten stimuli were averaged with this montage. This E-1/E-2 pair was then moved radially out to the rim of the volume conductor on this same line where 10 stimuli were again averaged. This same procedure was repeated for an E-1/E-2 interelectrode separation of 3 cm. Dual Volumes. The connection between the two volumes was initially positioned to restrict electrical continuity. The E-1 electrode was located at the 26 cm mark from the cathode in the volume containing the dipole generator just anterior to the conjoined region between the two volumes (Ef-lZ6,Fig. 1). Electrical continuity was then established by opening the gate, and 10 stimuli again delivered. The E-1 electrode was then placed, in the circular volume without the dipole generator, at four equidistant locations 27.5 cm from the center and directly in the volume's center (E-la-e, Fig. 1). Ten stimuli were averaged for each location in the second circular volume conductor. RESULTS

All potentials recorded throughout the experiment had a negative polarity and waveform morphology that was reproducible, easily measured with negligible slope, and represen-

Single Volume.

Far-Field Potentials in Circular Volumes

tative of the square wave pulse delivered. The sequential Ef-12-26electrode placement revealed a consistent negative square wave potential whose amplitude decreased as the distance from the dipole generator increased from 2 to 26 cm (48.5, 29.0, 18.0, 12.0, and 11.0 mV, respectively) (Fig. 2). Rotating the sequential E-1 electrode array 30, 45, and 60", while maintaining E-2 as shown in Figure 1, resulted in a similar amplitude-distance relationship with progressively smaller amplitudes as the distance increased (Fig. 2). For the 30" angle, the sequential amplitude decrement was 48.5, 28.0, 15.5, 10.5, and 9.5 mV; the 45" angle revealed an amplitude decline of 46.0, 25.5, 14.5, 9.0, and 8.0 mV. Finally, the 60" angle incremental electrode placement resulted in the following decremental amplitude data: 39.5, 20.0, 12.5, 7.5, and 6.5 mV. When electrical continuity between the two circular volumes was established, identical results were obtained as prior to electrical continuity. T h e E'-1,,,,/90": E-2 montage recorded no potential. Recording electrodes with a separation of 1 cm, located 2 cm from the cathode, yielded a negative potential with an amplitude of 20 mV. This same electrode pair, positioned with the E-1 electrode 25 cm and E-2 electrode 26 cm from the cathode respectively, along the same previous axis, resulted in a potential with an amplitude of 0.056

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mV. T h e E-l/E-2 pair, with 3 cm of interelectrode separation, produced potentials at the same two previous locations with amplitudes of 21 mV and 0.098 mV, respectively. The potential obtained from 10 stimuli were identical when recorded just anterior to the connection between the two compartments (E'-lz6), whether the gate was opened or closed. Additionally, all recording sites in the companion circular volume without the dipole generator were equipotential (1 1.0 mV), with that potential just anterior to the gate (E'-12-26 = 11.0 mV), with the gate in the opened position (E-la-.-, Fig. 1).

Dual Volumes.

DISCUSSION

The complexities regarding far-field potential generation continue to elude investigators, despite their initial description over 30 years ago. A comprehensive theory to fully explain far-field potentials does not yet exist, however, a significant piece of this puzzle has at least been qualitatively conceptualized by the leadingltrailing dipole model.3," This model proposes that an action potential may be considered as a leading and trailing dipole back-to-back (+ -, - +) corresponding to the source-sink/sink- source model of a propagating action potential. When the propagating action potential encounters a boundary condition of the enveloping volume conductor, an asymmetry between the two dipole moments occurs. This dipolar imbalance is registered by electrodes located in the far field as a far-field potential. The dipole generator in this investigation simulates the transient maximal imbalance in the dipole moments when one of the dipoles is no longer balanced by the other, e.g., the termination of excitable t i ~ s u e A . ~pure dipolar constant current generator source has not been shown to generate far-field potentials under controlled in vitro conditions. In the single circular volume, the spatial gradient of the current field produced by the dipolar constant current generator was first investigated with a referential electrode montage. T h e E-127,5/ 90" and E-2 montage did not record a potential, suggesting that these two electrodes were located on an isopotential line near zero. E-1 was then sequentially displaced away from the cathode, while E-2 remained at the volume's rim on the line perpendicular to the cathode and anode. The results of the multiple E'-1 and E"-1 recordings, therefore, most likely represent the potential difference at the E-l recording point with

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little contribution from the E-2 electrode. T h e amplitude of the detected potential declined as the electrode was displaced radially from the source. This finding suggests that the spatial gradient near the dipole source was quite high. That is, for a small change in distance, the potential declined precipitously (Fig. 2). Beyond 16 cm, however, the potential declined minimally for the four different angular sets of multiple electrodes. The spatial gradient of potential difference close to the rim of the volume conductor, therefore, is rather low. In unbounded spherical volumes, the far-field potential's magnitude is expected to reach zero potential at infinitely large distances from the generator source.8 T h e magnitude of our far-field potentials, however, do not decline to a zero potential but approach a constant vaue near the volume's boundary. The calculated regression equation to best fit all of our angular data points is: V

=

3

+ 10.5 mV

127 cm*mV ---

where R , is the measured distance from the anode and R 2 is the distance from the cathode, trigonometrically derived from R, and its respective angle. The above equation is in the form utilized for far-field potential generators in infinite spherical volumes, with the modification of 10.5 mV required for the finite circular volume boundary effect. A bipolar electrode montage, interelectrode

0

00

0

30'

8 45'

so0

i 04

,

,

,

,

,

,

,

,

,

1

,

I

I

0

2

4

6

8

10

12

14

16

18

20

22

24

26

Distance (c rn)

FIGURE 2. Graphic representation of the potential spatial gradient within the circular volume conductor. Note how the magnitude of the recorded potential markedly declines close to the generator source, but changes little near the margins of the volume conductor. The four ewes for 0, 30, 45, and 60 degrees are depicted.

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separation of 1 and 3 cm, was also utilized to examine the spatial gradient of the dipolar source and to confirm the referential montage findings. A potential of relatively large amplitude (20 mV) was recorded close to the dipolar generator, but a rather small potential of less than 0.1 mV was noted near the volume’s rim with both electrode pairs. These two sets of independent findings confirm the low spatial gradient of the dipolar current source near the rim of the volume, as compared with the region near the cathode. A potential recorded in a region of low potential gradient may be referred to as a far-field potential.5’s T h e dipolar generator in this study, therefore, produced far-field potentials. The assertion of the leadingkrailing dipole model that dipole sources produce far-field potentials is substantiated by the two different recording montages documenting a far-field potential near the outer aspects of the circular volume conduct ~ r . ~ , ~ , ~ When the two circular volumes were connected, a far-field potential was observed in the circular volume not containing the dipole genera-

tor. This far-field potential was equipotential throughout the entire volume (E’-lP6and E-laPe). The circular volume, without the dipole generator, behaved as a wick electrode through the small gate area, detecting the far-field potential at that entrance point to the volume conductor containing the dipole g e n e r a t ~ r . ~ , ~ This finding substantiates the clinical observations that far-field potentials extend instantaneously and equipotentially throughout volume conductors. That portion of the body on either side of the boundary generating a far-field potential, therefore, can be thought to behave as a wick electrode and acquire the potential associated with the dipolar imbalance. Of course, one may expect that, unlike this investigation, the body is inhomogeneous and, in portions, anisotropic, which may produce some distortions over distance from these findings. This study supports the major proposals of the leading/trailing dipole model that: (1) dipole sources can produce far-field potentials, and (2) volume conductors may behave as wick electrodes with respect to far-field potentials.

REFERENCES

1. Cracco RQ: T h e initial positive potential of the human scalp-recorded somatosensory evoked potential response. Electroencephalogr Clin Neurophysiol 197232623- 629. 2. Cracco RQ, Cracco JB: Somatosensory evoked potential in man: far-field potentials. Electroencephulogr Clin Neurophysiol 1976;4 l:460-466. 3. Deupree DL, Jewett DL: Far-field potentials due to action potentials traversing curved nerves, reaching cut nerve ends, and crossing boundaries between cylindrical volumes. Electroencephalogr Clin Neurophysiol 1988;70:355362. 4. Jewett DL, Romano M N , Williston ,jS: Human auditory evoked potentials: possible brain stem components detected on the scalp. Science 1970;167:1517-1518. 5. Jewett DL, Williston JS: Auditory-evoked far-fields averaged from the scalp of humans. Brain 1971;94:681-696. 6. Jewett DL, Deupree DL: Far-field potentials recorded from action potentials and from a tripole in a hemicylindrical volume. Electroencephalogr Clin Neurophysiol 1989; 72:439-449. 7. Jewett DL: T h e leadinghailing dipole model as a means of understanding generators of far-field potentials. Clin Evoked Potentials 1990;7:9- 13. 8. Jewett DL, Deupree DL, Bommannan D: Far-field poten-

Far-Field Potentials in Circular Volumes

tials generated by action potentials of isolated frog sciatic nerves in a spherical volume. Electroencephalogr Clin NeuroPhysiol 1990;75:105- 117. 9. Kimura J, Yamada T : Short-latency somatosensory evoked potentials following median nerve stimulation. A n n NY Acad Sci 1982;388:689-694. 10. Kimura J, Yamada T, Shivapour E, Dickens QS: Neural pathways of somatosensory evoked potentials: clinical implications. Electroencephalogr Clin Neurophysiol 1982; 3 6 ( ~ ~ p p l ) : 3 2335. 811. Kimura J , Mitsudome A, Yamada T, Dickens QS: Stationary peaks from a moving source in far-field recordings. Electroenc~phalogrC h i Neurophysiol 1984;58:351-361. 12. Klee M, Rall W: Computed potentials of cortically arranged populations of neurons. J Neurophysiol 1977; 40:647- 666. 13. Stegeman DF, Van Oosterom A, Colon EJ: Far-field evoked potential components induced by a propagating generator. Electroencephalogr Clin Neurophysiol 1987;67: 176-187. 14. Wiederholt WC, Iraqui-Madoz VJ: Far-field somatosensory evoked potentials in the rat. Electroencephalogr Clin Neurophysiol 1977;42:456-465.

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trailing dipole model.

The leading/trailing dipole model explains the production of far-field potentials as an asymmetry in the leading and trailing dipole moments of a prop...
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