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Specific impedance of cerebral white matter
EXPERIMENTAL
NEUROLOGY
Specific
13,
386-401
Impedance
of
PAUL
Space Biology
(1965)
W.
Cerebral
White
NICHOLSON'
Laboratory, Brain Research Institute, Cniversity Center for Health Sciences, Los Angeles, California Received
June
30, 1965;
Matter
Revision,
August
of California
18, 1965
A method is described for measuring specific impedances in deep brain structures. It was used to determine the impedance in a large fiber tract in the cat internal capsule. The impedance along an axis normal to the fiber direction was + 800 B cm and predominantly resistive at frequencies between 20 cycle/set and 20 kc/set. The impedance along the fiber direction was approximately one-ninth of this. Possible mechanisms of current conduction are examined and it is concluded that unless the oligodendrocyte membrane resistance is < 8 (;2 cm” an extracellular space of -10% is indicated. Introduction
The electrical impedance of cerebral tissue is of interest for several reasons. For instance, there have been severa observations (1, 18, 42) on the variations of impedance in several regions of the brain under different physiological conditions and it is plainly desirable to form some hypothesis concerning the contributory factors. Attempts to correlate the impedance with the anatomy have been mainly confined to cerebral cortex. Van Harreveld et al. (41-43) made extensive measurements under different physiological conditions and interpreted their results as indicating a large extracellular space. Ranck (26) made comprehensive investigations of resistive and reactive components of impedance over a wide range of frequencies. It is apparent from the analysis 1 This investigation was supported in part by U.S. Air Force Office of Scientific Research grant AF-AFOSR 246-63 and Public Health Service Research Grant MH 3708 from the National Institute of Mental Health. The author is indebted to Raymond T. Kado and Dr. W. Ross Adey for their help and encouragement and to Miss Hiroko Kowta for preparing the illustrations. The author’s present address is Chemistry Dept., University College, London W.C. 1. 2 The author is greatly indebted to Dr. D. C. Pease for making these electron micrographs available to him. 386
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
387
by Ranck (27) that the anatomical complexity of cortex makes a detailed interpretation of the impedance data difficult. The present measurements were made in a simpler structure, white matter, in the hope that they would yield more definite information about the histology, and in particular, about the more or less hypothetical CNS extracellular space (6, 14, 19, 40). The exact location in the internal capsule of the cat was chosen on the basis of a large content of wellaligned fibers and ease of accessibility for the experimental technique. Recently Ranck and BeMent (28) reported similar measurements for the white matter of the cat dorsal column. Anisotropic
Conductivity
Subject to certain conditions, the properties of an anisotropically conducting material can be characterized (20, 30) by three principal conductivities in three mutually perpendicular directions (the principal axes) fixed relative to the material. Maxwell (20; Sec. 297) showed that the necessary conditions are satisfied by any material which can be regarded as an assembly of elements each of which are ordinary linear conductors. If, therefore, the elements of biological tissue (membranes, intra- and extracellular components, etc.) conduct linearly the conditions are satisfied. The only serious source of nonlinearity would be expected to arise in conduction across membranes, and in the present work care was taken to use current magnitudes which precluded nonlinear behavior. Although Maxwell’s treatment was confined to materials having only resistive conductivity it is a simple matter to extend the validity to include reactive conductance also. The principal axes for resistive conductance may be different from those for reactive conductance. In the present case, however, the principal axes stem from a geometrical (i.e., anatomical) anisotropy and will therefore be common to both reactive and resistive conductances. The problems of determining the specific impedance of biological tissue have been well summarized by Schwan (33, 34). The only way of adequately eliminating the effects of electrode impedance (“polarization”) in the present case was to use separate electrodes to apply the current and to sample the voltage. The configuration used was similar in principle to that used by Ranck (28) and consisted of an approximately point source of current placed in the brain region of interest, a remote (or “indifferent”) electrode as current sink, and a movable probe electrode to sample the voltage in the vicinity of the point source.
388
NICHOLSON
For a point source of current I in an infinite medium having principal conductivities oz, cry, op, the voltage V at any point x, y, z (expressed in coordinates of the principal axes of the medium) is given by (30) V=
I
-. 4X ( o,lsyx2 + oco,,.y2 + oUo,rz’)) 4
[ll
If measurements of voltage V1 and Vi, are made at distances d, and dz along the z axis, from Eq. [I] we obtain
[21 and correspondingly for voltages along the other two axes. While in general it would be necessary to make voltage measurements in three directions to determine the three conductivities uniquely, due to the cylindrical symmetry of a fiber tract, in the present case it is immediately possible to identify one of the principal axes as the fiber axis (say the z axis) and also to infer that es = ey. This means that measurements of voltage need only be made along the z axis and one of the other axes. In other words the voltage must be sampled along the fiber axis and also normal to it. Equations [I] and [ 21 are valid for the in-phase current or the out-ofphase current depending on whether the in-phase conductivity (i.e., the resistive conductivity) or the out-of-phase conductivity (i.e., the reactive conductivity) is used in the equations. Since the experimental arrangement was only approximately a point source, Eq. [2] could not be used to calculate the conductivities directly. Large scale models (40X ) of the electrode assembly were therefore constructed and subsidiary measurements were made in saline. Although saline is an isotropic conductor it is possible to exactly simulate the anisotropic conductor case by using a model with distorted (i.e., anisotropic) dimensions (22). In effect these subsidiary measurements served to compute a correction to Eq. [ 21. Method The “point” electrode was the uninsulated tip of an insulated stainless steel tube of 0.7-mm diameter. Inside this tube and protruding 1 to 2 mm beyond the end was a sliding wire probe, insulated except at its tip, which could be driven by a micrometer to take up different positions and thus sample the voltage at different places. The remote electrode was a plati-
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
389
nized stainless steel plate placed sagittally between the cerebral hemispheres. Figure 1 shows the complete experimental arrangement. The “up” position of the coupled switches enabled the cathode follower to measure the voltage across Rr and hence the current flowing from the “point” source. The “down” position enabled the cathode follower to measure the voltage on the inner probe electrode. The signal from the cathode follower was amplified, filtered and displayed as a Lissajous figure on the oscilloscope. Since the combined capacity of the interelectrode capacity and the cathode follower input capacity tended to load the voltage probe (inner I
I
FIG.
1.
Complete
experimental
arrangement.
electrode) at the higher frequencies it was necessary to apply a correction to the measured voltages. The value C1 was made equal to this combined capacity (- 12 pf) and, by observing the change in voltage when the calibration switch (Fig. 1) was thrown, it was possible to calculate back and determine what the voltage would have been had there been no loading of the voltage probe. In practice this correction constituted less than 10% of the calculated impedances at 20 kc/set and was negligible at the other frequencies used. Consideration of other effects of stray capacitances showed that they were negligible or differenced out. The entire procedure was checked by making measurements in various standard solutions of KCI. At all frequencies used the measured impedances agreed well with accepted values and showed no reactive component as expected. It was necessary to make measurements in a region of white matter where the fibers ran mutually parallel over at least a cube of 2-3 mm side. Such a region exists in the internal capsule of the cat at coordinates ante-
390
NICHOLSON
rior 7.0 mm, lateral 10.5 mm, height 8.0 mm, and is located dorsal to the anterior end of the lateral geniculate body. The fibers run in a sagittal plane, at 45” to the coronal plane, between posterodorsal and anteroventral. It was therefore possible to insert the probe assembly into this region so as to be parallel to the fiber axis or alternatively perpendicular to the fiber axis. All measurements were made with a maximum rms voltage and current of 4 mv and 1 t-1amp, respectively. This corresponds to a peak electric field of 10 mv per mm. Before nonlinear (active) characteristics of neuroglial or neuronal membrane would set in, it would be necessary to have at least 20 mv (17, 39) across a single membrane. This possibility can therefore be safely excluded in the present measurements. Six cats weighing between 2.1 and 3.6 kg were anesthetized with Nembutal (35-50 mg/kg, ip). Four holes were drilled in the skull to allow placements of the probe assembly. The skull was removed over the midline a total width of 5 mm, the dura was cut along a sagittal plane 2 mm from the midline and the remote electrode placed between the hemispheres. This electrode was held in position by its lead wire which was cemented to the skull with dental acrylic resin. The probe assembly and cathode follower were mounted on a conventional electrode carrier and stereotaxically located with the tip in the region of interest, All probe placements were subsequently checked histologically. Measurements
Longitudinal and transverse measurements were made in each internal capsule (two probe placements each side), and of the resulting twentyfour sets of data, four sets were discarded due to inaccurate placement and a further two discarded because of extensive subcortical bleeding. The calculated specific impedances are given in Table 1. The errors in Table 1 TABLE
1
VALUES OF SPECIFIC IMPEDANCE OF WHITE MATTER EXPRESSED AS THE EQUIVALENT SERIES RESISTANCE AND CAPACITIVE REACTANCE IN
OHMS
Normal to Frequency 20 200 2 20
cycle/set cycle/set kc/set kc/m
Resistive 850 770 770 750
-cin k 2
150 160 140 150
FOR A I-CM
CUBE
fibers
Parallel
Reactive 67 39 55 140
2 5 -c c
32 18 27 110
Resistive 89 89 78 78
-c e k zk
40 39 33 33
to fibers Reactive 7k8 425 6-c6 15 & 22
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
391
were assessedfrom the standard deviations in the measuredvoltages and an additional 10% error from the measurementsmade with the model electrode. No&finite Medium. The method assumesthe measurementsare made in an infinite homogeneousmedium, whereas the region of measurement was limited by structures of different specific impedance. This was investigated in the work on the large-scale model electrode. Large nonconducting and well-conducting masseswere introduced into the vicinity of the model. The somewhat qualitative conclusion was that the effect of the structures near the region of internal capsule where measurementswere performed would be small compared to the errors quoted above. Bleeding. The bleeding that often took place around the probe assembly would offer a preferential path for the current and thus modify the voltagesobserved. Assuming a value of 150 Q cm for the resistivity of blood (34), it was calculated that a O.l-mm film of blood over the electrode surface would alter the calculated impedancesin Table 1 by less than 5%. No measurementsof blood film were made but it was thought that the error introduced into the measuredimpedancewould be negligible. Other evidence for this assumption came from the consistency of the experimental data from experiment to experiment. In two cases,the measurementswere discarded due to extensive subcortical bleeding but, in fact, the measured impedanceswere indistinguishable from those taken with minimal bleeding conditions. Other Sources of Error. Similar arguments may be applied for tissue damage local to the surface of the electrodes. The problem of tissue reaction to the mechanical disturbance of the electrode is more imponderable. Van Harreveld, Murphy and Noble (41) found the impedance of cerebral cortex very sensitive to mechanical trauma. In white matter, no histologically evident effects of injury have been reported within 6 hours of injury (12, 16). This is too long a time for any such effects to make themselves felt in the duration of an experiment. The above finding does not, of course, preclude many other possible consequencesof mechanical trauma. However, impedance measurementsmade at the beginning of an experiment were indistinguishable from those made 2 hours later, so that if the tissue were in someway modified so as to effect its impedance, the modification must occur completely ,within the first few minutes of placement of the electrode. Other Measurements. Several other measurementsof the impedance of white matter have been made. The data of Van Harreveld, Murphy and
392
NICHOLSON
Noble (41) taken at 1 kc/set in rabbit white matter adjacent to the cortex, yields the value of 960 _t 70 Q cm. Freygang and Landau (9), using pulses, reported a much lower value of 330 !J cm for the cat. Both these determinations lie within the range reported in the present determination (Table 1) . The work of Ranck (28) on the cat dorsal column indicates values for longitudinal and transverse resistive components of approximately 138 52 cm and 12 11 Q cm, respectively. A small reactive component was observed in both cases. In view of the uncertainties it appears that these are in quite good agreement with the values in Table 1. For frog sciatic nerve Tasaki (38) has reported a transverse specific impedance of 7 000 to 13 000 Q cm and longitudinal specific impedance of 245 Q cm. It is rather surprising that the difference between central and peripheral white matter should apparently account for such a difference in the impedances. Histology
of
Cerebral
White
Matter
The histology of cerebral white matter has been reviewed by several authors (6, 11, 19). The myelinated nerve fibers are found in bundles in close proximity to the fine processes of oligodendrocytes, and occasionally astrocytes and transitional forms of neuroglia cells have been seen. The classic work of Rio Hortega (29) and Penfield (23) showed the oligodendrocytes to be arranged in rows parallel to the myelinated nerve fibers, and with long processes running in the same direction. Electron microscopy (6, 19, 31, 32) has confirmed the early observations and shown that the oligodendrocyte has a large number of very fine processes or sheets. In all cases the extracellular space in normal material has been observed to consist of the usual 120-200 A gap between membranes. Correction
for
Conduction
by
Capillaries
A small part of the experimentally determined impedances would be accounted for by conduction due to blood capillaries. This will now be calculated and a correction applied to give the impedance of the rest of the tissue. Let 2 be the measured value of the specific impedance N 800 CJcm, 2 be the corrected value of the specific impedance N 850 Q cm, Rn be the resistivity of blood 150 Q cm (34), and V be the fractional volume of blood in white matter N 2%.
IMPEDANCE
OF
CEREBRAL
WHITE
393
MATTER
The maximum contribution due to blood conduction would be when the blood vessels all run in the direction of current flow. In this case: z=
Z’(l--li)
1-
VZ’/Rs
N Z’[l
+
V(Z’/R,
-
[3]
l)]-z’[l+4.3v].
The minimum contribution would be when the vessels all run normal to the direction of current flow. An expression for this has been given by Bozler and Cole (2) which gives 1 +
z = 2’ 1
-
V(z--R,)/(Z
V(Z-
+RB)
RB)/(Z
+ RB) N Z’(
1 +
2v
Z-RB )P
Z’(1
+
1.4V).
[4]
Z+RB
If the vessels are assumed to run in equal numbers in the X, y and z directions, then for current flow down any of these axes, twice as many vessels would be normal to the conduction axis as there are parallel to it. After weighting [3] and [4] accordingly we obtain as the final correction expression : Z = Z’( 1 + 2.4V). [51 The most critical factor in the above analysis is V. For the cat the only source of information is from capillary counts. For white matter in the cat, Campbell (3) gives a total capillary length of 4.5 X 104 cm per cc and Dunning and Wolff (7) give a value of 3.74 X lo4 cm per cc. If we assume these figures include a linear shrinkage of 15% through fixation and the capillary diameter is 10 p then this gives a mean value of V of 2.3%. The corrected values of impedance are given in Table 2. TABLE VALUES OF TRANSVERSE SPECIFIC CAPILLARY CONDUCTION. THE RESISTANCE
Frequency
Resistance 890
200
cycle/set 2 kc/set 20 kc/set
AF~ZR CORRECTION ARE THE EQUIVALENT
AND CAPACITIVE RFMITANCE OHMS FOR A I-CM CUBE
20
cycle/set
2
IMPEDANCE NUMBERS
k
EXPRESSED,
FOR BLOOD SERIES IN
Reactance 160
810 f 170 810 k 150 790 2 160
80 k 35 50 -+ 20 66 & 30 160 f
120
394
NICHOLSON
Mechanisms
of
Transverse
Conduction
At this point it is useful to anticipate a conclusion of this analysis: It is difficult to reconcile the low value of the transverse impedance, 2, with a small ( 5%) extracellular space. The interest in every case will therefore be to cakulate the maximum possible contributions from conduction in the neuroglia1 and axonal compartments. Conduction in these two compartments may be envisaged by two parallel paths. First, conduction may take place across individual fibers and processes. Second, the neuroglia cells could conduct by current entering the processes on, say, the left-hand ramifications of a cell, through the cytoplasm, and out via the processes of the same cell on the right-hand side. It is possible to dispense safely with the first possibility as contributing significantly. If we assume the axons and neuroglial processes are cylindrical, an expression for the impedance of a suspension of cylinders has been given by Bozler and Cole (2). Generalizing their expression to include two different species of fiber we obtain Z,/UI + RI - Ro (F)(l
-PP)
Z,/al
+ RI -k &,
+ &/a2 + R2 -
(1 --p>(1
-PP)
R.
Z-/a:! + R2 + R,, =
Z--o Z+Ro
’
161
where Zi, Z,, RI, R2, al, a, are the membrane impedances, cytoplasm resistivities and radii of the two fiber species, the suspending medium has resistivity R,, and occupies a fraction p of the total volume, and F is the fraction of total fiber volume occupied by specie 1. Since this expression was derived on the assumption p -+ 1 it was necessary to examine its validity for p 3 0. By assuming the fibers to have a simple regular geometry it is shown in the Appendix that (rather surprisingly) the Bozler and Cole type of expression predicts the impedance within a small percentage even at P = 0, and therefore is assumed valid for the present consideration. For myelinated axons the membrane resistance and capacity may be taken as IO5 52 cm” and 5 X 10P3 uF cm-2, respectively (37). For neuroglial membrane the lowest resistance reported is 3 f2 cm2 (13) and the capacity may be assumed to be about 1 uF cm-2 in common with most cells (33, 34). To a high degree of accuracy for all frequences used Eq. [6] therefore reduces to 1-P
= (Z - Ro)/(Z
+ Ro),
[71
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
395
which represents the conduction due to the extracellular space only. If we assume the extracellular medium has a specific impedance similar to that of cerebrospinal fluid this gives a value (5) of 51 Q cm for RO. The necessity for osmotic equilibrium would appear to exclude a value much lower than this. On substituting the appropriate quantities into Eq. [7] it will be seen that the calculated impedance for a small (5%) extracellular space is inconsistent with the observed values of N 850 Q cm. The second method of conduction by the neuroglial compartment can be simplified by making the following assumptions. The neuroglial cytoplasm resistivity will be neglected. Each oligodendrocyte is assumed to have its processes closely apposed to the processes of adjacent oligodendrocytes thus giving the lowest over-all impedance. The oligodendrocyte processes will be assumed to ramify over a mean lateral distance D (Fig. 2). There are N oligodendrocytes per unit volume each having membrane of total area A and impedance 2,.
FIG.
2.
Idealized
oligodendrocyte;
D
is the mean
lateral
spread
of the processes.
Half the membrane area of each cell will present an impedance 2ZJA so that the minimum impedance associated with the current flow through one cell will be 4ZJA. A block of cells filling the volume D X I X 1 would therefore present an impedance of 4ZJNAD. The minimum impedance across a unit cube would therefore be Z zz 4Z1/NAD2. The product of N and A was determined from electron micrographs. By means of a map measurer the total length of neuroglial membrane
396
NICHOLSON
per unit area in several different sections was measured. Provided a large enough number of random sections are measured in this way it can be simply shown that the mean length of membrane per unit area is equal to the mean glial membrane area per unit volume of tissue, i.e., N X d. Ten randomly selected electron micrographs of the rat corpus callosum’ sectioned normal to the fiber direction and each of area 16 u3 were analyzed. For the purpose of measurement, membrane was only included when at least one side apposed to neuroglial cytoplasm. The value obtained and standard deviation were (4.2 i 1.2)X lo4 cm per cm3. If the neuroglial processes are assumed to be cylindrical and occupying 3070 of the total tissue volume then the above figure indicates the mean radius would be 0.1 y. The value for D was found from light microscopy. Although the silver carbonate method will only impregnate major processes the assumption was made that these would be a fair index of the spread of all fibers. An estimate of size was made from Penfield’s (23) illustration of cat cerebellar white matter, a scale being added by the present author on the basis of observations3 on material from cat internal capsule stained with Dockrill’s modification of Rio Hortega’s method (24). Further evidence came from the observation of nuclear size of 3 1~ (Nissl stain) and also reported by Hosokawa and Mannen ( 15). Thus it was determined that in Fig. 2 the maximum value of D is 20 u. Kuftler and Potter (17) reported the neuroglial membrane resistance as 1000 Q cmS for the leech ganglion cells. It appears that the values of 3-10 Q cm2 obtained by Hild and Tasaki ( 13) for cultured neuroglia cells are subject to considerable uncertainty as the contribution of the cell processes could not be assessed. Discussion
The minimum total impedance would be made up by extracellular (Eq. [ 71) and neuroglial impedance (Eq. [S] ) electrically in parallel. For an extracellular space of 5% the membrane impedance would have to be < 70 Q cmZ. Since Eq. [8] was formulated on the basis of perfect cellto-cell contact and zero resistivity cytoplasm, the effect of taking these into account might reasonably be guessed as introducing a large factor into Eq. [8] so that unless the membrane resistance is much lower than 3 The author the histological
would like to thank Miss Arlene preparation of this material.
Koithan
and Miss
Cora
Rucker
for
IMPEDANCE
OF
CEREBRAL
WHITE
397
MATTER
70 Q cm2 it seems that a space of ~10% must account for the observed impedance. Additional evidence for this came from the fact that at 20 kc/set no significant drop in the resistive component, or large increase in the reactive component of the tissue impedance was observed. This indicates that the low membrane reactance (8 Q cmZ) at this frequency is not a significant factor and implies one of two things. Either the membrane resistance is < 8 Q cmB and hence “shorts out” the reactance, or the neuroglial conduction is negligible anyway. It may be noted that this reasoning is still valid even if the cytoplasmic resistivity and cable conduction of the processes (25) is taken into account. Kuffler and Potter ( 17) have reported a low resistance connection between adjacent neuroglia cells of the leech central nervous system. However, there appearsto be no reasonto assumethis cannot be explained on the basis of the close apposition of the large areas of membrane of the the leech cells. The existence of the small amount of reactivity in the measured impedance at the lower frequencies is not easy to account for. To explain it on the basis of membrane capacity would require the assumption of a very large capacity at low frequencies only. Similar effects have been reported in suspensionsof glassand polyethylene spheresand have been interpreted in terms of a tangential conductivity at the interface (10, 35, 36). However, it can be shown that the effect would be too small to account for the present observations. A similar problem was encountered in striated muscle by Fatt (8) and extensively examined. Fatt attributed the low frequency reactivity to a diffusion-limited current flow through the sarcoplasmictubules, but presumably no such system exists in neuroglia cells. There is a possibility that the existence of an extracellular gel (4) and an electrostatically-held ionic atmosphere could explain the effect. Such effects are commonly observed in colloids (2 1, 35) but too little is known about the extracellular medium to draw any conclusionsin the present case. APPENDIX:
APPLICABILITY
OF BOZLER-COLE
REGULAR
CLOSE
PACKED
EXPRESSION
FOR
FIBERS
If the cytoplasm is assumedto have zero resistivity, for a single specie of fiber Eq. [6] can be re-expressed 1
-= z
P R,(2--p)
+
4(1
-P)
Rot2 - P)P f &(2
-P)%
[91
398
NICHOLSON
For the cases of interest &,p < Zl(2 - p)/al 0 < p < 0.2 correspondingly 1 > 4(1 --p)/(2
--)‘a
and in the range
0.987.
[lOI
Making these two approximations Eq. [9] simplifies to l/Z = P/&I(~ -P)
+ al/Zl.
[Ill
Hexagonal Fibers. Assuming a constant interstitial distance between membranes, perfectly conducting cytoplasm, and the current flow from left to right in Fig. 3 (upper), from consideration of symmetry it will be seen that the full lines are equipotentials and the broken lines are lines of current flow. This meansit is only necessaryto analyze the impedance across a region limited by broken and full lines and then combine impedances to yield the gross specific impedance Z. For small interstitial
VA
vB
FIG. 3. Upper: Array full lines are equipotentials alent circuit of an area broken and full lines.
of hexagonal and broken in upper part
fibers with a small interstitial space. Vertical lines are lines of current flsow. Lower: Equivof this figure bounded by adjacent pairs of
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
399
distances the impedance across such a region can be calculated from the equivalent circuit shown in Fig. 3 (lower). The gross specific impedance Z was found to be 1 -=z
3b 1 2Z1 I$,( 1 + 2 tanh 4,/$)
+ 11,
[I21
where 4’ = R,b/v’3Zlp and b is the side of the hexagon. The quantity in the square brackets is a function of + only and by direct numerical evaluation it can be shown that it differs from the quantity [l/3@ + l] by not more than 6% for any value of 4. Making this approximation and substituting for +,, Eq. [ 121 simplifies to l/Z = p/2Ro
$ a1/1.05Z1,
1131
where nal’) = a(3 Q 3) b2 so that al is defined as the radius of the “equivalent” cylinder, having the same cross-sectional area. It will be seenthat the agreement between Eqs. [ 1l] and [ 131 is surprisingly good. There is a maximum discrepancy of a few per cent for p < 0.1. The expression for the impedance was also worked out for current flowing from top to bottom in Fig. 3 (upper) and an identical expression to Eq. [ 131 was obtained. Square Fibers. Regardlessof whether the mean current flow is parallel to the side or to the diagonal of the square the following result was obtained : l/Z = P/2R, + a1/1.13Z1, [I41 where al, as before, is the radius of a cylindrical fiber of the samecrosssectional area. It will therefore be seen that the agreement between the circuit analysis approach (Eq. [ 141) and the Bozler-Cole expression (Eq. [ 111) is good for square fibers also. References 1.
2. 3.
4.
W. R., R. T. KADO, and D. 0. WALTER. 1965. Impedance characteristics of cortical and subcortical structures: evaluation of regional specificity in hypercapnea and hypothermia. Exptl. Neuuol. 11: 190-216. BOZLER, E., and K. S. COLE. 1935. Electrical impedance and phase angle of muscle in rigor. J. Cell. Camp. Physiol. 6: 229-241. CAMPBELL, A. C. P. 1939. Variations in vascularity and oxidase content in different regions of the brain of the cat. A.M.A. Arch. Neural. Psych&. 41: 223-242. COGGESHALL, R. E., and 0. W. FAWCETT. 1964. The fine structure of the central nervous system of the leech Hirudo Medicinalis. J. Neurophysiol. 27: 229-289. ADEY,
400 5.
6.
8. 9.
10. 11. 12. 13. 14.
15. 16.
17.
18.
19. 20. 21. 22. 23. 24.
NICHOLSON
CRILE, G. W., H. R. HOSMER, and A. F. ROWLAND, 1922. The electrical conductivity of animal tissues under normal and pathological conditions. Am. J. Physiol. 60: 59-106. DE ROBERTIS, E., and H. M. GERSCHENPELD. 1961. Submicroscopic morphology and function of glial cells. Znlevlz. Rev. Keuvobiol. 3: l-65. DUNNING, H. S., and H. G. WOLFF. 1937. The relative vascularity of various parts of the central and peripheral nervous system of the cat and its relation to function. J. Camp. Neural. 67: 433-350. FATT, P. 1964. An analysis of the transverse electrical impedance of striated muscle. PYOC. Roy. Sot. Ser. B159: 606-651. FREYGANG, W. H. JR., and W. M. LANDAU. 1955. Some relations between resistivity and electrical activity in the cerebral cortex of the cat. J. Cell Camp. Physiol. 45: 377-392. FRICKE, H., and H. J. CURTIS. 1937. The dielectric properties of waterdielectric interphases. J. Phys. Chem. 41: 729-745. “Neuroglia, Morphology and Function.” Blackwell, Oxford. GLEES, P. 1955. GONATAS, K, H. M. ZIMMERMAN, and S. LEVINE. 1963. Ultrastructure of inflammation with edema in the rat brain. Ant. J. Pathol. 42: 455-469. HILD, W., and I. TASAIX. 1962. Morphological and physiological properties of neurons and glial cells in tissue culture. J. Neurophysiol. 25: 277-304. HORSTMANN, A. L., and H. MEVES. 1959. Die Feinstruktur des moleku!aren Rindengraues und ihre physiologische Bedeutung. Z. Zellfousch. Mikroskop. Anat. 49: 569-604. HOSOKAWA, I,, and I. MANNEN. 1963. “Morphology of Ne;lroglia,” J. Nakai [ed.]. Igaku Shoin Ltd., Japan. KLATZO, I., A. PIRAUX, and E. J. LASKOWXI. 1958. The relationship between edema, the blood-brain barrier and tissue elements in a local brain injury. J. Neuropathol. Exptl. Newel. 17: 548-564. KUFFLER, S. W., and D. G. POTTER. 1964. Glia in the leech central nervous system: physiological properties and neuron-glia re!ationship. J. Seurophysiol. 27: 290-319. LEAO, A. A. P., and H. M. FERREIRA. 1953. Alteracao da impedancia electrica no decurso da depress0 abastrante da atvidade do cortex cerebral. Anais Acad. Brad Cienc. 25: 259-266. LUSE, S. A. 1962. Ultrastructure and metabolism of the nervous system. Res. Publ. Assoc. Res. Nervous Mental Disease. 40: l-26. MAXWELL, J. C. 1891. “A Treatise on Electricity and Magnetism.” 3rd ed., Vol. 1. Dover, New York (1954). MICHAELS, A. S., G. L. FALKENSTEIN, and N. S. SCHNEIDER. 1965. Dielectric properties of polyanion-polycation complexes. J. Phys. Chem. 69: 1456-1465. NICHOLSON, P. W. 1965. Experimental models for current conduction in an anisotropic medium (to be published). PENFIELD, W. 1924. Oligodendroglia and its relation to classical neuroglia. Brain 4’7: 430-452. PENFIELD, W. 1930. A further modification of Del Rio Hortega’s method of staining oligodendroglia. Am. J. Pathol. 6: 445-448.
IMPEDANCE
25. 26. 27. 28. 29. 30. 31. 32.
33. 34.
35.
36. 37.
38. 39.
40.
41. 42. 43.
OF
CEREBRAL
WHITE
MATTER
401
RALL, W. 1960. Membrane potential transients and membrane time constant of motoneurones. Exptl. Neural. 2: 503-532. RANCX, J. B. JR., 1963. Specific impedance of rabbit cerebral cortex. Exptl. Neural. 7: 144-152. RANCK, J. B. JR. 1963. Analysis of specific impedance of cerebral cortex. Exptl. Neurol. 7: 153-174. RANCK, J. B. JR., and S. L. BEMENT. 1965. The specific impedance of the dorsal columns of cat: an anisotropic medium. Exptl. Neural. 11: 451-463. RIO HORTEGA, P. DEL. 1942. La neuroglia normal, conceptos de angliogliona y neurogliona. Archives Histol. Norm. Patol. Buenos Aires 1: 5. RUSH, S. 1962. Methods of measuring the resistivity of anisotropically conducting media in situ. J. Res. Natl. Bur. Stand. U.S. C66: 217-222. SCHULTZ, R. L. 1964. Macroglial identification in electron micrographs. J. Comp. Neurol. 122: 281-295. SCHULTZ, R. L., E. A. MAYNARD, and D. C. PEASE. 1957. Electron microscopy of neurons and neuroglia of cerebral cortex and corpus callosum. Am. J. Anat. loo: 368-407. SCHWAN, H. P. 1957. Electrical properties of tissue and cell suspensions. Advan. Biol. Med. Phys. 5: 148-209. SCHWAN, H. P. 1963. Determination of biological impedances, pp. 323-407. In “Physical techniques in biological research.” W. L. Nastuk [ed.]. Academic Press, New York. SCHWAN, H. P., G. SCHWARZ, J. MACZUK, and H. PAULY. 1962. On the lowfrequency dielectric dispersion of colloidal particles in electrolyte solution. J. Phys. Chem. 66: 2626-2635. SCHWARZ, G. 1962. A theory of the low-frequency dielectric dispersion of colloidal particles in electrolyte solution. J. Phys. Chem. 66: 2636-2642. TASAKI, I. 1955. New measurements of the capacity and the resistance of the myelin sheath of the nodal membrane of the isolated frog nerve fibre. Am. J. Physiol. 181: 639-650. TASAKI, I. 1964. A new measurement of action currents developed by single nodes of Ranvier. J. Neurophysiol. 27: 1199-1206. TASAKI, I., and K. MI~UCUCHI. 1949. The changes in the e!ectrical impedance during activity and the effect of alkaloids and polarization upon the bioelectric processes in the myelinated nerve fibre. Biochim. Biophys. Acta 3: 484-493. VAN HARREVELD, A., J. ,CROMWELL, and S. K. MALHMRA. 1965. A study of extracellular space in central nervous tissue by freeze-substitution. 1. Cell Biol. 25: 117-137. VAN HARREVELD, A., T. MURPHY, and K. W. NOBLE. 1963. Specific impedance of rabbit’s cortical tissue. Am. J. Physiol. 205: 203-207. VAN HARREVELD, A., and S. O~CHS. 1956. Cerebral impedance changes after criculatory arrest. Am. J. Physiol. 187: 180-192. VAN HARREVELD, A., and J. P. SCHADB. 1960. On the distribution and movements of water and electrolytes in the cerebral cortex, pp. 239-256. In “Structure and Function of the Cerebral Cortex.” D. B. Tower and J. P. Schade reds.]. Elsevier, Amsterdam.
NEUROLOGY
Specific
13,
386-401
Impedance
of
PAUL
Space Biology
(1965)
W.
Cerebral
White
NICHOLSON'
Laboratory, Brain Research Institute, Cniversity Center for Health Sciences, Los Angeles, California Received
June
30, 1965;
Matter
Revision,
August
of California
18, 1965
A method is described for measuring specific impedances in deep brain structures. It was used to determine the impedance in a large fiber tract in the cat internal capsule. The impedance along an axis normal to the fiber direction was + 800 B cm and predominantly resistive at frequencies between 20 cycle/set and 20 kc/set. The impedance along the fiber direction was approximately one-ninth of this. Possible mechanisms of current conduction are examined and it is concluded that unless the oligodendrocyte membrane resistance is < 8 (;2 cm” an extracellular space of -10% is indicated. Introduction
The electrical impedance of cerebral tissue is of interest for several reasons. For instance, there have been severa observations (1, 18, 42) on the variations of impedance in several regions of the brain under different physiological conditions and it is plainly desirable to form some hypothesis concerning the contributory factors. Attempts to correlate the impedance with the anatomy have been mainly confined to cerebral cortex. Van Harreveld et al. (41-43) made extensive measurements under different physiological conditions and interpreted their results as indicating a large extracellular space. Ranck (26) made comprehensive investigations of resistive and reactive components of impedance over a wide range of frequencies. It is apparent from the analysis 1 This investigation was supported in part by U.S. Air Force Office of Scientific Research grant AF-AFOSR 246-63 and Public Health Service Research Grant MH 3708 from the National Institute of Mental Health. The author is indebted to Raymond T. Kado and Dr. W. Ross Adey for their help and encouragement and to Miss Hiroko Kowta for preparing the illustrations. The author’s present address is Chemistry Dept., University College, London W.C. 1. 2 The author is greatly indebted to Dr. D. C. Pease for making these electron micrographs available to him. 386
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
387
by Ranck (27) that the anatomical complexity of cortex makes a detailed interpretation of the impedance data difficult. The present measurements were made in a simpler structure, white matter, in the hope that they would yield more definite information about the histology, and in particular, about the more or less hypothetical CNS extracellular space (6, 14, 19, 40). The exact location in the internal capsule of the cat was chosen on the basis of a large content of wellaligned fibers and ease of accessibility for the experimental technique. Recently Ranck and BeMent (28) reported similar measurements for the white matter of the cat dorsal column. Anisotropic
Conductivity
Subject to certain conditions, the properties of an anisotropically conducting material can be characterized (20, 30) by three principal conductivities in three mutually perpendicular directions (the principal axes) fixed relative to the material. Maxwell (20; Sec. 297) showed that the necessary conditions are satisfied by any material which can be regarded as an assembly of elements each of which are ordinary linear conductors. If, therefore, the elements of biological tissue (membranes, intra- and extracellular components, etc.) conduct linearly the conditions are satisfied. The only serious source of nonlinearity would be expected to arise in conduction across membranes, and in the present work care was taken to use current magnitudes which precluded nonlinear behavior. Although Maxwell’s treatment was confined to materials having only resistive conductivity it is a simple matter to extend the validity to include reactive conductance also. The principal axes for resistive conductance may be different from those for reactive conductance. In the present case, however, the principal axes stem from a geometrical (i.e., anatomical) anisotropy and will therefore be common to both reactive and resistive conductances. The problems of determining the specific impedance of biological tissue have been well summarized by Schwan (33, 34). The only way of adequately eliminating the effects of electrode impedance (“polarization”) in the present case was to use separate electrodes to apply the current and to sample the voltage. The configuration used was similar in principle to that used by Ranck (28) and consisted of an approximately point source of current placed in the brain region of interest, a remote (or “indifferent”) electrode as current sink, and a movable probe electrode to sample the voltage in the vicinity of the point source.
388
NICHOLSON
For a point source of current I in an infinite medium having principal conductivities oz, cry, op, the voltage V at any point x, y, z (expressed in coordinates of the principal axes of the medium) is given by (30) V=
I
-. 4X ( o,lsyx2 + oco,,.y2 + oUo,rz’)) 4
[ll
If measurements of voltage V1 and Vi, are made at distances d, and dz along the z axis, from Eq. [I] we obtain
[21 and correspondingly for voltages along the other two axes. While in general it would be necessary to make voltage measurements in three directions to determine the three conductivities uniquely, due to the cylindrical symmetry of a fiber tract, in the present case it is immediately possible to identify one of the principal axes as the fiber axis (say the z axis) and also to infer that es = ey. This means that measurements of voltage need only be made along the z axis and one of the other axes. In other words the voltage must be sampled along the fiber axis and also normal to it. Equations [I] and [ 21 are valid for the in-phase current or the out-ofphase current depending on whether the in-phase conductivity (i.e., the resistive conductivity) or the out-of-phase conductivity (i.e., the reactive conductivity) is used in the equations. Since the experimental arrangement was only approximately a point source, Eq. [2] could not be used to calculate the conductivities directly. Large scale models (40X ) of the electrode assembly were therefore constructed and subsidiary measurements were made in saline. Although saline is an isotropic conductor it is possible to exactly simulate the anisotropic conductor case by using a model with distorted (i.e., anisotropic) dimensions (22). In effect these subsidiary measurements served to compute a correction to Eq. [ 21. Method The “point” electrode was the uninsulated tip of an insulated stainless steel tube of 0.7-mm diameter. Inside this tube and protruding 1 to 2 mm beyond the end was a sliding wire probe, insulated except at its tip, which could be driven by a micrometer to take up different positions and thus sample the voltage at different places. The remote electrode was a plati-
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
389
nized stainless steel plate placed sagittally between the cerebral hemispheres. Figure 1 shows the complete experimental arrangement. The “up” position of the coupled switches enabled the cathode follower to measure the voltage across Rr and hence the current flowing from the “point” source. The “down” position enabled the cathode follower to measure the voltage on the inner probe electrode. The signal from the cathode follower was amplified, filtered and displayed as a Lissajous figure on the oscilloscope. Since the combined capacity of the interelectrode capacity and the cathode follower input capacity tended to load the voltage probe (inner I
I
FIG.
1.
Complete
experimental
arrangement.
electrode) at the higher frequencies it was necessary to apply a correction to the measured voltages. The value C1 was made equal to this combined capacity (- 12 pf) and, by observing the change in voltage when the calibration switch (Fig. 1) was thrown, it was possible to calculate back and determine what the voltage would have been had there been no loading of the voltage probe. In practice this correction constituted less than 10% of the calculated impedances at 20 kc/set and was negligible at the other frequencies used. Consideration of other effects of stray capacitances showed that they were negligible or differenced out. The entire procedure was checked by making measurements in various standard solutions of KCI. At all frequencies used the measured impedances agreed well with accepted values and showed no reactive component as expected. It was necessary to make measurements in a region of white matter where the fibers ran mutually parallel over at least a cube of 2-3 mm side. Such a region exists in the internal capsule of the cat at coordinates ante-
390
NICHOLSON
rior 7.0 mm, lateral 10.5 mm, height 8.0 mm, and is located dorsal to the anterior end of the lateral geniculate body. The fibers run in a sagittal plane, at 45” to the coronal plane, between posterodorsal and anteroventral. It was therefore possible to insert the probe assembly into this region so as to be parallel to the fiber axis or alternatively perpendicular to the fiber axis. All measurements were made with a maximum rms voltage and current of 4 mv and 1 t-1amp, respectively. This corresponds to a peak electric field of 10 mv per mm. Before nonlinear (active) characteristics of neuroglial or neuronal membrane would set in, it would be necessary to have at least 20 mv (17, 39) across a single membrane. This possibility can therefore be safely excluded in the present measurements. Six cats weighing between 2.1 and 3.6 kg were anesthetized with Nembutal (35-50 mg/kg, ip). Four holes were drilled in the skull to allow placements of the probe assembly. The skull was removed over the midline a total width of 5 mm, the dura was cut along a sagittal plane 2 mm from the midline and the remote electrode placed between the hemispheres. This electrode was held in position by its lead wire which was cemented to the skull with dental acrylic resin. The probe assembly and cathode follower were mounted on a conventional electrode carrier and stereotaxically located with the tip in the region of interest, All probe placements were subsequently checked histologically. Measurements
Longitudinal and transverse measurements were made in each internal capsule (two probe placements each side), and of the resulting twentyfour sets of data, four sets were discarded due to inaccurate placement and a further two discarded because of extensive subcortical bleeding. The calculated specific impedances are given in Table 1. The errors in Table 1 TABLE
1
VALUES OF SPECIFIC IMPEDANCE OF WHITE MATTER EXPRESSED AS THE EQUIVALENT SERIES RESISTANCE AND CAPACITIVE REACTANCE IN
OHMS
Normal to Frequency 20 200 2 20
cycle/set cycle/set kc/set kc/m
Resistive 850 770 770 750
-cin k 2
150 160 140 150
FOR A I-CM
CUBE
fibers
Parallel
Reactive 67 39 55 140
2 5 -c c
32 18 27 110
Resistive 89 89 78 78
-c e k zk
40 39 33 33
to fibers Reactive 7k8 425 6-c6 15 & 22
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
391
were assessedfrom the standard deviations in the measuredvoltages and an additional 10% error from the measurementsmade with the model electrode. No&finite Medium. The method assumesthe measurementsare made in an infinite homogeneousmedium, whereas the region of measurement was limited by structures of different specific impedance. This was investigated in the work on the large-scale model electrode. Large nonconducting and well-conducting masseswere introduced into the vicinity of the model. The somewhat qualitative conclusion was that the effect of the structures near the region of internal capsule where measurementswere performed would be small compared to the errors quoted above. Bleeding. The bleeding that often took place around the probe assembly would offer a preferential path for the current and thus modify the voltagesobserved. Assuming a value of 150 Q cm for the resistivity of blood (34), it was calculated that a O.l-mm film of blood over the electrode surface would alter the calculated impedancesin Table 1 by less than 5%. No measurementsof blood film were made but it was thought that the error introduced into the measuredimpedancewould be negligible. Other evidence for this assumption came from the consistency of the experimental data from experiment to experiment. In two cases,the measurementswere discarded due to extensive subcortical bleeding but, in fact, the measured impedanceswere indistinguishable from those taken with minimal bleeding conditions. Other Sources of Error. Similar arguments may be applied for tissue damage local to the surface of the electrodes. The problem of tissue reaction to the mechanical disturbance of the electrode is more imponderable. Van Harreveld, Murphy and Noble (41) found the impedance of cerebral cortex very sensitive to mechanical trauma. In white matter, no histologically evident effects of injury have been reported within 6 hours of injury (12, 16). This is too long a time for any such effects to make themselves felt in the duration of an experiment. The above finding does not, of course, preclude many other possible consequencesof mechanical trauma. However, impedance measurementsmade at the beginning of an experiment were indistinguishable from those made 2 hours later, so that if the tissue were in someway modified so as to effect its impedance, the modification must occur completely ,within the first few minutes of placement of the electrode. Other Measurements. Several other measurementsof the impedance of white matter have been made. The data of Van Harreveld, Murphy and
392
NICHOLSON
Noble (41) taken at 1 kc/set in rabbit white matter adjacent to the cortex, yields the value of 960 _t 70 Q cm. Freygang and Landau (9), using pulses, reported a much lower value of 330 !J cm for the cat. Both these determinations lie within the range reported in the present determination (Table 1) . The work of Ranck (28) on the cat dorsal column indicates values for longitudinal and transverse resistive components of approximately 138 52 cm and 12 11 Q cm, respectively. A small reactive component was observed in both cases. In view of the uncertainties it appears that these are in quite good agreement with the values in Table 1. For frog sciatic nerve Tasaki (38) has reported a transverse specific impedance of 7 000 to 13 000 Q cm and longitudinal specific impedance of 245 Q cm. It is rather surprising that the difference between central and peripheral white matter should apparently account for such a difference in the impedances. Histology
of
Cerebral
White
Matter
The histology of cerebral white matter has been reviewed by several authors (6, 11, 19). The myelinated nerve fibers are found in bundles in close proximity to the fine processes of oligodendrocytes, and occasionally astrocytes and transitional forms of neuroglia cells have been seen. The classic work of Rio Hortega (29) and Penfield (23) showed the oligodendrocytes to be arranged in rows parallel to the myelinated nerve fibers, and with long processes running in the same direction. Electron microscopy (6, 19, 31, 32) has confirmed the early observations and shown that the oligodendrocyte has a large number of very fine processes or sheets. In all cases the extracellular space in normal material has been observed to consist of the usual 120-200 A gap between membranes. Correction
for
Conduction
by
Capillaries
A small part of the experimentally determined impedances would be accounted for by conduction due to blood capillaries. This will now be calculated and a correction applied to give the impedance of the rest of the tissue. Let 2 be the measured value of the specific impedance N 800 CJcm, 2 be the corrected value of the specific impedance N 850 Q cm, Rn be the resistivity of blood 150 Q cm (34), and V be the fractional volume of blood in white matter N 2%.
IMPEDANCE
OF
CEREBRAL
WHITE
393
MATTER
The maximum contribution due to blood conduction would be when the blood vessels all run in the direction of current flow. In this case: z=
Z’(l--li)
1-
VZ’/Rs
N Z’[l
+
V(Z’/R,
-
[3]
l)]-z’[l+4.3v].
The minimum contribution would be when the vessels all run normal to the direction of current flow. An expression for this has been given by Bozler and Cole (2) which gives 1 +
z = 2’ 1
-
V(z--R,)/(Z
V(Z-
+RB)
RB)/(Z
+ RB) N Z’(
1 +
2v
Z-RB )P
Z’(1
+
1.4V).
[4]
Z+RB
If the vessels are assumed to run in equal numbers in the X, y and z directions, then for current flow down any of these axes, twice as many vessels would be normal to the conduction axis as there are parallel to it. After weighting [3] and [4] accordingly we obtain as the final correction expression : Z = Z’( 1 + 2.4V). [51 The most critical factor in the above analysis is V. For the cat the only source of information is from capillary counts. For white matter in the cat, Campbell (3) gives a total capillary length of 4.5 X 104 cm per cc and Dunning and Wolff (7) give a value of 3.74 X lo4 cm per cc. If we assume these figures include a linear shrinkage of 15% through fixation and the capillary diameter is 10 p then this gives a mean value of V of 2.3%. The corrected values of impedance are given in Table 2. TABLE VALUES OF TRANSVERSE SPECIFIC CAPILLARY CONDUCTION. THE RESISTANCE
Frequency
Resistance 890
200
cycle/set 2 kc/set 20 kc/set
AF~ZR CORRECTION ARE THE EQUIVALENT
AND CAPACITIVE RFMITANCE OHMS FOR A I-CM CUBE
20
cycle/set
2
IMPEDANCE NUMBERS
k
EXPRESSED,
FOR BLOOD SERIES IN
Reactance 160
810 f 170 810 k 150 790 2 160
80 k 35 50 -+ 20 66 & 30 160 f
120
394
NICHOLSON
Mechanisms
of
Transverse
Conduction
At this point it is useful to anticipate a conclusion of this analysis: It is difficult to reconcile the low value of the transverse impedance, 2, with a small ( 5%) extracellular space. The interest in every case will therefore be to cakulate the maximum possible contributions from conduction in the neuroglia1 and axonal compartments. Conduction in these two compartments may be envisaged by two parallel paths. First, conduction may take place across individual fibers and processes. Second, the neuroglia cells could conduct by current entering the processes on, say, the left-hand ramifications of a cell, through the cytoplasm, and out via the processes of the same cell on the right-hand side. It is possible to dispense safely with the first possibility as contributing significantly. If we assume the axons and neuroglial processes are cylindrical, an expression for the impedance of a suspension of cylinders has been given by Bozler and Cole (2). Generalizing their expression to include two different species of fiber we obtain Z,/UI + RI - Ro (F)(l
-PP)
Z,/al
+ RI -k &,
+ &/a2 + R2 -
(1 --p>(1
-PP)
R.
Z-/a:! + R2 + R,, =
Z--o Z+Ro
’
161
where Zi, Z,, RI, R2, al, a, are the membrane impedances, cytoplasm resistivities and radii of the two fiber species, the suspending medium has resistivity R,, and occupies a fraction p of the total volume, and F is the fraction of total fiber volume occupied by specie 1. Since this expression was derived on the assumption p -+ 1 it was necessary to examine its validity for p 3 0. By assuming the fibers to have a simple regular geometry it is shown in the Appendix that (rather surprisingly) the Bozler and Cole type of expression predicts the impedance within a small percentage even at P = 0, and therefore is assumed valid for the present consideration. For myelinated axons the membrane resistance and capacity may be taken as IO5 52 cm” and 5 X 10P3 uF cm-2, respectively (37). For neuroglial membrane the lowest resistance reported is 3 f2 cm2 (13) and the capacity may be assumed to be about 1 uF cm-2 in common with most cells (33, 34). To a high degree of accuracy for all frequences used Eq. [6] therefore reduces to 1-P
= (Z - Ro)/(Z
+ Ro),
[71
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
395
which represents the conduction due to the extracellular space only. If we assume the extracellular medium has a specific impedance similar to that of cerebrospinal fluid this gives a value (5) of 51 Q cm for RO. The necessity for osmotic equilibrium would appear to exclude a value much lower than this. On substituting the appropriate quantities into Eq. [7] it will be seen that the calculated impedance for a small (5%) extracellular space is inconsistent with the observed values of N 850 Q cm. The second method of conduction by the neuroglial compartment can be simplified by making the following assumptions. The neuroglial cytoplasm resistivity will be neglected. Each oligodendrocyte is assumed to have its processes closely apposed to the processes of adjacent oligodendrocytes thus giving the lowest over-all impedance. The oligodendrocyte processes will be assumed to ramify over a mean lateral distance D (Fig. 2). There are N oligodendrocytes per unit volume each having membrane of total area A and impedance 2,.
FIG.
2.
Idealized
oligodendrocyte;
D
is the mean
lateral
spread
of the processes.
Half the membrane area of each cell will present an impedance 2ZJA so that the minimum impedance associated with the current flow through one cell will be 4ZJA. A block of cells filling the volume D X I X 1 would therefore present an impedance of 4ZJNAD. The minimum impedance across a unit cube would therefore be Z zz 4Z1/NAD2. The product of N and A was determined from electron micrographs. By means of a map measurer the total length of neuroglial membrane
396
NICHOLSON
per unit area in several different sections was measured. Provided a large enough number of random sections are measured in this way it can be simply shown that the mean length of membrane per unit area is equal to the mean glial membrane area per unit volume of tissue, i.e., N X d. Ten randomly selected electron micrographs of the rat corpus callosum’ sectioned normal to the fiber direction and each of area 16 u3 were analyzed. For the purpose of measurement, membrane was only included when at least one side apposed to neuroglial cytoplasm. The value obtained and standard deviation were (4.2 i 1.2)X lo4 cm per cm3. If the neuroglial processes are assumed to be cylindrical and occupying 3070 of the total tissue volume then the above figure indicates the mean radius would be 0.1 y. The value for D was found from light microscopy. Although the silver carbonate method will only impregnate major processes the assumption was made that these would be a fair index of the spread of all fibers. An estimate of size was made from Penfield’s (23) illustration of cat cerebellar white matter, a scale being added by the present author on the basis of observations3 on material from cat internal capsule stained with Dockrill’s modification of Rio Hortega’s method (24). Further evidence came from the observation of nuclear size of 3 1~ (Nissl stain) and also reported by Hosokawa and Mannen ( 15). Thus it was determined that in Fig. 2 the maximum value of D is 20 u. Kuftler and Potter (17) reported the neuroglial membrane resistance as 1000 Q cmS for the leech ganglion cells. It appears that the values of 3-10 Q cm2 obtained by Hild and Tasaki ( 13) for cultured neuroglia cells are subject to considerable uncertainty as the contribution of the cell processes could not be assessed. Discussion
The minimum total impedance would be made up by extracellular (Eq. [ 71) and neuroglial impedance (Eq. [S] ) electrically in parallel. For an extracellular space of 5% the membrane impedance would have to be < 70 Q cmZ. Since Eq. [8] was formulated on the basis of perfect cellto-cell contact and zero resistivity cytoplasm, the effect of taking these into account might reasonably be guessed as introducing a large factor into Eq. [8] so that unless the membrane resistance is much lower than 3 The author the histological
would like to thank Miss Arlene preparation of this material.
Koithan
and Miss
Cora
Rucker
for
IMPEDANCE
OF
CEREBRAL
WHITE
397
MATTER
70 Q cm2 it seems that a space of ~10% must account for the observed impedance. Additional evidence for this came from the fact that at 20 kc/set no significant drop in the resistive component, or large increase in the reactive component of the tissue impedance was observed. This indicates that the low membrane reactance (8 Q cmZ) at this frequency is not a significant factor and implies one of two things. Either the membrane resistance is < 8 Q cmB and hence “shorts out” the reactance, or the neuroglial conduction is negligible anyway. It may be noted that this reasoning is still valid even if the cytoplasmic resistivity and cable conduction of the processes (25) is taken into account. Kuffler and Potter ( 17) have reported a low resistance connection between adjacent neuroglia cells of the leech central nervous system. However, there appearsto be no reasonto assumethis cannot be explained on the basis of the close apposition of the large areas of membrane of the the leech cells. The existence of the small amount of reactivity in the measured impedance at the lower frequencies is not easy to account for. To explain it on the basis of membrane capacity would require the assumption of a very large capacity at low frequencies only. Similar effects have been reported in suspensionsof glassand polyethylene spheresand have been interpreted in terms of a tangential conductivity at the interface (10, 35, 36). However, it can be shown that the effect would be too small to account for the present observations. A similar problem was encountered in striated muscle by Fatt (8) and extensively examined. Fatt attributed the low frequency reactivity to a diffusion-limited current flow through the sarcoplasmictubules, but presumably no such system exists in neuroglia cells. There is a possibility that the existence of an extracellular gel (4) and an electrostatically-held ionic atmosphere could explain the effect. Such effects are commonly observed in colloids (2 1, 35) but too little is known about the extracellular medium to draw any conclusionsin the present case. APPENDIX:
APPLICABILITY
OF BOZLER-COLE
REGULAR
CLOSE
PACKED
EXPRESSION
FOR
FIBERS
If the cytoplasm is assumedto have zero resistivity, for a single specie of fiber Eq. [6] can be re-expressed 1
-= z
P R,(2--p)
+
4(1
-P)
Rot2 - P)P f &(2
-P)%
[91
398
NICHOLSON
For the cases of interest &,p < Zl(2 - p)/al 0 < p < 0.2 correspondingly 1 > 4(1 --p)/(2
--)‘a
and in the range
0.987.
[lOI
Making these two approximations Eq. [9] simplifies to l/Z = P/&I(~ -P)
+ al/Zl.
[Ill
Hexagonal Fibers. Assuming a constant interstitial distance between membranes, perfectly conducting cytoplasm, and the current flow from left to right in Fig. 3 (upper), from consideration of symmetry it will be seen that the full lines are equipotentials and the broken lines are lines of current flow. This meansit is only necessaryto analyze the impedance across a region limited by broken and full lines and then combine impedances to yield the gross specific impedance Z. For small interstitial
VA
vB
FIG. 3. Upper: Array full lines are equipotentials alent circuit of an area broken and full lines.
of hexagonal and broken in upper part
fibers with a small interstitial space. Vertical lines are lines of current flsow. Lower: Equivof this figure bounded by adjacent pairs of
IMPEDANCE
OF
CEREBRAL
WHITE
MATTER
399
distances the impedance across such a region can be calculated from the equivalent circuit shown in Fig. 3 (lower). The gross specific impedance Z was found to be 1 -=z
3b 1 2Z1 I$,( 1 + 2 tanh 4,/$)
+ 11,
[I21
where 4’ = R,b/v’3Zlp and b is the side of the hexagon. The quantity in the square brackets is a function of + only and by direct numerical evaluation it can be shown that it differs from the quantity [l/3@ + l] by not more than 6% for any value of 4. Making this approximation and substituting for +,, Eq. [ 121 simplifies to l/Z = p/2Ro
$ a1/1.05Z1,
1131
where nal’) = a(3 Q 3) b2 so that al is defined as the radius of the “equivalent” cylinder, having the same cross-sectional area. It will be seenthat the agreement between Eqs. [ 1l] and [ 131 is surprisingly good. There is a maximum discrepancy of a few per cent for p < 0.1. The expression for the impedance was also worked out for current flowing from top to bottom in Fig. 3 (upper) and an identical expression to Eq. [ 131 was obtained. Square Fibers. Regardlessof whether the mean current flow is parallel to the side or to the diagonal of the square the following result was obtained : l/Z = P/2R, + a1/1.13Z1, [I41 where al, as before, is the radius of a cylindrical fiber of the samecrosssectional area. It will therefore be seen that the agreement between the circuit analysis approach (Eq. [ 141) and the Bozler-Cole expression (Eq. [ 111) is good for square fibers also. References 1.
2. 3.
4.
W. R., R. T. KADO, and D. 0. WALTER. 1965. Impedance characteristics of cortical and subcortical structures: evaluation of regional specificity in hypercapnea and hypothermia. Exptl. Neuuol. 11: 190-216. BOZLER, E., and K. S. COLE. 1935. Electrical impedance and phase angle of muscle in rigor. J. Cell. Camp. Physiol. 6: 229-241. CAMPBELL, A. C. P. 1939. Variations in vascularity and oxidase content in different regions of the brain of the cat. A.M.A. Arch. Neural. Psych&. 41: 223-242. COGGESHALL, R. E., and 0. W. FAWCETT. 1964. The fine structure of the central nervous system of the leech Hirudo Medicinalis. J. Neurophysiol. 27: 229-289. ADEY,
400 5.
6.
8. 9.
10. 11. 12. 13. 14.
15. 16.
17.
18.
19. 20. 21. 22. 23. 24.
NICHOLSON
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