The relationship between axon diameter, myelin thickness and conduction velocity during atrophy of mammalian peripheral nerves

Brain Research, 259 (1983) 41-56 Elsevier Biomedical Press

41

The Relationship Between Axon Diameter, Myelin Thickness and Conduction Velocity During Atrophy of Mammalian Peripheral Nerves M. J. GILLESPIE* and R. B. STEIN

Department of Physiology, University of Alberta, Edmonton, T6G 2H7 (Canada) (Accepted May 24th, 1982)

Key words: axon diameter - - myelin thickness - - conduction velocity - - nerve atrophy

The atrophy of cutaneous (sural) and muscle (medial gastrocnemius) nerves proximal to a ligation were studied in cats for periods up to 9 months, using light and electron microscopy, conduction velocity measurements and computer simulations. As atrophy proceeds, nerve fibres become increasingly non-circular. Cross-sectional areas of axons and fibres (axon + myelin) were measured. The diameters of equivalent circles (having the same axon and fibre cross-sectional area) were then calculated. A linear relation was found between axon diameter and fibre diameter, but the slope decreased as atrophy continued. This indicates that the axon cross-sectional area decreases relatively more than the total fibre area. Reduction in conduction velocity correlates more closely with reduction in axon diameter than fibre (axon + myelin) diameter. The ratio of the inner (axon) perimeter to the outer (myelin) perimeter remains constant at or near the optimal value of 0.6 for conduction in all groups of fibres at all periods of atrophy. Furthermore, the thickness of the myelin remains constant for a given perimeter over the entire period of atrophy studied. This suggests that the number of turns of myelin and the length of each turn remain unchanged during peripheral nerve atrophy. A simple geometric model explains how this can occur without gaps developing between the axon and myelin or between the turns of myelin. The Frankenhaeuser-Huxley equations for conduction in myelinated nerve fibres predict changes in conduction velocity similar to those observed, if the axons atrophy without changes in myelin. The advantages of this mode of atrophy are discussed, INTRODUCTION

few

E a r l y investigators all a g r e e d t h a t the fibres in the p r o x i m a l p o r t i o n o f a nerve b e c o m e r e d u c e d in size following nerve section (axotomy)15,~8,31,82,48, 50, 51,59. Several studies also i n d i c a t e d t h a t a x o n a l diameters were m o r e r e d u c e d t h a n fibre ( a x o n ÷ m y e lin) diameterl,Xs, 48. H o w e v e r , there was disagreem e n t in these light m i c r o s c o p e studies w h e t h e r myelin thickness increased15, 4s o r decreased 31. M o r e recent studies with the e l e c t r o n m i c r o s c o p e have failed to settle the issue10,54, 56. It is clear t h a t as a nerve fibre atrophies, the myelin occupies an increasing fraction o f the cross-sectional area, b u t it is u n c e r t a i n w h e t h e r this represents an a b s o l u t e increase in myelin thickness o r a relative increase in c o m p a r i s o n to the r e d u c e d a x o n size. R e d u c e d c o n d u c t i o n velocity p r o x i m a l to a lesion a c c o m p a n i e s the r e d u c t i o n in d i a m e t e r over the first

W h e n r e i n n e r v a t i o n o f the p e r i p h e r y is prevented, the loss o f fibre d i a m e t e r a n d r e d u c e d c o n d u c t i o n velocity persist indefinitelylS,17,32,4a,4s. A f t e r a p e r i o d o f two m o n t h s or m o r e following ligation, large sensory fibres a p p e a r to a t r o p h y m o r e t h a n o t h e r fibres in the nerve, when m e a s u r e d as charge g e n e r a t e d at the d o r s a l a n d ventral rootsaa, a4 or as a r e d u c t i o n in c o n d u c t i o n velocity 43. M i l n e r a n d Stein 43 cited p r e l i m i n a r y o b s e r v a t i o n s t h a t the decline in c o n d u c t i o n velocity in a t r o p h i c uerves a p p e a r e d to be m u c h greater t h a n the decline in fibre d i a m e t e r m e a s u r e d histologically. T o t a l fibre d i a m e t e r has often been used as the relevant p a r a meter d e t e r m i n i n g c o n d u c t i o n velocity o f p e r i p h e r a l nerve fibres 3-5A4,~5,37,47. H o w e v e r , m a n y o t h e r p a r a m e t e r s have been suggested to affect c o n d u c t i o n velocity: a x o n d i a m e t e r 2a, a x o n p e r i m e t e r a n d crosssectional arealZ, as, i n t e r n o d a l distance14,15,47, 49 a n d

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* Present address: Department of Occupational Therapy, University of Alberta, Edmonton, Canada. 0006-8993/83/0000-0000/$03.00 © 1983 Elsevier Biomedical Press

42 the ratio of axon diameter to total fibre diameter 27, 47,49,52,57,60. There is considerable uncertainty about the relationship of this last ratio to fibre diameter in n o r m a l n e r v e s 9,1s,27,28,36,47,48,57,60 and almost no information for atrophic nerves. Fortunately, Frankenhaeuser and Huxley 25 developed an extensive series of equations for the conduction in myelinated fibres so that it is possible to compute the dependence of conduction velocity on a variety ofparametersZg,41,45,52, 5s. We therefore thought it would be worthwhile to re-examine the relationship between axon diameter, myelin thickness and conduction velocity in normal and atrophic nerves using anatomical measurements, physiological measurements and computer calculations based on the Frankenhaeuser-Huxley equations. Interestingly, all our observations turned out to be compatible with the simplest possible model of nerve fibre atrophy, namely that the cross-sectional area of the axon declines without any marked change in the thickness or other properties of the myelin. Under these conditions the axon gradually becomes less circular in such a way that no gaps develop between the axon and the myelin. Conduction velocity then varies to a first approximation as axon diameter and not as fibre (axon + myelin) diameter. MATERIALS AND METHODS The sural nerve and the medial gastrocnemius (MG) nerves were cut and ligated in one hind limb of 26 cats. The ligature was sewn to a small silastic pad placed between the nerve and its former target organ to prevent reinnervation. At intervals of about 20 days between 29 and 273 days after nerve section, the compound action potentials generated by the cut nerves were studied in acute experiments and conduction velocity distributions were determined as previously described 4~,44. The contralateral nerves were used as controls.

Light microscopy U p o n completion of the physiological studies, samples 7-15 m m in length of each experimental and control nerve were taken about I0 m m proximal to the cut end or to the recording site. Several experimental nerves were also sampled 2-3 cm proximal to the cut. The nerves were fixed in 3 ~ gluteraldehyde

in 0.1 M phosphate buffer for 2-6 h, post-fixed in osmium tetroxide for 1 h, dried in ascending alcohols and embedded in Araldite. Sections of 1 p m thickness were stained with 2.5 ~ p-phenylenediamine and photographed with phase-contrast microscopy using a green filter. Photographs were printed at 1000 times magnification and every myelinated fibre in each nerve was compared with circles of standard sizes. The equivalent diameter assigned to each fibre was that of the standard circle having the closest area. Histograms of fibre size were prepared and the cumulative percentage of fibres at or below a given total diameter was plotted against logx0 fibre diameter (see for example Fig. 3).

Electron microscopy Electron micrographs of ultra-thin (silver-gray) sections of nerves were printed at a magnification of 9240. Per nerve 50-150 fibres were measured by tracing the perimeter s, of the axon adjacent to the innermost myelin lamella and the perimeter S of the outermost myelin lamella using a cursor with a cross-hair. The rectangular coordinates of the position of the cross-hair during the tracing were recorded using a microcomputer (LSI-11; Digital Equipment Corp.) and from these values, the perimeters and the areas enclosed by them (a and A respectively) were calculated. Further quantities were then derived as follows. deq = equivalent axon diameter, calculated as the diameter of a circle of the same area (i.e. deq

2 ~/a/~z). O e q = equivalent fibre (axon + myelin) diameter, calculated in the same way from the total area (i.e. D e q = 2x/A/~). Our measurements were reproducible to within 2 ~ and provide the most accurate means of determining true fibre sizeS,24,39,61. g -- ratio of inner perimeter to outer perimeter z (siS). The same letter g is often used for the ratio of inner to outer diameter (deq/Oeq), but the two ratios are only the same for a circle and are affected quite differently as nerves atrophy (see Appendix). m -- myelin thickness, calculated from the perimeters, m -- (S-s)/(2~t). This simple formula turns out to be remarkably accurate (see the Appendix) and far less time consuming than trying to measure the thickness directly at a number of points around the fibre or counting the turns of myelin.

43 q0 = circularity of axon, calculated as the axon cross-sectional area divided by the area of a circle with the same perimeter (~v = 4z~a/s2). Clearly, for a circle q~ ~- 1, and for a slit with no area ~0 = 0. Thus, q~ provides a convenient normalized measure of circularity 2. Dyck and his collaborators have used an index which is the square root of ~0 aboveZ0,21,sL = circularity of fibre (axon q- myelin), calculated as above from the outer measurements

(qb = 4z~A/S2). Computed action potentials Action potentials were computed from the Frankenhaeuser-Huxley equations 25 for a frog node, using the cable equations for the internodal regions, as first described by Goldman and Albus 29. The equations were implemented for ten nodes using a F O R T R A N program modified from Smith and Koles 5z and Koumarelas 42 to run on a PDP 11-34 computer. The standard conditions were for a fibre at 20 °C with an outer diameter of 15 # m and an inner diameter of 10.5/zm. For a circular fibre this corresponds to a ratio g = diD = 0.7. If the axon atrophied without any changes in the properties of the axon or myelin membranes (see Appendix), the only change would be in the axoplasmic resistance per unit length of cable:

Ra = Q/a = (4Q)/(~d 2) = 12.7 Mf2/mm where Q = the axoplasmic resistivity = 1100 f~mm and d = 10.5 #m. During atrophy the axon area a will decrease as the axon becomes non-circular, as will the diameter deq of an equivalent circle. The axoplasmic resistance Ra will therefore increase, while all other parameters of the equation remain constant. An 8-fold increase in Ra, corresponding to a 2.8-fold decrease in the equivalent axon diameter was studied. With the higher resistivity the stimulus at the first node was adjusted to ensure that an action potential was propagated. In computing conduction velocity the first node was ignored because of possible stimulus artefacts, and the last node was ignored because of the termination of the cable in a short circuit. Conduction velocity was determined as the time for the half-maximum point on the rising edge of the action potential to propagate over the intermediate nodes (internodal distance =

1.38 ram).

RESULTS Fig. 1 shows cross-sections of cat cutaneous (sural) nerves and muscle (medial gastrocnemius) nerves. Control nerves and nerves 228 days after axotomy are shown. The majority of axons in the experimental nerves showed altered profiles compared to the control nerves, and became progressively less round with time after ligation. The myelin sheaths became wavy and involuted, but always remained intact and in contact with the axolemma. They gradually collapsed into the axonal space as the axon shrank in volume and cross-sectional area. Similar changes have been observed and quantified in diabetic nerves 55 and in hyperosmolarity20, 21. In general, the sural nerves became more oval and flattened and were almost completely closed at times greater than 200 days after axotomy (Fig. 1B). The axons of the muscle nerve showed a regularly indented, crenated profile and many axons resembled normal para-nodal fibres. Some fibres remained relatively round, perhaps due to the lesser atrophy of the motor axons in these nerves 34.

Fibre diameter histograms Histograms of fibre diameter (axon q- myelin) measured from light micrographs of control sural nerves had a bimodal distribution with. peaks at 6 and 10 # m (Fig. 2). The bimodal nature of the histogram was generally lost for the experimental sural nerves. The major peak occurred at 4 #m and very few fibres were observed greater than 12 /,m in diameter. Similarly, the control muscle nerves had bimodal distributions with peaks at about 6 and 16 #m. After atrophy only a single peak was observed at 4--6 # m and again very few large fibres were observed. Note that the total numbers of nerve fibres were almost identical, even more than 200 days after ligation. Thus, the relative absence of large fibres was not due to loss of fibres, but rather to atrophy to smaller values. Since the bimodal character of the distribution was soon lost during atrophy, the relative changes in large and small fibres were difficult to determine from conventional histograms. There was also considerable variability in the degree of atrophy from nerve to nerve. We therefore grouped the nerves into 4 categories as previously43: control, early experi-

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Fig. 3. Cumulative fibre histograms giving the percentage of fibres less than or equal to the equivalent diameters indicated for control nerves and nerves grouped according to time periods after ligation as indicated. Each cumulative histogram is the average obtained for 2-5 nerves. Note that fibre diameters are plotted on a logarithmic scale so that the histograms would shift uniformly to the left, if all fibres atrophied to the same relative extent. Further details in the text.

3). For example, if two nerves had 50 and 60 °/o of the fibres less than or equal to 8 ffm in total diameter, a value of 55 ~ would be plotted in Fig. 3 for a diameter of 8 ffm. A logarithmic scale has been used in this figure so that the entire curve would be shifted to the left an equal amount if all fibres atrophied to the same relative extent (i.e. reduction from 20 to 10 ffm would produce the same shift as a reduction from 10 to 5 #m). The control nerves ( x ) in each part of Fig. 3 are the contralateral nerves from the same cats from which the experimental nerves were taken. In the early period ( V ) little atrophy is evident, although the diameters have shifted a few per cent. Atrophy in all fibre groups is seen in the later periods, but still only represents a change of 2030~. Milner and Stein 48 measured the changes in conduction velocity for the nerves in the same animals and observed larger changes for all fibre groups when these were plotted in the same manner (see Fig. 6). Since axons appear to atrophy more than the myelin (see Fig. 1 and the Introduction), more detailed measurements were made from electron micrographs to determine if changes in axon diameter corresponded more closely to the changes in conduction velocity.

Axon size and total fibre size Fig. 4 shows electron micrographs from control and experimental cutaneous and muscle nerves. F r o m these micrographs the inner and outer myelin perimeters were traced and the coordinates automatically entered into a computer (see Materials and Methods). The area within each perimeter was then calculated together with the equivalent axon and fibre diameters. Since the experimental nerves are quite irregular and non-circular (see particularly the large sural nerve fibre in Fig. 4), the equivalent diameter was taken to be the diameter of a circle having the same cross-sectional area. The relationships between axon diameter and fibre diameter are plotted in Fig. 5 for control nerves and after middle and later periods of atrophy. Although there is some scatter in the measurements, the data are all reasonably well fitted by straight lines, which were computed according to a least mean squares criterion. It is also evident that the slope of the fitted lines de-

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Fig. 4. Electron micrographs of fibres in the sural and MG nerves over 200 days after the nerves had been ligated. Some fibres had atrophied relatively little (MG) whereas in others the axoplasm had almost disappeared (large sural fibre). The scale shown on the right is 1/~m. creases as the period following ligation increases. Thus, for a fibre diameter of 8/~m, the axon would have a diameter 4-5 # m in a control sural nerve, but only about 2 # m during a later period of atrophy. The changes were somewhat less in the M G nerve, consistent with the less extreme atrophy noted in the light micrographs. The fitted straight lines also offer a means of converting the fibre diameter histograms of Fig. 3 to corresponding axon diameter histograms. In Fig. 6A the cumulative fibre diameter histograms are replotted for control ( × ) nerves and nerves after a middle (@) or later (&) period of atrophy. In Fig. 6B cumulative axon diameter histograms are seen after converting from fibre diameter to axon diameter using the fitted straight lines in Fig. 5. A much greater relative shift (atrophy) is seen in axon diameter than in fibre diameter.

Conduction velocity In Fig. 6C the corresponding cumulative conduction velocity distributions are plotted (see Milner and Stein 43 for details). It can be seen, for example, that in the control sural nerve only 20 % of the fibres had conduction velocities less than or equal to

20 m/s, but that the percentage rose to 50 % by the late period of atrophy. Conduction velocity histograms were computed separately for motor and sensory fibres 4a but were combined for comparison with fibre diameter histograms assuming 45 % of the fibres in the M G nerve are sensory G. Since the same ten-fold range of conduction velocities is used in Fig. 6C as for diameters in Fig. 6A and 6B, the data can be directly compared. Clearly, the change in conduction velocity is substantially greater than for fibre diameter, but similar or somewhat less than the change in axon diameter. We will return to the relationship between changes in axon size and conduction velocity later, but now turn to examining the changes, if any, in the myelin sheath during atrophy.

Myelin Since both increases and decreases have been reported in the thickness of myelin during atrophy (see the Introduction), the most appropriate hypothesis to test is the null hypothesis, namely that no change occurs in the number of turns (i.e. the thickness of myelin) or in the length of each turn. The lengths of the innermost and outermost turns are available from the perimeter measurements on elec-

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Fig. 5. Equivalent axon diameter is linearly related to equivalent fibre (axon + myelin) diameter in (A) control nerves and at various periods after ligation (13 and C), but the slope of the straight line decreases during atrophy. The symbols are the same as in Fig. 3. The straight lines were calculated from the standard methods for least mean square deviation and had the following slopes (and standard errors) for the sural nerves: (A) 0.66 4- 0.02, (B) 0.31 4- 0.02, and (C) 0.17 4- 0.03, and the M G nerves (A) 0.70 40.03, (B) 0.51 ± 0.06, and (C) 0.41 -4- 0.02.

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tron micrographs such as in Fig. 4, and the average thickness can be easily computed from the difference between these inner and outer perimeters (see Materials and Methods and Appendix). The myelin thickness is plotted against the fibre perimeter for control nerves and for the middle and

late periods of atrophy in Fig. 7. Arbuthnott et al. a rePorted a logarithmic relationship between myelin thickness and perimeter so a logarithmic scale has been used in plotting perimeter measurements. The relatively linear relationships in the semilogarithmic plots for the control data confirm the earlier results.

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Fig. 7. Myelin thickness m varies approximately as the logarithm of the outer (myelin) perimeter (S) for (A) control nerves and at various stages of atrophy (B and C). The slopes of the fitted lines did not show any statistically significant changes over the entire period studied or between the two nerves (cf. Fig. 5). The time periods are as defined in Fig. 3. Furthermore, the same relationship appears to hold out to the longest period of atrophy studied. There is no obvious decrease in slope, and there may if anything be a slight increase in the latest period (Fig. 7C). Another parameter of interest is the ratio g of inner perimeter to outer perimeter. For a circle the ratio g measured f r o m perimeters would be the same as the ratio diD of axon and fibre diameters. Since the axon atrophies substantially for a given fibre diameter (Fig. 5) the ratio g measured from perimeters would decline a similar amount if the axons remained relatively circular and the myelin got thicker to fill in the space vacated by the axon. This is clearly not the case, as shown in Fig. 8. Arbutlmott et al. 2 found that the myelin was rela-

tively thinner for the largest fibres (the ratio g was increased). Although there is a wider scatter of values for small compared to large fibres, our data show m a n y values o f g near 0.6 for all sizes of fibres at all stages of atrophy. The value 0.6 is of interest, since Rushton 47 calculated theoretically that this value would optimize conduction of nerve impulses for a circular fibre of a given total diameter. There are a few fibres with very low values of g in the late sural nerves, which is associated with the almost complete closure of some of the sheaths in these nerves (Fig. 4), with little or no axon remaining. Another possibility is segmental demyelination i9, but in this case values o f g should approach 1, which was not observed. Thus, our measurements of mye-

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Fig. 8. Most fibres have a ratio of inner (axon) to outer (myelin) perimeter (g = s/S) close to 0.6 in (A) control nerves and nerves at various periods of atrophy (B and C). The scatter of values tends to be greater for small fibres and after long periods of atrophy. The time periods are as defined in Fig. 3. lin thickness a n d the ratio g are consistent with the null hypothesis that there are n o substantial, morphometric changes in the myelin, except for a few fibres showing the most extreme degree of atrophy.

Non-circularity A r b u t h n o t t et al. 2 f o u n d that nerve axons are n o r m a l l y n o n - c i r c u l a r a n d provided a n o r m a l i z e d index o f circularity, which can be applied to axons or fibres (axon + myelin), as indicated in Materials a n d Methods. Our average values for control fibres were all between 0.7 a n d 0.8 (Fig. 9), while the circularity of control axons was a b o u t 0.2 lower for

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Fig. 9. The circularity of axons and fibres (axon + myelin) in sural and MG nerves decreases steadily with time after nerves are ligated. No major differences in the circularity of large ( x, +) and small (A, ~ ) fibres were seen at any time period, but the circularity of the axons was always less than that of the fibres.

51 all groups of fibres. These values are lower than reported by Arbuthnott et al. 2 who also observed differences between large and small fibres. The differences could be due to differences in fixation techniques or to the fact that they excluded axons which were sectioned in paranodal regions or near Schmitt-Lanterman clefts. No such correction was made in the present study because it was impossible to determine such features accurately from crosssections of atrophied fibres. The circularity decreases steadily with time after the nerve was ligated, with average values near 0.3 for axons at the longest period studied. For comparison, the schematic diagram in Fig. 11C shows an axon with a long axis 2.5 times the short axis. This represents a circularity index of 0.76. Thus, during atrophy axons become extremely non-circular.

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Computed conduction velocity Changes in conduction velocity were calculated based on the Frankenhaeuser-Huxley equations for frog Nodes of Ranvier (see Materials and Methods). Although there are important differences between frog nodes and mammalian nodes12, 46, analogous equations are not available for mammalian nodes. Changes in longitudinal axoplasmic resistance were implemented which would occur during atrophy, and the action potentials, computed for the fifth (of ten) nodes, are shown in Fig. 10A. A wide variation in axoplasmic resistance or equivalent axon diameter had relatively little effect on the size or duration of the spike, but produced a marked increase in the time for the action potential to propagate to this point in the fibre. As the axoplasmic resistance increases the spread of the action potential becomes less secure, and with the highest value shown, the voltage just reaches threshold. Conduction failed when the axoplasmic resistance was increased further (more than 8 times the standard value). Although nodal membrane properties were assumed to remain unchanged, conduction velocity is relatively insensitive to nodal properties as compared to internodal properties 4~. I f there were segmental demyelination these calculations would not apply. Conduction velocity was computed from the propagation time between nodes (see Materials and Methods) and the observed values are plotted for different values of the equivalent axon diameter deq

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i

20 (ms)

Fig. 10. A: action potentials computed from the Frankenhaeuser-Huxley equations for a frog myelinated fibre. As axoplasmic resistance per unit length of cable is increased (resulting from atrophy of the axon), the time to conduct to the fifth node (whose action potentials are shown here) slows markedly. From left to right the values of axoplasmic resistance are 1, 4, 6 and 8 times the standard values (see Materials and Methods), corresponding to a decrease in equivalent axon diameter to 1, 0.5, 0.41 and 0.35 times the standard value of 10.5 pm. For smaller values of diameter (higher axoplasmic resistances) conduction failed altogether. Note the slow depolarization which just reaches threshold for the action potential on the far right of A. B: the conduction velocity of the action potentials shown in A varies approximately linearly with the equivalent axon diameter (×) or fibre (axon + myelin) diameter (O). However, a three-fold change in conduction velocity (from 15 to 5 m/s) is associated with about a two-fold change in axon diameter, but less than an 20~ change in fibre diameter (from 13 to 11 l~m). and fibre diameter D e q (Fig. 10B). I f conduction velocity changed proportionally with equivalent diameter a straight line through the origin would result. Although neither fitted straight line goes precisely through the origin, the approximation is much better for the axon diameter d than the fibre diameter D (the intercept on the y-axis is about 10 ffm for D and only 2/zm for d). Thus, functional

52 changes should be more nearly correlated with changes in axon diameter than total fibre diameter, as found experimentally.

-I

c)

X ~

X

~

DISCUSSION

Our results indicate clearly that measurements of fibre diameter underestimate the changes taking place in axons during atrophy. As the axons became more shrunken with time after ligating a nerve, the intact tube formed by the Schwann cell and myelin appeared merely to collapse inwardll,z0,21, 53. Thus, the changes taking place in fibre diameter were much less than those occurring in axon diameter (Fig. 6). The conduction velocity of the nerve fibres was related more closely to axon diameter than total fibre diameter during the course of atrophy. Thus, to determine the functional effects of nerve atrophy from anatomical measurements, electron microscopy must be used to measure axon size accurately. Early work using light micrographs was inconclusive as to whether the myelin thickness increased or decreased during atrophy (see Introduction). From the present measurements from electron micrographs, there is no evidence that the myelin changes in average thickness (Fig. 7), despite the substantial surface changes19, 53. Furthermore, the relationship between inner (axon) perimeter and outer (myelin) perimeter remains unchanged. Friede and Bischhausen 26 recently reported a precise correlation between the surface area of axon covered per node and myelin thickness. Our data sugge3t that these parameters and relationships are quite insensitive to the gross morphological changes taking place in the axonal geometry. How can the myelin thickness (or the number of turns) and the perimeter (or the length of individual turns) remain invariant without buckling and separating from the axon during the course of atrophy? In the Appendix a simple scheme is presented in which it is possible for the axon to disappear completely without the inner perimeter, the outer perimeter or the myelin thickness changing. The circular shape in Fig. l 1A evolves to the slit in Fig. 11B via a series of increasingly fiat shapes of the type shown in Fig. 11 C. The result is generalized in Fig. l i D to any arbitrarily shaped polygon as long as extreme in-

/

b2

bn

b2

.-.

b1

bl

Fig. 11. A myelinated nerve fibre could pass f r o m a circular shape (A) to a slit (B; no axon) via a series of flattened shapes (C) without c h a n g i n g the myelin thickness m, the inner perimeter or the outer perimeter. D -- outer fibre diameter, d = inner a x o n diameter, x -- length o f flattened portion a n d y -- short axis o f the axon. A n y arbitrarily shaped a x o n (D) can be a p p r o x i m a t e d by a polygon to a desired degree of accuracy. Myelin o f c o n s t a n t thickness m c a n then be laid d o w n such that the outer perimeter S is related to the inner perimeter s by the relation S = s -t- 2m, as long as the angles o f the polygon are all less t h a n ~z radians. If there are obtuse angles greater t h a n ~t r a d i a n s (E), t h e n myelin of c o n s t a n t thickness c a n n o t be laid d o w n a n d the relation between the inner a n d outer perimeters (s a n d S) cited above no longer holds exactly. See the A p p e n d i x for definition of s y m b o l s a n d derivation of these relationships.

folding does not occur (Fig. llE). Thus, there is a broad class of paths an axon could follow during atrophy without requiring the myelin to alter its thickness or perimeter. Only in the terminal stages of atrophy, when the axons show severe enough infolding to produce sharp angles, must reorganization of the myelin occur in a way which will produce a separation in the myelin layers. Our experimental results strongly support this process for atrophy. A further consequence of this argument is that the axon could regrow to its original size by simply reversing the geometric changes, if regeneration is permitted. Gordon and Stein 30 have recently shown that the size and conduction velocity of axons proximal to a lesion does return precisely to control levels if and only if regeneration to a suitable end organ occurs (see also Cragg and Thomas15). In contrast, ;f the myelin were to expand and replace the space vacated by the axon, it would be much more difficult for the axon to regain its former size. Finally, because the axon is not normally circular, it could hypertrophy so that circularity increases from 0.7 or 0.8 to 1.0, without causing any severe

53 complications in the myelin. This also provides a margin of safety for the axon against osmotic shock or other local changes and provides a rationale for the normal non-circularity of axons. Thus, our experimental and theoretical results are all consistent with a simple, reversible mode of nerve axon atrophy and hypertrophy. As an axon loses volume during atrophy, tissue fluid pressure outside the myelin flattens it in such a way that there is no change in the number of turns of myelin or the length of each turn. The axonal conduction velocity decreases approximately in parallel with the decrease in effective axonal diameter (but not the external fibre diameter). If regeneration occurs, the process reverses along a similar geometric path to its original size or possibly to a more circular, hypertrophied size. APPENDIX

The two extremes of circularity are shown in Fig. 11A and 1 lB. The notation is the same as indicated in Materials and Methods, with the addition of the variable x to represent the length o f axon which has become flattened. In the extreme case of Fig. llB, where the axon has disappeared completely, x represents the length of the slit which remains. A number

of parameters which are defined in Materials and Methods and used throughout the Results can be readily calculated for the two extremes A and B and are listed in Table I. We assume that the unflattened portions remain circular so that the myelin thickness is the same at all points. Note that the axon perimeter will be the same in A and B if x = red/2. The external perimeter will also remain the same since: 2(x q- z~m) = ~r(d + 2m) -----z~D

(1)

The length of all layers of myelin intermediate between the internal and external perimeter can also be shown to remain unchanged. Finally, the myelin thickness can be obtained in both extremes from the formula: m = (S - - s)/(2z0

(2)

To see how an axon could atrophy from A to B consider the intermediate example C which again retains a constant thickness of myelin. The variable y represents the short axis of the axon. The corresponding values of the parameters are also listed in Table I for the intermediate case C. The axon perimeter will always be the same as that of the original circle if

y : d--2x/z~

0 < y <_ d

(3)

TABLE I

Computed parameters for the simple geometric shapes o f Fig. 11 T h e c o l u m n s A, B a n d C c o r r e s p o n d to the parts of Fig. 11, while the c o l u m n D c o r r e s p o n d s to the simplifying a s s u m p t i o n given by Eqn. 3, which c o r r e s p o n d s to a particular m o d e of a t r o p h y for the axon. F u r t h e r discussion of this table is f o u n d in the A p p e n d i x a n d the text. A

B

C

D

s = a x o n perimeter

z~d

2x

zty + 2x

ztd

S = external perimeter

ztD

2(x + z~m)

nO" + 2m) + 2x

ztD

g = sis

diD

x/(x + z~m)

a = a x o n area

1/4~d2

0

1/4zcy~ q- y x

A = total fibre area

1/4gD2

z~m2 + 2mx

x/4z~(y + 2m) 2 + (y -}- 2m)x

~p = 4na/s 2 ~ a x o n circularity

1

0

diD

0

a/a/A = ratio o f equivalent a x o n a n d outer diameters

deq/Deq :

z~y + l x z~(y + 2m) + 2x

diD ll 4zgd2 - - X2/:Tr.

~(z~y e + 4yx)

(zty + 2x)2 ~i

z~Y2+4yx (y q- 2m)2 -b 4(y -r 2m)x

~2D2 __ 4x 2

54 The two extremes are (A) : y = d and x = 0, which corresponds to the circle, and (B): y = 0 and x = z~d/2, which corresponds to the completely atrophied axon. Substituting condition (3) into the formulae in the Table under column C gives the simplified expressions of column D. Thus, by using this particular geometric transformation an axon could atrophy from a circle to a slit, while the myelin thickness m, the inner and outer perimeters s and S, and the ratio g = s i S remain unchanged, as found experimentally. Note that the ratio of equivalent diameters deq/Deq will not remain constant, but will decline to 0. It is this ratio which corresponds to the slopes calculated in Fig. 5. The particular geometric transform in Fig. 11, in which m, s, S and g remain constant during atrophy is in fact one of a large class of such transforms. Any arbitrarily shaped axon can be approximated by a polygon, the level of approximation increasing with the number of sides of the polygon. If there is an nsided polygon as shown in Fig. 11D with angles/31, fi2 . . . . . fin and sides bl, bz . . . . . bn, then we can construct another figure a distance m from the original polygon by drawing parallel lines of length bl, b2, . . . . bn a distance m outside the original lines and connecting them with arcs (subtending angles 71, )% . . . . 7n from circles of radius m). The perimeter of the inner figure is then: s=bl+b2+...

+bn

(4)

S = bl @ b2 . . . q- bn 4- m ( 7 1 + Tz + . . . yn)(5)

-~- fin =

(t7-

2)zr

(6)

Also: y.

fii = 3zr, fil :

fi2 ~

z/2 and

i=1

/33 = fis, so 2fia = 2zr - - fia Then: 5 S

Z bi

and:

s=2

5

b ~ - - 2 l q-m(71 + 7 2 + 7 8 + 7 5 )

7~-

fin

if fin <- 7t

(7)

Thus, if all the angles are less than zr: 7~ + 7z + . • • + 7- = n~ - - fil --/32 - - . . . - - fin 2= (8) =

Thus, S = s + 2~m. So long as the inner layer of myelin remains the same length and the number of layers remain constant (which determines m), the outer perimeter will remain unchanged as will the

(10)

/=1

Since 71 = 72 = az/2 and 7a = ~r - - t3a = 75 : S = s + 3~zrn--21--2flam

(11)

However, tan ,t -- l/m and 2 = zt/2 - - /38 = (fi~ - - ~ ) / 2 .

Substituting these expressions and

simplifying: (12)

where tan ;t ~ 2 + 2a/3, for small values of 2. Thus: (13)

In the example shown where//4 < 3zr/2, 2 < zr/4 and (tan 2) - - 2 < 0.215; s if- 2zrm (0.93) < S < s q- 2zrm

=

(9)

i=1

S ~ s + 2zcm - - 2m;ta/3

For an n-sided polygon 11 " ' "

5

S = s q- 2zrm - - 2m (tan 2 - - 2)

and the perimeter of the outer figure is:

fil -~- fi2

ratio g = s/S. This general result is also the rationale for calculating the average thickness of the myelin from the formula given in Materials and Methods. However, if one or more of the angles is greater than zr, the simple relations derived above break down. For example, consider the pentagon of Fig. 11E in which one angle, f14, is greater than zr. We have drawn a symmetric pentagon in which two angles, fix and fiz = Jr/2 and fia = fis, but the same results will apply qualitatively to more general shapes. Use of a symmetric shape merely simplifies the calculations. For this pentagon:

(14)

Thus, the difference is only a few percent, but as the myelin becomes more infolded, the difference will increase and the outer layers will either have to shrink relative to the inner ones or separate from them. It is not the infolding per se which leads to a breakdown of the relation S = s q- 2:~m, but the oblique angle. I f the myelin becomes wavy with segments composed of arcs of circles in which the

55 inner and outer layers are always separated by a distance m, it can easily be shown that the relation between S, s and m will still hold. ACKNOWLEDGEMENTS

We thank Dr. A. Aguayo and Mrs. S. Prasad for their valuable help with the histological measure-

ments and preparations, respectively and Dr. Z. Koles and Mr. R. Rolf for their assistance in computer programming. Dr. T. Gordon and Mr. T. Milner made helpful suggestions on the manuscript. This work was supported in part by grants from the Medical Research Council of Canada and the Muscular Dystrophy Association of Canada.

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