Input conductance, axonal conduction velocity and cell size among hindlimb motoneurones of the cat

Brain Research, 204 (1981) 311-326

311

© Elsevier/North-Holland Biomedical Press

I N P U T C O N D U C T A N C E , A X O N A L C O N D U C T I O N VELOCITY A N D CELL SIZE A M O N G H I N D L I M B M O T O N E U R O N E S OF T H E CAT

D. KERNELL and B. ZWAAGSTRA Department o f Neurophysiology, Jan Swammerdam Institute, University of Amsterdam, 1054 B W Amsterdam (The Netherlands)

(Accepted July 3rd, 1980) Key words: motoneurone-size - - input conductance -- recruitment order

SUMMARY Input conductance and axonal conduction velocity were measured for hindlimb motoneurones that were anatomically labelled by substances injected through the intracellular microelectrode (Procion dyes, horseradish peroxidase). We confirmed that there is a good correlation between the axonal conduction velocity of a hindlimb motoneurone and the size of its cell body. Furthermore, we confirmed that the power relation between neuronal input conductance and axonal conduction velocity has an exponent of about 3-4. If large motoneurones were simply scaled-up versions of the smaller ones, this exponent should have been between 1.5 and 2.0. We showed that the unexpectedly high input conductance of fast-axoned motoneurones, compared to that of the more slow-axoned ones, was not due to a corresponding disproportion between the axonal conduction velocity and the size of the cell body. Neither could it be explained by differences between large and small cells with respect to the relative sizes and numbers of dendritic stems. The unexpectedly high input conductance of large cells seems likely to be largely caused by a lower average value for the specific membrane resistance among these cells than among the smaller ones. Hitherto unknown differences in dendritic architecture between large and smaller cells might conceivably be of some importance as well. Our results are consistent with the view that, in muscle contractions evoked by the central nervous system, thin-axoned motoneurones might be recruited more easily than more thick-axoned ones even if all the cells were activated by the same density of equipotent synapses.

INTRODUCTION It is known from a number of investigations, that hindlimba-motoneurones

312 with slow (thin) axons tend to have a greater input resistance than those with faster (thicker) axonsl,2,~L This means that less activating current will generally be needed to start a discharge in a thin-axoned cell than in a thick-axoned one~L It is also known that, in many kinds of reflex or voluntary motor acts, motoneurones with thin axons tend to be more easily recruited than those with thicker ones within the same pool (e.g. ref. 9). Due to differences in input resistance between the cells, such a recruitment order might be simply produced if, for instance, all cells of a population were simultaneously influenced by the same absolute amount of synaptic current (e.g. same absolute number of equivalent active synapses per cell). No ascending-size-order of recruitment would be expected to occur, however, among motoneurones that differ only in size and not in membrane properties if the cells were all synaptically influenced in proportion to their surface area (e.g. same density of equivalent active synapses per cell; cf. refs. 3 and 18). Hence, it is of interest to know to what an extent it would in fact be correct to regard large motoneurones simply as scaled-up versions of the smaller ones. No significant potential gradients would be expected to occur within the cell body of a motoneurone (cf. ref. 15). Thus, the input conductance (reciprocal of resistance) of the soma membrane should be proportional to its surface area, which would be directly related to the square of the mean soma diameter. The input conductance of a uniform cable (finite or infinite length) 15 is proportional to its diameter raised to the power of 1.5. As measured with an electrode in the soma, the input conductance of a whole neurone is, of course, equal to the sum of the input conductance of its dendrites and its cell body membrane. Thus, if all motoneurones had the same membrane properties and shape (constant ratio between the diameters of soma and dendrites, constant number of dendrites) their resting input conductance would be expected to be proportional to a linear measure of their size (e.g. soma diameter) raised to a power of between 1.5 and 2.0. The more dominating the dendrites, the closer would this exponent approach 1.5. Previous studies of spinal motoneuronesl, 13 have indicated that the dendrite to soma conductance ratio might be of the order of 10-20:1. With a ratio of 10:1 an exponent of about 1.6 would be expected for the power relation between input conductance and soma diameter. As is well known, the axonal conduction velocity of myelinated fibres is proportional to their diameted 0. Thus, in the case of simple linear scaling (i.e. axon diameter proportional to soma and dendrite diameters), an exponent of about 1.6 would also be expected for the power relation between motoneuronal input conductance and axonal conduction velocity. However, as has recently been pointed out by Stein and Bertoldi ~6, the previous experimental data of Kernel112 and Burke 2 would actually be best fitted by power relations with exponents of 3-4. Thus, in comparison with a thinaxoned motoneurone, the input conductance of a typical thick-axoned one is greater than it should have been if the respective cells differed only in size. In the present investigation we have subjected these differences between large and smaller motoneurones to a further experimental analysis. Axonal conduction velocity and neuronal input conductance were measured in cells that were individually labelled by the injection of Procion dye or horseradish peroxidase via the microelectrode. Later on,

313 the dimensions of the cell body and the proximal dendrites were measured in all the labelled cells20. Our results show that the unexpectedly high input conductance of thick-axoned motoneurones is not caused by the presence of unexpectedly large cell bodies and/or stem dendrites among these cells. METHODS The results were obtained from cats (2-5 kg) anaesthetized with pentobarbitone. A wide laminectomy was performed and thelhindlimb was denervated. Body and leg temperature were kept close to 38 °C. The material of neurones and the general experimental and anatomical procedures were the same as those described in a preceding paper z°. Motoneurones of the lumbosacral.spinal cord were penetrated with single-barrelled glass microelectrodes filled with Procion yellow (ICI), Procion red (ICI) or horseradish peroxidase (HRP, Sigma). With respect to present kind of anatomical measurements, all these types of intracellular labelling gave very similar results 20. The motoneurones were identified by their antidromic response to the stimulation of a motor nerve. The number of cells belonging to various muscles and muscle groups (number of cells for input conductance, axonal speed and anatomy/ number of cells for axonal speed and anatomy only) were: intrinsic foot muscles of the tibial nerve (2/13 cells), m.soleus (6/8 cells), m.triceps surae (6/17 cells), other post-tibial muscles (1/5 cells), muscles of the common peroneal nerve (2/8 cells), hamstring muscles (i.e.m.semitendinosus, m.semimembranosus, m.biceps femoris, 10/23 cells), m.quadriceps (8/9 cells), other proximal muscles (i.e.m.gluteus maximus, m.sartorius; 2/4 cells), all muscles together (37/87 cells). Axonal conduction velocity was calculated from measurements of conduction distance and antidromic latency (stimulation at two times antidromic threshold). For all motoneurones to muscles below the knee, stimulation of the sciatic nerve just above the knee was used for the determination of conduction speed. In order to avoid problems with the precise balance of stimulus artifacts, input resistance was determined by measuring the effects of steady currents, injected through the intracellular microelectrode, on the amplitude of antidromic spikes 7 (see also refs. 2, 3, 12 and 14). When used with weak currents and healthy-looking spikes, this indirect method is known to give very similar estimates of neuronal input conductance as more direct methods 12,14. In order to avoid the complication of changing rectifying events 11, the measurements were performed at about 0.45 sec after the onset of a current step. A number of weak deand hyperpolarizing test currents (_< 5 nA) were used in each cell, and the neuronal input conductance (reciprocal of input resistance) was determined from plots of current intensity versus changes in spike amplitude 7. About the same conductance value was obtained with either polarity of test current (mean value used). The cells used for conductance measurements all had healthy-looking spikes with an amplitude of 59 mV or more (mean 77 4- 10 mV), and their resting membrane potential was 62 4- 9 mV. There was no statistically significant correlation between the measured input conductance of a cell and its spike amplitude or resting membrane potential. After completion of the physiological measurements, Procion dye or HRP was

314

Fig. 1. Drawing showing the cell body and the proximal dendrites of a motoneurone belonging to m. triceps surae. The cell was stained by Procion yellow that was injected via an intracellular microelectrode. The drawing represents a reconstruction from serial sections. Dorsal is upwards and lateral to the right. Arrows indicate the site of measurements of dendritic stems. The projected area of the soma and the approximate maximum and minimum soma diameters are indicated by separate lines. injected into the cell by the passage of a suitable current through the microelectrode. At the end of an experiment, the cat was perfused by a suitable fixative (2 formaldehyde for Procion dyes, 1 ~ p a r a f o r m a l d e h y d e and 1.25 ~o glutaraldehyde for HRP). Serial transverse sections of 50 # m were cut from the lumbosacral spinal cord on a freezing microtome. Procion-stained cells were studied with a fluorescence microscope. Sections with HRP-injected cells were stained by histochemical methods 17 and studied with ordinary transmitted light. Ceils showing signs of damage (e.g. fragmentation) were discarded. The microscope was equipped with a split-beam drawing tubus. The outlines of the cell b o d y and the proximal dendrites were carefully drawn with a fine pencil. Drawings from adjacent sections were aligned by aid of visible landmarks in the preparation, and all drawings from one cell were combined into one final drawing showing the whole cell body and all dendritic stems in transverse projection (Fig. 1). On transparent g r a p h - p a p e r the projection area of the cell body was drawn by a smooth line that closely followed the cell border but excluded the sharply tapering 'hillocks' of the emerging dendrites or axon (Fig. 1). The total surface area of the cell body was taken to be 4 times its measured projection area (cf. f o r m u l a for sphere). An 'equivalent' soma diameter (D) was calculated from the measured projection area of the soma (A) according to the equation D = 2 ~ / A / ~ (cf. formula for circle). A 'direct' soma diameter was obtained by taking the mean of two cell body diameters, directly measured at right angles to each other and roughly corresponding to the m a x i m u m and m i n i m u m diameters of the s o m a (cf. Fig. 1). The two figures for soma diameter were strongly correlated to each other (r -= 0.90, n ~:,~87), and the direct diameter was, on an average, a b o u t 6 ~o larger than the equivalent one. Due to the irregular shape of the cell body in m a n y motoneurones, we felt that the

315 equivalent soma diameter was somewhat more reliable than the direct one as an indication of cell body size. The correlation to axonal conduction velocity did, for instance, show somewhat less scatter for the equivalent diameter than for the direct one (correlation coefficients 0.78 and 0.69 respectively, n = 87). Unless otherwise specified, all soma diameters referred to in text, tables and illustrations are the equivalent ones, derived from measurements of the cross-sectional area of the cell body. Diameters of dendritic stems were measured distal to the initial tapering and proximal to the slight swelling associated with the first branching site. The accuracy of measurement was about -q- 0.5 /~m. Whenever possible, these measurements were made at about 40/~m from the soma--dendrite junctionl,13,14, 20. Care was taken not to include stained axons into measurements referring to dendrites. More complete information concerning our anatomical and general experimental techniques may be found elsewhere 2°. Unless otherwise specified, average values are given ~: S.D. RESULTS

Neuronal input conductance and axonal conduction velocity The triangles of Fig. 2 show the relation between neuronal input conductance and axonal conduction velocity for the 37 cells of the present study. For comparison, the corresponding 50 measurements of Kernel112 are included into the same diagram (circles). Both coordinates of Fig. 2 are logarithmic, and for both populations of data there is a good linear correlation between log (y) and log (x) (see Table I). Thus, the relation between these y- and x-values may be expressed by a power relation of the type y = ax b, where a is a constant and b is equal to the slope for the linear regression of log (y) versus log (x). These slopes are, for both populations of data, at least twice as large as the expected figure of about 1.6 (Table I; cf. Introduction). A similarly high exponent was obtained if the comparisons were limited to motoneurones of re.triceps

INPUT CONDUCTANCE ~i 0 -~ S A

4

A zxx^ S o O ~

o ~-~o

~

1 60 CONDUCTION

60

~ i SPEED

0

o

o

' 120

M/S

Fig. 2. Neuronal input conductance (Siemens) plotted versus axonal conduction velocity (miser) for 37 intracellularly labelled motoneurones from the present study (triangles) and for the 50 cells of KernellTM (circles). Logarithmic coordinates. Regression line in this and subsequent graphs calculated by method of least squares (slope b = 3.6; correlation coefficient r = 0.84, n = 87). For further statistical data, see Table I. The cells that are signified by triangles in this graph are represented by the same symbol throughout all subsequent illustrations.

316 TABLE I Interrelations between neuronal input conductance and parameters o f neuronal size within mixed samples o f hindlimb motoneurones

b = slope of regression line, calculated by method of least squares, for plot of log(y) versus log(x). Thus, b is equal to the exponent of the power relation y = ax b, where a is a constant, r -- correlation coefficient for log(y) versus log(x). P = probability that correlation equals zero (t-test). G = input conductance. CV = axonal conduction velocity. D = soma diameter, d = stem dendrite diameter. Variable

Source

Y

X

G

CV

D

CV

G ~d 3/2

D D

Kernell t" present data Cullheim 6 present data present data present data

(Fig. 2) (Fig. 3) (Fig. 4) (Fig. 7)

b

r

n

P

4.0 3.6 0.4 0.9 3.3 2.1

0.83 0.87 0.69 0.78 0.83 0.83

50 37 24 87 37 87

<0.001 .<0.001 <:0.001 -: 0.001 <0.00l - :0.001

surae ( T a b l e II). T h e s e o b s e r v a t i o n s m a y m e a n : (a) t h a t cell size increases faster w i t h a x o n a l s p e e d t h a n e x p e c t e d f r o m s i m p l e scaling a r g u m e n t s , a n d / o r (b) t h a t i n p u t c o n d u c t a n c e increases faster w i t h cell size t h a n e x p e c t e d f r o m scaling a r g u m e n t s (see I n t r o d u c t i o n ) . I f the first e x p l a n a t i o n were c o r r e c t , the e x p o n e n t for the p o w e r r e l a t i o n b e t w e e n s o m a d i a m e t e r a n d a x o n a l c o n d u c t i o n v e l o c i t y s h o u l d be significantly larger t h a n 1.0. I f the s e c o n d e x p l a n a t i o n w e r e true, the e x p o n e n t for the p o w e r r e l a t i o n b e t w e e n i n p u t c o n d u c t a n c e a n d s o m a d i a m e t e r s h o u l d be significantly l a r g e r t h a n 1.6 (cf. I n t r o d u c t i o n ) . S o m a size a n d a x o n a l conduction velocity

I n the d o u b l e - l o g a r i t h m i c d i a g r a m o f Fig. 3, s o m a d i a m e t e r s h a v e b e e n p l o t t e d versus a x o n a l c o n d u c t i o n velocity. A s has r e c e n t l y b e e n s h o w n also by C u l l h e i m 6, t h e r e is a g o o d c o r r e l a t i o n b e t w e e n these t w o m e a s u r e s o f n e u r o n a l size 19. F o r all the TABLE lI Interrelations between neuronal input conductance and parameters o f neuronal size within samples of motoneurones belonging to re.triceps surae (including m. soleus)

For explanations, see Table I. Variable Y

Source

b

r

n

P

Kernel112 Burke 2 present data present data present data present data

4.2 3.0 2.9 0.7 3.2 1.7

0.86 0.59 0.77 0.64 0.73 0.72

18 56 12 25 12 25

< 0.001 <0.001 < 0.01 <0.001 <0.0t < 0.001

X

G

CV

D G )2d3/2

CV D D

317 SOMA D IAM MICRON 70

o

A

A

(J O

A

AA

o

A(~

AO

A

30

6 0- 8~0 CONDUCTION

i 0~0 SPEED

120 M/S

Fig. 3. Soma diameter (#m) plotted versus axonal conduction velocity (m/sec) for all the intracellularly labelled motoneurones. Logarithmic coordinates. For statistical information, see Table I. cells of Fig. 3, the mean ratio between the equivalent soma diameter (/am) and axonal conduction velocity (m/sec) was 0.54 -4- 0.07. The slope of the regression line of Fig. 3 was rather close to 1.0 (95 ~o confidence range: 0.7-1.0), i.e. there was on an average nearly a direct proportionality between axonal conduction velocity and 'equivalent' soma diameter. With 'direct' soma diameters, the corresponding plot had a slope of 0.95. For motoneurones of m.triceps surae, the slope of a double-logarithmic diagram like that of Fig. 3 tended to be somewhat lower than 1.0 (Table II), and a rather low value was also calculated from the published data of Cullheim 6 (Table I). Thus, the relationship between soma size and axonal conduction velocity does not provide any explanation for the unexpectedly high input conductance of thick-axoned motoneurones. Input conductance and soma size

In the double-logarithmic plot of Fig. 4, a direct comparison is made between the measured values for input conductance and soma size of the present 37 hindlimb motoneurones. The slope of the regression line is 3.3 (95 ~o confidence range: 2.5-4.0), i.e. it is about twice as great as the value of 1.6 that would be expected from simple scaling arguments (see Introduction). The disproportionately high input conductance INPUT

CONDUCTANCE

0 -~ S A A

ZXy

AZIZIA

A

2~

A

zx

13~0 SOMA

L DIAM

5hO

~ 7~0 M I C'RON

Fig. 4. Neuronal input conductance (Siemens)plotted versus soma diameter ~m) for 37 intracellularly labelled motoneurones. Logarithmic coordinates. For statistical information, see Table I.

318 r (~!qI)kJ(} I-AN( L : S I Z E '; 0 As

A

20 A

~0

80

100

CONDUCTION SP~ED

i20

N/S

Fig. 5. Conductance:size ratio ( x 10 s Siemens cm s) plotted versus axonal conduction velocity for

the same cells as those of Fig. 4. The figures for the conductance: size ratio were, for each cell, obtained by dividing the neuronal input conductance by the surface area of the soma. The somatic surface area was taken to be 4 times as large as the measured cross-sectional area of the cell body. Linear coordinates. Correlation coefficient r ~ 0.61 (n -- 37, P < 0.001). o f f a s t - a x o n e d l a r g e m o t o n e u r o n e s is f u r t h e r i l l u s t r a t e d b y t h e d i a g r a m o f Fig. 5. H e r e the ratio between input conductance and soma area (conductance: plotted versus axonal conduction velocity. For motoneurones

size) h a s b e e n

w i t h a x o n s p e e d s o f :2

90 m / s e c , t h e m e a n c o n d u c t a n c e : size r a t i o w a s a b o u t t w i c e as l a r g e as t h a t o f m o r e s l o w - a x o n e d cells ( T a b l e I11). A c c o r d i n g t o s i m p l e s c a l i n g a r g u m e n t s (cf. I n t r o d u c t i o n ) , t h e c o n d u c t a n c e : s o m a size r a t i o o f f a s t - a x o n e d cells w o u l d a c t u a l l y h a v e b e e n e x p e c t e d t o b e s m a l l e r t h a n t h a t o f m o r e s l o w - a x o n e d cells. T h u s , o n a n a v e r a g e , t h e i n p u t c o n d u c t a n c e o f t h e f a s t - a x o n e d cells w a s a t l e a s t a b o u t t w i c e as l a r g e as o n e w o u l d h a v e e x p e c t e d f r o m t h e d i f f e r e n c e in s o m a size b e t w e e n fast- a n d s l o w - a x o n e d motoneurones. For the

12 i n v e s t i g a t e d m o t o n e u r o n e s

of

m.triceps

surae,

the

correlation

TABLE Ill Conductance:size ratio (G/S) and dendrite:soma ratio ()2dZ/2/S) fi~r motoneurones with slow and fast OXOllS

Mean :k S.D. Statistical significance of differences tested by t-test (P =: probability ; n.s. = not significant, i.e. P > 0.1). The motoneurones have been arbitrarily subdivided into a group with slow axons (conduction velocity < 9 0 m/sec) and a group with fast axons (conduction velocity ~ 9 0 m/sec). G = neuronal input conductance, d == diameter of stem dendrite. S, - non-corrected soma area ( = 4 times the measured cross-sectional area). $2 soma area including dendritic cones, correction type A (see text). Sa - soma area including dendritic cones, correction type B (see text). Parameter

G/S1 G/S2 G/Sa (S,d:~/~)/S1 (5",d3/'2)/S~ (~d3/z)/S3 n

Cells with

Cells with

SIo W aXOHS

l a s t (iXOllS

6.6 4.9 3.6 1.9 1.4 1.0

3 3.1 ± 2.2 ~ 1.6 i: 0.6 ± 0.4 ~ 0.2 17

12.8 9.5 7.8 1.9 1.4 I.I

± 5.1 m S c m z ± 3.8 mS cm 2 -~: 3 . 1 m S c m 2 ± 0.4 cm '/~ ~ 0.4 cm ~/2 -~: 0.2cm '/~ 20

P

- 0.001 ~0.001 ~-0.001 n.s. n.s. n.s.

319 coefficient was 0.76 (P < 0.01) for the relation between axonal conduction velocity and the conductance: size ratio. Thus, the main findings of Figs. 4 and 5 remained valid also if the comparisons were limited to gastrocnemius-soleus motoneurones (cf. Table II). Furthermore the observations of Figs. 4 and 5 are fully consistent with those of Figs. 2 and 3. The results mean that, in comparison to smaller motoneurones, the large ones might have: (i) soma sizes that have been systematically underestimated in the present kind of measurements; (ii) disproportionately many and/or thick dendritic stems; (iii) disproportionately extensive dendritic trees, or (iv) a comparatively low mean value for the specific membrane resistance. These various groups of possible explanations are, of course, not mutually exclusive. Factors relating to explanations (i) and (ii) have been further analyzed by aid of the present experimental material. Factors relating to explanations (iii) and (iv) will be dealt with in the Discussion. The measurement o f soma size: 'projection errors'

Most cell bodies of the present study were more or less elongated (cf. Fig. 1). The mean ratio between maximum and minimum diameters was 1.67 4- 0.62 ('shape index'; n ~ 87). Many of these elongated cells were presumably oriented with their long axis in another plane than the one of our transversal sections. Thus, many such cells will have had their soma size underestimated by various amounts. If the orientation of the cell bodies varied randomly with respect to the plane of projection, somas with a more elongated shape would, on an average, tend to become more seriously underestimated than those whose real shape index was smaller. Within the present material of hindlimb motoneurones, there was no evident correlation between the measured shape index of a cell and the 'equivalent' diameter of its soma (r -- 0.06 for all the 87 ceils studied anatomically, and r : 0.09 for the 37 cells with measurements of input conductance; in both cases P > > 0.1). Thus, with respect to 'projection errors', there was no evidence indicating that the sizes of large somas would have been more seriously under-estimated than those of smaller cell bodies. 'Projection errors' of measurement will, however, have added significantly to the spread of values referring to soma sizes (Figs. 3, 4 and 7). The measurement o f soma size: dendritic hillocks

As is evident from Fig. 1, dendrites are commonly joined to the soma by a sharply tapering, cone-shaped 'hillock'. These hillocks were typically fairly short (mean 13/zm) in comparison to the probable space constant of dendrites. If, for instance, the internal specific resistance is 100 f~ cm and the specific membrane resistance 2500 ~ sq.cm (cf. refs. 1, 13 and 14), the space constant of a typical dendrite of 5/zm diameter would be 559 #m. Thus, it would seem fairly reasonable to add the membrane areas of dendritic hillocks to the one of the cell body (cf. ref. 13). From the 87 labelled motoneurones of the present study there emerged totally 1051 stem dendrites. Cone-shaped hillocks were seen for only 528 of these dendrites. Dendrites with no visible hillocks often seemed to emerge from cell areas central to the projected rim of the soma. For all visible hillocks, the height and the proximal and distal diameters were measured. The membrane surface area of each hillock was then

320

CONE A~£~ NICR(}N ~ 2000 f • '

: )

'1

lO'~O

O0 10 DEND~IT£ DIAN

_

_

t

20 NICDON _

•

Fig. 6. Average membrane areas of dendritic hillocks (sq./~m) plotted versus corresponding dendritic stem diameters (~m). The number of hillocks per plotted average value was, from left to right: 6, 11, 64, 98, 148, 60, 54, 28, 34, 4, 6, 4, 6, 2, 1, 1, 1. Vertical bars give ± S.D. for the first 13 values and total range for the 14th average value from the left. Linear coordinates. Regression line calculated by method of least squares and drawn according to equation: y 53.7x -~ 147.9 (r - 0.88, n - 17, P < 0.001).

calculated by aid o f the standard formula for truncated cones. The height of the cones showed no strong correlation to dendrite diameter. As demonstrated in Fig. 6 there was, however, an evident relationship between the m e m b r a n e area o f a hillock cone and the diameter of the same dendrite distal to the initial tapering. By aid o f these measurements, we made two different calculations of corrected soma areas: (i) c o r r e c t i o n A - - m e m b r a n e areas o f all visible dendrite-cones o f a cell were added to the non-corrected total soma area and (ii) c o r r e c t i o n B - - for each dendritic stem o f a cell, a 'corresponding' cone area of membrane was added to the non-corrected soma area. These latter cone-areas were calculated from the respective dendrite diameters by aid of the regression line o f Fig. 6. Irrespectively of whether corrected or non-corrected soma areas were used, the mean conductance :size ratio o f fast-axoned motoneurones remained much larger than that o f more slow-axoned ones (Table III). Thus, even if the surface areas of the cell bodies were taken to include dendritic hillocks, the input conductance o f fast-axoned motoneurones was at least twice as large as one would have expected from the differences in soma size between fast- and slow-axoned motoneurones. For all the 87 motoneurones that were studied anatomically, the mean ratio between corrected and non-corrected soma areas was 1.31 _~ 0.15 for correction A and 1.69 :k 0.19 for correction B. Cell size and dendrite:soma relation

As a first approximation, one might reasonably assume that the input conductance o f a motoneuronal dendrite is proportional to the 3/2 power o f its diameter 15. The total input conductance o f all the dendrites o f a cell would then be proportional to Zda/2, where d is dendritic stem-diameter. The input conductance of the soma membrane would be proportional to soma diameter raised to the power o f 2. The double-logarithmic diagram o f Fig. 7 shows the relation between Zd 3/2 and soma diameter. The slope o f the regression line is almost exactly 2 (95 ~ confidence range: 1.8-2.4) and a somewhat smaller slope was obtained if the calculations were limited to

321 SUN D E N D D :'I I C RrgN :~'':> ,5;0 r

D i ~ r ¢3"'~ A

o

6°/

t

-] L] i

~;~"A %

'

0 O0

i

~>0 t

30 S0"1~

bO

D!AN

Y0 MICRON

Fig. 7. 'Combined dendrite parameter' Cuma/~) plotted versus soma diameter Ohm) for all the intra' cellularly labelled motoneurones. The values along the ordinate are, for each cell, equal to Y,d3/~ where d is dendritic stem diameter. Logarithmic coordinates. For statistical data, see Table I.

the motoneurones o f m.triceps surae (Table II). In the case o f simple linear scaling, a slope o f 1.5 would have been expected. Thus, the results o f Fig. 7 suggest a small departure from a simple anatomical scaling. The results also suggest, however, that the dendrite :soma conductance ratio is nearly the same for all hindlimb m o t o n e u r o n e s irrespectively o f their sizO z. Thus, the disproportionately high input conductance o f large m o t o n e u r o n e s (Figs. 2, 4 and 5; Tables I - I I I ) was not caused primarily by the presence o f extraordinarily m a n y a n d / o r thick stem dendrites a m o n g these cells. The data o f Fig. 8 demonstrate directly that there was no evident correlation between neuronal input conductance and the presumed anatomical correlate o f the dendrite :soma conductance ratio (Z(d3/2)/S, where S is s o m a surface area). Table I I I shows that, irrespectively of whether non-corrected or 'cone-corrected' soma areas were used, fast-axoned m o t o n e u r o n e s showed nearly the same dendrite :soma ratio as more slow-axoned cells.

INPUT ~ 1 0 -7

50

CONDUCTANCE S A

2O

z}xz~ A a

A AA

o

AA~ I

A I

~a

A

£

,~!~L~

1 2 DENDR I TE : SOMA

A I

A

Fig. 8. Neuronal input conductance (Siemens) plotted versus anatomical dendrite: soma ratio (cm-t). The values along the abscissa are, for each cell, equal to (Y.da/~)/S,where d is dendritic stem diameter and S the surface area of the soma. Linear coordinates. There was no statistically significant correlation between the y- and x-values of this diagram (r = 0.009, n = 37, P ~ 0.1).

322 DISCUSSION We have confirmed that the input conductance of hindlimb motoneurones increases more rapidly with axonal speed than one would have expected from simple scaling arguments 16 (see Introduction, Fig. 2, Table I). In this respect, motoneurones of the hindlimb seem to differ from those of external eye muscles 8. In the present analysis, we have shown that the unexpectedly high input conductance of fast-axoned hindlimb motoneurones was not due to: (i) disproportionately large cell bodies of fast-axoned ceils (Figs. 3 and 4, Table I), or (ii) unexpectedly thick and/or many dendritic stems of large motoneurones (Figs. 7 and 8, Table l). Furthermore, the main conclusions remained valid also if the comparisons were limited to the motoneurones of one single muscle group (m. triceps surae) (Table II). There remain two possible explanations for the disproportionately high input conductance of the fast-axoned large-bodied motoneurones: (i) dendrites of the same stem diameter might be much more extensive in large than in smaller motoneurones and/or (ii) the specific membrane resistance might have a lower mean value for large than for smaller cells.

D~fferences in dendritic architecture? Anatomical as well as physiological measurements have indicated that a typical dendrite is about 1.5 space constants long in large as well as in smaller motoneurones 1, 4,14 (uniform membrane resistance assumed). In the anatomical studies concerned, relatively few cells were included that had an axon slower than 90 m/sec (ref. 1) or an input conductance smaller than 5 × 10 -7 Siemens 1,1a. Thus, it still cannot be excluded that further studies might reveal some differences in dendritic branching patterns between large low-resistance and smaller high-resistance hindlimb a-motoneurones. Existing evidence clearly does not, however, give any support for the view that the high conductance :size ratio of large motoneurones (Fig. 5, Table III) is due to disproportionately extensive dendrites among these cells. It should be noted that the input conductance of a hindlimb motoneurone would be expected to be relatively insensitive to mere changes in dendrite length. For a uniform sealed-end cable, the input conductance is equal to G ~ tanh (L), where G ~ is the input conductance of an infinite cable and L is the actual electrotronic length of the cable 15. The factor tanh (L) has the value of 0.76 for L -- 1, 0.905 for L 1.5 and 1.00 for L : c~. Thus, a decrease of dendrite length from 1.5 to 1.0 space constants might be expected to decrease the dendritic input conductance by about 16 },/,. An increase from 1.5 space constants to infinite length might enhance input conductance by about 10 ?,~. The axon would, of course, add to the total neuronal input conductance. This axonal contribution would, however, be expected to be very modest and insignificant in comparison to the total dendritic conductance. Each motoneurone had, on average, about 12 dendritic stems, and the mean dendritic diameter was about 5/zm 2°. Such an average dendrite is somewhat thicker than the average initial segment of an a-axon (range about 2.5-4.5 #m)5, 6. Furthermore, at distances greater than about 25-35 #m the axon is enveloped by an insulating myelin sheathL Thus, as seen from the soma, the axonal input conductance might well be substantially smaller than that of a single dendrite of the same small diameter.

323

Differences in specific membrane resistance? It is well-known from the morphophysiological studies of Lux et a1.14 and Barrett and Crill 1 that neuronal input resistance varies with neuronal size. Among the 7 cells of Lux et al. 14 the neuronal input resistance varies over a rather limited range (all below 2 M f~) and there is for these neurones no significant correlation between input resistance and computed specific membrane resistance. The sample of Barrett and Crill 1, however, includes a number of cells with an input resistance above 2 Mf~, and within this sample of neurones there is indeed a significant positive correlation between the measured input resistance and the computed specific membrane resistance (r = 0.64 for condition of infinite terminal dendrites, r = 0.65 for condition closed terminal dendrites, P < 0.05 in both cases, calculations based on data from Table I of ref. 1). Thus, among these cells, the presence of a comparatively high input resistance was in fact partly caused by a high specific membrane resistance and partly by a small neuronal size. For 9 of the cells of Barrett and Crill 1 a figure was reported for the axonal conduction velocity. For the 3 cells with an axonal speed slower than 90 m/s the specific membrane resistance was 3000 ~ 577 f2 sq.cm and for the remaining 6 cells with faster axons it was 2112 ~ 445 f2 sq.cm (calculations based on data for condition of infinite terminal dendrites from Table I of ref. 1). The difference between these mean values is statistically significant (t-test, P < 0.05). Thus, this analysis of the published data of Barrett and Crill 1 gives considerable support for the hypothesis that the difference in conductance:size ratio between o.ur slow- and fast-axoned motoneurones is caused, to an important extent, by differences in specific membrane resistance. The membrane time constant of a neurone is equal to the product of its specific membrane resistance and specific membrane capacitance. The latter factor is generally assumed to be constant between the cells. Hence, the presence of a greater specific membrane resistance among small motoneurones than among larger ones would be expected to lead to correspondingly large differences of membrane time constant between the differently sized cells. Previous measurements on hindlimb motoneurones do indeed suggest that small motoneurones tend to have a longer membrane time constant than that of larger cells a,4. However, such systematic differences were, on an average, quite small compared to the total range of values observed (range about 3-10 msec). The lack of experimental evidence for any substantial differences in membrane time constant between large and smaller hindlimb a-motoneurones may, however, to some extent be due to the complexities and difficulties involved in this kind of measurements on multipolar neurones3,4,15. Among physiologically investigated motoneurones that also have been anatomically reconstructed, the correlation is not always very evident between the experimentally determined values for specific membrane resistance and membrane time constant respectively. When calculated for the 7 cells o f L u x et al. 14 this correlation was r -- 0.185 (P >> 0.1) and for the 10 cells of Barrett and Crill 1 it was r = 0.636 (P < 0.05, condition closed terminal dendrites) or r ---- 0.412 (P > 0.1, condition infinite terminal dendrites).

324

Consequences for the control of recruitment order within motoneurone pools We will briefly discuss this question by aid of some very simple neurone models. Consider a pool of motoneurones that all have the same resting membrane potential (Vm), threshold potential for spike initiation (Vt) and equilibrium potential for the post-synaptic excitatory current (Ve). The effects of membrane capacitance and of subthreshold rectifying processes will be neglected. Furthermore, we will at first only consider synapses located to the soma. We will assume that the cell body is uniformly covered by a number of equipotent excitatory synapses, and that the total excitatory post-synaptic conductance (ge), which is present at a given moment, will be proportional to fS, where fis the fraction of the soma area (S) over which the currently active synapses are distributed. The factor f is then equivalent to the somatic density of synaptic excitation, and f oc ge/S

(1)

The minimal current (i~t) needed for just bringing the membrane potential of a silent neurone to the threshold for spike initiation (Vt) will be i~t -- gL (Vt - - Vm)

(2)

where gL is the passive neuronal input conductance ('leakage conductance') as seen from the soma. In the absence of other current sources, the current iLt must be equal to, and opposite in direction to, the current flowing through the post-synaptic permeability channels of activated somatic excitatory synapses. Hence, the intensity of post-synaptic excitatory current (iet) and conductance (get) needed for just activating the cell will be let -= get ( V t get = gL

--

We) := --

gL (Vt --

Vt - - Vm

VIII)

(3) (4)

Ve - - Vt

The density of somatic excitation (fi) needed for just activating the neurone will be ftoc:

get S

::-

gL

Vt - - Vm

S

Ve ---Vt

(5)

Thus, the somatic density of synaptic excitation required for recruiting a given cell within the population would be proportional to its 'conductance :size ratio' gL/S. Our experimental results have shown that this ratio is larger for fast-axoned motoneurones than for more slow-axoned ones (Fig. 5, Table III). Thus, our results suggest that, even if all the individual cells of a motoneurone population were simultaneously activated by the same somatic density of synaptic excitation, slow-axoned motoneurones would tend to be more easily recruited (i.e. activated at a lower 'excitation density') than more fast-axoned ones. In the hypothetical case of a uniform membrane and a uniform distribution of active excitatory synapses over the whole cell, including the dendrites, we can take get and gL of Eqn. 4 to represent conductances per unit area. The overall density of

325 synaptic excitatory c o n d u c t a n c e (get) n e e d e d for j u s t discharging the cell would, in this special case, be directly p r o p o r t i o n a l to the specific m e m b r a n e c o n d u c t a n c e o f the n e u r o n e (gL). As we have p o i n t e d o u t above, the differences in c o n d u c t a n c e :size ratio between the cells o f Fig. 5 (Table I I I ) are p r o b a b l y to an i m p o r t a n t extent due to the existence o f a greater specific m e m b r a n e c o n d u c t a n c e (lower m e a n specific m e m b r a n e resistance) for f a s t - a x o n e d m o t o n e u r o n e s t h a n for m o r e s l o w - a x o n e d ones. Thus, if all the h i n d l i m b m o t o n e u r o n e s o f a p o o l were s i m u l t a n e o u s l y activated by the same overall density o f u n i f o r m l y d i s t r i b u t e d excitation, the s l o w - a x o n e d cells w o u l d still be likely to be m o r e easily recruited t h a n m o r e f a s t - a x o n e d ones. Thus, o u r results help to explain why an ascending-size-order o f r e c r u i t m e n t occurs so c o m m o n l y within m o t o n e u r o n e p o o l s in response to various kinds o f synaptic inputs 9. It should n o t be forgotten, however, t h a t the r e c r u i t m e n t o r d e r never d e p e n d s only on n e u r o n a l properties. W h i c h e v e r p r o p e r t i e s the n e u r o n e s have, the r e c r u i t m e n t o r d e r in response to a given synaptic i n p u t will always d e p e n d on the c o m b i n e d effects o f the d i s t r i b u t i o n o f n e u r o n a l p r o p e r t i e s over a m o t o n e u r o n e p o o l a n d the d i s t r i b u t i o n o f the relevant synaptic effects within the s a m e pool.

REFERENCES 1 Barrett, J. N. and Crill, W. E., Specific membrane properties of cat motoneurones, J. Physiol. (Lond.), 239 (1974) 301-324. 2 Burke, R. E., Motor unit types of cat triceps surae muscle, J. Physiol. (Lond.), 193 (1967) 141-160. 3 Burke, R. E., Group Ia synaptic input to fast and slow twitch motor units of cat triceps surae, J. PhysioL (Lond.), 196 (1968) 605-630. 4 Burke, R. E. and Ten Bruggencate, G., Electrotonic characteristics of alpha motoneurones of varying size, J. PhysioL (Lond.), 212 (1971) 1-20. 5 Conradi, S., Functional anatomy of the anterior horn motor neuron. In D. N. Landon (Ed.), The Peripheral Nerve, Chapman and Hall, London, 1976, pp. 279-329. 6 Cullheim, S., Relations between cell body size, axon diameter and axon conduction velocity of cat sciatic a-motoneurons stained with horseradish peroxidase, Neurosci. Lett., 8 (1978) 17-20. 7 Frank, K. and Fuortes, M. G. F., Stimulation of spinal motoneurones with intracellular electrodes, J. Physiol. (Lond.) , 134 (1956) 451-470. 8 Grantyn, R. and Grantyn, A., Morphological and electrophysiologicalproperties of cat abducens motoneurons, Exp. Brain Res., 31 (1978) 249-274. 9 Henneman, E., Organization of the motoneuron pool. The size principle. In V. B. Mountcastle (Ed.), MedicalPhysiology, 14th edn., C. V. Mosby, St. Louis, 1980, pp. 718-741. 10 Hursh, J. B., Conduction velocity and diameter of nerve fibers, Amer. J. Physiol., 127 (1939) 131-139. 11 Ito, M. and Oshima, T., Electrical behaviour of the motoneurone membrane during intracellularly applied current steps, J. PhysioL (Lond.), 180 (1965) 607-635. 12 Kernell, D., Input resistance, electrical excitability and size of ventral horn cells in cat spinal cord, Science, 152 (1966) 1637-1640. 13 Lux, H. D. and Schubert, P., Some aspects of the electroanatomy of dendrites. In Advances in Neurology, Vol. 12, Raven Press, New York, 1975, pp. 29-44. 14 Lux, H. D., Schubert, P. and Kreutzberg, G. W., Direct matching of morphological and electrophysiological data in cat spinal motoneurones. In P. Andersen and J. K. S. Jansen (Eds.), Excitatory Synaptic Mechanisms, Universitetsforlaget, Oslo, 1970, pp. 189-198. 15 Rall, W., Core conductor theory and cable properties of neurons. In E. Kandel (Ed.), Handbook of Physiology, Section I, Vol. I, Part 1, American Physiological Society, Bethesda, 1977, pp. 39-97. 16 Stein, R. B. and Bertoldi, R., The size principle: a synthesis of neurophysiological data. In J. E. Desmedt (Ed.), Progress in Clinical Neurophysiology, Vol. 9: Recruitment Patterns of Motor Units and the Gradation of Muscle Force, Karger, Basel, in press.

326 17 Streit, P. and Reubi, J. C., A new and sensitive staining method for axonally transported horseradish peroxidase (HRP) in the pigeon visual system, Brain Research, 126 (1977) 530-537. 18 Zucker• R. S.• The•retica• imp•icati•ns •f the size princip•e •f m•t•neur•ne recruitment• J. the•ret. Biol., 38 (1973) 587-596. 19 Zwaagstra, B., Morphofunctional correlations of 78 a-spinal-motoneurons of the cat, Neurosci. Lett., Suppl. 1 (1978) S107. 20 Zwaagstra, B. and Kernell, D., Sizes of soma and stem dendrites in intracellularly labelled amotoneurones of the cat, Brain Research, 000 (1980) 000-000.