A quantitative estimate of infolding and of its effect on electrical characteristics in axons of different sizes is given by expressions based on the allometric equation, y = axβ. These expressions can be used to describe the effect of stretching the axon and the size dependence of the axo-somatic conductance ratio.
The parameters in the allometric equation are determined for a range of axon sizes in three species of opisthobranchs from original and from published data. Analysis of variance for these data suggests that interaxonal variability of infolding is less for axons of an identifiable neurone from different specimens of the same species than for axons of different neurones from the same specimen. With the exception of the Aplysia R2 axon where the infolding size dependence may be less, the ratio of axon volume to surface in these neurones is given approximately by the equivalent axon diameter to the one-half power. For this relation of infolding to size (and if it is assumed that infolding in the soma varies with size in the same way as in the axon), the axo-somatic conductance ratio is inversely proportional to soma diameter to the three-fourths power.
These results suggest that conduction velocity increase with size is small, and because of the penalty in the concomitant increase in axon surface membrane with size, selection for larger axons probably does not occur to satisfy a requirement of decreased response time.
In certain species of gastropod molluscs, neurone DNA content (Bezruchko et al. 1969; Coggeshall, Yaksta & Swartz, 1970; Lasek & Dower, 1971) and size (Arvanitaki & Tchou, 1942) continue to increase during adult life. The axon membranes of those neurones in which growth persists develop a complex morphology with elaborate invaginations. These surface indentations, which vary in amount for different neurones, affect synaptic integration and axon electrical properties (Mirolli & Talbott, 1972; Gorman & Mirolli, 1972).
For axons of irregular shape, the ratio of volume to surface area is the geometral variable of interest; it has been suggested that this ratio should be independent of changes in axon length if the axolemma is not deformable at the ultrastructural level (Hodgkin, 1954). This analysis has been used to explain the independence of conduction velocity (and, more generally, of length constant) on stretching in infolded axons. Mirolli & Talbott (1972) measured volume and surface area for G-cell axons in the nudibranch Anisodoris nobilis and developed equations to relate these measurements to the axon passive electrical characteristics. They estimated axon volume and surface area from cross-sectional area (A) and perimeter (P), respectively. Area and perimeter were combined to give a geometric factor for the length constant, H = (A/P)1/2, and for the input conductance, M = (AP)1/2. They concluded that, although axon area and perimeter varied widely in different nerve cross-sections, both H and M could be approximated by values averaged along the axon length, and that H remains constant when the axon is stretched, while M decreases.
Although infolding has been described in some detail for specific gastropod neurones (Mirolli & Talbott, 1972; Pinsker, Feinstein, Sawada & Coggeshall, 1976) and a correlation with size has been shown in Aplysia (Graubard, 1975), volume and surface have not been analytically related for axons of different sizes. This report shows that infolding can be predicted by an empirical formula which gives the ratio of axon volume to surface area as a function of equivalent axon diameter in the form of the allometric equation, y = αxβ (Huxley, 1932). Such a relation can be used to describe the size-dependent effect of infolding on the passive electrical properties of axons; infolding can be quantitatively related to axon size through the parameters α and β; but, because of the empirical nature of this relation, α and β must be determined for each axon type or group of axons considered. These parameters are estimated from measurements given here for Tritonia axons and from the available data for Aplysia and Anisodoris fibres (Mirolli & Talbott, 1972; Graubard, 1975; Pinsker et al. 1976).
MATERIALS AND METHODS
Tritonia axon measurements
The lateral buccal nerve (Longley, 1976) from a 40 g Tritonia diomedia (Bergh) obtained from East Sound in the San Juan Islands of northwest Washington State was prepared by immersing the buccal mass, with ganglia and nerves exposed, for 12 h in fixative (4% glutaraldehyde, 3% formaldehyde, and 0·2M-NaH2PO4 in distilled water; pH adjusted to 7-3 with NaOH). The nerve was then dissected, postfixed for 2h in 2% OsO4, dehydrated in ethanol, and embedded in Epon. A silver section of the whole nerve cross-section was placed on a support film, stained with uranyl acetate and lead citrate, and photographed at a magnification of 1800 using a Philips 300 electron microscope. In a montage of the nerve enlarged 4800 times, axon area, A, was measured with a planimeter and by weighing axon tracing cutouts, while the perimeter, P, of each axon was measured with a measuring wheel. These measurements were repeated at least two times. Uncertainty in measurements of axon cross-sections did not seem to be an important source of variation, since changes in area and perimeter with repeated measurements of axon tracings were small compared to the interaxonal scatter of the data. No correction was made for artifacts introduced by fixation and handling of the tissue (see Fig. 1).
Aplysia and Anisodoris data conversion
Data on infolding analysed here from other papers were obtained from graphs (Mirolli & Talbott, 1972; Graubard, 1975) and from tables and figures of axon cross-sections (Pinsker et al. 1976). When necessary, these materials were enlarged photographically to increase accuracy in translating the results of these authors to axon area and perimeter for comparison to results obtained from Tritonia (Fig. 3). Data for axons of the Aplysia right abdominal connective (Fig. 4A and filled squares in Fig. 4B) are from Graubard (1975). For the Aplysia R1 and R2 axons (Fig. 4B, open and filled inverted triangles), the data are from Pinsker et al. (1976). Area and perimeter values of the G-cell axon of Anisodoris used in plotting Figs 5 and 6 are taken from Mirolli & Talbott (1972).
The constants a and b (in equation 1) required to solve the allometric equation were estimated by fitting a straight line with the least squares method to logarithmic transformations of the data. The validity of this straight line model was tested using linear Egression statistics.
Biological variability in interaxonal results is estimated from the residual mean squares of the infolding variable about the regression line. By using logarithmic transformations of the data, the size dependence of the variance is removed (Lewontin, 1966). In Figs 3A and 4A, values of log U are symmetrically located about the regression lines and tend to form a normal distribution (D’Agostino’s test, P> 0·2). On this basis, the F-test is used in comparing the residual mean squares of log U in different axon groups.
Different techniques and procedures used by the different authors also affect the variability of the data. Variations may occur in shrinkage and distortion because of different histological techniques. If swelling or shrinkage occurs uniformly throughout the tissue without changing axon perimeter, the exponent b in equation (1) will not be affected, but the constant a will be changed by the factor f2-b, where f is the fractional change in the dimensions of the tissue. Such a uniform change in size would not affect the variability of the results for axons in a single nerve, but where axons in several nerves are compared, differences in/may increase the variance of the data. The extent of non-linear distortion, which could change both b and a and also increase the variance of the data, cannot be estimated except from the overall consistency of the results.
Where an identified axon is measured in more than one nerve, differences in the amount of stretching of the nerves before fixation will cause variations in the size of the axons (see Appendix). For the G-cell axons stretched over a range of 1·5–1·8 times, as reported by Mirolli & Talbott (1972), possible effects on the variance of the data are negligible. In R1 and R2 of Aplysia, where stretching of the nerve was based on an electrophysiological criterion (Pinsker et al. 1976), the apparent interaxonal variability of infolding (in different nerves) may have been increased by different amounts of stretching.
Analytical relation of axon size to infolding
A plot of the equivalent axon diameter and ratio of volume to surface area used in the above equation shows the effect of infolding, but the importance of this relation is in defining the parameters a and b. When a and b are known, H, M, and other infolding variables (see Appendix) can also be expressed as a power function of D. If the least squares method is used in the calculation of a and b in this power function, the choice of variable is arbitrary. However, U, the variable used here, retains the magnitude of the biological variability in measurements of surface area and perimeted to a greater extent than other possible variables and is less misleading in its relation to D when presented graphically (e.g., compare the scatter of Tritonia data for U in Fig. 3A and M in Fig. 3B).
Comparison of infolded axons in different species
Tritonia buccal nerve
The buccal nerve axons outlined in Fig. 2 are primarily from monopolar neurones and one pair of bipolar neurones whose somata are found in the buccal ganglion of Tritonia, but one or two branches of the giant cerebral neurones are also present (Longley, 1976). The number of mitochondria in these infolded axons tends to be Proportional to axon perimeter but not to axon area (see Fig. 1). Values of U for these axons are plotted in Fig. 3A against their equivalent diameters, and estimates of the parameters of the regression line through the log-transform of these data are given in Table 1. For the value b = 0·42 of the exponent in equation (1), from the slope of this line, the geometric factor for the axon length constant H (= U1/2) would vary as D0·21 instead of as D0· 5 in cylindrical axons with no infolding. The fit achieved with a power function of D to the geometric factor for axon conductance (M) is shown in Fig. 3B.
M was calculated from area and perimeter of the axons outlined in Fig. 2 using M = (AP)1/2. The line through these points (Fig. 3B) can either be drawn by using the values of a and b in Table 1 for these axons and the expression for M in the Appendix or calculated directly from log M and log M using least squares.
Aplysia abdominal ganglion connective
Axon infolding in the right abdominal ganglion connective of Aplysia califomica has been examined by Graubard (1975). Values of U for axons with D < 25 μm (a size range comparable to the Tritonia buccal nerve axons) are shown in Fig. 4A. In these axons, neither of the estimated parameters a and b (Table 1) differ significantly (P>0·2) from those found for Tritonia axons. Infolding in the two largest axons in this connective from neurones R1 and R2 has been described by Pinsker et al. ( 1976). Measurements were made by these authors on both R1 and R2 axons, in sections cut from 14 nerves taken from animals ranging in size from 5 to 300 g. The regression line for their R1 axon data (Fig. 4B, open inverted triangles) overlies the results for the above groups of Tritonia and Aplysia axons (see estimated parameters in Table 1).
The R2 axons of smaller diameter in Fig. 4B (filled inverted triangles) were shown by Pinsker et al. (1976) to have relatively greater infolding than R1 axons from the same connectives. This finding is illustrated quantitatively in Fig. 4B and Table 1 as a displacement to a lower elevation and slope of the regression line for R2 data relative to results obtained for Rl. Measurements on larger R2 axons in Aplysia (Fig. 4B, filled squares) from Graubard (1975) have an entirely different slope, with a value of b near 2. If b = 2, the perimeter of the axon remains constant as its equivalent diameter, D, increases and the relative infolding factor, F, varies inversely with D (see equations for P and F in Appendix). This is not consistent with any of the other results presented here and indicates in this axon a marked change in growth relations for infolding in larger animals. Changes in growth of Aplysia abdominal ganglion neurones with animal size have been reported previously (Arvanitaki & Tchou, 1942), and a decrease in efficacy with age has also been reported in an Aplysia abdominal ganglion neurone (Peretz, Ringham & Wilson, 1982). The above results suggest an age-related infolding change in R2 axons.
G-cell axon in Anisodoris
The data presented on infolding in the Anisodoris G-cell axon by Mirolli & Talbott (1972) are unique in that these authors examined histologically the effect of stretching on an identifiable fibre. It is possible from these data to show a proportionality between axon size (D) and their estimate of G-cell soma size (d), so that D = kd. In Fig. 5, this proportionality constant, k, is calculated for both the unstretched and stretched axons by regression through the origin. Stretching of the axons leads to a reduction in k.
The data of Mirolli & Talbott (1972) on the G-cell for both unstretched and stretched axons (Fig. 6A, open and filled circles, respectively) give values of the exponent in equation (1) which are similar to results from other axon groups shown in Table 1. For the regression line through values of logU, stretching should produce a shift with constant slope towards smaller D and an increase in the elevation, a, consistent with that shown in Fig. 6A, although the value of U for a particular axon should not change with stretch (see Appendix). Even though the exponent b of the G-cell axons is similar to that for other axons in the Tritonia buccal nerve and the Aplysia connective, the elevation of the regression line is lower, placing this line near the R2 results from Pinsker et al. (1976). For the other four groups of Aplysia and Tritonia axons in Table 1, the mean value of a is 0·30 with a range of 0·24–0·34, but for the unstretched G-cell axons, a = 0·10. This difference may indicate relatively greater shrinkage of this axon in histological procedures, which could reduce a and leave b unchanged, or it may represent a real difference in the G-cell in this size range.
Mirolli & Talbott (1972) also give measurements on infolding along the length of a single G-cell axon. A correlation between U and D based on their results is shown in Fig. 6B. Although equation (1) is used to relate U and D for these data where changes in cross-sectional area and perimeter are occurring along the length of a single axon, the parameters a and b in this case are not comparable to results from other axons in Table 1, where a and b are derived from an interaxonal correlation (i.e. the average of the values of log U and log D in Fig. 6B would appear as one data point Fig. 6A). A strong correlation is present between U and D measured at different points along the length of this axon, even though H (= U1/2) was previously thought to be constant for these data (Mirolli & Talbott, 1972). Estimates of the parameters are a = 0·013 and b = 1·15, ±S.E. 040 (33). If b is close to one, as in this case, the axon perimeter tends to be proportional to the equivalent diameter, but the relative amount of infolding along the axon is independent of size variations (see expressions for P and F in Appendix). The exponent b for this intra-axonal correlation is significantly greater than that for other axons analysed as a group (P < 0·001 ). This result for the size dependence of infolding along this axon suggests that local control of axon surface differs from regulation of the average volume to surface ratio in the axon during growth.
Biological variability of infolding
The principal variation of U in different axons results from its dependence on axon size, as shown in Figs 3A, 4 and 6A, but differences may also exist in the variance of U in neurones of the same size. This latter possibility, which can be examined with data assembled here, would suggest an intrinsic difference in the infolding mechanism in these neurones. Results for the interaxonal variation of log U of axon groups whose size dependence has been shown above are given in Table 2 as the residual mean squares.
The residual mean squares also contains the intra-axonal variance, which is a major source of uncertainty in estimating the variance of log U between axons. In the case of the G-cell axon (Fig. 6B), the intra-axonal variance of log U can be calculated and is shown in Table 2 for comparison to the residual mean squares of axon groups. However, this intra-axonal variance would not add directly to the variance of log U between axons because of the intra-axonal correlation of log U to log D. The values of logU from intra-axonal sampling in this axon would lie in an ellipse with a major axis slope of about TO. When the variance of these values is projected on the residual mean squares of a regression line through interaxonal data with a slope of 0·5, the contribution of this intra-axonal variance will be reduced eight times over that which would be present if intra-axonal values of logU were independent of axon size. For the two groups of G-cell axons, the intra-axonal variance present in the residual mean squares is further reduced an additional tenfold by averaging, and the residual mean squares shown in Table 2 for these axons should be almost entirely free of intraaxonal sampling effects. In other neurones, if the magnitude of the intra-axonal variation of log U and correlation of U to log D is similar to that shown for the G-cell in Fig. 6B, then intra-axonal variability will add less than 8% to the residual mean squares of other groups of axons shown in Table 2, even in those cases where there is no averaging.
Although the residual mean squares of the stretched G-cell axon group and the Rl and R2 axon groups are larger than that of the unstretched G-cell axons, these differences are not significant [for the unstretched G-cell axons and Rl axons, F( 13,7) = 3·13; 0·1 >P> 0·05, one tailed test]. The residual mean squares from these identifiable neurone groups are lower than those for the axon groups from different neurones in the Aplysia connective and the Tritonia buccal nerve. When tested between the Rl and Aplysia connective groups where the values are closest, this difference is significant [F(34,13) = 2·49; P<0·05, one tailed test]. Overall, these results suggest that the observed variability in U, after removal of size dependence, does not arise from random sampling along the axon, but is due to differences between individual neurones. This variability is greater in a group of axons from different neurones than in axons from an identifiable neurone.
Size dependence of axo-somatic conductance ratio
Values for the calculation of the axo-somatic conductance ratio using equation (4) are available for the G-cell. For the unstretched axon, using μm as the length unit, k = 0·15, a = 0·10 and b = 0·59. From Gorman & Mirolli (1972), Rm and Ri are 106Ωcm2 and 102Ωcm, respectively. For soma diameters varying from 133–333 –m (corresponding to axon diameters from 20–50μm), ϱ from equation (4) decreases from 8·6 to 4·5. If infolding in the G-cell soma has a constant value of 7·5, independent of size (Gorman & Mirolli, 1972), (lover the above size range would vary from 9·8 to 7·5. These values are in the range given by Gorman & Mirolli (1972) for their anatomical estimates of the axo-somatic conductance ratio in the G-cell.
Where infolding can be described by the allometric equation, there must be a mechanism which maintains the ratio of increments in volume and surface in proportion to the axon volume/surface ratio as the axons increase in size, even though the cons may be growing at different rates to reach their various sizes (Rosen, 1967). The variability of the data suggests that there are differences in this process and that a range of parameter sets may be required to describe infolding in different neurone types. In the neurones considered here, infolding variability is less for axons from an identifiable neurone in different animals than it is for axons from neurone groups in the same animal. This suggests that the developmental history of the neurones is a more important factor in regulating infolding during growth than the identical genetic endowment of neurones from one individual. However, with the observed variability, it is still feasible to specify the parameters a and b in equation (1) both for axons from identified neurones at different growth stages and for axons of different sizes from the same animal. This demonstrates that the allometric equation, with its interpretative significance (Savageau, 1979), can be used to describe axon infolding, and the parameters of this equation, when determined empirically for a group of axons, can be used to predict the size dependence of geometric factors of axon passive electrical characteristics. The range of these parameters for other infolded axon groups in opisthobranch and pulmonate species remains to be investigated.
Increasing axon size is generally correlated with an increase in axon spike velocity, but the constants associated with this relation vary widely in different species (Bullock & Horridge, 1965). In the squid giant axon, spike velocity was found to be proportional to the square root of axon diameter (Pumphrey & Young, 1938), although a linear relation has also been reported (Hodes, 1953). The square root relation is predicted analytically (Hodgkin, 1954) and is widely accepted for spike velocity in unmyelinated cylindrical axons. In axons with infolding, the geometric factor for spike velocity is not the diameter but the square root of the axon volume to surface ratio, or H, as developed by Mirolli & Talbott (1972). In spike velocity comparisons in cases where there is grossly different infolding (axons of R1 and R2 in the Aplysia abdominal ganglion), the ability of H to predict relative spike velocity was verified, although other factors may be present since agreement between morphological and electrophysiological measurements was not exact (Pinsker et al. 1976). Using H as a correlate, spike velocity for the infolded axons described here can be expected to increase about as the fourth root of their equivalent diameter, although there may be exceptions, such as the R2 axon, where the size dependence is much less. In order to double spike velocity, it would be necessary to increase the perimeter 64 times and the cross-sectional area 256 times in these axons. This enormous increase in membrane required for infolding would seem to rule out selection of larger axons solely to obtain an increased spike velocity.
Infolded membrane may have a role in facilitating stretching of the larger axons in these gastropod species, but it may also offer an advantage of increased oxygen supply, since the number of mitochondria in the Tritonia buccal nerve axons is correlated with this disproportionate axon surface increase with size. Such a nutritive role for infolding has been repeatedly proposed (Pinsker et al. 1976). The increasing size of these axons during the animal’s growth may be associated with an increased transport of proteins and smaller molecules from the neurone soma to the periphery. This possibility is supported by continuing DNA synthesis, which may increase up to 75 000 times the haploid amount of DNA in the Aplysia R2 neurone (Coggeshall et al. 1970).
It is striking that axons from three different species evolving in different environments have parameters for infolding which are very similar. Membrane infolding may occur in an axon in a manner analogous to that proposed for formation of convolutions in the vertebrate brain where growth is limited by the skull (Clark, 1945). In such a model axon volume increases would be constrained by the nerve sheath, and excess axolemma production would result in infolding. Thus it is possible that determining the parameters for infolding, in addition to being useful in predicting size related electrical characteristics of these gastropod axons, will also allow separation of neurones having different metabolic activity.
Other infolding variables
Stretching of infolded axons
The expected displacement of the regression line of U for the stretched Anisodoris G-cell axons (Fig. 6A) can be calculated from the effect of axon stretching on D. There is a proportionality between axon and soma size in this neurone (Fig. 5) and since soma size should be unaffected by stretching the axon, the amount of stretching can be estimated from the change in the proportionality constant k (from D = kd) through the inverse relation of D to s1/2. This leads to s = (k/k′)2, where k′ is the constant for the stretched axons. For these data in Fig. 5, s = 1-58, in agreement with the range of stretch values of 1·5–1·8 times indicated by Mirolli & Talbott (1972). From this result the values of U of the stretched axons in Fig. 6A would be expected to be shifted to a smaller D by the factor s1/2, or 1·26. The expected increase in the elevation of the regression line for these stretched axons is 13 %, calculated from the expression for a′ above.
The effect of axon stretch on other infolding variables above can be determined directly from the stretch related change expected in area and perimeter or by substituting into these equations modified values of a and D as indicated above. The length constant H should be independent of axon stretching, as is U, but P and M should be inversely proportional to s, and F inversely proportional to s1/2.
I thank Drs Douglas Junge, Bill Moody, and Kathy Graubard for helpful criticisms of earlier versions of this manuscript and the anonymous referee for numerous helpful comments, many of which were included in the manuscript. I also thank Dr Alan Grinnell, director of the Jerry Lewis Center for Neuromuscular Research, and Dr Dennis Willows, director of Friday Harbor Laboratories, for use of facilities during the preparation of this paper. This work was supported in part by NIH Special Research Fellowship 5 F03 GM45283.