Brain Research, 108 (1976) 413-417 © ElsevierScientificPublishingCompany,Amsterdam- Printed in The Netherlands
413
Short Communications
Determination of axon terminal density in the central nervous system
M. PALKOVITS
First Department of Anatomy, Semmelweis University Medical School, Budapest (Hungary) (Accepted February 18th, 1976)
The direct counting of various types of axon terminal profiles in electron micrographs would yield highly misleading results due to their differences in size. The probability of their being cut by the plane of sectioning is obviously greater for large particles than for smaller ones. A quantitative comparison of synaptic terminals that makes any claim of reliability, therefore, has to be based on the density (number of synaptic terminals per unit volume) instead of the relative number (per unit surface area). On the basis of synaptic terminal density and of cell number per unit volume (cell density) the number of synaptic terminals per cell can be calculated. Due to the difference in the dimensions (electron versus light microscopic measurements) only densities are comparable. In order to compare light microscopic and electron microscopic data quantitatively, the shrinkage due to the different histotechnical procedures has to be determined and all data have to be calculated for the original tissue volume. Since fixation by perfusion in itself does not change the volume of the brain significantly, all reference to original volume refers in practice to the volume in the living state s. The following measuring procedures and calculations may be used in order to determine the density of synaptic terminals (SD). (1) Determination of synaptic density by measuring the diameters and the sectional surfaces of the synaptic terminals (Saltykov methodS; see refs. 11 and 12). There is a continuous size distribution of particles in a material, but Saltykov11 divides them into a finite number of classes. Experience has shown that the diameters of sectional areas of particles usually follow a log-normal distribution. It is therefore desirable to use a logarithmic scale (log 10-0.1) for the class intervals. If the particles are not spherical their size distribution has to be characterized by the distribution of sectional areas (A). In order to specify the size of a section, Saltykov11 adopted a relative ratio (A/Amax) instead of the absolute values of sectional areas. The scale factor used to determine the class intervals for A/Amax is based on a logarithmic scale of diameters of particles with the factor 10-°.1 ( = 0.7943). Consequently, for sectional areas a logarithmic scale with a factor (10-0.1)2 = 10-0.3 ( = 0.6310) can be used.
414 The volume density of a particle (number of particles per umt volume), SD, is given by SD -- N/T: where N is the number of particle sectmn areas per umt area of the sectioning plane, and T is the thickness. The longest diameter Tmax : 1 and the others are classified in log 10-°.1 classes. The largest sectional area Amax ~-- I : the others are m classes of log -° 2. From the number of particle secnon profiles per unit area for each individual class (NAx) the number of particles per umt volume in the same class (Nvx) can be determined by the formula Nv -- NA/T; Nv values for all classes should be calculated using the formula and table given by Saltykov (see refs. 11 and 12). The synaptic terminals for each class have to be calculated separately and the sum of the values obtained corresponds to the synaptic terminal density. This method is suitable for the determination of synapt~c terminal densities using electron micrographs. However, due to the need to measure both diameters and sectional areas of all axon terminal profiles, and to the long and cumbersome process of calculation and classification, the application of this technique proved to be extremely laborious, even to the degree of becoming prohibitively uneconomical. The other two (second and third) methods are simple enough to be used routinely, and their results did not differ significantly from those obtained by the application of the more sophisticated Saltykov method (Palkovits, Mezey, H~tmori and Szentfigothai, in preparation).
(2) Determination of axon terminal density by measuring the diameters of synaptic profiles. For the determination of volume (V) of the sample in which the terminals are located the knowledge of the profile section surface (F) and the section thickness (T) are necessary. Section thickness in electron microscopy, however, is practically zero. The extension of the synaptic terminals in the third dimension can, nevertheless, be specified by the 'theoretical thickness', which is equal to the distance between two tangent planes which include the whole of the terminal averaged for all orientations of synaptic boutons (it is also called the cahper diameter). In order to determine T one has to know the average diameter (2r) of the synapt~c bouton profiles as well as the diameters of the smallest profiles (a) that would still be identified as a bouton of this kind (Fig. 1). From these data T can be calculate& on the basis of the formula T=2
]~ /'r e -
(-~a ) z "
The product of section surface of the electron micrographs (F) and the 'theoretical thickness' (T) gives the volume of the brain area sampled. By counting the synaptic terminals in electron micrographs (n) the synaptic density (SD) can be calculated from SD = n/F.T. (In view of the different sizes of the types of synaptic boutons within the same brain area, T has to be determined for each type separately and calculations have to be made accordingly - - Fig. 1.)
(3) Determination of axon terminal densities by measuring their volume fractions in electron micrographs. The volume fraction is one of the most important parameters required in quantitative histological analysis. The volume fraction of synaptic terminals can be estimated by the measurement of either linear or planar fractions of
415
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Fig. 1.
the synaptic profiles (using a planimeter or 'point counting method'). The average planar (surface) fraction determined on sections through a volume represents an estimate of the volume fractionL There is also an equivalence of linear and volume fractions lo. The systematic point count technique using a two-dimensional point grid should be the most efficient method of volume fraction analysisl,3-5,12. A 100-grid (10 × 10 lines)7, 9 drawn upon a clear plastic sheet was superimposed on the electron
416 mlcrograph, taking care to apply random positions. (The optimum density for the grid roughly corresponds to an average of one point per profile to be measured.) The number of crossing points that lay over the synaptic terminals was counted. Then the grid was repositioned 30 times randomly and the same count was made. The average number of points was then divided by 100 (because of the 100-grid) and the value obtained corresponded to the planar (surface) fraction of synaptlc terminals on the electron mlcrographs stud~ed. According to the rule of Delesse 2 the surface fraction of a given particle (Zf/F) is equivalent to the volume fraction of the particle (Zv/V). Furthermore, the volume fraction is equal to the product of the volume of a single particle and of its density (Zv/V ----- v.SD). The density can be calculated on the basis of the formula SD -: (Zf/F)/v. Since v is equal to ~/(Zf/n) 3, then SD = (f/F)/~/(Zf/n) a ( f = total surface areas of randomly measured synaptic terminals; F -~ total area of the electron micrographs; n = number of synaptic terminals in F area). For the application of this technique the section surface of synaptic terminals should be measured by aid of a planimeter. According to Rosiwa's rule (see ref. 8), the linear fraction of a particle profile is characteristic of the volume fraction of that particle. Axon terminal density can, therefore, also be determined on the basls of the formula SD--
~l/L (Zl/n) 3
(L = total lengths of lines tracted through on the electron micrographs; 1 ---- linear fractions of particle, i.e., total lengths of lines passing through the particle profiles; n = number of particle profiles crossed by the random lines on the micrographs). Since the synaptic terminals are usually irregular and do not resemble any simple geometrical shape, measuring in one dimension only results in a somewhat higher technical error than in the case of two-dimensional (surface determination by platometer) measurements. The surface fraction (SF) corresponding to Zf/F can be measured by the point counting method; by this means the measurement of section surface can be eliminated. By so doing, three simple measurements: (1) the area of electron micrographs, (ii) counting the synaptic terminals and (iii) using a 10 × 10 grid for surface fraction determination, are sufficient to determine the density of the synaptic terminals in a given brain region. The steps of this technique in practice are as follows. (1) Measurement of the area of the electron micrographs in sq./zm (using a correction factor for shrinkage). (2) Determination of the surface fraction by a 100-grid superimposed randomly 30 times per micrograph. (3) Counting the synaptic terminals. (4) Calculation of Z f ( = SF.F/100) and SD ( = SF/~/(Zf/n)3). Further simplifications are possible by direct measurement under the electron microscope. After determination of SF on electron micrographs it proved to be
417 sufficient to count synaptic terminals directly on the electron microscope screen when F is equal to the areas of the grids investigated. The two above simplified techniques give similar results, with acceptably low standard errors (Palkovits, Mezey, Hgtmori and Szent~gothai, in preparation). For example, the number of synaptic terminals and of synaptic terminals per cell in a hypothalamic nucleus could be determined both under normal and experimental conditions 6, showing the usefulness of this new quantitative approach to synaptic connectivity. The author is indebted to Professor J. Szent~gothai for his valuable advice and critical comments.
1 CHALKLEY,H. W., Methods for the quantitative morphologic analysis of tissue, J. nat. Cancer Inst., 4 (1943) 47-53. 2 DELESSE, A., Pour d6termine la composition des roches, Ann. des Mines, 13 (1848) 379-388. 3 GLAGOLEV,A. A., On the geometrical methods of quantitative mineralogic analysis of rocks, Trans. Inst. Econ. Min. (Moscow), 59 (1933) 162-194. 4 GLAGOLEV,A. A., Quantitative analysis with microscope by the point method, Engng. Min. J., 135 (1934) 399-410. 5 HILLIARD,J. E., AND CAHN, J. W., An evaluation of procedures in quantitative metallography for volume fraction analysis, Trans AIME, 221 (1961) 344. 6 L~R/~rcrn,Cs., ZAnORSZKY,L., MARTON,J., AND PALKOVITS,M., Studies on the supraoptic nuclei in the rat. I. Synaptie organization, Exp. Brain Res., 22 (1975) 509-523. 7 PALKOVrrs, M., Histologische Fiirbung und quantitative Methodik zum Nachweiss der Blutfiillung der endokrinen Organe, Microscopic, 17 (1962) 300-303. 8 PALKOVITS,M., MAGYAR,P., AND SZENT/~GOTHAI,J., Quantitative histological analysis of the cerebellar cortex in the rat. I. Number and arrangement in spaces of the Purkinje cells, Brain Research, 32 (1971) 1-13. 9 PALKOVITS,M., MAGYAR,P., AND Sz~wr/~GOTHAI,J., Quantitative histological analysis of the cerebellar cortex in the cat. II. Cell numbers and densities in the granular layer, Brain Research, 32 (1971) 15-30. 10 ROSIWAL,A., (~ber geometrische Gesteinsanalysen, l/erh, tier K.-K. geol. Reichanstalt, 5 0898) 143. 11 SALTYKOV,S. A., The determination of the size distribution of particles in an opaque material from a measurement of the size distribution of their sections. In H. ELIAS (Ed.), Stereology, Springer, Berlin, 1967, pp. 163-173. 12 UNDERWOOD, E. E., Quantitative Stereology, Addison-Wesley, Reading, Mass., 1970, 274 pp.
413
Short Communications
Determination of axon terminal density in the central nervous system
M. PALKOVITS
First Department of Anatomy, Semmelweis University Medical School, Budapest (Hungary) (Accepted February 18th, 1976)
The direct counting of various types of axon terminal profiles in electron micrographs would yield highly misleading results due to their differences in size. The probability of their being cut by the plane of sectioning is obviously greater for large particles than for smaller ones. A quantitative comparison of synaptic terminals that makes any claim of reliability, therefore, has to be based on the density (number of synaptic terminals per unit volume) instead of the relative number (per unit surface area). On the basis of synaptic terminal density and of cell number per unit volume (cell density) the number of synaptic terminals per cell can be calculated. Due to the difference in the dimensions (electron versus light microscopic measurements) only densities are comparable. In order to compare light microscopic and electron microscopic data quantitatively, the shrinkage due to the different histotechnical procedures has to be determined and all data have to be calculated for the original tissue volume. Since fixation by perfusion in itself does not change the volume of the brain significantly, all reference to original volume refers in practice to the volume in the living state s. The following measuring procedures and calculations may be used in order to determine the density of synaptic terminals (SD). (1) Determination of synaptic density by measuring the diameters and the sectional surfaces of the synaptic terminals (Saltykov methodS; see refs. 11 and 12). There is a continuous size distribution of particles in a material, but Saltykov11 divides them into a finite number of classes. Experience has shown that the diameters of sectional areas of particles usually follow a log-normal distribution. It is therefore desirable to use a logarithmic scale (log 10-0.1) for the class intervals. If the particles are not spherical their size distribution has to be characterized by the distribution of sectional areas (A). In order to specify the size of a section, Saltykov11 adopted a relative ratio (A/Amax) instead of the absolute values of sectional areas. The scale factor used to determine the class intervals for A/Amax is based on a logarithmic scale of diameters of particles with the factor 10-°.1 ( = 0.7943). Consequently, for sectional areas a logarithmic scale with a factor (10-0.1)2 = 10-0.3 ( = 0.6310) can be used.
414 The volume density of a particle (number of particles per umt volume), SD, is given by SD -- N/T: where N is the number of particle sectmn areas per umt area of the sectioning plane, and T is the thickness. The longest diameter Tmax : 1 and the others are classified in log 10-°.1 classes. The largest sectional area Amax ~-- I : the others are m classes of log -° 2. From the number of particle secnon profiles per unit area for each individual class (NAx) the number of particles per umt volume in the same class (Nvx) can be determined by the formula Nv -- NA/T; Nv values for all classes should be calculated using the formula and table given by Saltykov (see refs. 11 and 12). The synaptic terminals for each class have to be calculated separately and the sum of the values obtained corresponds to the synaptic terminal density. This method is suitable for the determination of synapt~c terminal densities using electron micrographs. However, due to the need to measure both diameters and sectional areas of all axon terminal profiles, and to the long and cumbersome process of calculation and classification, the application of this technique proved to be extremely laborious, even to the degree of becoming prohibitively uneconomical. The other two (second and third) methods are simple enough to be used routinely, and their results did not differ significantly from those obtained by the application of the more sophisticated Saltykov method (Palkovits, Mezey, H~tmori and Szentfigothai, in preparation).
(2) Determination of axon terminal density by measuring the diameters of synaptic profiles. For the determination of volume (V) of the sample in which the terminals are located the knowledge of the profile section surface (F) and the section thickness (T) are necessary. Section thickness in electron microscopy, however, is practically zero. The extension of the synaptic terminals in the third dimension can, nevertheless, be specified by the 'theoretical thickness', which is equal to the distance between two tangent planes which include the whole of the terminal averaged for all orientations of synaptic boutons (it is also called the cahper diameter). In order to determine T one has to know the average diameter (2r) of the synapt~c bouton profiles as well as the diameters of the smallest profiles (a) that would still be identified as a bouton of this kind (Fig. 1). From these data T can be calculate& on the basis of the formula T=2
]~ /'r e -
(-~a ) z "
The product of section surface of the electron micrographs (F) and the 'theoretical thickness' (T) gives the volume of the brain area sampled. By counting the synaptic terminals in electron micrographs (n) the synaptic density (SD) can be calculated from SD = n/F.T. (In view of the different sizes of the types of synaptic boutons within the same brain area, T has to be determined for each type separately and calculations have to be made accordingly - - Fig. 1.)
(3) Determination of axon terminal densities by measuring their volume fractions in electron micrographs. The volume fraction is one of the most important parameters required in quantitative histological analysis. The volume fraction of synaptic terminals can be estimated by the measurement of either linear or planar fractions of
415
_
4
_]i
IN
i
I
t i
I
J
>
II
I Jl .
I
I. . . .
I[11
.[11
i t
•
!
!
II
Fig. 1.
the synaptic profiles (using a planimeter or 'point counting method'). The average planar (surface) fraction determined on sections through a volume represents an estimate of the volume fractionL There is also an equivalence of linear and volume fractions lo. The systematic point count technique using a two-dimensional point grid should be the most efficient method of volume fraction analysisl,3-5,12. A 100-grid (10 × 10 lines)7, 9 drawn upon a clear plastic sheet was superimposed on the electron
416 mlcrograph, taking care to apply random positions. (The optimum density for the grid roughly corresponds to an average of one point per profile to be measured.) The number of crossing points that lay over the synaptic terminals was counted. Then the grid was repositioned 30 times randomly and the same count was made. The average number of points was then divided by 100 (because of the 100-grid) and the value obtained corresponded to the planar (surface) fraction of synaptlc terminals on the electron mlcrographs stud~ed. According to the rule of Delesse 2 the surface fraction of a given particle (Zf/F) is equivalent to the volume fraction of the particle (Zv/V). Furthermore, the volume fraction is equal to the product of the volume of a single particle and of its density (Zv/V ----- v.SD). The density can be calculated on the basis of the formula SD -: (Zf/F)/v. Since v is equal to ~/(Zf/n) 3, then SD = (f/F)/~/(Zf/n) a ( f = total surface areas of randomly measured synaptic terminals; F -~ total area of the electron micrographs; n = number of synaptic terminals in F area). For the application of this technique the section surface of synaptic terminals should be measured by aid of a planimeter. According to Rosiwa's rule (see ref. 8), the linear fraction of a particle profile is characteristic of the volume fraction of that particle. Axon terminal density can, therefore, also be determined on the basls of the formula SD--
~l/L (Zl/n) 3
(L = total lengths of lines tracted through on the electron micrographs; 1 ---- linear fractions of particle, i.e., total lengths of lines passing through the particle profiles; n = number of particle profiles crossed by the random lines on the micrographs). Since the synaptic terminals are usually irregular and do not resemble any simple geometrical shape, measuring in one dimension only results in a somewhat higher technical error than in the case of two-dimensional (surface determination by platometer) measurements. The surface fraction (SF) corresponding to Zf/F can be measured by the point counting method; by this means the measurement of section surface can be eliminated. By so doing, three simple measurements: (1) the area of electron micrographs, (ii) counting the synaptic terminals and (iii) using a 10 × 10 grid for surface fraction determination, are sufficient to determine the density of the synaptic terminals in a given brain region. The steps of this technique in practice are as follows. (1) Measurement of the area of the electron micrographs in sq./zm (using a correction factor for shrinkage). (2) Determination of the surface fraction by a 100-grid superimposed randomly 30 times per micrograph. (3) Counting the synaptic terminals. (4) Calculation of Z f ( = SF.F/100) and SD ( = SF/~/(Zf/n)3). Further simplifications are possible by direct measurement under the electron microscope. After determination of SF on electron micrographs it proved to be
417 sufficient to count synaptic terminals directly on the electron microscope screen when F is equal to the areas of the grids investigated. The two above simplified techniques give similar results, with acceptably low standard errors (Palkovits, Mezey, Hgtmori and Szent~gothai, in preparation). For example, the number of synaptic terminals and of synaptic terminals per cell in a hypothalamic nucleus could be determined both under normal and experimental conditions 6, showing the usefulness of this new quantitative approach to synaptic connectivity. The author is indebted to Professor J. Szent~gothai for his valuable advice and critical comments.
1 CHALKLEY,H. W., Methods for the quantitative morphologic analysis of tissue, J. nat. Cancer Inst., 4 (1943) 47-53. 2 DELESSE, A., Pour d6termine la composition des roches, Ann. des Mines, 13 (1848) 379-388. 3 GLAGOLEV,A. A., On the geometrical methods of quantitative mineralogic analysis of rocks, Trans. Inst. Econ. Min. (Moscow), 59 (1933) 162-194. 4 GLAGOLEV,A. A., Quantitative analysis with microscope by the point method, Engng. Min. J., 135 (1934) 399-410. 5 HILLIARD,J. E., AND CAHN, J. W., An evaluation of procedures in quantitative metallography for volume fraction analysis, Trans AIME, 221 (1961) 344. 6 L~R/~rcrn,Cs., ZAnORSZKY,L., MARTON,J., AND PALKOVITS,M., Studies on the supraoptic nuclei in the rat. I. Synaptie organization, Exp. Brain Res., 22 (1975) 509-523. 7 PALKOVrrs, M., Histologische Fiirbung und quantitative Methodik zum Nachweiss der Blutfiillung der endokrinen Organe, Microscopic, 17 (1962) 300-303. 8 PALKOVITS,M., MAGYAR,P., AND SZENT/~GOTHAI,J., Quantitative histological analysis of the cerebellar cortex in the rat. I. Number and arrangement in spaces of the Purkinje cells, Brain Research, 32 (1971) 1-13. 9 PALKOVITS,M., MAGYAR,P., AND Sz~wr/~GOTHAI,J., Quantitative histological analysis of the cerebellar cortex in the cat. II. Cell numbers and densities in the granular layer, Brain Research, 32 (1971) 15-30. 10 ROSIWAL,A., (~ber geometrische Gesteinsanalysen, l/erh, tier K.-K. geol. Reichanstalt, 5 0898) 143. 11 SALTYKOV,S. A., The determination of the size distribution of particles in an opaque material from a measurement of the size distribution of their sections. In H. ELIAS (Ed.), Stereology, Springer, Berlin, 1967, pp. 163-173. 12 UNDERWOOD, E. E., Quantitative Stereology, Addison-Wesley, Reading, Mass., 1970, 274 pp.
