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NSL 08008

Neuroscience Letters, 130 ( 1991) 112- l 16 © 1991 Elsevier Scientific Publishers Ireland Ltd. 0304-3940/91/$ 03.50

ADONIS 0304394091004923

A fractal analysis of pyramidal neurons in mammalian motor cortex

R. P o r t e r , S. G h o s h * , G . D a v i d L a n g e * * a n d T . G . S m i t h J r .***

Faculty of Medicine, Monash University, Clayton, Vic. (Australia)

(Received 18 April 1991; Revised version received 31 May 1991; Accepted 3 June 1991)

Key words. Motor cortex; Pyramidal neuron; Fractal dimension

Pyramidal neurons in the mammalian cerebral cortex can be described by a fractal dimension (Mandelbrot, 1982), which is an objective, quantita- tive measure of the complexity of their soma/dendritlc borders. In the cat, the fractal dimensions of lamina V cells, which include pyramidal tract neurons (PTN), indicate that these cells are more complex than other pyramidal neurons (PN) in the same region of motor cortex. The lamina V cells of the cat are also more complex than those in motor cortex of the monkey. Moreover, lamina III neurons in the monkey are more complex than monkey lamina V neurons. The fractal dimension of the intracortical axon collateral arborizations of the same pyramidal neurons indicated, in all cases, that the branching of these terminals is less complex than the branching of the dendrites of the same cells. In line with the observation that the fractal dimensions of some homologous cellular populations are different in different species, it is suggested that the fractal dimension and the degree of morphological complexity may relate to the requirement for the number of separable functions to be accommodated within one neuron. For example, as the size of the cortex and the number of neurons in a region increase, the opportunity exists within a given cortical zone, for individual functions to be segregated and for functional specialization to be accommodated with less morphological complexity of the individual neurons performing each of these functions.

The morpho log ica l classif icat ion o f neurons in the ver-

tebra te nervous system has been examined at a number

o f levels. Pe rhaps the mos t c o m m o n defini t ion is accord-

ing to gross ana tomica l locat ion, (dorsal roo t gangl ion

neurons) . Other classif icat ions are based on the connect i -

vity o f the neurons , ( sp ino- tha lamic t ract neurons).

Some are re la ted to a character is t ic shape o f the neu ron ' s

soma, (pyramida l neurons) . The names m a y indicate

phys io logica l funct ion, (spinal motoneurons) . The dis-

t inctive shapes or loca t ions o f the neurons m a y be used

and are of ten eponymous , e.g. Purkinje neurons o f the

cerebellum. M o r e general schemes have been p r o p o s e d

on the basis o f the overal l s t ruc tura l character is t ics or

gestal t o f soma-dendr i t i c b ranch ing pa t te rns [12].

M o s t quant i ta t ive analyses o f indiv idual neurons have

involved measurements o f the number , d iameters and

lengths o f b ranches o f dendr i t ic trees. Such measure-

ments have been useful in relat ing cer ta in physiologica l

*Present address: Department of Physiology and Pharmacology, University of Queensland, Qld. 4072, Australia. **Present address: Instrumentation and Computer Section NINDS, National Institutes of Health, Bethesda, MD 20892, U.S.A. ***Present address: Laboratory of Neurophysiology, National Insti- tutes of Health, Bldg. 36, Room 2C 02, Bethesda, MD 20892, U.S.A. Correspondence: R. Porter, Faculty of Medicine, Monash University, Clayton, Vic. 3168, Australia.

funct ions, e.g. the spread o f m e m b r a n e poten t ia l changes

th rough dendr i t ic trees, to morpho log ica l s t ructure [6,

13]. There are, however , cer ta in morpho log ica l charac-

teristics o f indiv idual neurons that are general ly recog-

nised as i m p o r t a n t but are charac te r i sed with a relat ive

'more ' or ' less ' o f the a t t r ibute . One such a t t r ibu te is a

cell 's morpho log ica l complexi ty , with, for example , a

Purkinje cell being recognised as 'more complex ' than,

say, a ret inal gangl ion cell.

Recently, however , the concepts o f M a n d e l b r o t ' s frac-

tal geomet ry have afforded a means o f quant i ta t ive ly

measur ing the complexi ty o f the borders or geometr ica l

outl ines o f neurons and glia in a comple te ly object ive

and unbiased manne r [9, 16, 17]. The 'measure ' is the

fractal d imens ion (D) o f the cell 's border , which in-

creases in value with increase in complexi ty . Here we re-

po r t on such measurements made on camera lucida re-

cons t ruc t ions o f cat and m o n k e y cor t ical p y r a m i d a l neu-

rons in a rea 4 (moto r cortex), some identif ied as py rami -

da l t rac t neurons (PTN) by sending axons into the basis

peduncul i or the py ramida l tract . W e find stat is t ical ly

significant differences among the fractal d imens ions o f

several different neurona l types, which conceivably may

have relevance to their phys io logica l function.

As to the in te rp re ta t ion o f D as a measure o f complex-

ity, it is the f rac t ional pa r t o f D tha t is re levant for com-

par i son purposes . The integer (1 .) denotes a d imens iona-

lity for the object that lies between a straight line and a flat plane (2. means a dimensionality between a plane and a volume). The fractional part is obtained from the slope of a line in a log-log plot and is a unitless ratio of two log values (Fig. 1). It is this ratio that expresses the relative complexity of two objects. An increase in the fractional part of D from 0.1 to 0.2 represents a doubling in complexity, and the range of 1.39 to 1.63 found in cat motor cortex cells in this study represents an approxima- tely 1.5 to 2-fold difference in complexity.

The experimental procedures for the initial experi- ments on the live animals have been reported completely elsewhere [3-5]. Basically, anaesthetized cats and mon- keys were studied, intracellular records were made from motor cortex neurons and these cells were then injected with horseradish peroxidase. Subsequently, camera lucida drawings of the completely filled cells were made from superimposed serial sections of fixed tissue on mic- roscope slides. This generated two dimensional illustra- tions of the outlines of the total branched surface of these cells. Even in the reconstructions there is little 'loss' of definition caused by the superimposition of dendritic profiles. The method of calculating the fractal dimension from an image of a cell's border or outline has also been reported extensively elsewhere [2, 16, 17].

Thirty-one cells from area 4 of the cat, and thirteen cells from area 4 of the monkey were analysed for their fractal dimensions. Thirteen cells from the cat and three from the monkey were identified as pyramidal tract neu- rons (PTN); the others were classified as pyramidal neu- rons (PN) by the shape of their somata, the arrangement and distribution of their apical dendrites and the exit of their axon from the cortex. They were assigned to the cortical laminae (II-VI) occupied by their somata. All PTN neurons were from lamina V. The fractal dimen- sion results of individual cells or parts of cells of each cell group were analysed for statistically significant differ- ences between them by employing analysis of variance and the Scheffe post hoc test (SuperAnova, Abacus Con- cepts, Inc., Berkeley, CA, U.S.A.). We have chosen a 95% confidence level ( P < 0.05) as our criterion for signif- icant difference.

Fig. 1. illustrates both the camera lucida projection of the most complex fast PTN in the cat (C20) and the gra- phical representation of its fractal dimension (1.63). As can be seen in Tables I and II the fractal dimensions (D) for the entire soma-dendritic arbor of individual cat py- ramidal neurons ranged from 1.39 to 1.63, while those for cells from the monkey were from 1.34 to 1.56. Taken as a whole, and using a Student's t-test, for example, the cat and monkey data are not significantly different. However, when an analysis of variance allowing for clas- sification according to species and lamina (two-way

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1000

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DIL : -0.63

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L E N G T H O F M E A S U R I N G E L E M E N T

Fig. 1. Upper panel: silhouette from camera lucida projection drawing of a cat motor cortex PTN neuron (C20), with cortical laminae indi- cated (I-VI). Lower panel: Richardson plot of the log of the perimeter of the cell's border vs. the log of the length of the measuring element (e.g., ruler, caliper), using the DILATION method [2, 17]. The fractal dimension is calculated from the measured slope (S; Dil: -0.63),

D = I - S = I-(-0.63) = 1.63

ANOVA) is employed, then significant differences are revealed. Post hoc analysis shows differences between homologous laminae in the two species. There are also differences among the various categories of cat neurons. The most interesting results are obtained when the fast and slow PTNs, which are not themselves significantly different, are lumped and are included with the other lamina V cells, rather than being considered in separate categories. All lamina V neurons project to relatively dis- tant targets and, in this sense, may have some common characteristics. Post hoc analysis indicates that the neu- rons of the various laminae fall into overlapping groups along a continuum of decreasing complexity. Although the laminar groups were not completely separable statis- tically in the cat, the rank order of the means of each group with decreasing complexity was: lamina V >

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lamina III > lamina II > lamina VI. It should be noted, however, that there were fewer than five cells in the sam- ples of laminae III and VI. However, in the monkey, D was statistically significantly greater in lamina III (mean 1.53) than in lamina V (mean 1.46). These results are indicated in Table I for cats and Table II for monkeys.

A correlation matrix examined the relationship between D and thirteen conventional physiological and anatomical parameters, e.g. conduction velocity of the axon, somatic diameters, number of apical dendrites, dendritic extent and orientation, density of dendritic spines and spread of intracortical axon collaterals. (These measures are reported in Ghosh et al. [3] and re- constructions of many of the cells are illustrated in this and associated publications.) No correlations greater than 0.65 were found. This would indicate that no more than 42% of the variance in any of these measures can be explained by D. Such analyses do not rule out a possi- ble non-monotonic or other nonlinear relationship of D with these other measures.

In order to test for the presence of the defining proper- ty of a fractal object, namely, self-similarity, the fractal dimension of a portion, say a quarter to a half, of the dendritic tree, was calculated for each of the cat neurons (Table I). Self-similarity means that the object looks

qualitatively the same, irrespective of scale or magnifica- tion. The average D of the entire cell was 1.52, while the average D of the part-cell images was 1.54 and the means were not significantly different (P<0.24). On the other hand, the D of the intracortical collateral arborization of the axon of each cell in the cat (mean of 1.39), when available (8 cells), was statistically different from its cor- responding whole cell - soma and dendrites (P < 0.0001 ). Moreover, as is indicated in both Table I and Table II, in each cell's case, for both the lamina III and the lamina V neurons of the monkey as well as the cat, the D of the intracortical axonal arborization was lower (less com- plex) than that of the soma-dendritic tree.

Perhaps the most interesting result came from a com- parison of the cells in laminae III and V of the cat and monkey. Somewhat surprisingly, the cat laminar V cells had a significantly larger D (mean of 1.55) than those of lamina V of the monkey (mean of 1.48). Moreover, the lamina III neurons of the monkey (mean 1.53) were sig- nificantly more complex than those of monkey lamina V (mean 1.48).

The fractal dimension of an object is generally taken to measure, quantitatively, the complexity of whatever aspect of the object is being measured. Here the borders of the branched profiles of cat and monkey motor cortex

TABLE I

F R A C T A L DIMENSIONS OF CAT P Y R A M I D A L N E U R O N S AREA 4

S.D., soma dendritic boundary; PART, fractal dimension for part of cell; I.A.C,, intracortical axonal arborizations; C.V., conduction velocity of

axon.

Neuron type ~ D Neuron type D

S.D. Part I.A.C. C.V. m/s S.D. Part I.A.C.

Fast PTN 1.60 1.66 - 35.7 Lamina II 1.48 1.48

Fast PTN ~ 1.63 1.66 1.59 27.0 Lamina II 1.49 1.52 Fast PTN b 1.54 1.56 1.36 27.5 Lamina IIe 1.48 1.50

Slow PTN 1.55 1.62 5.7 Lamina II f 1.41 1.45 Slow PTN 1.49 1.53 11.0 Lamina II 1.57 1.51

Slow PTN c 1.60 1.64 1.49 11.0 Lamina IIg 1.47 1.43 Slow PTN 1.60 1.66 8.8 Lamina III h 1.52 1.58

Slow PTN 1.53 1.55 5.0 Lamina III 1.48 1.49

Slow PTN 1.51 1.59 1.29 11.4 Lamina III 1.55 1.50 Slow PTN 1.49 1.43 1.37 19.2 Lamina III 1.50 1.50 Slow PTN 1.54 1.51 8.3 Lamina VP ! .47 1.49 Slow PTN 1.53 1.56 19.2 Lamina VIi 1.41 1.50 Slow PTN d 1.57 1.60 1.44 19.2 Lamina VI k 1.39 1.47 Lamina V 1.55 1.63

Lamina V 1.59 1.53 1.33 Lamina V 1.52 1.51 - Lamina V 1.44 1.48 1.26

Lamina V 1.52 1.48

IA number of these neurons have been illustrated in a previous paper [3] and may be identified in that paper as follows: (a) Fig. 9B; (b) Fig. 9A; (c) Fig. 7B; (d) Fig. 11; (e) Fig. 3B; (f) Fig. 3C; (g) Fig. 3E; (h) Fig. 5A; (i) Fig. 16A; (j) Fig. 16B; (k) Fig. 16C.

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TABLE II

F R A C T A L DIMENSIONS (D) OF M O N K E Y P Y R A M I D A L N E U R O N S A R E A 4

S.D., soma/dendritic complex; I.A.C., intracortical axon collateral arborization.

Lamina III D Lamina V

S.D. I.A.C.

D

S.D. I.A.C.

1 1.52 1.32 1 (Fast) 1.49

2 1.54 1.51 2 (Fast) 1.51

3 1.53 1.46 3 (Slow) 1.46 4 1.51 4 1.48

5 1.56 5 1.34 6 1.49 6 1.47

7 1.45

Mean 1.53 1.43 S.D. 0.02 0.09

1.39 1.27

1.30 1.42

1.46 1.34

0.06 0.07

neurons were examined. For these outlines, as for the edge traces of other complex geometric forms, the higher the D, the more complex the object.

With respect to what aspects of cellular morphology the D measures, we have found three characteristics to be important [16, 17]. The first is the degree of branching of the dendritic tree, with the greater branching giving a higher D. Second, is the ruggedness of the border of the cell, with a more rugged, jagged or uneven border leading to a higher D. And, third, is the degree to which the border fills the plane of the projection of the two- dimensional image. These categories are not indepen- dent. The distinction between branching and ruggedness is, to some extent, a question of scale, with increases in either, being more space filling [16, 17]. In the neurons studied here, the first aspect, the degree of branching, appeared to be the most important.

The importance of dendritic branching to the func- tioning of neurons is likely to arise because the number and modes of interaction of the inputs to a cell are criti- cal to the cell's output behaviour. It seems quite likely that degree of branching and the number and distribu- tion of inputs are correlated and that dendritic branch- ing affords a medium for complex interactions among these inputs. Specialisations, such as dendritic spines which accommodate synaptic inputs, would be expected further to increase the measure of complexity of the cell's border and also the degree of input/output interactions. Separation of different inputs on different dendritic branches may provide for different input/output charac- teristics for separate aspects of a neuron's functions. A high D in cat lamina V neurons, which includes the pyra- midal tract neurons (PTNs), may therefore signify a high degree of computational complexity in these cells. This level of complexity is revealed in other studies of such

cells by Deschenes et al. [1]. Here, a more complex geo- metry may be required for computationally greater re- quirements of neurons that control many other elements far removed from the somatic location of the cell, than for the cells that have only local intracortical connec- tions or synapses with relatively localised nearby targets in the thalamus or basal ganglia. The opposite may be true of monkey lamina III neurons, which must be pre- sumed to be ipsilateral cortico-cortical or cortico-callo- sal neurons [7], and which are more complex than the lamina V neurons, including fast and slow pyramidal tract neurons, whose axons innervate distant targets, including spinal motoneurons. We still need to under- stand the extent and functional implications of the terri- tory innervated by an individual pyramidal neuron in either lamina III or lamina V. For PTN in the cat and the monkey, it may be relevant that, in general, the more complex cells of the cat appear to have a wider spread of their spinal terminal arborizations than those of the monkey [8, 14, 15]. However, for individual cells, direct correlations between complexity of soma/dendritic mor- phology and extent of terminal arborization cannot be made.

The findings that cat and monkey have reciprocal de- grees of the complexity in laminae III and V are interest- ing and challenge explanation. In another study, of Bergman glial cells of the cerebellum of rodents, mon- keys and man (A. Reichenbach et al., personal commu- nication), it was found that D's were significantly differ- ent in the three species, in the order rodent > monkey > man. If it were known, for example, that D's for cere- bellar glial cells were inversely related to the motor repertoire of the various species, it might be an impor- tant development. We do not, however, have a well- defined measure of motor repertoire.

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The fact that no large correlations were found between D and conventional physiological and anatomi- cal measures indicates that measuring the D of a cell can- not substitute for measuring other aspects of function and form directly. For example, Spain et al. [18] have recently identified different membrane channel proper- ties in different pyramidal cells of lamina V of in vitro slices of cat's motor cortex.

The demonstration of self-similarity in motor cortex neurons is consistent with our findings in other neuronal and glial studies [16, 17] and indicates that the borders of these neurons may be considered to be fractal objects. Here, as elsewhere [16, 17], the borders were fractal over a range of scale of from 1.5 to 2 decades, a range com- mon for biological objects [11]. While the entire soma- dendritic complex demonstrated self-similarity, the same was not true of the intracortical axon terminals of their parent cells, where the terminals were considerably less complex. While 'true' fractals are self-similar everywhere in their structure [10] it is common in 'natural' fractal objects to find that D changes with either a large change of scale [11] or with different parts of the same object, e.g., in the leaves of a tree compared to the branches of the same tree.

One can raise the legitimate question as to the validity of measuring D from two-dimensional projections of what are, in fact, three-dimensional structures. In the field of fractal geometry, where the rules are still being developed, this is a matter of some controversy. Mandel- brot, the father of fractal geometry, says that such esti- mates of D can be considered a rough approximation of the 'true' complexity and are more likely to underesti- mate the latter [9]. The methods of reconstruction of the images of the cells from serial histological sections will certainly result in distortions of the length of processes which traverse the plane of the sections. However, the demonstration, provided here, of self-similarity, revealed by the same measure of the fractal dimension for parts of the dendritic arbor as for the whole cell, indicates that such non-uniform representations of lengths of dendrites in different planes have had little influence on D. Meth- ods for estimating D in higher dimensions are under de- velopment, but until we have those methods and the data in the proper format, we are limited to what we have done here. Such measures as we have performed may, nonetheless, be useful quantifiers of cellular mor- phology which could illuminate relationships between

cellular structure and function which are not revealed by other measures of cellular geometry and tell us some- thing about the importance and function of homologous cells across species.

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