Fractal Theoryin Neuroscience

8/18/2019 Fractal Theoryin Neuroscience

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F E R N ÁND EZ AND J ELINE K310

complexity of the borders of a neuron (6– 12) a nd to ma thema tica l fra cta ls, una voidably finite a nd l im-

ited in scale by their own nature. Thus natural pat-measure how completely the branches of a neuronterns display statistical self-similarity only betweenfill its d endritic field (13, 14) it should be noted th a tan upper and lower bound. Limitations are also im-th e fra cta l dimension is only a d escriptive pa ra meter,posed by recording and imaging techniques. The finall ike the dendritic field a rea or the size of the soma,va lue of th e a mount of deta il or irregula rity a t differ-and does not necessarily imply any biological process

ent scales associated with a natural object can thennor mechanism involved in their development. be determined by the use of fractal analysis.I n t h i s p a p er w e em p ha s i ze t h a t f r a ct a l a n a l y s is

is a useful tool for improving image description and

for ca tegorizing images representing morphologi-FRACTAL DIMENSIONScally complex objects ba sed on th e valu e of th e fra cta l

dimension. The fra cta l dimension for t his purpose isAn importa nt pa ra meter in fracta l ana lysis of bio-th erefore not intend ed to indica te wh ether th e ima ge

logical structures is the fractional or fractal dimen-is a fra cta l object. We also review some of the met hod-sion (D ) , which increases in value with increasingologies available for calculating the fractal dimen-structural complexity and describes the “fractured”sion, the a dva nta ges an d problems of fracta l geome-na ture of objects in na ture (10). D is called fractalt r y, a n d s om e of it s cu rr en t a p plica t i on s inbecause it usually is not an integer. It is called dimen-neuroscience.sion beca use it provides a mea sur e of how completely

a n object fills spa ce. When D ta kes a n integer value,i t is equal to the standard Euclidean dimension for

THEORETICAL CONSIDERATIONS OFwhich an ideal point has a dimension of 0, an ideal

FRACTAL GEOMETRY AND NATURALLY l ine ha s a dimension of 1, a n ideal plane ha s a dimen-OCCURRING FRACTALS sion of 2, a nd a perfectly solid volume ha s a dimen-

sion of 3.

An object is sa id to be fra cta l if cert a in criteria such

a s th e object being self-similar or scale inva ria nt a re

m et . Fi gur e 1 shows an a ppr oxim a t i on of a n i deal /

theoretical fractal with a fractal dimension of 1.26

t hat was desc r i bed by t he Swedi sh m at hem at i c i an,

Helge von Koch in 1904. Computer-generated frac-

ta ls, such a s th e Koch curve, are sometimes termed

prefra cta ls since th ey a re l imited resolution images

a nd t herefore do not rea lize th e deta il implicit in th e

complete mathematical formulation (15). The form

of t h is ob ject is com plex s in ce a n y ch a n g e in

ma gnifica tion/sca le w ill show m ore det a il to the reso-

lution l imit as the magnification is increased. This

addi t i o n o f det ai l r esul t s i n an i deal fr ac t al o bj ec tha ving a n infinite bounda ry lengt h (16, 17). Ma ndel-

brot has shown that the boundary length of a fractal

object can be mathematically expressed as a power

l aw. Thus fr ac t al s ar e al ways desc r i bed by po wer

functions since homogeneous power laws lack natu-FIG. 1. Construction of the Koch curve with a D of 1.26. Ther a l sca l es ; t ha t i s, t hey do not have a cha r ac t er ist i csequential construction of this f racta l begins w ith a stra ight l ine

unit of length, t ime, or ma ss (16). Ma ny pa tt erns in (A). Then the middle third is raised to produce an equilateraltr iangle (B). Raising equilateral tr iangles from the middle thirdbiology display a limited self-similarity or approxi-of each of the l ine segments in t he object produces t he ima ge inmate self-similari ty. They are generally held to be(C). At higher stages of construction (D, E, and so on) the f ine

statistically self-similar. Further i t should be kept detai l of the complex curve would be lost due to the resolutionlimits of the printing process.in mind tha t a l l na tura l objects a re, in contra st w ith


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U SE OF FRACTAL THE ORY IN NE U ROSCI ENC E 311

Since, for insta nce, nerve cells seen in tw o dimen- from digitized photogra phs, dra wings, or other ex-

perimental data obtained from presentations of natu-si o ns ar e no t s t r ai ght l i nes, and t hey do no t c o m -pletely cover the two-dimensional area, their D va l- ra l objects, a very good estima te of D ca n be a chieved

by di f fer ent fr ac t al anal ysi s m et ho ds ( Fi g. 3 ) . Al lues fall between 1 and 2. For example, neurons with

low D va lues, sa y 1.2, would ha ve rela tively few den- methods rely on the relationship betw een a measur-ing device and the object’s spatial distribution.drit ic branches a nd cover the tw o-dimensiona l a rea

less completely than neurons with higher D valueslike 1.45 (Fig. 2).It is not ea sy t o give a precise definition of a fr a cta l Hausdorff Dimension

(15), and there are in the l i terature many differentThe origina l intention of Ha usdorff w a s t o define

t ypes o f fr ac t al di m ensi o ns so t hat even r esear c ha parameter that was independent of the resolution

m at hem at i c i ans ar e no t agr eed o n t hei r nam es o ro f m e a s u r e m e n t a n d w a s a p p l i c a b l e t o a l l s h a p e s

equ iva lence (18, 19). Ta ble 1 lists some of t he m ost(16). It is ca lculat ed by covering a n object wit h count -

i m po r t ant fr ac t al di m ensi o ns wi t h t hei r syno nym sa b l e s p h e r e s w h o s e r a d i i a r e n o t g r e a t e r t h a n t h e

a nd context. S ince ma ny of these fra cta l dimensionsima ge but decreas e to zero. Mea suring a ny self-simi-

a r e u s e d m a i n l y i n p u r e m a t h e m a t i c s o r a p p l i e dlar set w ith spheres of integer dimension, the volume

physics, w e consider only th ose tha t a re potentia l lygoes either to zero or infinity. Hausdorff (32) sug-

useful in neuroscience. Various other aspects of frac- g e s t e d t h a t t h e v o l u m e o r m e a s u r e o f t h e s p h e r et a l a n a l ys is a n d D are discussed formally by othershould be e D wher e e equals the resolution of mea-

authors (15, 16, 18, 20– 31).surement . The D -dimensional Hausdorff measure ofa n ima ge is finite only w hen D (the dimension va lue)equal s t he di m ensi o n o f t he i m age. Fo r t he Ko c h

METHODS FOR DETERMINING FRACTALcurve shown in Fig. 1, this would be when D log

DIMENSIONS 4/log 3 1.26. This definition of dimension w a s ex-t ended and put i nt o a m o r e syst em at i c fr am ewo r k

by Besicovitch (33).Although the ma thema tica lly rigorous determina -

tion of D is impossible for a fra cta l point set obta ined Ca lculat ing the Ha usdorff dimension is generally

FIG. 2. Examples of di f ferent cat ganglion cells, drawn from throughout the retina, with their associated box counting dimensions.The beta cell (on the right) has a more profuse branching pattern than the gamma cell (on the left) or the alpha cell (in the center). The

fracta l dimensions a re suf f iciently dif ferent t o suggest t ha t t hey represent dist inct ga nglion cell types. A straight l ine is dra wn from t hecell si lhouette t o i ts va lue on the D axis.


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TABLE 1

Some of the Most Widely Used Fractal Dimensions with their Synonyms and Contexts

D im en sion S y m bol S y n on y ms C ont ext Refer en ce

F r a c t a l D G e n e r ic t e r m f or f r a c t a l d i m en s i on M a n d e lb r ot , 1 98 3 (5 )Hausdorf f D H H a u s dor ff – B e si cov it ch Wid ely u sed i n p ur e m a t h em a t ics , H a u s dor ff, 1919 (32)

dim en sion but it ca nnot be st rict ly a pplied B esicovit ch , 1935 (33)t o n a t u r a l ob je ct s d u e t o i t s f in i t e M a n d e lb r ot , 1 98 3 (5 )

range of f ractal structureMinkowski– D M Min kow sky sa usa ge E a sier t o eva lua te t ha n D H ; Ma ndelbrot , 1983 (5)

B ouliga n d dimen sion, d ila t ion usua lly is grea t er t h a n or S m it h et al., 1989 (10)dimen sion eq ua l t o t h e H a usdorff S ch r oeder, 1991 (29)

dimensionCalliper D C R ich a r ds on d im en s ion , d iv id er O ft en u se d in ca l cu la t i ng t h e R ich a r ds on , 1961

dim en sion, com pa ss fra ct a l d im en sion of out lin es Ma nd elbr ot , 1983 (5)dim en sion, perim et er Ta ka y a su, 1990 (16)dim en sion S m it h et al., 1989 (10)

B ox D B C a pa cit y d im en sion , b ox U s ed for ca l cu la t in g t h e fr a ct a l M a nd elb rot , 1983 (5)coun t in g dim en sion, K olm ogor ov dimension s of m a ny biologica l Ta tsum i et al., 1989 (45)

dim en sion, m osa ic st ruct ures in 2D a n d 3D S m it h et al., 1989 (10)

a m a lga m a t ion dimen sion, Ta ka y a su, 1990 (16)D (0) in m ult ifra ct a l a n a ly sis P eit gen et al., 1992 (48)

Ca serta et a l . , 1995 (7)M a s s D MR M a s s f ra c ta l d im en si on , m a s s U s ed i n t h e con t ex t of clu st er s a n d C a s er t a et al., 1990 (6)

radius dimension, D (2) in net works; ca n a lso be a pplied J elin ek a nd F ern an dez,mult ifra ct al a na ly sis t o surfa ces a nd biologica l 1998 (59)

structures

very difficult and a more practical parameter of D ,the capacity dimension, was introduced by Kolmo-

gorov (34, 35). The difference from the Hausdorff–Besicovitch dimension is that the set is now covered

with spheres of identical radius (16). The “capacity

dimension” ha s become th e funda ment a l definition offra cta l dimension in th e minds of ma ny. The capa city

dimension is relat ed to the box counting a nd m a ss–

ra dius methods th a t a re its a pplied, tw o-dimensiona l

embodiment and described below.

Calliper Method

An algorithm based on the Hausdorff dimensionFIG. 3. Some methods used for determina tion of f ractal dimen- is the ca lliper dimension (a lso known a s th e compa ss,sions of a Koch triad ic islan d wit h a D 1.50. (A)Calliper method.

divider, or ya rdst ick dimension). To det ermine D , aThis method is ba sed on counting t he number of steps tha t giveruler of decreasing size r i s u s e d t o m e a s u r e t h ea polygona l representa tion of an arbitra ry object using dif ferent

calliper span s.(B ) B ox count ing method. (C) Dila tion method. After bounda ry or coa stline of an ima ge. The length of th edilat ion w ith a disk kernel diameter of 16 pixels. Note loss of coa stl ine then equa ls the size of the ruler t imes theb or d e r d e t a i l s h o w n i n (A) a n d (B ) (D ) M a s s m e t h od e xa m p l e

number of steps r ha s t a ken t o t r ac e t he coast . Fi g-af ter application of six groups of concentr ic disks, with variousures 3A and 4 show examples of this method. Oned i a m e t e r s a n d c e n t e r e d o n t h e b o r d e r o f t h e K o c h i s l a n d . A l l

centers lie with in th e ra dius of gyra tion (la rge circle). See text for f i nds t hat t he bo undar y l engt h i s a func t i o n o f t hemore deta ils. Reprinted, w ith permission, from T. G . Sm ith, J r . , span of the call iper employed in the measurement.a nd G . D. La nge (1998) i n Fra cta ls in Biology a nd Medicine (Non-

Tha t is, the lengt h does not converge to a st a ble va luen e n m a ch e r, T. F. , L os a , G . A. , a n d We ib el , E . R . , E d s . ), B i r -hauser , Basel . but keeps increasing a s the ca lliper span decreases.


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U S E O F FRAC TAL TH E O RY IN NEU RO SC IE NC E 313

FIG. 4. Ca l li per method for a scertaing t he boundary length of a n ima ge. (A) Measuring t he length of the c oast l ine of the Austral ian

continent. (B) Graph of resulting log– log plot.


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I f t he l engt h of t he boundar y (coa st l ine) ver sus t he i s a c onst a nt . I n pr a ct i c e, t he i m age i s digi t i zed by

pixels having a given scaling factor r (F ig. 6A). Th ecalliper lengt h is plott ed on a log– log sca le, t he point sw i ll f a ll on a s t ra i g ht lin e b et w e en a n u p per a n d n u m be r N (r ) o f p i x e l s c o n s t i t u t i n g t h e i m a g e i s

cou n t e d a n d a t t h e s a m e t i m e t h e s ca l in g f a ct o r r lower bound with negative slope. The slope, S , orgr adi ent i s r el at ed t o t he fr ac t al di m ension by D of pixels is recorded. Mea sur ement of N (r ) a t l ar ger

scaling fa ctor (lower resolutions) is usua lly done by1 S (5). The calliper method has previously been

used t o cha r ac t er ize neur ons (10, 36). One m a jor zoo m ing down t he im a ge usi ng t he m em or y fr am ewi t h fo ur adj ac ent pi xel s m aki ng o ne pi xel ( Fi gs.dra w ba ck of the ca lliper method, is tha t ima ges com-

posed of m or e t ha n one si m pl e per i m et er ca nnot be 6 B , 6 C ). A M a c int o sh pr ogr a m for ca l cul at i ng D u s i n g t h i s m e t h o d c a n b e f o u n d a t t h e f o l l o w i n gprocessed a ccura tely (e.g., noncontiguous st ructures

or clos ed loops w it h in a s t ru ct u re). U R L : h t t p://pla n t ecoh os t .h a r va r d .ed u/g mb WWW/

APPL.ht m l .The following m eth ods ca n be used for noncont igu-

ous st r uc t ur es a s w ell a s for 2 D a nd 3 D i m a ges.

Box Counting Method Minkowski–Bouligand Dimension

To estim a te D , the Euclidean spa ce conta ining the The Minkow ski– B ouliga nd dimension is different

from the Hausdorff dimension (18). The method isimage is divided into a grid of boxes of size r , w i t hthe initial box size being the size of the image. r is i llustra ted in Fig. 3C. A circle is sw ept continuouslyal o ng t he l i ne and t he ar ea t hat i s c o ver ed, c al l edth en ma de progressively sma ller a nd th e correspond-

ing nu mber of nonempty boxes, N (r ), is counted (Fig. the Minkow ski “sa usa ge,” is determined. This valueis then plotted as a function of the circle diameter,3B). The logarithm of N (r ) versus t he loga rithm of r

gives a l ine whose gradient corresponds to D . The a nd t he sl ope (on t he usua l l og– l og pl ot ) gives t hedimension. The important difference between thissequence of box sizes for grids is usua lly reduced by

a fac t or of 1/2 fr om one gr i d t o t he next . Fi gur e 5 a nd t he ca l li per m et hod i s t ha t t he c ir cl e is m o ved

so that its center lies on every point of the line. Forsho ws an exam pl e o f t hi s c al c ul at i o n fo r a r et i nal

gangl i on cell . A m ac r o for t hi s m et hod c an be ob- a sm oot h, Euc li dea n cur ve, t he r esult wi l l be t he

l engt h o f t he c ur ve. Fo r a fr ac t al c ur ve t he l engt htained from the National Insti tutes of Health (NIH)

a t ftp://codon.nih.gov/pub/nih-ima ge/user-ma cros/ w ill continue t o increa se a s t he ra dius of th e cir-cles decrea ses.box count m ac r o. t xt , for use wi t h NI H I m age i m -

a ge processing softw a re.

Many research reports using this scheme to ana-l yze neur o n st r uc t ur es ar e fo und i n t he l i t er at ur e

Pixel Dilation Method(10, 12, 37– 42). The box coun tin g met hod a pplies toa ny st r uct ur e i n t he pla ne a nd c an be ada pt ed for The pixel di la t i on m et hod i s based on t he M inkow-

ski– B ouliga nd dimen sion, (5, 29). A common form ofst ruct ur es in t hr ee-dimen siona l spa ce (7, 10, 43, 44).This is equivalent to the “grid” method described by this a lgorithm, a s devised by Flook (50), ha s been

implemented by S mith et al. (10) and others (11, 36,Sm i t h et al. (10).A m et ho d si m i la r t o t he box c ount i ng t ec hni que 51 – 56). The pixel di la t i on m et hod, as r epor t ed by

the above articles, replaces each pixel of the borderis th e grid int ercept m ethod (45, 46). Note tha t t het e r m s b ox cou n t i n g m et h o d a n d g r i d i n t er c ep t by a circle w hose dia meter ra nges from 3 to 61 pixels(Fig. 7). This is done by application of a convolutionmethod r e f e r g e n e r a l l y t o t w o d i f f e r e n t m e t h o d s

but a r e used i nt er c hangea bl y i n t he l it er a t ur e (1 6, pr ocedur e whi ch i s par t of t he i m a ge ana l ysis pr o-gr am (di la t i on m a cr o fr om NI H). Thi s f il t er s out47– 49). The grid int ercept meth od relies on pro-

gr essi vel y coa r seni ng t he i m age r epr esent a t i o n (by st r uct ur es sm al ler t han t he cur r ent diam et er of t he

circle. The length of the border for each respectivepi xel s ha vi ng di f fer ent sca l i ng fa ct o r s) a nd c ount -

i n g t h e n u m b e r of p ix el s i n t er s e ct i n g a p or t i on o f d ia m e t er i s d et e r min ed b y t h e a r e a of t h e o ut l in edivided by the diameter. The fractal dimension isth e ima ge (Fig. 6). D i s t hen c al c ula t ed by f i t t i ng a

l i near r egr essi o n t o t he fo l l o wi ng equat i o n: l o g(r ) then estima ted from th e slope of the log– log plot ofl engt h agai nst di am et er . The NI H I m age pr o gr am D * log(n ) K , w h e re r resolution of image

(number of pixels per unit length), n

t he num ber a nd it s m any m a cr os can be fet ched i n a num ber ofw a ys. These a re deta iled a t ht tp://rsb.info.nih.gov/of pixels intersecting a portion of the image, and K


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nih-image/down loa d.html. There a re var ious ver- fal ls outside. This ca n happen with a known fra cta l

sions for various Macintosh computers. This particu- such as a Koch snow flake.l ar m a cr o (fr ac t al di la t i on. t xt ) and ot her user c on- F or s t r ict l y s el f-s im il a r m a t h e m a t i ca l f r a ct a l st ribu t ed ma cros a re foun d a t ft p://codon.n ih.g ov/pub/ such as th e Koch curve, a l l a ppropriat e fra cta l ana ly-n ih -im a ge/us er-ma cros/. sis methods a pproa ch t he sa me l imit , th e Ha usdorff

dimension (29, 57). Having decided which analysis

method to use, there ar e tw o further considera tions.Mass–Radius MethodOne is relat ed to how image presenta tion ma y influ-

The ma ss– ra dius dimension is defined by th e rela -ence the possible scaling relationship of the image

tionship betw een t he si tes of a n ima ge found w ithina n d t h e a s s o c i a t e d e s t i m a t e d D , a n d t h e o t h e r i sa spher e o r c i r c l e o f a c er t ai n r adi us c o ver i ng t her elat ed t o est i m at i ng D from the data points.image. The sites may be pixels obtained from box

c o unt i ng, s t eps o f a r ando m wal k, m o no m er s i n apolymer chain, a dsorption sites on a surfa ce, primar y

IMAGE PREPARATION AND DETERMINATIONpar t i c l es o f a c o l l o i dal aggr egat e, et c . U sual l y t hequa ntity of interest is the area of the ima ge, M , t h a t OF D i ncr ea s e s w i t h t h e i ncr ea s e i n t h e r a d i us r (see

Fig. 3D). Digit ized ima ges ca n be presented a s bina ry, skele-To implement t his m ethod for the a na lysis of 2Dtonized or border-only ima ges. When a na lyzing neu-i m ages, a c i r c l e o r spher e o f r adi us r is laid overrons, the cell body a nd/or the a xon ma y a lso be re-the image. The method first computes the center ofm oved fr om bina r y or out l ine r epr esent at i ons ofgr avi t y a nd t hen t he r adi us of gyr at i on. To l essenneurons. The choice of format is related to the space-com put at i on t i m e a fr ac t ion, sa y 0.6, o f t he r a di us

of gyra tion, can be used a nd every point w ithin th is f il li ng at t r i but es of t he im a ge and t he a t t r i but es oflimit is then chosen as a local origin and the cluster the image one deems to be importa nt. For insta nce,mass (number of pixels occupied) within a distance Man delbrot (5)st a ted tha t an object tha t fil ls a planer of th is loca l origin calcula ted. All possible choices completely has a dimension value of 2. With neuronsof local origin are averaged and the average cluster speci fi cal l y, t he cell body i nt er ior an d t ha t of t he

m a s s M (r ) is obtained. The double logarithmic plot dendrites do fil l a pla ne completely a nd hence ha veof M (r ) agai nst r gives a quantitative value for D . Aa D of 2. Therefore when calculating the D using

mult ipla tform version for computing the ma ss fra cta lcomplete binary images of neurons there may be a

dimension is a va ilable from ht tp://life.csu.edu.a u/space-fi l l ing effect that can lead to a higher D or afra ctop/a nd discussed by J ones a nd J elinek in thisD of 2, depending on the relationship between theissue. This version, named Fractop, has the addedinterna l a rea a nd t he contour. P revious results (58)a dvant a ge o f pr ovidi ng a choi ce fo r t he n um ber ofha ve demonstra ted no significa nt difference betw eenc e n t e r s a n d t h e f r a c t i o n o f t h e r a d i u s o f g y r a t i o nthe estima ted D of binary ima ges, binary ima ges wit hrequired. Note that it is necessary to sample all local

origins to sample as many data points belonging to cell body an d axon removed, or border-only ima ges ofth e imag e as possible. The ra dius of gyra tion is intro- ca t retina l ga nglion cells a s long as the dendrites a re

duced as a meth od of a voiding th e out er edges of the t hi n wi t h r espect t o t he cell body. Thi s i s due t ofigure ba sed on th e premise th a t th e periphera l par ts the area of the cell body and dendrites being muchof t he i m age t ha t r epr esent na t ur a l o bject s such a s smaller than the extent of the border (58, 59). How-neurons is incomplete. This premise st ems from t he

ever, this finding is dependent on the type of cellfa ct tha t t he computer screen ha s a limited resolution

one an a lyzes a nd d oes not hold for g lia cells (60). Ina nd ma y not be a ble to represent bra nching pat terns

addition, one could claim that i t is only the borderbelow the size of one pixel. The histologica l tech-

tha t is fra cta l ; the fi lled interior is solid, with a D ofniques used ma y a lso lea d t o incomplete st a ining of2 (60). Skeletonized images, on the other hand, hadth e periphera l par ts of the cell. H ow ever, wh en mostsignifica ntly low er D values (58) since they repre-of the ma ss is concentra ted in a convex out er bordersented only the dendrit ic bra nching a nd do not reflectt he m et ho d t o t al l y fai l s because t he r a di us of gyr a -

t h e ot h e r ch a r a ct e r is t ic of com pl ex it y, b or d ertion falls tightly within the border itself. If one takesa fra ction of the ra dius of gyra tion, the entire border roughness (Fig. 8).


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sam e i m age, when skel et o ni zed, was no t sc al e- i n-GRAPHIC DETERMINATION OF THEvariant under this transformation and method (58).

FRACTAL DIMENSIONHo wever, som e i nvest i gat o r s ha ve obt a i ned l inear

plots using skeletonized images of neurons. Mon-H ow t h e a c t ua l D va lue is obta ined from t he log– ta gue a nd Fr iedlander (14), for inst a nce, using a dif-

log da ta points can lea d t o differences in the ma gni- ferent implementation of the box counting methodtude of D . In such a plot, D is related to the slope of

( gr eat er num ber o f bo x si zes) and di f fer ent i m aget he l ine, t he num ber of da t a poi nt s being r elat edha ndling (rotat ion of ima ge a nd using multiple cen-

t o t he num ber of m ea sur ing st eps. The a ct ua l da t aters), obta ined linear log– log plots ( 2 generations)

points genera lly do not lie on a str a ight line for morewith skeletonized images of retinal neurons. Caserta

th a n one to tw o deca des. This limited self-similar ityet al. (6, 7) using t he m ass – r a di us m et hod, a s de-or sca le invar ia nce is char a cteristic of biologica l ma -scribed a bove, a lso obta ined linear log– log plots w ithteria l a nd is a focus of some controversy (51, 61).skeletonized images. This dependency on the analy-Di ffer ences i n t he l inear i t y of t he l og– l og dat asis met hod to produce linea r log– log plots w ith skele-points was observed between binary, outl ine-only,tonized imag es ma y expla in th e conclusions of Pa nicoa nd skeletonized images (58). Ana lysis of skele-and Sterling (61). These authors used two variantst o ni zed i m ages usi ng t he o r i gi nal NI H I m age bo xof t h e b ox cou n t in g m et h od a n d t h e m a s s – r a d i u scounting method (Version 1.2) led at times to a sig-

moid log– log da ta point distribution, indica ting t he method w ith skeletonized images a nd concluded tha t

FIG. 8. B ox counting a na lysis of the sa me tur tle ga nglion cell, using binary (A), outlined (B ), a nd skeletonized (C) images. The figureson the bottom are the associated graphs of their f ractal dimensions.


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ca t retina l ganglion cells are not fra cta l due to their use of the value w ith the longest linear ra nge is sug-

limited linearity. gested. Alterna tively, other methods included onlyBecause of the limited scale invariance of neurons poi nt s t ha t fell on t he st r a i ght pa r t o f t he l ine and

different a uthors ha ve used different methods to de- excluded da ta points obta ined from th e peripheraltermine D from log– log values. The simplest method pa rt s of the ima ge (41).of obta ining D i s t o f it a r egr ession l ine t o a l l dat a P a nico a nd St erling (61) also used t he loca l slope

points a nd det ermine t he slope of t his line; however, m et ho d t o det er m i ne t he D of their images. Theirfrom a statistical point of view such a method would c o nc l usi o n was t hat t he r egi o n o f t r ue l i near i t y o fnot be justified, a s biologica l objects display sta tist i-

t he l o c al s l o pes was l ess t han o ne gener at i o n andca l self-similari ty only betw een a short ra nge of di-

therefore th e images a na lyzed w ere not self-similarm en s ion s . S e ver a l p ub li ca t i on s h a v e u s ed t h i s

and could not be fractal. This method, however, hasm et ho d t o det er m i ne t he sl o pe o f t he dat a po i nt s

severa l fla ws. One of the ma in ones is tha t t he sensi-(10, 37, 52, 53, 61). The linear region can also be

t i vi t y c hanges as t he wi ndo w o ver whi c h t he l o c alca l cu la t ed b y d et e r min i ng t h e l oca l s lop es . O n e

slopes is obtained is decreased (62). Therefore themethod for this, described by Caserta et a l . (7) forlinearit y region increa ses as th e w indow is increasedthe mass– radius method, is to calculate the n -pointand makes this is a very subjective method.local slopes, as the difference in log N (r ) divided by

C l ear l y t hen, t he quest i o ns o f whet her an i m agelog (r ) for every n successive points. The region in is fra cta l and w hether a n ima ge belongs to a certa inwhich the local slopes are constant is then taken as

group based on the D va l ue ar e di f fer ent an d needt he linea r region (7). The use of a hiera rchical clust erto be disenta ngled. The ra nge of l inearity is not im-a na lysis to compute pa rt icula r subsets of the log– log

val ues t hat a chi eve t he best l i nea r f i t t i ngs (Fi g. 9) por t a nt i f t he D obta ined in th is wa y is used in differ-has also been reported (12). This technique allows entiat ing betw een different cell types (Ca serta , per-the detection of changes in D at different scales of sona l com m uni cat i on). Deci ding on t he r a nge ofmeasurement and compensates for the finite size ef- linearit y a nd especia lly if it is significa nce ha s beenfects induced by the limited resolution of the ima ges. a ddressed by Russ (49), w ho suggest ed th a t compa r-When this method produces multiple values of D , i ng t he f i t o f t he dat a po i nt s t o a s t r ai ght l i ne and

to a higher-degree polynomial can clarify whether a

stra ight-line fi t is a n a ppropriat e model of the da ta .The higher-degree polynomial will always, of course,

be able to fit the data better, but it uses up one more

degree of freedom in the process, a nd the improve-

m e n t i n t h e f i t m a y n o t b e t h a t g r e a t . T h e t e s t i s

based on the ratio of reduced 2 values, which will

h a v e a n F distribution (49). The test is performed

using a cri t ical value of F ( p 0.25). If the linear

fi t is accepted then the image is fractal .

ADVANTAGES AND POTENTIAL PROBLEMSOF FRACTAL DIMENSIONS

In a recent stu dy (12) w e posed th e follow ing ques-

t i o n : C a n t h e e s t i m a t e o f D resolve differences inFIG. 9. Method for the gra phic determination of f racta l dimen-sions. The lef t-hand side shows the digi t ized image of a retinal neuronal branching when simpler metrical analysisbipolar cell. The right -ha nd side sh ows a plot of t he box count ing

a lone ca nnot? Our results indica ted th a t a l though D measurements. A hierarchical cluster an alysis yielded t wo regres-alone does not completely specify a cell’s morphology,sion lines wit h tw o different D va lues: 1.41 (open circles) an d 1.07

(filled circles). This method considers the D of the cell draw ing to a nd indeed it should not be expect ed to, it is a st a t isti-be th e one wit h t he longest linea r ra nge (1.41, open circles). The

cally significant parameter for identifying and differ-D with the smallest l inear range (1.07, f i l led circles) could beat tr ibuted t o f ini te size ef fects at very low scales. entia ting neurona l cell cla sses. Thus fra cta l a na lysis


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ha s an i m por t a nt r ol e i n cha r a ct er iz i ng na t ur a l ob- Fur t her m or e whet her a hi gher fr ac t al di m ensi on

jects, a nd va rious a uthors ha ve discussed cla ssifica - w ould correlat e w ith a more complex physiologica ltion syst ems of neurons using fra cta l an a lysis (7, 12, response is still a n unresolved issu e (9, 12). Notw ith -39, 42, 59, 62, 63). s t a ndi ng t he abo ve-m ent ioned l im i t at i ons, i t r e-

Furthermore fractal geometry has some other ad- m a i ns t ha t i n m any si t uat i ons a s i ngle num ber, t heva nta ges over i ts integer-dimensiona l counterpa rts. fra cta l dimension, summar izes concisely an d mean-

Thus in a lmost a ll circumsta nces the fra ctiona l com- ingfully the a mount of deta il, space fil ling, or com-ponent of dimension is reta ined wh en a fr a cta l object plexity of neurons.is projected to a lower-order dimension (18, 19), an A basic considera tion is tha t most measurementsexample being the projection of three-dimensional cover only a relatively short range of dimensions.retina l ga nglion cells onto a tw o-dimensiona l film or U nl i ke m a t hem at i cal l y gener at ed fr ac t al s , r eal dat adra w ing (7). This cont ra sts w ith int eger-dimensiona l cannot be ideally fractal over all scales. Thus, somemeasurement of a nisotropic objects w hich require of t h e i m a g es a n a l yz ed u s in g f r a ct a l a n a l ys is m a ymult iple samples thr ough the thickness of the th ree- not demonstrate self-similari ty or scale invariancedimen siona l objects (1). over more than one or two levels of magnification

A f u r t h e r a d v a n t a g e o f f r a c t a l a n a l y s i s i s t h a t a nd ma y not be fra cta l (61, 64). Furt hermore wh ethershrinkage or expansion of a specimen will not affect

scale invariance is observed for a particular image

D a s long a s the a rti fa ct a cts equa lly in a l l directions is dependent on image presenta tion a nd th e ana lysisand the measured points sti l l l ie on the l inear seg-pr o gr am appl i ed t o o bt ai n t he f i nal D (59, 62). Fi-

ment of the graph (19). This means that D valuesna l ly, bi ol ogi cal dat a t ha t ha ve a l inear f it of m o r e

of specimens that have been processed in differentt h a n t w o o r d e r s o f m a g n i t u d e a r e e x t r e m e l y r a r e

bat c hes o r at di f fer ent l abo r at o r i es c an usual l y be(66– 69). Not even t he “coa stline of B rita in” exa mple

compared directly (as long as the same methodologyin Mandelbrot’s seminal work (5) has a power law

to calcula te fra cta l dimension is used).behavi o r spanni ng m o r e t hat o ne o r t wo o r der s o f

Although a ll a na lysis methods rely on th e rela tion-magnitude (69).shi p bet ween a m easur ing devi ce a nd t he object ’s

spatial distribution, not al l methods give identical

r esul t s fo r t he sam e fo r m . Fo r exam pl e, hi gher D CONCLUSIONS AND FUTURE

val ues ar e o bt a i ned by usi ng t he m a ss fr ac t al m et h- DEVELOPMENTSods tha n by using t he pixel dilat ion a nd box count ingprocedures. I t has thus become important to estab-

lish some criteria for choosing a particular method Fr a ct a l ana l ysi s has a l r ea dy found wi despr ead a p-and ho w t hese m et ho ds c o m par e i n o r der t o s t an-

plication in the field of neuroscience and is beingdardize the computation of D (59). Our r esult s us ing

u s e d i n m a n y o t h e r a r e a s . M a n y n e u r o n s d i s p l a ydifferent met hods to compute the D va lues show tha t

irregular shapes and discontinuous morphogenetica l t hough di ffer ent m easur em ent pr ocedur es a nd

pat t er ns i n suppor t an d i n c onnect i on wi t h t heireven the sa me a lgorith m performed by different com-

functiona l diversity. To capture a l l t his richness ofput er pr ogr am s a nd/or exper i m ent er s m ay gi ve

th is complex structure int o a t heoretical m odel is oneslightly different numerica l va lues of D , the results

of the major challenges of modern theoretical biology

a re a lwa ys consistent. These dat a reinforce the idea (64). Thus ma ny q ua ntita tive para meters ha ve beenthat comparison of measurements of different pro-used to cha ra cterize t he morphology of nerve cells.f i l es usi ng t he sam e m easur em ent m et ho d m ay beThese par am et er s r ange fr o m si m pl e m et r i c al de-useful and valid even i f the exact numeric value ofsc r i pt o r s , suc h us dendr i t i c f i el d ext ent and t o t althe dimension is not n ecessarily very a ccura te.dendrit ic length, t o more complica ted global, descrip-It should, however, be kept in mind that D is onlyt o r s s u c h a s D , t h a t c a n b e u s e d f o r a n o b j e c t i v ea descriptive par a meter, like the dendritic field a reaa ssessment of the degree of complexity (a conceptor the number of segments of a dendritic tree, andheretofore not rea dily qua nt ifiable)of developing an ddoes not necessarily imply any underlying mecha-mature neurons. Thus determining D of a neuron, innism of form generation. In general, the connectiona ddition to the oth er morphometr ic criteria ty pica llybet w een em pi r ica l val ues of D a n d a n y s peci fi c

used, could immensely a id in t he morphological dis-gr o wt h m ec hani sm sho ul d be avo i ded and r equi r et he answ er ing of fur t her exper i m ent a l quest ions. cer nm ent of dif fer ent neur on t ypes or neur ons t hat


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disease, or experimental treatments.12. Ferna ndez, E., Eldred, W. D., Ammermü ller, J ., B lock, A., von

A criticism th a t could be leveled a t a lmost a l l theB loh, W., a nd K olb, H. (1994) J . Comp. N eur ol . 347, 397– 408.

implementations of measuring D i s t h a t i t i s n o t13. Monta gue, P . R . , a nd Friedlander, M. J . (1989) Proc. Nat l .

always an adequate descriptor of a determined pro- Acad. Sci. U SA 86, 7223– 7227.file. Furt her a str ucture ca n be a mixtur e of different 14. Monta gue, P. R. , and Friedlander, M. J . (1991) J. Neurosci.

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fr ac t al s , eac h one w i t h a di ffer ent va l ue of D . This 15. Feder, J . (1988) Fra cta ls, P lenum, New York.mean s tha t a ny single number ca nnot be cha ra cteris-16. Taka yasu, H. (1990) Fra ctals in t he P hysical Sciences, Ma n-tic of the mixture (48). Some investigators are start-

chester Un iv. P ress, Ma nchester.i ng t o use m ul t i fr a ct a l s a s a m or e c om pr ehensive17. P eitgen, H.-O., an d Richter, P. (1986) The B eaut y of Fra cta ls,methodology which gives information about the dis-

Springer-Verlag, B erlin.tribution of fracta l dimensions in a structure. In a d-

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try, Springer-Verlag, New York.

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