Fractal Dimension and the 4/5 Allometric Scaling Law for the Human Brain.

Transcription

1 Fractal Dimension and the 4/5 Allometric Scaling Law for the Human Brain. Kodjo Togbey Advisor: Jack Heidel Department of Mathematics University of Nebraska at Omaha Omaha, NE 68182, USA Abstract One of the most ubiquitous laws in biology is allometric scaling, whereby a biological b variable Y is related to the mass of the organism by a power lawy = Y o M, where b is the so called allometric exponent. Empirical observations and recent researches showed that the basal metabolic rate of all organisms is allometrically scaled by 2/(Rubner Law), or /4 (Kleiber law). A fractal analysis of these exponents leads to a 4/5 law that empowers the human brain with a 5 th dimension. We will explain these dimensions in terms of fractal geometry. Introduction Living organisms span an impressive range in body mass, varying over 21 orders in 1 magnitude ranging from mycoplasma 10 g to the blue whale10 8 g. Biologists have described a large number of relationships between body size, organ size, rate of physiological processes and biological cycle times. These relations are usually modeled b by power scaling laws of the form Y = Y o M wherey o and b are constant and M is the body mass. Strikingly these relations hold true over many orders of magnitude. The most b pervasive of these allometric scaling relationships is B = B0M where B is the basal metabolic rate, M the mass of particular species and Bo is an experimentally determined constant and b is the scaling exponent. The power exponent has a simple mathematical interpretation in terms of fractal geometry. We will investigate the several known theories for the origin of allometric scaling and elucidate them in term of fractal geometry.

2 The relationship between body mass and metabolic rate was first noted by Rubner in 188. He proposed that the metabolic rate B of living organisms was proportional to their surface area S. Then the power exponent b has the value of 0.67 since the area of geometrically similar object is related to volume raised to the 2/ power. This can be simply derived by a Euclidean geometry analysis. L is a characteristic length, then 2 S ~ L and the volumev ~ L. If the mass density is constant ρ, then M ~ ρ V ~ L, implying that linear size and area scale with mass respectively, as L = a M l 1/ and S = a s M 2 / where a l and a s depend on the units used, and thus. 2 / B ~ S ~ M. The 2/ exponent is experimentally observed among many species especially small mammals and birds (Dodds et al 2001, White et al 200). However in 192, Keibler studying mammals ranging from rat (0.15 kg) to steer (679 kg) obtained b = This value is closer to / 4 = than to 2/ A few years after, Brody (1945) extended the mass range (mouse to elephant) and obtained approximately the same value (b = 0.74). Since the work of Keibler, an enormous quantity of papers have been published in biological scaling characterized by exponent that are multiple of 1/4. These include the metabolic rate (b /4), lifespan (b 1/4), growth rate (b-1/4), heart rate (b -1/4), length of aortas and height of trees( b1/4), radii of aortas and trees trunks (b /8) and cerebral gray matter (b5/4). According to West, Brown and Enquist (1999) the quarter power scaling is a consequence of a fourth dimension of life which appears due to the fractal nature of the transportation network and the existence of invariant units. The fractal structure implies that biological surface area S, volume V and length L have scaling description modified 2+ e v by the existence of invariant units. They found that s / + e S ~ V, where e j are corrections to the Euclidean dimensions. Moreover, they established that e = e + e from the relation V = SL and supposed that 0 e l 1. Since the metabolic rate was proportional to exchanges through a surface area, they wrote v s l

3 B 2+ es / + ev 2+ es / + ev 2+ es / + es + el ~ S ~ V ~ M ~ M. Now, assuming that organisms evolve to maximize their metabolism by maximizing their surface S, they maximized 2 + es and minimized + e s + el ( e l = 0, e s = 1) and that lead to / 4 B ~ M. Comparison Table Variable Conventional Euclidean Fractal Biological Length Area Volume 1 / 2 1 / 1 / L ~ A ~ V ~ M 1 / 1 / 4 1 / 4 A 2 2 / ~ L ~ V ~ M V ~ L ~ 2 / M l ~ a ~ V ~ M a / 4 ~ l ~ V ~ M 4 v ~ l ~ M / 4 The above argument also has been challenged (Dodds et al) and others scientists have attempted to solve the allometric scaling problem. Darveau et al (2002) proposed a multiple cause cascade model of metabolic allometry. The basic equation is MR = a i c M i b i where MR is the metabolic rate in any give state, M is the body mass, b i the scaling exponent of the processi, c i the control coefficient of the processi. This new approach has been questioned (West et al 200, Banavar et al 2002) and West et al (200) pointed out that this scaling is base on technical, theoretical and conceptual errors. Banavar et al showed that the cascade model is flawed and meaningless for control of metabolic rate in a given organism and for scaling of the metabolic rate. He and Chen (200) took into account ph values and then predicted respectively that the power exponent is1/2, 2/, /4 for one dimensional, two dimensional and three D / D+ 1 dimensional organisms: the pattern may then be B ~ M where D is the dimensionality, B the metabolic rate and M is the mass of the organism. He and Chen reconciled the 2/ and the /4 exponent. He and Zhang (2004) proposed another creative approach based on the fractal cell geometry and suggest that the fractal dimension of a brain cell is 4/5 concluding a fifth dimension of life and a 4/5 allometric scaling law for the human brain. Fractal approach to multicellular organisms.

4 First assume there are n basal cells. The metabolic rate B scales linearly with respect to the total surface A. 2 B ~ nr ~ A (1) since the surface area A is fractal then d A ~ r. He argued that, for dimensional organisms, d the fractal dimension tend to because the area is volume filling to allow nutrients to reach each cell. To explain this, He used the box counting dimension. He assumed that he has boxes to catch the cells. If he catches one cell with a box of dimension 2 r then a box of dimension 4 r will catch 8 cells ( 4r ) ( 2r) = 8). The fractal dimension ln( 2 n ) d = lim =. n ln(2 ) n 2 From the relation A ~ r and B ~ nr ~ A, we can see that n ~ r. (2) The mass M of an organism is linearly proportional to its volume then M ~ nr. () Combination of (2) and () leads to M 4 ~ r then 1/ 4 r ~ M (4) Substitution (4) into (1), 1/ 4 B ~ r B ~ ( M ) then / 4 B ~ M which is the Keibler law. For two dimensional organisms, we proceed by the same reasoning. d A ~ r where d the fractal dimension tend to 2. 2 B ~ nr ~ A (5)

5 2 A ~ r then 0 n ~ r (6) Using M ~ nr and (6) we can see that 1/ M ~ r r ~ M, (7) and the combination of (5) and (7) leads to 2 / B ~ M, the Rubner law. Fourth dimension of life and 4 5law. In 1977 Blum suggested the /4 dimension can be understood as a fourth dimension of time because the power exponents are of the form D / D + 1. Similar to the 4 th dimension of life the human brain cell may have a fifth dimension. That is the dimension of thought. B ~ nr ~ δ brain A brain. Consider the brain cell to be volume filling A brain ~ r We can deduct that δ n ~ r and then the total mass of the brain satisfies 6 δ M brain ~ nr ~ r. (8) From (8), r ~ M 1 6 δ And then

6 brain ~ brain 6 δ B M (9) He assumes that the human brain cells have similar construction to a snowflake, so they can receive as much information as possible from the environment. Therefore assume the δ / 4 as the surface of snowflake. The relation (9) will be: B brain M ~ brain 4 5 and here the fifth dimension is called the dimension of thought. Experimental results agree with the 4/5 law prediction of the brain cell.(wang et al 2001). Fractals with rational dimension He predicted that the developed brain cell has a surface dimension of 9 / 4 = by comparing it to the Koch surface. The Koch curve has a dimension ln( 4) / ln() One (1) dimension extension to a surface will give = We carry this analysis little further to generate a fractal surface with exactly the dimension of9 / 4. 9 ln( N) ln 2 9 To create a suitable surface, we use D = = =. 4 ln( R) ln = 512 = / , and the scaling factor is 2 4 = 16. We start with a square surface of side 16 and stack infinite sequences of cubes with side scaling each time by a factor of = 2 4.

7 Step I: The initiator is the square with side Note that the first surface, which is the square, can be covered by16 16 = 256 boxes. Step II: The generator is The side of the cube is 4 2. To cover the surface we only need to add 4 squares of side 4 2 which can be covered by 128 boxes ( 4(4 2) 2 = 128). Then at this step N is From step III and so on, each cube will be transformed into 5 cubes with side 2 4 of the previous one. It is important to understand that we still need to take in account only 4 new cubes out of the 5 newly created from each cube. N i / = 256 = 2.2 = = i= ,

8 and the scaling factor is 2 4 = 16, thus the surface has a dimension 9 ln( N) ln 2 9 D = = =. 4 ln( R) ln 2 4 Generating that surface from a cube will not change the dimension D. And the iterations to infinity of the whole process generate an object that looks schematically similar to the one below. A vertical or horizontal cross section looks like

9 By curiosity, we compute the fractal dimension of the 2 dimensional curve outlying the above figure:

10 For simplicity let the initiator have length 16 and then the generator has length = ln( ) Thus D = is irrational. ln(16) The idea of creating a fractal surface or curve with rational dimension is of great interest due to the fact that they are rare. Most known fractals (Sierpinski

11 triangle D = ln( ) / ln(2) , the Cantor middle third set D = ln( 2) / ln() 0. 61, the Menger sponge D = ln( 20) / ln() ) have irrational dimension. We look at a curve that have rational dimension. The Quadratic Von Koch curve with dimension D = ln( 8) / ln(4) = / 2.

12 Conclusion The pervasive metabolic allometric scaling law is still one of the most debated laws in biology. However a fractal geometry analysis led to the 4/5 scaling law of the human brain also known as the 5 th dimension of thought. This predicts that the human brain cell fractal dimension is rational 9/4, which is a rare dimension for a fractal surface. We have elaborated a fractal surface of 9/4 and have realized that its 2 dimensional cross section curve has an irrational dimension. We, further, hope to expand our research to a more realistic shape similar to brain cells, and establish algorithms to create fractal surfaces and curves with rational dimension. References: 1. Jayanth Banavar et al, Supply-demand balance and metabolic scaling, Proc Nat Acad Sci 99 (2002), Anthony Barcellos,Tthe Fractal geometry of Mandelbrot, The College Mathematics Journal, Mar J.J Blum, On the geometry of four-dimensions and the relationship between metabolism and body mass. Journal of Theoretical Biology 64 (1977) Adrian Bejan, Shape and Structure from Engineering to Nature, Cambridge F. Bokma, Evidence against universal metabolic allometry, Functional Ecology, 18 (2004) S. Brody, Bioenergetics and growth, New York: Van Nostrand Reinhold 7. James Brown and Geoffrey West, Scaling in Biology, Oxford Eric Charnov, Life History Invariants, Oxford Lloyd Demetrius, The origin of allometric scaling laws in biology, Journal of Theoretical Biology, 24(2006), C.A Darveau, R.K Suarez, R.D Andrews, P.W Hochachka, Allometric cascade as unifying principle of body mass effects on metabolism. Nature 417 (2002) Olaf Dreyer, Allometric scaling and central source systems, Phys Rev Lett 87(2001), PS Doods, DD Rothman, JS Weitz, Re-examination of the /4-law of metabolism. Journal of Theoretical Biology, 209(2001), J. Grasman, J.Brascamp, J.Van Leeuwen, and B.Van Putten, The Multifractal Structure of Arterial Trees, J. Theor Biol, 220(200), Ji-Huan He, H Chen, Effects of size and ph on metabolic rate, International Journal of Nonlinear Sciences and Numerical Simulation 4, Ji-Huan He, J. Zang, Fifth dimension of life and the 4/5 allometric scaling law for the human brain. Cell Biology International. 28(11) (2004), M. Kleiber, Body size and metabolism, Hilgardia 6 (192) 15-5.

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