Scaling principle of brain weight and metabolism

Abstract

The same universal 4/3 law, itself a consequence of the principle of dimensional capacity, underlies 3/4 scaling of brain weights and metabolisms.

Full text

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Scaling principle of brain weight and metabolism2
3
4
Robert Shour5
6
16th January, 20217
Toronto, Canada8
Abstract9
The same universal 4/3 law, itself a consequence of the principle of10
dimensional capacity, underlies 3/4 scaling of brain weights and metabol-11
isms.12
Keywords dimension, dimensional capacity, invariance, Kleiber’s Law13
1 Scaling: brains, respiration rates, metabolisms14
1.1 Scaling in relation to body mass15
As the size of a mammalian species increases, the animal’s16

Brain weight relative to body weight decreases;17

Respiration rate decreases;18
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Metabolism decreases.19
1.2 Brain weight scaling20
Otto Snell in 1892 suggested that a mammal’s brain weight scaled relative to21
its body weight [Snell, 1892]:22
h∝ks.(1)
In German, Hirn is brain, and Hirngewicht is brain weight, hence the hin (1),23
K¨orper is body and K¨orpergewicht is body weight, hence the kin (1). Snell24
denoted sthe somatic exponent and suggested that perhaps s= 2/3.25
1.3 Respiration scaling26
Sarrus and Rameaux in 1838 suggested [Sarrus and Rameaux, 1838, 2017] that27
an animal’s respiration rate scaled by exponent 2/3 relative to its body’s mass,28
or weight.29

Scaling brains and metabolisms 2
1.4 Metabolic scaling30
An animal’s respiration rate is proportional to its metabolic rate Y, since the31
oxygen is used to burn nutrients it consumes in order to provide energy to the32
animal. The 1838 work of Sarrus and Rameaux about mammalian respiration33
implies that a mammal’s metabolic rate34
Y∝Ma(2)
where Mis the animal’s mass, for some exponent a.35
1.5 Scaling ranges, universality and generality36
The largest mammalian brain weights, found in whales, are about 130,000 bigger37
than the smallest mammalian brain weight, that of a species of bat, while the38
largest mammalian body weights are 50 million times bigger than the smallest39
[van Dongen, 2008, Section 23.1.1].40
A study of metabolic scaling similarly remarks that body size varies by 2141
orders of magnitude West et al. [1997].42
Scaling of brains and metabolisms over such a vast size range implies that43
underneath is a general universal scaling principle. If a principle of physics44
applies over 21 orders of magnitude relating to animal size, its generality and45
universality implies it might very well also apply at the atomic and cosmological46
scales.47
Thus study of brain weight scaling and metabolic scaling may, likely will,48
reveal a principle of physics significant for a wide range of scientific disciplines49
because of its universality, that can explain a wide variety of phenomena because50
of its generality. Such a variety of phenomena, otherwise apparently unrelated,51
might all be related if all are manifestations of the same universal general phys-52
ical principle.53
2 Scaling based on surface areas54
2.1 Metabolic rate55
Sarrus and Rameaux in 1838 inferred [Sarrus and Rameaux, 1838, 2017] that56
an animal’s respiration rate relative to its body’s mass — or weight — by an57
exponent equal to 2/3.58
Here is a synopsis of their reasoning. A mammal has a constant body tem-59
perature, so the same amount of heat produced by the animal’s body must be60
emitted via the animal’s skin surface area for the animal to maintain a constant61
body temperature. A larger animal increases its volume V≡L3, by exponent62
3 of length L, while heat emission scales in the same way as surface area: as63
L2. Hence a larger animal must slow the production of heat by exponent 2/364
relative to animal mass to maintain constant body temperature. Heat emission65
must equal heat production, so heat production capacity must scale down to66
the the capacity of skin surface area to emit heat.67

Scaling brains and metabolisms 3
In view of the relationship between rates of respiration and of metabolism68
Y, the reasoning of Sarrus and Rameaux in 1838 implies that in Y∝Ma, with69
a= 2/3.70
Max Rubner’s data suggested to him that Y∝M2/3[Rubner, 1883, 1902].71
His data was consistent with, the 2/3 ratio of dimensions, implied by the 183872
reasoning of Sarrus and Rameaux.73
2.2 Brain weight74
Snell presented an argument in his 1892 paper that “the brain weight . . . is75
proportional to the surface area of the body” which implies s= 2/3 but remarks76
incidentally that “probably the somatic exponent is greater than” 2/3.77
Snell’s 1892 article suggests two possible bases for 2/3 scaling of brain weights78
[Snell, 1892].79
First he infers 2/3 scaling of brain weights relative to body weights, “Presup-80
posed always that the somatic functions of the brain directly depend on body81
surface”. This supposition is that the animal brain has to manage its somatic82
sensory input received at the boundary between animal body and its physical83
environment; that boundary is the animal skin surface. He qualifies the 2/384
scaling implication of his supposition by noting that, in effect, not all of the85
somatic function of the brain deals only with sensory input from the skin sur-86
face area but also “the mass of the animal body is a component in the amount87
of work that the metabolism [that is, the metabolism of the brain] has to do”.88
That’s why he finds it “therefore probable that the somatic exponent is greater89
than 0.666. . . ”.90
Snell does not expressly write in his article that he has an explanation alter-91
native to somatic surface area sensory input for the brain that he includes near92
the end of his article. In effect he observes that a small brain has more surface93
area relative to its volume — and hence relative to its weight — than a large94
brain. That implies that a small brain accomplishes more mental work relative95
to brain weight than does a large brain.96
Thus Snell’s 1892 article contains two possible reasons for 2/3 scaling of97
brain size, one being that a brain’s mental capacity varies with sensory input98
proportional to skin surface area, and the other that the brain’s surface does99
most of the brain’s thinking, and that the thinking surface area part of the brain100
is relatively bigger for smaller brains.101
Despite Snell’s reservation that the somatic exponent was larger than 2/3,102
for many years scientific consensus favored s= 2/3.103
Based on earlier research [von Bonin, 1937, Count, 1947], Jerison in 1955104
concluded that for modern mammals the somatic exponent s= 2/3 [Jerison,105
1955]. His quantitative analysis suggested that s= 2/3 for mammals in the106
Eocene and Oligocene as well as for modern mammals [Jerison, 1961].107

Scaling brains and metabolisms 4
3 3/4 not 2/3 surface law108
3.1 Brain weight109
A 1981 examination of brain/body weight data for 547 mammalian species con-110
cluded that brain weights scale by 0.74 relative to animal body weights [Eisen-111
berg, 1981, p. 275-283]. Given the larger data base, the consensus became112
s= 3/4 [Armstrong, 1983], [van Dongen, 2008, p. 2101].113
3.2 Metabolism114
Metabolic rate, based on measurement, more likely scales by 3/4 relative to115
animal mass [Kleiber, 1932]. An animal does not eat to produce heat for skin116
surface heat emission [Brody, 1945, p. 361]. Instead, an animal’s circulatory117
system distributes energy to its tissues.118
The empirical finding of a 3/4 exponent for metabolic scaling is known as119
Kleiber’s Law.120
3.3 General laws121
The generality and universality of 4/3 scaling laws favor 3/4 as an exponent122
for metabolic rate and brain weight scaling. The same 4/3 scaling appears in a123
variety of settings including gas molecular mean path lengths [Clausius, 1860,124
Shour, 2017], Richardson’s scaled wind eddies [Richardson, 1926], Kolmogorov’s125
analysis of Richardson’s empirical inference about wind eddies [Kolmogorov,126
1991a,b, Batchelor, 2008], the 4/3 Brownian motion fractal envelope [Lawler127
et al., 2001], cosmological energy densities [Wang, 2010, p. 17] among other128
instances.129
4 Brain weight scaling, no causal link to metabolism 3/4130
4.1 Correlation hypothesis131
Coincidence of a 3/4 exponent for, relative to animal mass or weight, brain132
weight scaling and metabolic scaling suggests the possibility a causal connection.133
One conjecture is the brain uses energy for its information and control functions134
[Armstrong, 1983, p. 1302]. This conjecture continues to attract attention [Isler135
and van Schaik, 2006, Karbowski, 2011].136
4.2 Against metabolism being causal for brain weight scaling137
1. Relative to body mass, a larger mammal has a smaller brain and a slower138
metabolism than a smaller mammal but has a larger problem solving rate.139
A mammal’s metabolic rate proportional to its problem solving rate scales140
inversely to brain weight. Hence mammalian brain weight and metabolism141
cannot be linearly related.142

Scaling brains and metabolisms 5
2. If a slower metabolism causes smaller mammalian organs such as the brain,143
then a mammal increasing in size must at the same time be smaller because144
all its organs decrease in size, a logical contradiction, a proof by reductio145
ad impossibile that metabolism cannot cause brain weight scaling.146
4.3 A common physical principle?147
Logically there is no linear correlation between brain weight relative to body148
weight and brain cognitive capacity. But both the brain and an animal’s cir-149
culatory system are physical systems governed by the laws of physics. The150
exponent 3/4 being common to both types of scaling leaves open the possibility151
that the same physical principal accounts for scaling of the cognitive capacity of152
the brain and scaling of the energy distribution capacity of a circulatory system153
relative to animal mass or weight.154
5 Dimension capacity and 4/3 scaling155
5.1 Dimensional capacity156
The principle of dimensional capacity is that, for a given amount of weight, or a157
given amount of flow of heat, energy, or information per dimension, more dimen-158
sions can contain a greater total amount or quantity, and provides a conceptual159
point of view different than the scaling point of view.160
For example, an animal’s weight contained in its volume V∝L3scales161
by exponent 3 of length Lwhile cross-sectional area of weight-bearing animal162
bone scales by exponent 2 of L. Therefore, when animal size increases, animal163
bones, relative to animal size or weight, thicken, an example given by Galileo164
in discussing the strength of materials [Galilei, 1638, 1914].165
Consider a three dimensional system with a flow inside it, such as the tubes of166
a mammal’s circulatory system with blood flow. Denote a set of three orthogonal167
lines as [a1, a2, a3]. A point in 3-space can be designated by assigning a number168
to each of a1,a2, and a3. A particle in the 3-space has 3 independent degrees169
of freedom so far as direction is concerned.170
In particular, using particular numbers for a1,a2, and a3, [a1, a2, a3] can171
designate a point in an animal’s circulatory system. An empty circulatory sys-172
tem tube has 3 degrees of freedom. What if we add in something flowing at173
speed v, such as blood? Then taking the general case, [a1, a2, a3, v] has, by gen-174
eralizing observations about [a1, a2, a3], four degrees of freedom, one of which is175
based on v. In the case of the circulatory system, the four dimensioned system176
[a1, a2, a3, v] transmits energy to animal tissue, which is three dimensional.177
5.2 4/3 scaling178
In the case of the circulatory system, [a1, a2, a3, v], with four degrees of freedom,179
transmits energy to animal tissue, which has three degrees of freedom.180

Scaling brains and metabolisms 6
Thus the circulatory system with 4 degrees of freedom, scales for length of181
time twith exponent 4t, whereas animal tissue with three degrees of freedom182
scales contemporaneously with exponent 3t. The circulatory system, relative to183
the animal’s body, scales by 4t/3t= 4/3.184
The foregoing can be generalized. Any flow within a three dimensional185
system that is transmitted to a three dimensional system scales in time or with186
size change by 4/3 as much as the recipient three dimensional system.187
Therefore, a circulatory system has 4/3 energy distribution capacity of the188
capacity of animal tissue to use received energy. This observation relates to189
the principle of dimensional capacity. For a given fixed amount of heat, energy190
or information, a system with three dimensions through which heat, energy or191
information flows, an increase in size scales the system with flow by exponent 4192
and scales the homologous recipient system with exponent 3.193
6 Metabolic scaling194
6.1 Cellular metabolism195
Assume that the average cell in a mammal, regardless of species, has similar196
architecture, biochemistry and optimal operating temperature. If the cell’s in-197
ternal temperature is too low or too high, the chemical processes internal to the198
cell will not proceed optimally. Temperatures may be so high or so low that199
the cell sickens and dies. The mammal’s body is designed to operate optimally200
within a narrow temperature range. The animal’s survivability is otherwise201
impaired.202
If an animal species is larger, then the distribution of energy via its circula-203
tory system scales by 4/3 as much as the scaling of its mass (or weight). There204
would be too much heat if the rate at which energy is supplied to cells remained205
unaltered. To maintain as invariant the internal operating temperature of the206
animal’s cells, the rate at which energy is supplied to the cell must scale by207
exponent 3/4.208
6.2 Metabolic invariance209
Consider an index (subscript i) mammaliwith metabolic rate Yiand mass Mi
210
with its circulatory system distributing energy Eito its tissues. Then Yi∝Mi.211
Suppose mammal1with mass M1such that M1=Mb
i. Then the amount212
of energy that mammal1can distribute at mammali’s metabolic rate Yif the213
metabolic rate does not change is Y(4/3)b
i, because mammali’s circulatory system214
capacity scales by 4/3 with increasing size. Energy supplied to mammal1’s cells215
at the rate Y(4/3)b
iwould overheat its cells. Thus scale Y(4/3)
ibby 3/4, so that216
(Y(4/3)b
i)3/4=Yb
iin proportion with Mb
i.217
Since Y∝M, scaling Yias in (Y(4/3)b
i)3/4also scales Mias a proportional218
measurement of metabolism by 3/4, which is Kleiber’s Law.219

Scaling brains and metabolisms 7
6.3 Metabolic scaling as a paradigm220
The derivation of metabolic scaling forms a template for deriving brain weight221
scaling.222
7 Brain weight scaling223
7.1 Mammalian brain signal processing invariance224
Suppose that mammal brains process information in a way common to all mam-225
mals. Restricting analysis to mammalian brains is an indispensable step since226
animals that are not mammals likely have different brain architectures. For227
example, a reptile has a brain around one tenth the size of a bird or a mammal228
of a similar size [Font et al., 2019].229
Suppose further that the architecture and biochemistry of mammal brains230
are so similar that brain size relative to body size is a significant factor in231
determining the mammal’s cognitive capacity.232
We seek an invariance. If a mammal is larger in size than some other mam-233
mal, its processing of information is likely similar at the level of neuron, synapse234
and chemistry. Nature likely proceeds conservatively in modifying brain struc-235
tures as mammals vary in size as they evolve. The ratio of mammal brain236
processing capacity relative to animal weight or mass would have to be invari-237
ant. Expressly, for the mammal brain to function in accordance with its general238
design features, the ratio of brain weight to body weight must be invariant,239
as otherwise design features (architecture, chemistry, neurons) would have to240
be altered. But alteration of a mammal brain architecture would disable the241
advantages of brain engineering shared by all mammals; redesigning an already242
good functional design has a high evolutionary energy cost.243
If brain processing capacity relative to animal weight is invariant, what fol-244
lows from the 4/3 scaling of brain processing capacity relative to animal weight?245
Adapt the reasoning applicable to metabolic scaling to brain weight scaling.246
7.2 Relative information processing capacity invariance247
Consider an index mammaliwith problem solving rate (problem solving capac-248
ity) riand mass Miwith metabolism Yi∝Mi.249
Suppose mammal1with problem solving rate r1, mass M1=Mb
iand metabolism250
Y1. Then mammal1’s problem solving rate r1is Y(4/3)b
i, because mammali’s cir-251
culatory system capacity scales by 4/3 with increasing size. Changing brain252
problem solving (processing) capacity relative to body weight would alter an253
already working design. Thus scale r(4/3)
ibby 3/4, so that (r(4/3)b
i)3/4=rb
i,254
which is in proportion to Mb
i. In this way, brain processing design is preserved255
over a wide range of animal sizes.256

Scaling brains and metabolisms 8
7.3 How metabolism relates to brain weight257
The brain of mammal1larger than index mammaliwould require an amount of258
energy disproportionate to its increase in size because mammal1would have, as259
a result of being bigger, a problem solving rate (r(4/3)b
i). But if metabolism for260
mammal1slows by a 3/4 power, then the problem solving rate receives energy at261
a rate that is scaled by 3/4 exactly neutralizing the brain’s 4/3 scaled increase262
in problem solving capacity. Per processing event, the same amount of energy263
is used for mammals at all scales.264
8 Concluding remarks265
So far as I can tell, as of the end of 2020 there was no physical explanation266
for mammalian brain weight scaling relative to body weight. Perhaps without267
deploying 4/3 scaling laws no physical explanation is possible.268
If the concepts of dimensional capacity and 4/3 scaling validly solve brain269
weight scaling then:270
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That would solve brain weight scaling, a problem that goes back to Snell271
in 1892, or as Snell observes, originated implicitly as far back as Aristotle.272

The derivations above explain why the exponent 3/4 applied to body273
weight appears in connection with both metabolic scaling and brain weight274
scaling.275

Both scaling situations require recognition that a supply system capacity276
scales with size by 4/3, in the case of energy distributed by a mammalian277
circulatory system and in the case of the information processing capacity278
of a mammalian brain.279

Since the same 4/3 scaling concepts resolve metabolic scaling and brain280
weight scaling, the derivations above are evidence of the validity of 4/3281
scaling concepts.282

The generality and universality of the concepts of dimensional capacity283
and 4/3 scaling are further supported.284
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