  Brain is one of the most energy demanding organs in mammals, and its total
metabolic rate scales with brain volume raised to a power of around 5/6. This
value is significantly higher than the more common exponent 3/4 relating whole
body resting metabolism with body mass and several other physiological
variables in animals and plants. This article investigates the reasons for
brain allometric distinction on a level of its microvessels. Based on collected
empirical data it is found that regional cerebral blood flow CBF across gray
matter scales with cortical volume $V$ as $CBF \sim V^{-1/6}$, brain capillary
diameter increases as $V^{1/12}$, and density of capillary length decreases as
$V^{-1/6}$. It is predicted that velocity of capillary blood is almost
invariant ($\sim V^{\epsilon}$), capillary transit time scales as $V^{1/6}$,
capillary length increases as $V^{1/6+\epsilon}$, and capillary number as
$V^{2/3-\epsilon}$, where $\epsilon$ is typically a small correction for medium
and large brains, due to blood viscosity dependence on capillary radius. It is
shown that the amount of capillary length and blood flow per cortical neuron
are essentially conserved across mammals. These results indicate that geometry
and dynamics of global neuro-vascular coupling have a proportionate character.
Moreover, cerebral metabolic, hemodynamic, and microvascular variables scale
with allometric exponents that are simple multiples of 1/6, rather than 1/4,
which suggests that brain metabolism is more similar to the metabolism of
aerobic than resting body. Relation of these findings to brain functional
imaging studies involving the link between cerebral metabolism and blood flow
is also discussed.
