We consider that the surface of the WM–GM interface, with total area *A*_{W}, is crossed nearly perpendicularly by most axons leaving or entering the WM, of an average cross-sectional area *a*, and which, together with the ensheathing glial cells, comprise the entirety of the WM surface. *A*_{W} can thus be quantified as the product of the number of cortical neurons, *N*; the fraction *n* of these neurons that are connected through the WM; and their cross-sectional area, *a* (Figure ). Thus,

where γ is the average cosine of the incidence angle of axons at the WM–GM interface. The value of γ is of course 1 in the simplified case of perfectly orthogonal incidence, and presumably close to 1 in reality. The WM volume *V*_{W} is the sum total of the volumes of all fibers and is thus equal to one half of the product of *A*_{W} and the average axonal length in the WM, *l*, such that

Note that the average value of axonal length is given by

*l*=

*2V*_{w/}γ

*A*_{w}. A direct measurement of γ would provide us with a direct measurement of

*l* for each species. Assuming γ

=

1, then

*l* can be obtained from the existing measurements of

*V*_{w} and

*A*_{w}. Strictly speaking, these quantities are only lower bounds on the average axonal lengths; if however γ

≈

1 as we postulate, they can be taken as good approximations of their actual values.

If the WM scales under tension, the cubic root of its volume should increase more slowly than the square root of its surface area, leading to deformation of the latter, that is, to folding of the GM–WM surface. To quantify the extent of WM folding, we define a folding index *F*_{W}, which is the ratio between the actual WM surface, *A*_{W}, and the exposed surface expected from its volume (9π/2)^{1/3}
*V*_{W}^{2/3}.

Thus, a *F*_{W} value of exactly one implies a spherical WM, and larger values imply more convoluted forms. Importantly, notice that it is not necessary to model the cerebrum as a sphere for the 2/3 scaling relationship between its surface area and volume to hold; a volume of any shape that scaled isometrically would have the same scaling relationship of 2/3. In this case of isometric growth, which would ensue if the WM did not scale under tension, then we would expect *F*_{W} to be invariant as function of *N*.

Now, considering that

*a*,

*n*, and

*l* are themselves related to

*N* as power functions such that

*a*~

*N*^{α},

*n*~

*N*^{c}, and

*l*~

*N*^{λ}, the relationships above can be entirely rewritten as power functions of

*N*:

Note that if we took into account a systematic variation of the incidence angle of fibers at the GM–WM interface as a power law of *N*, we would have to introduce a (non-zero) new coefficient at the expression for *A*_{w}. There is unfortunately currently no experimental way of estimating the value of such coefficient. We have assumed throughout that it is small enough to be disregarded, but should it prove to be otherwise we will have to recalculate the other coefficients accordingly, and revisit the conclusion obtained.

Simultaneously, for cortices with average GM thickness T much smaller than the cortical characteristic length so that the internal and external areas of GM scale linearly (that is,

*A*_{G}*A*_{W}), T can be defined as simply the ratio between the volume of the GM,

*V*_{G}, and area of the GM–WM interface,

*A*_{W}. Given that

*V*_{G} scales as a power function of

*N*, with

*N*^{v}, then

and therefore cortical thickness T scales with *N*^{t}, such that

Instead of

*v*, we can use a more biologically meaningful parameter

*d*, which is the exponent relating neuronal density D in the GM to

*N*, given that

*V*_{G}=

*N*·D

^{−1} and that D

~

*N*^{d}. Because

*V*_{G}~

*N*^{1−d}, then

*v*=

1

−

*d*, and the equation above becomes

Average cortical thickness, therefore, scales a function of a GM-related variable (the scaling of neuronal density with *N*), and two WM-related variables that, together with *N*, determine WM folding (the scaling of connectivity and of axonal cross-sectional area with *N*). Remarkably, cortical thickness is therefore not itself a function of *N*, but rather of how exponents *d*, *c*, and α are interrelated.

Finally, the extent of GM folding,

*F*_{G}, can be expressed as the ratio between its actual surface, A

_{G} (which can be written as proportional to

*V*_{G}/T, or

*N*^{1−d−t}), and the exposed surface expected from the total volume (

*V*_{G}+

*V*_{W})

^{2/3}:

Note that although this equation is not an exact power law, it can in practice be well approximated by one since

*V*_{W} and

*V*_{G} scale in fairly similar ways with

*N*. Another way to express

*F*_{G}, now as an exact power law, is by writing the total volume

*V*_{T}=

*V*_{G}+

*V*_{W} itself as a measurable power function of

*N*, varying with

*N*^{z}. In this case,

*F*_{G} becomes

Returning to the first equation of

*F*_{G}, and recalling that

*t*=

−

*d*−

*c*−α, GM folding is thus a combined function of the number of cortical neurons; the fraction of these neurons that are connected through the WM; and the average cross-section area of the axons in the WM. Further, the thickness of the GM is thus a consequence of some of the same parameters that determine how the cortex folds, and not a determinant of it. A schematic of the model is depicted in Figure .