Cortical Composition Hierarchy Driven by Spine Proportion Economical Maximization or Wire Volume Minimization

The structure and quantitative composition of the cerebral cortex are interrelated with its computational capacity. Empirical data analyzed here indicate a certain hierarchy in local cortical composition. Specifically, neural wire, i.e., axons and dendrites take each about 1/3 of cortical space, spines and glia/astrocytes occupy each about (1/3)2, and capillaries around (1/3)4. Moreover, data analysis across species reveals that these fractions are roughly brain size independent, which suggests that they could be in some sense optimal and thus important for brain function. Is there any principle that sets them in this invariant way? This study first builds a model of local circuit in which neural wire, spines, astrocytes, and capillaries are mutually coupled elements and are treated within a single mathematical framework. Next, various forms of wire minimization rule (wire length, surface area, volume, or conduction delays) are analyzed, of which, only minimization of wire volume provides realistic results that are very close to the empirical cortical fractions. As an alternative, a new principle called “spine economy maximization” is proposed and investigated, which is associated with maximization of spine proportion in the cortex per spine size that yields equally good but more robust results. Additionally, a combination of wire cost and spine economy notions is considered as a meta-principle, and it is found that this proposition gives only marginally better results than either pure wire volume minimization or pure spine economy maximization, but only if spine economy component dominates. However, such a combined meta-principle yields much better results than the constraints related solely to minimization of wire length, wire surface area, and conduction delays. Interestingly, the type of spine size distribution also plays a role, and better agreement with the data is achieved for distributions with long tails. In sum, these results suggest that for the efficiency of local circuits wire volume may be more primary variable than wire length or temporal delays, and moreover, the new spine economy principle may be important for brain evolutionary design in a broader context.


THEORETICAL MODELS
Below, the details of calculaions are provided for the three principles considered in the main text.

The system of basic equations for optimal solution.
Explicit form of the fitness function.
The basic optimal equations.
Reduction of dimensionality in the system of basic equations.
Next, we show that we can decrease the number of basic equations from 4 to 3. First, we can get rid of λ 1 , since it always appears in the first power. The parameter λ 1 can be determined from Eq. (3) and it reads: Next, we can insert λ 1 into Eqs. (2) and (4). As a result we obtain Eqs. (29) and (30) in the main text.

Proof of the local minimum for optimal solution related to wire minimization.
Let us introduce the following notation: Then the function F w (Eq. 1) can be rewritten as: where g(x 1 , x 2 , x 3 ) denotes the constraint term present in Eq. (1). Let us define partial derivatives: F ij = ∂ 2 F w /∂x i ∂x j and g i = ∂g/∂x i , which are determined at the critical point represented by optimal values of x 1 , x 2 , x 3 . Using these definitions we can construct a matrix called bordered Hessian for our constraint optimization problem as [2]: This is a symmetric matrix, i.e. F ij = F ji .
A sufficient condition for F w to have a local minimum at the critical point represented by the optimal values x 1 , x 2 , x 3 is that two principal minors, i.e. determinants of the upper-left sub-matrices 3x3 (called D 1 ) and 4x4 (determinant of the entire bordered Hessian called D 2 ), have negative signs [2]. The explicit forms of these determinants are as follows: and Exact numerical values of the minors D 1 and D 2 are presented in Table A (below) together with the values of F ij . These results indicate that indeed we have local minima at the critical points.  2 Spine economical maximization principle.

The system of basic equations for optimal solution.
Explicit form of the fitness function.
The explicit dependence of the benefit-cost function F s on the three parameters x, y, u is given as The basic optimal equations.
The optimal values of x, y, u, and λ 2 are found by differentiating the benefit-cost function F s (Eq. 10) with respect to x, y, u, and λ 2 , and requiring that appropriate derivatives are zero.
As a result, we obtain the following set of four nonlinear equations: and x + y + P xy + a(P xy) 2/3 Note that from Eqs. (11) and (12) it follows that λ 2 must be negative, since all other terms on the left hand side are positive. This observation is used below for determination of the type of extremum.
Proof that x = y.
First, we show that for optimal x and y we have x = y. To do this, we subtract Eqs. (11) and (12). As a result we get: where the expression in the [...] bracket is equal either to −λ 2 u γ 2 /y (from Eq. 11) or to −λ 2 u γ 2 /x (from Eq. 12). Thus, Eq. (15) is equivalent to the following equation: which implies that for nonzero λ 2 and u we must have x = y. (The benefit-cost function F s is defined only for u > 0, see Eq. 10). If however, λ 2 = 0, then from Eqs. (11) and (12) we get that P x = P y = 0. The case P = 0 implies u = 0 (see eqs relating P and u in the Methods), which however is forbidden. Thus P = 0, and in this case we must have x = y = 0, i.e. x and y are still equal to each other.

Reduction of dimensionality in the system of basic equations.
Next, we show that we can decrease the number of basic equations. Because x = y, we can reduce the system of 4 equations to the system of 3 equations with unknowns x, u, λ 2 (Eqs. 11 and 12 are in fact the same equation). Moreover, we can get rid of λ 2 , since it always appears in the first power, which additionally allows us to reduce the system dimensionality to 2. The parameter λ 2 can be determined from Eq. (11) (with the substitution y = x) and it reads: Next, we can insert λ 2 into Eq. (13). After this procedure Eq.(13) becomes and Eq. (14) after the substitution y = x becomes 2x + P x 2 + aP 2/3 x 4/3 u 2/3 + aP 5/3 x 10/3 u 2/3 = 1.
The derivatives of P with respect to u have different forms depending on the type of density probability of spine volumes H(u) (see the main text).
Eqs. (18) and (19) constitute the reduced system of basic equations, which is used for computations of two independent variables x and u. This two-dimensional system can be solved by a handful of numerical techniques (e.g. [1]).

Proof of the local maximum for optimal solution related to spine economy.
As before, let us introduce the following notation: x 1 ≡ x, x 2 ≡ y, x 3 ≡ u. Then the fitness function F s (Eq. 10) can be rewritten as: where the probability P is a function of x 3 , and g(x 1 , x 2 , x 3 ) denotes the constraint term present in Eq. (10). Let us define partial derivatives: F ij = ∂ 2 F s /∂x i ∂x j and g i = ∂g/∂x i , which are determined at the critical point represented by optimal values of x 1 , x 2 , x 3 . Using these definitions we can construct a matrix called bordered Hessian for our constraint optimization problem as [2]: This particular matrix has a high degree of symmetry, since: g 1 = g 2 , F ij = F ji , and A sufficient condition for F s to have a local maximum at the critical point represented by the optimal values x 1 , x 2 , x 3 is that two principal minors, i.e. determinants of the upper-left sub-matrices 3x3 (called D 1 ) and 4x4 (determinant of the entire bordered Hessian called D 2 ), alternate in sign. Specifically, the principal minors must have respectively positive (D 1 ) and negative (D 2 ) signs [2]. Using the high symmetry in the Hessian matrix, the explicit forms of these determinants are as follows: and where ǫ ≡ g 3 /g 1 . It is relatively easy to show that g 1 ≥ 1, and the expression for ǫ reads In general for all considered distributions of spine volume, the numerical value of ǫ is very small at the critical point, i.e. |ǫ| ≪ 1. Typical values of F ij are in the range (−1.7, 1). Thus, approximately the sign of D 2 is determined by the sign of the product F 33 (F 12 − F 11 ), since other terms in Eq. (22) are much smaller and thus can be neglected. Of these two factors, F 33 is always negative (which comes from a numerical calculation) and (F 12 − F 11 ) = −λ 2 /x is always positive (λ 2 < 0). This implies that D 2 is negative, and D 1 is positive, which is sufficient for the benefit-cost function F s (Eq. 10) to have maximum. Exact numerical values of the rescaled minors D 1 /g 2 1 and D 2 /g 2 1 are presented in Table B (below) together with the values of ǫ and F ij .  3 Combined "wire minimization" and "spine economy maximization" principle.

The system of basic equations for optimal solution.
The optimal values of x, y, u, and λ are found by differentiating the meta fitness function F (Eq. 1 in the main text) with respect to x, y, u, and λ, and requiring that appropriate derivatives are zero. As a result, we obtain the following set of four nonlinear equations: and x + y + s + g + c = 1.

(27)
Reduction of dimensionality in the system of basic equations.