Global optimization of cerebral cortex layout

Abstract

Functional areas of mammalian cerebral cortex seem positioned to minimize costs of their interconnections, down to a best-in-a-billion optimality level. The optimization problem here, originating in microcircuit design, is: Given connections among components, what is the physical placement of the components on a surface that minimizes total length of connections? Because of unfeasibility of measuring long-range "wire length" in the cortex, a simpler adjacency cost was validated. To deal with incomplete information on brain networks, a size law was developed that predicts optimization patterns in subnetworks. Macaque and cat cortex rank better in this connection optimization than the wiring of comparably structured computer chips, but somewhat worse than the macroeconomic commodity-flow network among U.S. states. However, cortex wiring conforms to the size law better than the macroeconomic patterns, which may indicate cortex optimizing mechanisms involve more global processes.

Fig. 1.

Adjacency rule conformance, vs. total wire cost, of 100,000 C. elegans ganglion layouts randomly sampled from the set of all 11! possible layouts (2). Correlation between adjacency rule performance and wire cost is not strong (r² = 0.051); in general, the adjacency rule is not an effective means to good wire cost. However, the small set of layouts best fitting the adjacency rule (the points at the far left) behave markedly differently: they correspond closely to the best wire cost layouts. The larger point at the far left represents the actual, minimum wire cost layout. Thus, good adjacency rule scores seem worth exploring as a surrogate for layout wire cost (see Fig. 7).

Fig. 6.

Size law for three layouts of the AMI49 chip. In each case, the system of components is 15 contiguous central blocks, as in Fig. 5 (connections and adjacencies for Lin and Chang are as in Table 5). Optimality measure is conformance of the system to the adjacency rule, with a layout scored in terms of its number of all-or-nothing violations. A series of nested compact subsets of the set of blocks was generated, each consisting of from 5 to the full 15 areas. Each subset of the actual layout was compared with all possible alternative layouts of that subset for adjacency-rule optimality (14- and 15-element sets were each compared only with random samples of 10⁹ alternative layouts). The curve for the Lin and Chang layout (C) shows a similar but weaker size law trend as the cortex networks earlier; the full 15-component subset only attains an optimality rank of 1.5 × 10^–3. Both the Esbensen and Kuh (30) layout (A) and the Hong et al. (31) layout (B) show no size law pattern.

Fig. 2.

Global vs. local optimization. A simple illustration shows that connection-minimization of a total system does not entail connection-minimization of its subsets. The total system here consists of a 1D array of movable components, 1–3, with fixed edge-terminal (vertical bar) at left. All connections are of equal cost per unit length (horizontal only). Besides internal connection 2–3, 1 and 2 go to the left edge. (A) A globally optimal layout (cost: 4). However, if the system-subset is restricted only to components 2 and 3 with their outgoing connection to the left edge, then the 2 and 3 layout is (locally) suboptimal (cost: 3) compared with a layout with positions of 1, 2, and 3 swapped (cost: 2), as in B. In contrast, the complete layout (B) is locally optimal for subsystem 2 and 3, but at the expense of a higher cost for the total layout (cost: 5).

Fig. 3.

Parcellation of macaque cerebral cortex. Connection-cost optimization analysis of layout of 17 core areas of the visual cortex (white), along with 10 immediately contiguous “edge” areas (dark gray). Placement of the inter-connected functional areas is evaluated for how well total interconnection costs are minimized. A total of 120 connections are reported among the core areas and with the edge areas. Core and edge areas are listed in Table 1. Rostral is to right. Figure is after Felleman and Van Essen (figure 2 in ref. and figure 6 in ref. 24); areas MIP and MDP have been included in PO.

Fig. 4.

Size law for cortex areas. In each case, a series of nested compact subsets of the set of cortical areas was generated, each consisting of from four to the full set of areas. Each subset of the actual layout was compared with all possible alternative layouts of that subset for optimality; optimality-measure is conformance of the system to the adjacency rule (2). Sixteen- and 17-element sets were each compared only with random samples of 10⁹ alternative layouts. (A) The system of components is 17 contiguous macaque visual cortex areas as in Fig. 3, with connections and adjacencies as in Table 1, and order of successive elements added as in Table 1. (B) Similar analysis for 15 cat visual cortex areas. (C) Fourteen cat cortex metamodules composed from 40 Brodmann areas of visual, auditory, and somatosensory regions (see Figs. 8, 10, and 11, which are published as supporting information on the PNAS web site). In each case, the actual layout curve (diamonds) shows that smaller subsets rank approximately in the middle of their group of alternative layouts. But, as subset size increases, optimalityranking of the actual layout consistently improves (with one or two exceptions in each series, P < 0.02). E.g., for macaque, fewer than one in a million of all alternative layouts conform to the adjacency rule better than the actual layout of the complete macaque set. For comparison, each scrambled layout curve (circles) shows the corresponding analysis for layouts of the areas with their adjacencies randomly shuffled; no size law trend toward improving optimality is now evident.

Fig. 5.

Integrated circuit network for calibration of optimality analysis: AMI49 microchip, the largest of the MCNC set of benchmark circuits, with 49 modules, Lin and Chang layout. Cost to be minimized is total wire length. The central 15 blocks (white), along with the surrounding edge-zone of immediately contiguous blocks (dark gray), were analyzed. Again, placement of the interconnected areas is evaluated for how well total interconnection costs, adjacency rule violations, are minimized (see Figs. 12 and 13 and Table 5).