Nonoptimal component placement, but short processing paths, due to long-distance projections in neural systems

Abstract

It has been suggested that neural systems across several scales of organization show optimal component placement, in which any spatial rearrangement of the components would lead to an increase of total wiring. Using extensive connectivity datasets for diverse neural networks combined with spatial coordinates for network nodes, we applied an optimization algorithm to the network layouts, in order to search for wire-saving component rearrangements. We found that optimized component rearrangements could substantially reduce total wiring length in all tested neural networks. Specifically, total wiring among 95 primate (Macaque) cortical areas could be decreased by 32%, and wiring of neuronal networks in the nematode Caenorhabditis elegans could be reduced by 48% on the global level, and by 49% for neurons within frontal ganglia. Wiring length reductions were possible due to the existence of long-distance projections in neural networks. We explored the role of these projections by comparing the original networks with minimally rewired networks of the same size, which possessed only the shortest possible connections. In the minimally rewired networks, the number of processing steps along the shortest paths between components was significantly increased compared to the original networks. Additional benchmark comparisons also indicated that neural networks are more similar to network layouts that minimize the length of processing paths, rather than wiring length. These findings suggest that neural systems are not exclusively optimized for minimal global wiring, but for a variety of factors including the minimization of processing steps.

Figure 1. Projection Length Distribution and Total Wiring Length for Original and Rearranged Neural Networks

(A–C) Approximated projection length distribution in neural networks. Macaque monkey cortical connectivity network with 95 areas and 2,402 projections (A). Local distribution of connections within rostral ganglia of C. elegans with 131 neurons and 764 projections (B). Global C. elegans neural network with 277 neurons and 2,105 connections (C). (D–F) Reduction in total wiring length by rearranged layouts yielded by simulated annealing for Macaque cortical network (D), C. elegans local network (neurons within rostral ganglia) (E), and global C. elegans network (F). (G–I) Approximated projection length distribution in neural networks with optimized component placement. Macaque monkey cortical connectivity network (G). Local distribution of connections within rostral ganglia of C. elegans (H). Global C. elegans neural network (I). For all optimized networks, the number of long distance connections is reduced compared to the original length distribution in (A)–(C).

Figure 2. Original and Optimally Rearranged Macaque Cortical Networks

(A) Original placement of 95 cortical areas. (B) Network layout after evolutionary rearrangement of areas to minimize total wiring. A larger version of this figure is available at http://www.biological-networks.org.

Figure 3. Original and Optimally Rearranged Layouts of Local and Global Neural Networks of C. elegans

(A) Original placement of neurons within rostral ganglia. (B) Optimized wire-saving component placement of rostral ganglia neurons. (C) Original layout of global C. elegans network (lateral view). (D) Global C. elegans neuronal network, rearranged to minimize total network wiring. A larger version of this figure is available at http://www.biological-networks.org.

Figure 4. Network Properties of Original Cortical Networks and Minimally Rewired Networks of the Same Size Lacking Long-Distance Connections

Total wiring length (A) is substantially reduced in minimally rewired networks. Average metric length of the shortest path between any two nodes (B) is also lowered in the rewired networks. However, average path length (C), corresponding to the number of processing steps in the shortest path between components, is considerably smaller in the original than in the minimally rewired networks.

Figure 5. Wiring Arrangement of Neural Networks Compared to Minimum and Maximum Case Benchmark Networks

(A) Actual total wiring length relative to the minimum wiring length solution (value “0,” yielded by simulated annealing of component positions) and to networks optimized for maximum total wiring length (value “1,” also yielded by simulated annealing). The wiring of the different neural networks lies close to the middle between minimum and maximum case component arrangements. (B) Average shortest path length (characteristic path length) in neural networks relative to networks optimized for minimum path length (value “0,” yielded by simulated annealing of wiring organization) and maximum path length (value “1”). Actual path lengths in the neural networks are close to the lower bound of networks optimized for minimum paths.