Spatially constrained adaptive rewiring in cortical networks creates spatially modular small world architectures

Abstract

A modular small-world topology in functional and anatomical networks of the cortex is eminently suitable as an information processing architecture. This structure was shown in model studies to arise adaptively; it emerges through rewiring of network connections according to patterns of synchrony in ongoing oscillatory neural activity. However, in order to improve the applicability of such models to the cortex, spatial characteristics of cortical connectivity need to be respected, which were previously neglected. For this purpose we consider networks endowed with a metric by embedding them into a physical space. We provide an adaptive rewiring model with a spatial distance function and a corresponding spatially local rewiring bias. The spatially constrained adaptive rewiring principle is able to steer the evolving network topology to small world status, even more consistently so than without spatial constraints. Locally biased adaptive rewiring results in a spatial layout of the connectivity structure, in which topologically segregated modules correspond to spatially segregated regions, and these regions are linked by long-range connections. The principle of locally biased adaptive rewiring, thus, may explain both the topological connectivity structure and spatial distribution of connections between neuronal units in a large-scale cortical architecture.

Keywords: Adaptive rewiring; Coupled chaotic maps; Modular small world networks; Spatial self-organization; Synchronization.

Fig. 1

Cost functions of spatial distance: linear in blue, exponential in green, logarithmic in red. (Color figure online)

Fig. 2

Evolution of a the clustering coefficient values $C$ averaged over five runs; and b the average shortest path length values $L$ averaged over five runs; for the non-spatial and spatial rewiring processes, regular lattice on the sphere, and random network

Fig. 3

Evolution of a the clustering coefficient; and b the average shortest path length; for the non-spatial and spatial rewiring processes, regular lattice on the sphere, and random network. Individual runs in blue and their average value in red. (Color figure online)

Fig. 4

Evolution of a the spatially-weighted clustering coefficient vaues $C^w$ averaged over five runs; and b the network wiring cost values $M$ averaged over five runs; for the non-spatial and spatial rewiring processes, regular lattice on the sphere, and random network

Fig. 5

Evolution of a the spatially-weighted clustering coefficient values $C^w$; and b the network wiring cost values $M$; for the non-spatial and spatial rewiring processes, regular lattice on the sphere, and random network. Individual trials in blue and averaged value of five runs in red. (Color figure online)

Fig. 6

Linear correlation coefficient $\mathit{\rho }$ between edge betweenness and spatial distance averaged over five runs versus rewiring iterations. a The full course of rewiring; and b early course of rewiring

Fig. 7

Scatter plots of edge betweenness versus spatial distance. Betweenness values presented here were obtained by uniformly randomly selecting 4 % of nodes from the combined five runs and plotting the betweenness values of all their connections. Rows top to bottom for rewiring steps, $1,0.75\times {10}^3,1.5\times {10}^4,5\times {10}^4,3\times {10}^5$, columns are for different rewiring processes

Fig. 8

Evolution of the value of the modularity $Q$. a The average of five runs; and b the individual runs; for the non-spatial and spatial rewiring processes, regular lattice on the sphere, and random network. Individual runs in blue and their average values in red. (Color figure online)

Fig. 9

Permuted adjacency matrices that correspond to the module structure of the non-spatial, linear, exponential, and logarithmic rewiring processes. A point with coordinates $(i^{\prime },j^{\prime })$ is white if $i^{\prime },j^{\prime }$ are are the permuted indices of nodes $i$, $j$ that are connected; otherwise it is black

Fig. 10

Final community structure of one run of the linear rewiring process. a, b Opposite hemispheres. Nodes are coloured according to the module to which they belong

Fig. 11

Final community of the logarithmic rewiring process. a, b Opposite hemispheres. Nodes are coloured according to the module to which they belong

Fig. 12

Clustering coefficient, $C$, and average shortest path length $L$, for a, b non-spatial; c, d linear; and e, f exponential cost functions as function of edge density. Maximum, average, and minimum values from the five independent runs are shown, along with values for the corresponding random and regular graphs

Fig. 13

Small-worldness measure $\mathrm{\Sigma }$ averaged over five runs for non-spatial, linear, and exponential rewiring processes as a function of edge density

Fig. 14

Average value of absolute error between the cumulative number of points (normalised to unity) and the surface integral (normalised to unity) as a function of azimuthal angle