The non-random brain: efficiency, economy, and complex dynamics

Abstract

Modern anatomical tracing and imaging techniques are beginning to reveal the structural anatomy of neural circuits at small and large scales in unprecedented detail. When examined with analytic tools from graph theory and network science, neural connectivity exhibits highly non-random features, including high clustering and short path length, as well as modules and highly central hub nodes. These characteristic topological features of neural connections shape non-random dynamic interactions that occur during spontaneous activity or in response to external stimulation. Disturbances of connectivity and thus of neural dynamics are thought to underlie a number of disease states of the brain, and some evidence suggests that degraded functional performance of brain networks may be the outcome of a process of randomization affecting their nodes and edges. This article provides a survey of the non-random structure of neural connectivity, primarily at the large scale of regions and pathways in the mammalian cerebral cortex. In addition, we will discuss how non-random connections can give rise to differentiated and complex patterns of dynamics and information flow. Finally, we will explore the idea that at least some disorders of the nervous system are associated with increased randomness of neural connections.

Keywords: complex systems; connectome; networks; neural dynamics; neuroanatomy; neuroimaging.

Figure 1

Key graph measures and their definitions. The measures are illustrated in a rendering of a simple undirected graph with 12 nodes and 23 edges. (A) Node degree corresponds to the number of edges attached to a given node, shown here for a highly connected node (left) and a peripheral node (right). (B) The clustering coefficient is shown here for a central node and its six neighbors. These neighbors maintain 8 out of 15 possible edges, for a clustering coefficient of 0.53. (C) Each network can be decomposed into subgraphs of motifs. The plot shows two examples of two different classes of three-node motifs. (D) The distance between two nodes is the length of the shortest path. Nodes A and B connect in three steps, through two intermediate nodes (shown in gray). The average of the finite distances for all node pairs is the graph's path length. (E) The network forms two modules interconnected by a single hub node.

Figure 2

The Watts–Strogatz model of the small world. The network at the upper left hand corner represents a ring lattice with circular boundary conditions. Starting from this configuration connections are randomly rewired with a given rewiring probability p. For p = 0 (no rewiring), the network retains its regular lattice topology. For p = 1 the network is completely random and all lattice-like features have disappeared. Intermediate values of p result in networks that consist of a mixture of random and regular connections. The plot at the bottom shows the clustering coefficient C_p and the path length L_p, both normalized by their values for the regular network (P₀, L₀). Note that there is a broad range for the rewiring probability p where networks have clustering that is similar to that of the regular network, and a path length that is similar to that of the random network. Within this range, networks exhibit small-world attributes. Data computed following the procedure described in Watts and Strogatz (1998), with networks consisting of 1,000 nodes and 10,000 edges (data points represent averages of 400 rewiring steps).

Figure 3

The small-world topology of the macaque neocortex. (A) A structural connectivity matrix of 47 regions of macaque visual and somatomotor cortex, described in detail elsewhere (Sporns et al., 2007). Connections that are present are shown as black squares, absent connections are shown as white squares. (B) A sample of eight random networks with equal number of nodes and edges, and preserved node degrees. These networks were constructed by thoroughly randomizing the network shown in (A) using a random switching algorithm (Maslov and Sneppen, 2002). (C) Clustering coefficient and path length for a population of 250 random networks as well as the real macaque cortex. Networks along the dotted line would have clustering and path length exactly proportional to the random population, and therefore a small-world index of 1 (“no small world”). Networks that fall into the region to the lower right have far greater clustering than path lengths, relative to the random population, and thus a small-world index that is much greater than 1 (“small world”).

Figure 4

Small-world topology and wiring cost in the human cerebral cortex. Connection matrices at the top correspond to the empirically determined connection topology of the right hemisphere of the human cerebral cortex, as reported in Hagmann et al. (2008). The anatomical position of the nodes on the cortical surface is indicated by a gray scale at the left of the matrix. The plot at the right shows an example of a randomized network with equal number of nodes, edges, and equal node degrees. The plot at the bottom shows the small-world index and the wiring cost for the empirical network and a population of 100 randomized networks. Only the empirical network has a small-world index that is much greater than 1, due to high clustering and a path length that is approximately equal to the random case. Progressive randomization (curve) reduces the small-world index while at the same time incurring greater wiring cost. Wiring cost is approximated as the sum of all the Euclidean distances between connected brain nodes. The small-world index is the ratio of the normalized clustering coefficient and the normalized path length (both relative to randomized networks).

Figure 5

Graph evolution for neural complexity. The initial population of graphs in generation 1 consisted of 10 randomized graphs similar to the ones shown in Figure 3B, with 47 nodes and 505 edges. Simple linear dynamics (Galán, 2008) was run on these graphs and the graph generating the highest neural complexity (Tononi et al., 1994) was selected and copied forward to the next generation, as described in Sporns et al. (2000). Then, small random “mutations” were introduced in the graph's “offspring” and the process of selecting for complex dynamics was continued for a total of 50,000 generations. (A) Plots show the increase in complexity and a parallel increase in modularity. (B) Examples of graphs obtained at the end of the simulations exhibit non-random topologies, including higSh modularity and hub nodes.