Functional and spatial rewiring principles jointly regulate context-sensitive computation

Abstract

Adaptive rewiring provides a basic principle of self-organizing connectivity in evolving neural network topology. By selectively adding connections to regions with intense signal flow and deleting underutilized connections, adaptive rewiring generates optimized brain-like, i.e. modular, small-world, and rich club connectivity structures. Besides topology, neural self-organization also follows spatial optimization principles, such as minimizing the neural wiring distance and topographic alignment of neural pathways. We simulated the interplay of these spatial principles and adaptive rewiring in evolving neural networks with weighted and directed connections. The neural traffic flow within the network is represented by the equivalent of diffusion dynamics for directed edges: consensus and advection. We observe a constructive synergy between adaptive and spatial rewiring, which contributes to network connectedness. In particular, wiring distance minimization facilitates adaptive rewiring in creating convergent-divergent units. These units support the flow of neural information and enable context-sensitive information processing in the sensory cortex and elsewhere. Convergent-divergent units consist of convergent hub nodes, which collect inputs from pools of nodes and project these signals via a densely interconnected set of intermediate nodes onto divergent hub nodes, which broadcast their output back to the network. Convergent-divergent units vary in the degree to which their intermediate nodes are isolated from the rest of the network. This degree, and hence the context-sensitivity of the network's processing style, is parametrically determined in the evolving network model by the relative prominence of spatial versus adaptive rewiring.

Fig 1. Principles of network rewiring.

(A) Adaptive rewiring. The lightness of a node’s color represents the intensity of its communication with the white node. The darker the color, the more intense the communication is (B) Minimization of wiring distance. (C) Alignment to an external vector field. The red and green arrows indicate the rewiring link and the direction of the vector field respectively.

Fig 2. Schema of a convergent-divergent unit.

In a convergent-divergent unit, a convergent hub collects inputs and passes the information to a divergent hub through a subnetwork of intermediate nodes. The nodes sending information to the convergent hub are referred as source nodes, and those receiving information from the divergent hub as target nodes. Note that typically the source and target nodes can show overlap, i.e., a node can be both a source and a target node.

Fig 3. Schema of the adjacency matrix.

The elements of the adjacency matrix are the weights of links. Each row of the adjacency matrix contains the weights of in-links for the corresponding node, and the number of nonzero entries is its in-degree. Similarly, each column carries the weights of out-links, and the number of nonzero entries is the out-degree.

Fig 4. Rewiring based on the functional principle develops winner-take-all configurations, based on the distance principle forms clusters, and based on the wave principle aligns the connections with the latent field.

(A) Evolution of the adjacency matrix driven by the functional principle only. (B) Evolution of the network spatial layout driven by the distance principle only. (C) Evolution of the network spatial layout driven by the wave principle only when the wave propagates laterally. In all cases, we either rewire the out-links (p_in = 0 case) or the in-links (p_in = 1 case). Link weights follow the normal distribution.

Fig 5. Random rewiring enhances connectedness and increases the number of hubs when rewiring includes both advection and consensus (0<p_in<1).

(A) The proportion of connected node pairs, (B) average efficiency, (C) proportion of convergent hubs, and (D) proportion of divergent hubs as a function of p_random, for different p_in.

Fig 6. p_random controls the formation, connectedness, and degree of isolation of convergent-divergent units.

(A) Proportion of steps with no convergent-divergent unit in the network, (B) number of convergent-divergent units in rewired networks, (C) proportion of source nodes, target nodes and their overlap, (D) proportion of nodes in intermediate subgraphs and (F) density of intermediate subgraphs as a function of p_random. The black horizontal line is the density of the whole digraph.

Fig 7. Distance-based rewiring has similar effects on the connectedness and the number of hubs as random rewiring.

(A) Proportion of connected node pairs, (B) average efficiency, (C) proportion of convergent hubs, and (D) proportion of divergent hubs as a function of p_distance, for different probabilities of in-link rewiring, p_in.

Fig 8. p_distance, controls the formation, connectedness, and degree of isolation of convergent-divergent units.

(A) Proportion of steps with no convergent-divergent unit in the network, (B) number of convergent-divergent units in rewired networks, (C) proportion of source nodes, target nodes and their overlap, (D) proportion of nodes in intermediate subgraphs and (E) density of intermediate subgraphs as a function of p_distance. The black horizontal line represents the density of the whole digraph.

Fig 9. The way the wave principle affects the formation of convergent-divergent units depends on the underlying field.

(A) Spatial layout of a network evolved with a lateral field and (E) with a radial field. Green arrows indicate the direction of the underlying field. The proportion of in-link rewiring is 0.5, and (p_function, p_distance, p_wave) is (0.4,0.3,0.3). (B-D) The proportion of connected node pairs, average efficiency, and the proportion of steps with no convergent-divergent unit in the network, as a function of the distance-based principle, p_distance, with a lateral field, and (F-H) with a radial field.

Fig 10. Stochastic adaptive rewiring reduces the number of steps with no convergent-divergent unit is in the network.

The proportion of in-link rewiring, p_in, is 0.5. The proportion of steps with no convergent-divergent unit in the network, as a function of (A) p_random, and (B) p_distance.