Adaptive rewiring evolves brain-like structure in weighted networks

Abstract

Activity-dependent plasticity refers to a range of mechanisms for adaptively reshaping neuronal connections. We model their common principle in terms of adaptive rewiring of network connectivity, while representing neural activity by diffusion on the network: Where diffusion is intensive, shortcut connections are established, while underused connections are pruned. In binary networks, this process is known to steer initially random networks robustly to high levels of structural complexity, reflecting the global characteristics of brain anatomy: modular or centralized small world topologies. We investigate whether this result extends to more realistic, weighted networks. Both normally- and lognormally-distributed weighted networks evolve either modular or centralized topologies. Which of these prevails depends on a single control parameter, representing global homeostatic or normalizing regulation mechanisms. Intermediate control parameter values exhibit the greatest levels of network complexity, incorporating both modular and centralized tendencies. The simulation results allow us to propose diffusion based adaptive rewiring as a parsimonious model for activity-dependent reshaping of brain connectivity structure.

Figure 1

Example of an initially random network at progressive stages of rewiring.

Figure 2

Networks with binary, normal and lognormal weight distributions develop into small worlds. (A) From left to right, small world values (S) for networks with binary, normal and lognormal weight distributions for different rewiring rates (τ) and random rewiring probabilities (p_random). ε = 10⁻¹⁵, δ = 10¹⁵. (B) S as a function of τ for different values of p_random. The s.e.m lines are not visible since they do not exceed the markers.

Figure 3

Networks can have the same small world value but diverging connectivity pattern. (A) Example of a modular network (τ = 3) where nodes within clusters are densely intraconnected but sparsely interconnected between clusters. (B) Example of a centralized network (τ = 5) where there is a degree imbalance between nodes, with the vast majority of the nodes having very few connections and the rest being heavily connected, acting as hubs. Both networks have the same small world value (S = 3.4) despite their different topological characteristics. In both cases p_random is 0.2 and the weight distribution is normal. The color scaling indicates the weights of the connections.

Figure 4

Modularity profiles between networks with binary, normal and lognormal weight distributions have similar shape. (A) From left to right, Modularity index Q for networks with binary, normal and lognormal weight distributions for different values of the system parameters τ and p_random. (B) Q as a function of τ for different p_random. The s.e.m. lines do not exceed the markers.

Figure 5

Degree distribution for modular networks is close to the mean value; for centralized networks it is heavy tailed. (A) Proportion of nodes with outlier degrees as a function of τ for binary, normal and lognormal networks. The s.e.m. lines do not exceed the markers. (B) From left to right, the distribution of degrees for binary, normal and lognormal networks (τ_binary = 2, τ_normal = 3, τ_lognormal = 4.5) which are modular. Inset plots show the strength distributions. C. Same as in B, but for centralized networks (τ_binary = 5, τ_normal = 5, τ_lognormal = 7). In all cases p_random = 0.2.

Figure 6

Weighted networks show the most prominent topological rich club for rewiring rates in the modular and transition ranges, binary networks only in the modular range. Topological normalized rich club, Φ_norm(k), for binary, normal and lognormal networks in the (A) modular (τ_binary = 2, τ_normal = 3, τ_lognormal = 4.5) (B) transition (τ_binary = 4.1, τ_normal = 4.15, τ_lognormal = 5.5) and (C) centralized (τ_binary = 5, τ_normal = 5, τ_lognormal = 7) state. In all cases p_random = 0.2. Vertical lines indicate s.e.m.

Figure 7

Topological and weighted rich club coefficients diverge similarly to physiological data. Topological and weighted rich club coefficient for the normal network at τ_transition = 4.15, p_random = 0.2. Vertical lines indicate s.e.m.