Optimal hierarchical modular topologies for producing limited sustained activation of neural networks

Abstract

An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable regimes of network activation, typically arising from a limited parameter range. In this range of limited sustained activity (LSA), the activity of neural populations in the network persists between the extremes of either quickly dying out or activating the whole network. Hierarchical modular networks were previously found to show a wider parameter range for LSA than random or small-world networks not possessing hierarchical organization or multiple modules. Here we explored how variation in the number of hierarchical levels and modules per level influenced network dynamics and occurrence of LSA. We tested hierarchical configurations of different network sizes, approximating the large-scale networks linking cortical columns in one hemisphere of the rat, cat, or macaque monkey brain. Scaling of the network size affected the number of hierarchical levels and modules in the optimal networks, also depending on whether global edge density or the numbers of connections per node were kept constant. For constant edge density, only few network configurations, possessing an intermediate number of levels and a large number of modules, led to a large range of LSA independent of brain size. For a constant number of node connections, there was a trend for optimal configurations in larger-size networks to possess a larger number of hierarchical levels or more modules. These results may help to explain the trend to greater network complexity apparent in larger brains and may indicate that this complexity is required for maintaining stable levels of neural activation.

Keywords: brain connectivity; cerebral cortex; functional criticality; modularity; neural dynamics; neural networks.

Figure 1

Overview of variation of granularity and scales in the explored hierarchical modular networks. The plots show the outcome of 100 realizations of networks with 128 nodes and 4,096 directed edges. Gray level shading of the adjacency matrix indicates the frequency with which an edge was established (white: never established; black: established in all 100 generated networks). (A) Random networks without hierarchical structure, resulting from h = 0 (number of hierarchical levels) and m = 0 (number of sub-modules); (B) Flat modular networks with four modules, resulting from h = 1, m = 4; (C) Hierarchical modular networks with h = 2, m = 4. Note that each hierarchical level contains the same number of edges, resulting in 16 modules at the lowest hierarchical level in (C), which possess the highest edge density.

Figure 2

Determining the parameter range of limited sustained activity (schematic overview). (A) For several trials (shown here: 30 runs), it was tested whether activity spread through the whole network (here: activating 80% of all nodes), died out (all nodes becoming inactive), or was sustained at an intermediate level (here: activating 10 or 20% of all nodes). Note that even during complete spreading, not all the nodes were constantly active, due to the inactivation probability v specified in the dynamic model. (B) Simulations were run for different combinations of the number of initially activated nodes i and the localization parameter i₀. For each run, the simulated activity died out (), spread through the whole network (o) or was sustained within a limited compartment of the network (). (C) The parameter space of simulations was further explored for different combinations of deactivation probability v and activation threshold k. Gray levels for each parameter combination in the diagram reflect the percentage of cases giving rise to LSA (from subplot B). The average value across all entries was taken as the final measure of the parameter range of LSA for a particular network topology. It reflects the average proportion of limited sustained activation cases obtained across all parameter settings for a given hierarchical modular network.

Figure 3

Examples of neural dynamics for different simulation outcomes. Gray shading represents modules and individual gray levels represent different sub-modules. Nodes which are active at a time step are represented as blue dots. (A) Expiring (dying-out) activity. (B) Limited sustained activity. Although some modules appear completely activated, nodes can be inactive at various time steps due to the inactivation probability (inset). (C) Completely spreading activity.

Figure 4

Range of limited sustained activity for different hierarchical configurations of a small network. Shown is the parameter range of limited sustained activation and of topological features for a network with 512 nodes and average node degree of 50. Regions blocked by horizontal lines indicate configurations that were not admissible (see Materials and Methods). Parameters were explored for 1,000 runs of each set of spreading parameters k and v, while the number of initially activated nodes i and the localization parameter i₀ varied for each run. (A) Average of the number of parameter settings leading to LSA. (B) Maximum edge density based on the most highly connected modules (modules at the lowest level of the respective hierarchy). (C) Characteristic path length of the networks. (D) Average clustering coefficient of the networks. (E) Normalized characteristic path length (divided by the value for Erdös–Rényi random networks). (F) Normalized average clustering coefficient (divided by the value for Erdös–Rényi random networks).

Figure 5

Scaling of optimal configurations with network size for constant global edge density. Proportion of cases showing LSA (averaged over 200 generated networks for each configuration) in (A) “rat-size” networks with 300 columns, (B) “cat-size” networks with 4,150 columns, (C) “macaque-size” networks with 11,000 columns.

Figure 6

Scaling of optimal configurations with network size for constant average node degree (〈k〉 = 50). Proportion of cases showing LSA (averaged over 200 networks generated for each configuration) in (A) “rat-size” networks with 300 columns, (B) “cat-size” networks with 4,150 columns, and (C) “macaque-size” networks with 11,000 columns.

Figure 7

Varying the number of edges per hierarchical level. (A) Decreasing number of edges for higher hierarchical levels [E_i ∼ (2/3)ⁱ]. (B) Number of edges independent from hierarchical level (E_i = const.). (C) Increasing number of edges for higher hierarchical levels [E_i ∼ (3/2)ⁱ].

Figure 8

Varying the parcellation (number of sub-modules per module) for hierarchical levels. (A) Decreasing number of sub-modules m_i for higher hierarchical levels (m_i ∼ 0.9ⁱ). (B) Parcellation into sub-modules independent from hierarchical level (m_i = m = const.; see main text). (C) Increasing number of sub-modules m_i for higher hierarchical levels (m_i ∼ 1.1ⁱ).

Figure 9

Final activity for n = 20 runs classified as sustained activity. (A) Minimal final activity level. (B) Average final activity level. (C) Maximum final activity level.

Figure 10

Proportion of cases resulting in one of three scenarios of final activity level. (A) Activity dying out. (B) Limited sustained activity. (C) Activity spreading through the network (above 50% activation threshold).

Figure 11

Varying the edge density in a network with 512 nodes. (A) Decreased edge density d = 5%, average node degree 〈k〉 = 25. (B) Original edge density d = 10%, average node degree 〈k〉 = 50. (C) Increased edge density d = 20%, average node degree 〈k〉 = 100.