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

Marcus Kaiser, Claus C. Hilgetag

arXiv:1003.3081v1q-bio.NCcond-mat.dis-nnphysics.soc-ph

TL;DR

Complex neural networks need activity regimes that persist without dying out or spreading everywhere. This study simulated hierarchical modular networks across hierarchy, granularity, size, and connectivity constraints, finding that optimal configurations depended on scaling choice: constant edge density favored few configurations, whereas constant node degree preserved broader scalable ranges and favored greater hierarchy in larger networks.

  • Problem

    The study asks which hierarchical network structures support limited sustained activity, an essential precondition for stable functional patterns in neural systems.

  • Method

    The authors simulated activity spreading in hierarchical modular networks while varying hierarchical levels, sub-modules, network size, and connectivity constraints.

  • Results

    Constant edge density yielded few optimal configurations, whereas constant average node degree enabled up to 30% sustained-activity cases across network sizes and favored more hierarchical levels in larger networks.

  • Takeaways & Limitations

    The results suggest that increasing hierarchical complexity may help maintain stable neural activation as network size increases when average node degree remains constant.

  • Takeaways & Limitations

    The model assumed uniform cortical-column building blocks, omitted explicit internal column organization, and used phenomenological deactivation rather than detailed inhibitory mechanisms.

Abstract

from arXiv · show

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.

1. Introduction

The study asks which structural and functional parameters let neural networks sustain activity without either rapid extinction or whole-network activation. It focuses on hierarchical modular organization across scales and tests how hierarchy, modularity, and network size relate to limited sustained activity (LSA).

  • Critical neural dynamics require activation patterns that persist between rapid die-out and excitation of the whole network.
  • Brain networks exhibit modular and hierarchical organization across cellular, mesoscopic, and global scales, although its precise organization remains unsettled.
  • Earlier work found that hierarchical multi-modular networks supported limited sustained activity better than same-size random or simple small-world networks.
  • The present study varies hierarchical levels, sub-modules, and network size while preserving the model’s core topology and dynamic mechanisms.
  • The model represents cortical columns with exclusively excitatory connections between columns.

2. Materials and Methods

The researchers generated hierarchical modular networks representing cortical-column systems of different sizes and simulated threshold-based activity spreading. They varied hierarchy, modular granularity, initialization, and dynamic parameters under alternative connectivity constraints.

  • Networks approximated rat-like 300-node, cat-like 4,150-node, and macaque-like 11,000-node hemispheric cortical systems.The estimates assumed each macro-column occupied 1 mm^2.
  • The model distributed the total edge count equally across hierarchical levels, preserving a constant number of edges when hierarchy or granularity changed.For h+1 levels, each level received E_i = E / (h+1) edges.
  • Hierarchical levels ranged from zero random-network levels to four, while each module’s number of sub-modules was varied in steps of two.
  • A discrete-time threshold model activated randomly selected nodes initially, activated inactive nodes when at least k neighbors were active, and deactivated active nodes with probability v.
  • The simulations varied initial activation and localization, then evaluated LSA across k and v using repeated runs observed after 200 time steps.A sustained trial ended with at least one and at most 50% of nodes active.
  • Some configurations were inaccessible when the smallest module required more edges than could exist between its members.The condition was N_c(N_c - 1) < E_c.

3.1. Expiring, limited sustained and completely spreading activity patterns

Simulations produced three activity regimes: rapid extinction, limited sustained activity within network compartments, and activity spreading across the network. Their differences arose from whether active neighbors could maintain or propagate activation.

  • Figure 3 encodes modules with gray shading, sub-modules with distinct gray levels, and active nodes at each time step with blue dots.
  • Expiring activity quickly died out because deactivation left too few active neighbors to sustain it.
  • Limited sustained activity activated modules and sub-modules while remaining contained within a limited network compartment.A critical number of active neighbors could re-activate nodes, while the inactivation probability still left some nodes inactive at individual time steps.
  • Completely spreading activity escaped its initially activated module or sub-modules and rapidly produced whole-network activation.

3.2. Topological and small-world properties of hierarchical networks

Hierarchical networks displayed small-world characteristics across tested sizes and connectivity constraints. Their clustering exceeded random-network values while path lengths remained comparable or changed with network size and constraint.

  • For every tested network size, hierarchical networks had much higher clustering coefficients than same-size Erdős-Rényi random networks while retaining comparable characteristic path lengths.Networks with one hierarchical level were the special case of simple modular networks.
  • Under constant edge density, the tested networks used 1.2% density with two hierarchical levels and four sub-modules per module.Table 1 reports C, C_rand, L, L_rand, and SW.
  • Under constant edge density, characteristic path length was particularly high for the 300-node network because its low density provided fewer alternative pathways.
  • Under constant average node degree, path length increased with network size, while the small-world index was lower for 300- and 512-node networks than under constant edge density.The average number of connections per node was held at 50.

3.3. Optimal hierarchical configurations for LSA in a small network

In a 512-node network, LSA was most extensive for shallow hierarchies with many sub-modules, while topological measures and density did not map simply onto sustained activity.

  • The maximum LSA range occurred with one hierarchical level and the largest possible number of sub-modules per module.
  • Increasing sub-modules generally increased the LSA parameter range, but this pattern was less consistent as the number of hierarchical levels grew.
  • Some hierarchical configurations were inadmissible when required edges exceeded the number of possible edges within the smallest module.The condition was Nc (Nc-1) < Ec.
  • Bottom-module edge density showed no clear overall relationship with sustained activation across hierarchical configurations.
  • LSA classifications were based on activity outcomes after 200 time steps across tested spreading parameters, with additional runs varying initial activation and localization.
  • Characteristic path lengths for hierarchical networks were comparable to Erdös-Rényi random networks, while clustering increased with hierarchical levels and sub-modules.Path lengths ranged from 5.3 to 6.4.

3.4. Scaling of optimal hierarchical configurations for LSA with network size

The study compared LSA across rat-, cat-, and macaque-size hierarchical networks under constant global edge density versus constant average node degree. Scaling favored few robust configurations under fixed density, but broader optimal configurations under fixed degree.

  • 3.4.1. Constant edge density: For constant global density, within-module connection density was higher and between-module density lower, while the global average remained 1.2%.
  • 3.4.1. Constant edge density: Up to 25% of tested parameter settings produced LSA in cat-size networks, compared with up to 3% in rat-size networks.
  • 3.4.1. Constant edge density: Under constant global edge density, the variety of configurations producing LSA decreased with network size, leaving few robust configurations.
  • 3.4.1. Constant edge density: Only configurations combining an intermediate number of hierarchical levels with many sub-modules appeared suitable for LSA across all network sizes.
  • 3.4.2. Constant average node degree: The constant-degree scenario kept 50 connections per node while allowing edge density to vary across network sizes.
  • 3.4.2. Constant average node degree: With constant average node degree, the maximum LSA range was comparable across sizes at 15-30%, and broad configuration ranges persisted as networks grew.

4. Discussion

The study shows how hierarchical levels, module counts, network size, and connection constraints shape limited sustained activity (LSA). Constant average degree preserves more viable configurations as networks grow, while model assumptions delimit biological interpretation.

  • Study contribution: Hierarchical modular networks supported LSA across multiple combinations of hierarchical levels and sub-modules, extending earlier topology comparisons.The study varied levels, sub-modules, network size, and either constant edge density or constant average node degree.
  • Scaling constraints: Constant edge density reduced the number of sustained-activity configurations as network size increased.For large networks, two levels with the maximum possible number of sub-modules often performed best, but could require high lowest-level edge density.
  • Scaling constraints: For larger networks, increasing hierarchical levels can help balance the number and size of modules when module counts also rise.The modeling results support adding levels rather than only increasing modules in large networks.
  • Biological interpretation: The topology reflects distributed, nested modularity across biological neural scales, from cortical columns to large-scale brain divisions.The model focuses on modular and hierarchical organization rather than relying on global hubs.
  • Limitations: The model simplifies biology by using uniform column nodes, implicit self-loops, symmetric module partitioning, phenomenological deactivation, and no external inputs.The findings may therefore apply especially to situations with limited external input, such as sleep or early development.
  • Scaling constraints: Constant average degree preserved a wider range of hierarchical configurations, with up to 30% sustained-activity cases across network sizes.Larger networks under this constraint tended toward more hierarchical levels, suggesting a role for intricate topology in dynamical stability.
  • Interpretive scope: LSA is a necessary condition for criticality but does not guarantee criticality, and alternative dynamic constraints may also matter.The study contrasts LSA with trivial all-or-none activation and mentions synchronization, functional complexity, information propagation, and processing speed as additional possibilities.

Disclosure/Conflict of Interest

The authors report no commercial or financial relationships that could be construed as a conflict of interest.

  • The research was conducted without commercial relationships that could be construed as a conflict of interest.
  • The research was conducted without financial relationships that could be construed as a conflict of interest.
  • The authors declare no relationships meeting the stated conflict-of-interest criterion.

Appendix A: Control experiments

Control experiments varied edge allocation, parcellation, edge density, and classification conditions in hierarchical networks. These perturbations generally preserved sustained-activity patterns, while increased parcellation substantially raised the maximum proportion of LSA cases.

  • Control-experiment setup: The simulations used 512-node networks with 50 connections per node for the additional control experiments.
  • Varying the number of edges: Configurations with high sustained-activity probabilities remained comparable when the number of edges per hierarchical level decreased or increased.Decreased edge counts lowered the absolute sustained-activity level, whereas increased edge counts raised it and produced cases across two or more levels.
  • Varying the parcellation: Increasing parcellation extended the maximum proportion of sustained activity cases from 0.23 to 0.42.The absolute sustained-activity parameter range was almost twice as high as in the original setting.
  • Classification controls: Final activity levels were around 10-20% in configurations reporting many sustained-activity cases, indicating that the 50% threshold did not drive those results.Cases with extensive spreading had few sustained-activity outcomes.
  • Outcome distributions: For configurations with few LSA cases, dying-out and complete spreading each occurred in about 50% of cases under the reported conditions.With more than one hierarchical level, complete spreading occurred more often than dying-out.
  • Varying the edge density: Varying edge density from 5% to 20% preserved the overall sustained-activity pattern, with maximum LSA ranging from 0.172 to 0.238.The original maximum level was 0.238.
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