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Criticality of spreading dynamics in hierarchical cluster networks without inhibition
Marcus Kaiser, Matthias Goerner, Claus C. Hilgetag
TL;DR
The paper asks how neural networks sustain activity between rapid extinction and global spread, and tests whether topology can support that balance without inhibition. Simulations find persistent, contained activation in clustered networks but not random networks of the same size, with hierarchical clusters supporting it across a wider parameter range than small-world networks. The authors propose hierarchical cluster architecture as a possible structural basis for stable, scalable cortical activity.
Problem
The paper asks what structural and functional parameters let neural networks maintain persistent activity without either dying out quickly or spreading across the whole network.
Source: §1 · p. 2Method
Results
Persistent but contained activity occurred in clustered networks but not random networks of the same size, and hierarchical clusters had a substantially larger critical parameter space than small-world networks.
Source: §4 · pp. 7–8Takeaways & Limitations
The authors conclude that hierarchical cluster architecture may provide a structural basis for stable, scalable functional patterns in cortical networks.
Source: §4 · p. 8The model omits explicit inhibition, so its results address topology's contribution without that mechanism.
Source: §1 · p. 2
Abstract
from arXiv · showhide
An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable network activations within a limited critical range. In this range, the activity of neural populations in the network persists between the extremes of quickly dying out, or activating the whole network. The nerve fiber network of the mammalian cerebral cortex possesses a modular organization extending across several levels of organization. Using a basic spreading model without inhibition, we investigated how functional activations of nodes propagate through such a hierarchically clustered network. The simulations demonstrated that persistent and scalable activation could be produced in clustered networks, but not in random networks of the same size. Moreover, the parameter range yielding critical activations was substantially larger in hierarchical cluster networks than in small-world networks of the same size. These findings indicate that a hierarchical cluster architecture may provide the structural basis for the stable and diverse functional patterns observed in cortical networks.
1. Introduction
The paper asks how neural networks maintain persistent but contained activity, and whether network topology can help sustain this balance without inhibition. It investigates whether hierarchical modular structure supports such dynamics.
1. Introduction
- The central question is which structural and functional parameters keep neural activity between rapid extinction and whole-network spread.The introduction also asks what limits persistent activity from spreading across the brain.Source: §1 · p. 2
- Most models rely on inhibitory and excitatory interactions, while this study examines topology's contribution without explicit inhibition.Cortical networks are described as hierarchically modular, with denser connections within modules than between them.Source: §1 · p. 2
- The basic spreading model tests whether clustered topologies support persistent, contained activations, with possible relevance beyond neural networks.The authors note that the model may also apply to spreading in social or computer networks.Source: §1 · p. 2
2. Materials and Methods
The simulations compare random, small-world, and hierarchical cluster networks using a basic spreading model without inhibition. They vary initial activity and its localization to examine network activation.
2. Materials and Methods
- Simulations compared random, small-world, and hierarchical cluster networks, each with 1,000 vertices and 12,000 edges.The small-world networks varied in the fraction of randomly wired links; the hierarchical networks divided nodes into clusters and sub-clusters.Source: §2 · pp. 2–3
- Hierarchical networks allocated 4,000 edges within sub-clusters, 4,000 within clusters, and 4,000 across the network.Each network had 10 clusters of 100 vertices, each divided into 10 sub-clusters of 10 vertices.Source: §2 · p. 3
- The basic spreading model updated active and inactive nodes in discrete time, without explicit inhibition.Inactive nodes activated when connected to at least k active nodes; active nodes deactivated with probability ν.Source: §2 · p. 3
- Initialization varied both the number of active nodes and their localization, using consecutive node numbering to target sub-clusters or clusters.Smaller i0 values produced more localized initial activity.Source: §2 · p. 3
- In small-world topology comparisons, sustained intermediate activity—neither dying out nor spreading through the whole network—occurred only for p between 0.4 and 0.6, so simulations used p = 0.5.Source: §2 · p. 3
3. Results
Hierarchical cluster networks produced persistent, contained activity across a broader range of initial conditions than the comparison networks. Activity patterns also depended on where activation began and on the distribution of connections across network levels.
3. Results
- Hierarchical networks showed more varied behavior than random or small-world networks, including sustained activity limited to one or two clusters.Across 30 simulations per network with 10% of nodes initially active, random networks usually died out or spread quickly; small-world networks eventually tended toward all-or-nothing activation.Source: §3 · p. 4
3.1. Critical Range of Nonlocalized Activation
- For unlocalized initial activity, the critical range was around i = 65 to i = 105 in small-world and hierarchical networks; random networks showed only all-or-none activation.The estimate was based on 100 simulations for each network and initial activation value.Source: §3.1 · p. 4
3.2. Influence of Localization
- With i = 60, random-network activity always died out, while localization affected spreading in small-world and hierarchical networks.More focused initial activation could yield whole-network spread or persistent subregion activity, especially in hierarchical networks; small-world intermediate activation could partly reflect slow spread.Source: §3.2 · pp. 4–5
3.3. Delay of Activation Spreading
- Random-network spreading, when it occurred, was typically under 10 time steps; hierarchical networks reached maximum delay with initial activity localized within two or three clusters.For hierarchical networks, a delay around 40 steps was reported for i0 = 200, i = 90; overly strict localization did not lead to global spread.Source: §3.3 · p. 5
3.4. Parameter Range for Persistent Contained Activity
- Hierarchical networks had sustained activity in a larger share of tested cases than small-world networks, with maximum ratios of 67% and 30%, respectively.Reducing within-cluster or sub-cluster connections reduced sustained activity, while increasing those densities raised it; fewer inter-cluster connections could permit containment across more than two clusters.Source: §3.4 · p. 6
- Persistent contained activity was robust across a large range of conditions in hierarchical networks, whereas small-world networks required a relatively small parameter range.The authors report this as the overall parameter-range comparison.Source: §3.4 · p. 6
3.5. Exhaustion
- Persistent contained activity remained possible under different node-exhaustion limits, with an average sustained-activity ratio of 0.272±0.068 across spreading-parameter pairs.As the consecutive-active-step limit fell, a lower deactivation parameter ν was sufficient to maintain balance.Source: §3.5 · pp. 6–7
4. Discussion
The discussion attributes persistent, contained activation to hierarchical cluster topology and identifies a larger critical parameter space than in small-world networks. It presents this architecture as a possible structural basis for stable, scalable activity patterns.
4. Discussion
- Denser within-cluster connections facilitate local activation, while sparser between-cluster connections limit global spread and expand the critical functional range.The authors describe the simulations as a proof of concept for persistent but contained activation without inhibitory nodes.Source: §4 · pp. 7–8
- Hierarchical cluster networks had a substantially larger critical parameter space than small-world networks for limited, sustained activity.The parameter space concerns both initial activation and its localization.Source: §4 · p. 7
- Similar activation patterns in integrate-and-fire simulations suggest the findings may reflect network topology rather than the specific neuronal model.The paper also reports that systematic variation of model parameters did not change the principal behavior.Source: §4 · p. 7
- The studied hierarchical cluster architecture differs from hub-centered hierarchical modular networks, and the authors note no current algorithms for producing it.Single-level clustered architectures are described as possible starting points for modeling multi-level development.Source: §4 · pp. 7–8
- The authors propose that hierarchical architecture may provide a structural basis for stable, scalable cortical patterns, while inhibition and other controls may further extend the critical range.They also state that activity can persist without external random input and that inter-cluster bottlenecks limit spread while dense local connections sustain recurrent activity.Source: §4 · p. 8
- The study abstracts from specific biological neural-network characteristics and may also inform analysis of spreading in other hierarchical systems.The discussion suggests testing connectivity changes in epilepsy and evaluating inhibition at lower organizational levels.Source: §4 · p. 8
Appendix
Sustained activity was less frequent when connections were removed from sub-clusters or clusters than in the original network, but remained more frequent than in small-world networks. Increasing within-cluster connection density was associated with higher sustained-activity rates.
Appendix
- 43.6% of cases had sustained activity in the original network, versus 1.96% for small-world networks and 18.4% or 7.9% in the two reduced-connection arrangements.The reduced arrangements had 1,000 sub-cluster and 7,000 global connections, or 1,000 cluster and 7,000 global connections, respectively.Source: Appendix · pp. 8–9
- 85.1% of cases had sustained activity when within-cluster edge density rose from 17% to 21%, while raising sub-cluster edge density from 90% to 98% yielded 53.7%.Source: Appendix · p. 9