Activation Pattern Perspective in Neural Systems

• Activation pattern perspective is an approach that analyzes the spatial and temporal configuration of activations in neural and artificial networks to explain stability and efficient information processing.
• It emphasizes the role of hierarchical modular architectures in balancing localized activation with global connectivity, thereby preventing runaway excitation and promoting robustness.
• The analysis integrates simulation, theory, and visualization to demonstrate how parameters such as modular levels and connectivity influence limited sustained activity in complex systems.

An activation pattern perspective examines neural, cognitive, or artificial systems through the spatial and temporal configuration of activations—whether in biological substrates (e.g., cortical columns), cognitive states, or artificial networks (e.g., DNNs, GNNs). This approach interprets the functional and dynamical capabilities of networks in terms of how patterns of local or modular activations emerge, percolate, and stabilize, with a particular focus on the constraints imposed by network topology, parameter distribution, and modular organization. It is central to explaining criticality, information processing, stability, and generalization capabilities in both natural and artificial intelligence systems.

1. Limited Sustained Activity and Localized Activation Patterns

Limited Sustained Activity (LSA) is defined as the regime in which, post-perturbation, network activity neither decays to quiescence nor spreads pathologically through all nodes, but instead stabilizes at an intermediate, localized level (usually 10–20% active nodes, but below a global spread threshold, e.g., below 50%). This condition is operationalized in simulations by tracking the global fraction of active nodes following an initial excitation and classifying activation as limited and sustained if it plateaus at an intermediate value over time.

LSA is crucial because it enables complex information integration and processing by preventing both under-responsiveness (all activity dies) and catastrophic over-excitation (activation avalanche) (Kaiser et al., 2010). In biological settings, LSA reflects the criticality inherent in cortical function and is mathematically associated with the emergence of stable functional clusters or avalanches, as observed in empirical neural data.

2. Hierarchical Modular Network Topologies

Hierarchical modular networks comprise nested modular subdivisions (modules within modules, forming multiple levels), each characterized by high internal (intra-module) connectivity and sparse inter-module links. Construction adheres to the principle:

$E_{(i)} = \frac{E}{h + 1}$

where $E$ is total edge count and $h$ is the number of hierarchical levels, distributing connectivity evenly across layers.

Random networks lack modularity; small-world networks possess high local clustering but do not exhibit explicit multi-level organizational structure. Hierarchical modular networks, in contrast, exhibit high clustering coefficients without significant changes to characteristic path length, enabling sustained local activations while still preserving rapid long-distance information propagation. The modular subtending of strong within-module connection density serves as a dynamical buffer—activations remain localized, and global cascades are inhibited unless parameters cross critical thresholds.

3. Topological Determinants of Activation Patterns

Key topological variables governing LSA include the number of modular levels ($h$) and the number of sub-modules ($m$); for constant network size and connectivity, maximizing the number of sub-modules in a shallow hierarchy (e.g., $h=1$) often maximizes LSA parameter range. For constant $h$, increasing modular granularity (higher $m$) further enlarges the LSA domain. The relationship between connectivity and LSA is encapsulated in:

$d = \frac{E}{N(N-1)}$

$\langle k \rangle = \frac{E}{N}$

$d = \frac{\langle k \rangle}{N-1}$

where $N$ is node count and $\langle k \rangle$ is mean node degree.

Hierarchical modular architectures thus balance the tradeoff between integration and segregation, harmonizing local information persistence and global communication potential. Enhanced clustering supports local pattern emergence, while preserved path length maintains system-wide accessibility.

4. Scaling Behavior and Complexity Constraints

Analysis under two scaling regimes—constant edge density versus constant node degree—reveals divergent effects on LSA maintenance (Kaiser et al., 2010):

• Constant edge density ($d$ fixed): As $N$ increases, mean degree $\langle k \rangle$ rises, making system-wide activation more probable by default; only specific configurations with intermediate hierarchies and numerous modules per level support robust LSA at all sizes.
• Constant average degree ($\langle k \rangle$ fixed): The parameter space supporting LSA expands with network size; larger systems tend towards increased hierarchical complexity, i.e., more levels or greater modularity is required to sustain stable activations.

These findings align with comparative neuroanatomical evidence: evolutionary enlargement of mammalian brains (e.g., from rodent to primate cortex) is accompanied by increased hierarchical modularity, interpreted here as a robustness feature for LSA scaling.

5. Implications for Artificial and Biological Systems

The hierarchical modular design principle, grounded in activation pattern dynamics, suggests routes for architectural improvement in artificial neural networks:

• Incorporating configurable modular divisions (multilevel module nesting).
• Preserving or regulating node degree as scale increases.
• Exploiting dense intra-module and sparse inter-module connectivity to achieve robust LSA.

Algorithmic advantages include improved failure tolerance (avoiding global runaway activation), better information compartmentalization, and greater parallelism in information processing.

For neuroscientific interpretation, these results provide a topological rationale for the hierarchical organization of the cerebral cortex: stable processing states, persistent activity, and criticality required for cognitive function are natural consequences of the modular, hierarchical wiring found in biological systems. The translation from network architecture to accessible activation patterns explains observed trends toward more intricate hierarchical modularity in larger mammalian brains and suggests that this complexity is required for maintaining functional regime balance.

6. Integration of Theory, Simulation, and Visualization

Parameter sweeps and dynamic simulations within hierarchical modular networks closely recapitulate empirical features of brain activity and provide a framework for tuning artificial systems toward desirable dynamical regimes. The division of edge budget across hierarchical levels, combined with simulations of spreading activation, allows direct quantitative assessment of LSA regimes and their stability boundaries.

Conceptual diagrams typically illustrate hierarchical modular networks as nested clusters—large modules split into submodules, with visual density indicating local clustering and sparsity of cross-connections. This structural arrangement highlights the pathways through which limited sustained activation patterns can arise and persist.

7. Concluding Synthesis

The activation pattern perspective, as formalized in (Kaiser et al., 2010), establishes that hierarchical modular network topologies are optimally suited for generating and maintaining regimes of limited sustained activity. The number of hierarchical levels, the granularity of module subdivision, and the scaling law for connectivity critically determine the breadth and stability of LSA, offering insight into both biological evolution and artificial system design. The analytical relationships connecting topology to activation regime underpin a unified understanding of how neural systems avoid both under- and over-activation, and provide explicit criteria for architectural optimization in large-scale neural circuits.