Avalanches of Activation and Spikes in Neuronal Complex Networks

• ER, BA, PN, and PA networks yield abrupt activation after initial steps, whereas WS and GG networks show more gradual dispersion from the source.Source: §I · p. 1
• PN dynamics can contain a later secondary avalanche, while GG dynamics shows only a gradual and smooth increase in spiking.Source: §I · p. 1, Figure 1 · p. 2
• Activation timing is related to community organization, supporting community identification through the first non-zero activation times received by neurons.Source: §I · pp. 1–2
• The paper characterizes source-centered concentric neighborhoods using hierarchical node counts, hierarchical degrees, and intra-ring degrees.Source: §I · p. 1
• A chain complex neuronal network provides a mean-field equivalent whose avalanche behavior follows the number of nodes at each concentric level.Source: §I · p. 2

• An h-neighborhood contains nodes at shortest-path distance h from reference node i, forming successive concentric or hierarchical levels.Source: §II · p. 2
• Hierarchical degree k_h(i) counts edges between levels h and h+1, while intra-ring degree a_h(i) counts edges within level h.Source: §II · pp. 2–3
• The concentric representation can include node counts n_0(i)=1, n_1(i)=3, n_2(i)=7, and n_3(i)=4, alongside hierarchical and intra-ring degrees.Source: §II · p. 3, Figure 2 · p. 3
• The study compares ER, BA, WS, GG, PN, PA, and PI network models, with PI modifying PN so all nodes have identical degrees.Source: §II · p. 3
• Each integrate-and-fire neuron has an integrator, memory state, and hard-limiter threshold, firing synchronously and distributing accumulated activation across outgoing axons.Source: §II · p. 3, Figure 3 · p. 3
• Undirected network edges are converted into directed synapses and axons, with a constant-intensity source driving activation while spike patterns are monitored over time.Source: §II · pp. 3–4

Hub configuration

• At t = N −1, all neurons attached to the source spike simultaneously, producing a total of N spikes in the hub network.The continuously firing source accumulates activation linearly in the other neurons until they reach threshold T = 1.Source: §III · p. 4
• Avalanches arise from hubs of hierarchical degree in the source-centered concentric organization, not exclusively from ordinary high-degree hubs.They also occur in degree-regular and highly degree-regular networks, including PI, ER, and PN models.Source: §III · p. 4

Chain configuration

• In the chain, backward activation produces saw oscillations whose peaks increase almost linearly after a brief transient, with slopes varying across nodes.The alternating high and low states are separated by one time step and correspond to activation exchange between neighboring nodes.Source: §III · p. 4, Figure 5 · p. 5
• The chain transfers activation progressively toward the right-hand nodes while concentrating activation in the initial nodes through backward flow.The corresponding spikes remain quite regular despite the complex activation shapes.Source: §III · pp. 4–5
• Alternating chain oscillations persist because spiking high-state neurons redistributes their activation to neighboring low-state neurons, reversing the pattern on the next step.This alternating exchange supports a stable oscillatory attractor.Source: §III · p. 5

Measurement procedure

• The study computes hierarchical node counts, hierarchical degrees, and intra-ring degrees for every node across seven theoretical network models.The measurements are averaged over nodes to illustrate the overall hierarchical structures of the network types.Source: §IV · p. 5

Hierarchical number of nodes

• Hierarchical node counts generally rise from the source, peak, and then decline to zero, with most peaks near the middle hierarchical level.The hierarchical number of nodes starts at one for the initial node and varies by network topology.Source: §IV · p. 6
• Topology determines the detailed node-count signatures: PN is most regular, ER and PA are homogeneous, while GG and WS have the longest signatures.GG also shows the greatest signature diversity, and longer signatures imply lower y-axis values because node counts sum to N.Source: §IV · p. 6

Hierarchical degrees

• Hierarchical degrees typically peak one level before the maximum number of nodes, except for the GG structure.PN has the most similar signatures, while GG and WS again produce the longest and lowest profiles.Source: §IV · p. 6

Intra-ring degrees

• Intra-ring-degree signatures show properties similar to the other hierarchical measurements, while the highly regular PI network yields the most uniform signatures.For PI, the node-count, hierarchical-degree, and intra-ring-degree profiles are uniformly distributed.Source: §IV · p. 6

Activation timing

• The concentric representation also estimates random-walk path diversity and first-activation timing, whereas random-walk activation proceeds gradually without abrupt changes.Hierarchical degrees approximate the number of self-avoiding paths, with intra-ring edges preventing exactness.Source: §V · p. 6

Avalanche mechanism

• When the largest concentric level spikes, intense activation to preceding and following levels can recruit most or all network neurons within one or a few time steps.The mechanism follows from strong activation received by the next level and prior activation accumulated closer to the source.Source: §V · p. 7

Limitations and assumptions

• The model assumes sufficiently regular degrees; non-homogeneous degrees and intra-ring connections can desynchronize or delay spiking, but their full treatment is left for future work.Degree heterogeneity is especially relevant to the BA case, while intra-ring edges retain activation within a level and delay adjacent levels.Source: §V · pp. 7–8

Equivalent mean-field model

• The equivalent model represents all neurons in each concentric level with one node and weights inter-level connections to capture asymmetric activation distribution.This reduction is justified by the relative uniformity of degree distributions in most considered network models.Source: §V · p. 8

Avalanche predictions

• Avalanche initiation is estimated at the concentric level containing the largest number of neurons.The critical level is identified from the hierarchical number of nodes.Source: §V · p. 8
• Beginning activation time for a level is estimated by summing the numbers of nodes from the source level through that level.This connects first activation timing to the hierarchical distribution around the source.Source: §V · p. 8
• The first avalanche time is estimated by summing nodes from level 1 through the level with the largest node count, while its intensity is estimated by that critical level's size.These estimates use the hierarchical number of nodes as the principal predictor.Source: §V · pp. 8–9

Avalanche evolution

• Adjacent-level node counts, hierarchical degrees, and intra-ring connections determine avalanche dispersion, simultaneity, intensity, and width.Connections within a ring and degree variation can delay or spread spiking across neighboring levels.Source: Figure 9 · p. 8, §V · p. 9

VI. ILLUSTRATIVE EXAMPLES

• The original and equivalent network curves differ only slightly because the equivalent model omits intra-ring degrees, producing small activation delays and spread.Source: §VI · p. 9
• Most ER, BA, WS, PN, and PA networks show similar avalanche patterns near 80 steps, whereas GG exhibits a more gradual transition.WS and PN also display secondary avalanches in the full-network results.Source: §VI · p. 9
• Equivalent models reproduce a clear transition near 80 steps, with stronger oscillation amplitudes in the knitted PN and PA models.The PA model has a lower transition of nearly 40 units, while the other models are close to 50.Source: §VI · p. 9
• The highly regular PI model exhibits a pronounced avalanche, a long stabilization plateau, and a secondary avalanche near 210 steps.Its equivalent model identifies the timing and intensity of the first avalanche.Source: §VI · p. 10
• The equivalent models agree with the original networks on the main avalanche properties, capturing their timing and intensity.The simple chain model does not reproduce secondary avalanches observed in the WS and PN models.Source: §VI · p. 10

VII. HOW UNIVERSAL ARE AVALANCHES?

• Avalanche positions and intensities in PI networks do not depend on average degree, consistent with the equivalent model using hierarchical-degree ratios.Source: §VII · p. 11
• Avalanche positions increase monotonically but nonlinearly with network size, with an abrupt increase near N = 100.Source: §VII · p. 11
• Avalanche intensities show a similar size-dependent pattern and a less abrupt increase near N = 100.Source: §VII · p. 11

VIII. A REAL-WORLD EXAMPLE: C. ELEGANS NETWORK

• The analyzed C. elegans largest component contains 246 nodes after thresholding the original matrix at 4.Source: §VIII · p. 11
• For C. elegans with source node 50, the maximum concentric level predicts an avalanche at 200 steps with intensity 123, closely matching the full-network dynamics.Source: §VIII · p. 11
• Hierarchical node counts and degrees can predict C. elegans avalanche features without simulating the corresponding dynamics.Source: §VIII · p. 11
• Changing the activation source among C. elegans neurons produces different avalanche intensities and onset positions.This source-location dependence was experimentally verified.Source: §VIII · p. 12

Network models and extreme topologies

• The highly PI model provides regular concentric degrees for analyzing avalanche dynamics, while hub and chain networks produce maximally simultaneous versus gradual activation patterns.The two extreme topologies corroborate the duality between star-like and path-like connectivity.Source: §IX · p. 12

Limitations and future directions

• Secondary avalanches occur in some models, especially WS and PN, but the equivalent model does not capture them because it omits intra-ring connections and degree non-uniformities.The authors propose larger computational experiments, augmented models, and extensions to other dynamics and real-world networks.Source: §IX · pp. 12, 13, 16

Equivalent-model predictions

• Avalanche intensity and timing are estimated from concentric node counts, with the largest level determining intensity and the cumulative levels determining when the main avalanche occurs.The equivalent model captures these features across seven theoretical models and C. elegans, except the geographical model, which does not yield avalanches.Source: §IX · pp. 14–15

Universal features of avalanches

• Avalanche properties in the PI model are independent of average degree but increase monotonically and sub-linearly with network size, suggesting possible universality across many networks.This extension is presented as applying mainly to models with similar concentric organization, excluding Watts-Strogatz and geographical structures.Source: §IX · pp. 13–15, Figure 17 · p. 14

C. elegans neuronal network

• C. elegans exhibits an avalanche near time step 200 for source node 50, but other source neurons can produce markedly different avalanche features.This source dependence suggests a specific heterogeneous topology not captured by the seven theoretical models.Source: Figure 18 · p. 14, §IX · pp. 15–16, Figure 19 · p. 15

Relating network structure and dynamics

• The study identifies activation avalanches as consequences of concentric network structure and develops a chain equivalent model to predict their main features.The model represents concentric levels as chain nodes and uses hierarchical measurements to estimate avalanche intensity and timing.Source: §IX · pp. 14, 16