Hierarchical networks, power laws, and neuronal avalanches

Friedman, Eric; Landsberg, Adam S.

doi:10.1063/1.4793782

Cited by 45 publications

(44 citation statements)

References 26 publications

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“…In the case of HMNs, structural disorder arises from the stochastic nature of the processes by which modules at diverse scales are connected. The structural origin of critical-like behavior is in agreement with renormalization arguments, which show how in hierarchical networks even standard percolation produces generic power-law distributions of connected component sizes, a feature otherwise normally ascribed to the critical point (or percolation threshold) [25,26].…”

Section: Introductionsupporting

confidence: 76%

Topological dimension tunes activity patterns in hierarchical modular networks

2017

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Connectivity patterns of relevance in neuroscience and systems biology can be encoded in hierarchical modular networks (HMNs). Recent studies highlight the role of hierarchical modular organization in shaping brain activity patterns, providing an excellent substrate to promote both segregation and integration of neural information. Here, we propose an extensive analysis of the critical spreading rate (or 'epidemic' threshold)-separating a phase with endemic persistent activity from one in which activity ceases-on diverse HMNs. By employing analytical and computational techniques we determine the nature of such a threshold and scrutinize how it depends on general structural features of the underlying HMN. We critically discuss the extent to which current graph-spectral methods can be applied to predict the onset of spreading in HMNs and, most importantly, we elucidate the role played by the network topological dimension as a relevant and unifying structural parameter, controlling the epidemic threshold.

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Section: Introductionsupporting

confidence: 76%

Topological dimension tunes activity patterns in hierarchical modular networks

2017

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“…The parabolic profile also differentiates neuronal avalanches from stochastic processes with no memory, which typically display a semicircle motif 74 . Specific graph-theoretical constructs such as a hierarchical topology 40 can also mimic scale-free size distributions but fail to produce an inverted parabola profile. Network and biophysical models have demonstrated avalanches to emerge with oscillations at a particular E/I balance or topology; however, the corresponding avalanche profile was not reported 66,75,76 .…”

Section: Figs 1 2)mentioning

confidence: 99%

“…Variable and asymmetric profiles have been reported for neuronal cultures 35,36 , and profiles seem to depend on avalanche duration in humans 18 . Identifying the correct profile of neuronal avalanches will provide insights into the temporal evolution of brain activity, will distinguish between different models of avalanche generation [37][38][39][40] and, importantly, might provide a biomarker given recent findings that profiles predict recovery from brain insults 41,42 . A second prediction from critical theory is that the avalanche parabola can be collapsed over many avalanche durations with a scaling exponent, χ, larger than 1.5 (Refs.…”

Section: Introductionmentioning

confidence: 99%

The scale-invariant, temporal profile of neuronal avalanches in relation to cortical γ–oscillations

Miller

Plenz

2019

Preprint

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Miller et al.The temporal profile of neuronal avalanches 2 ABSTRACT Activity cascades are found in many complex systems. In the cortex, they arise in the form of neuronal avalanches which capture ongoing and evoked neuronal activities at many spatial and temporal scales. The scale-invariant nature of avalanches suggests that the brain is in a critical state, yet predictions from critical theory on the temporal unfolding of avalanches have yet to be confirmed in vivo. Here we show in awake nonhuman primates that the temporal profile of avalanches, which describes how avalanches initiate locally, extend spatially and shrink as they evolve in time, follows a symmetrical, inverted parabola spanning up to hundreds of milliseconds. Parabolas of different durations can be collapsed with a scaling exponent close to 2 supporting critical generational models of neuronal avalanches. Spontaneously emerging, transient γ-oscillations coexist with and modulate these avalanche parabolas thereby providing a temporal segmentation to inherently scale-invariant, critical dynamics. Our results identify avalanches and oscillations as dual principles in the temporal organization of brain activity. Significance StatementThe most common framework for understanding the temporal organization of brain activity is that of oscillations, or "brain waves". In oscillations, distinct physiological frequencies emerge at well-defined temporal scales, dividing brain activity into time segments underlying cortex function. Here, we identify a fundamentally different temporal parsing of activity in cortex. In awake Macaque monkeys, we demonstrate the motif of an inverted parabola that governs the temporal unfolding of brain activity in the form of neuronal avalanches. This symmetrical motif is scale-invariant, that is, it is not tied to time segments, and exhibits a scaling exponent close to 2 in line with prediction from theory of critical systems. We suggest that oscillations provide a Miller et al.The temporal profile of neuronal avalanches 3 transient "wrinkle" of regularity in an otherwise scale-invariant temporal organization pervading cortical activity at numerous scales.

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“…These findings have important implications for understanding the robustness of networks and in quantifying epidemic outbreaks in the susceptible-infected-recovered (SIR) model of disease spread. Percolation theories are among the most studied in network science [1], as well as in several other areas [2,3], providing insights for a broad range of applications such as robustness of a network to random failures or attacks [4], epidemics in contact processes [5], vaccination strategies [1], neuronal avalanches [6], and stability of gene regulatory networks [7]. In the simplest case of bond (or site) percolation, a fraction p of the links (nodes) are randomly chosen to be occupied and the rest of the links (nodes) are removed from the network [1].…”

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confidence: 99%

Emergence of coexisting percolating clusters in networks

et al. 2016

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It is commonly assumed in percolation theories that at most one percolating cluster can exist in a network. We show that several coexisting percolating clusters (CPCs) can emerge in networks due to limited mixing, i.e., a finite and sufficiently small number of interlinks between network modules. We develop an approach called modular message passing (MMP) to describe and verify these observations. We demonstrate that the appearance of CPCs is an important source of inaccuracy in previously introduced percolation theories, such as the message passing (MP) approach, which is a state-of-the-art theory based on the belief propagation method. Moreover, we show that the MMP theory improves significantly over the predictions of MP for percolation on synthetic networks with limited mixing and also on several real-world networks. These findings have important implications for understanding the robustness of networks and in quantifying epidemic outbreaks in the susceptible-infected-recovered (SIR) model of disease spread.

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Hierarchical networks, power laws, and neuronal avalanches

Cited by 45 publications

References 26 publications

Topological dimension tunes activity patterns in hierarchical modular networks

Topological dimension tunes activity patterns in hierarchical modular networks

The scale-invariant, temporal profile of neuronal avalanches in relation to cortical γ–oscillations

Emergence of coexisting percolating clusters in networks

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