Griffiths phases and the stretching of criticality in brain networks

Paolo Moretti; Miguel A. Muñoz

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Griffiths phases and the stretching of criticality in brain networks

Paolo Moretti, Miguel A. Muñoz

arXiv:1308.6661v2Aug 30, 2013cond-mat.dis-nnq-bio.NC

TL;DR

Critical-like behavior in brain networks is difficult to reconcile with the fine tuning required by a single critical point. The paper uses computational and analytical approaches on hierarchical-modular synthetic networks and empirical brain networks, finding Griffiths phases with extended critical-like behavior and anomalously large responses.

Problem
A single critical point requires fine tuning, motivating the question of how brain networks can exhibit critical-like dynamics over broader operating conditions.
Method
The paper combines computational and analytical analyses of activity propagation on hierarchical-modular synthetic networks and empirical brain networks.
Results
The study finds Griffiths phases in synthetic hierarchical networks and real brain networks, with power-law avalanches and responses that diverge across an extended region.
Takeaways & Limitations
Structural disorder in hierarchical-modular networks provides a generic basis for extended critical-like behavior and enhanced responsiveness.

Abstract

from arXiv · show

Hallmarks of criticality, such as power-laws and scale invariance, have been empirically found in cortical networks and it has been conjectured that operating at criticality entails functional advantages, such as optimal computational capabilities, memory, and large dynamical ranges. As critical behavior requires a high degree of fine tuning to emerge, some type of self-tuning mechanism needs to be invoked. Here we show that, taking into account the complex hierarchical-modular architecture of cortical networks, the singular critical point is replaced by an extended critical-like region which corresponds --in the jargon of statistical mechanics-- to a Griffiths phase. Using computational and analytical approaches, we find Griffiths phases in synthetic hierarchical networks and also in empirical brain networks such as the human connectome and the caenorhabditis elegans one. Stretched critical regions, stemming from structural disorder, yield enhanced functionality in a generic way, facilitating the task of self-organizing, adaptive, and evolutionary mechanisms selecting for criticality.

A. Hierarchical network architectures

The paper constructs synthetic hierarchical-modular networks that reproduce nested brain-network organization and span disconnected, finite-dimensional, and small-world regimes. Their architecture is generated by recursively grouping modules and adding hierarchy-dependent inter-module connections.

Biological organization: Brain networks contain nested modules across scales, from cortical columns and areas to systems-level brain regions.This organization forms a fractal-like hierarchical structure.
Model construction: The synthetic HMN models contain s hierarchical levels, N nodes, and L links, enabling systematic analysis of hierarchical network structure.
Model construction: HMN-1 builds fully connected local modules, recursively groups them, and adds stochastic inter-module links with hierarchy-dependent probabilities.HMN-2 instead uses a deterministic level-dependent number of connections.
Structural characterization: The topological dimension D measures how node neighborhoods grow with distance, with Nr ∼r^D for large r.D = 0 describes fragmented networks, finite D describes large-worlds, and D →∞ corresponds to small-world networks.
Structural characterization: The HMN models span the full spectrum of topological dimensions, from disconnected networks through finite-dimensional networks to small-world networks.Around p = 1/4, networks have finite D, finite connectivity, and finite connection density in the large-N limit.
Structural characterization: For p = 1/4, inter-module connectivity remains stable across hierarchical levels, whereas p > 1/4 and p < 1/4 produce increasingly dense and sparse networks, respectively.

B. Architecture-induced Griffiths phases

Hierarchical-modular structure creates rare regions that replace a singular critical point with a broad Griffiths phase. In these finite-dimensional networks, power-law activity and avalanche statistics persist across a range of spreading rates.

Rare-region mechanism: Rare regions can remain locally active while the global system is disordered, allowing activity to linger for long times before eventual decay.
Rare-region mechanism: A broad distribution of rare-region sizes and lifetimes produces anomalously slow system-wide dynamics through the convolution of heterogeneous contributions.
Griffiths-phase phenomenology: The HMN phase diagram contains an intermediate Griffiths phase with generic power-law behavior between ordered and disordered regimes.
Griffiths-phase phenomenology: For HMN-2 networks, avalanche sizes are power-law distributed across 2.60 ≤λ ≤2.79, demonstrating a broad Griffiths-phase interval.
Robustness: These Griffiths-phase conclusions are supported by finite-size scaling and generalize across network architectures and dynamical models.
Rare-region mechanism: Structural heterogeneity can act like quenched disorder, producing Griffiths phases in activity-propagation models on networks with finite topological dimension D.Small-world networks with D = ∞ lack sufficiently isolated rare regions for this mechanism.

C. Anomalous propagation dynamics in HMNs

Minimal activity-propagation models on finite-dimensional HMNs exhibit anomalous critical-like dynamics over extended parameter ranges. Rare active regions explain the resulting power-law decay and continuously varying avalanche exponents.

Dynamical models: The models represent each node with a binary active or quiescent state, spontaneous deactivation rate µ, and propagation rate λ.Here µ = 1, while activity propagates along direct connections at rate λ.
Dynamical models: Model A activates a randomly selected quiescent node across an active-quiescent synapse, whereas Model B tests activation of all neighbors of a selected neuron.The two variants correspond to the contact process and SIS model, respectively.
Conventional criticality: In conventional systems, activity separates into active and inactive phases at a critical spreading rate λc, where observables follow power laws.Examples include ρ(t) ∼t^-θ and avalanche distributions P(S) ∼S^-τ.
HMN dynamics: On finite-D HMNs, power-law decay of average activity extends across a broad λ interval rather than occurring only at one critical point.Finite-size scaling supports the existence of this broad interval.
HMN dynamics: Avalanche sizes are power-law distributed across the same broad region, with continuously varying exponents, and the behavior remains robust up to N = 2^20.Robustness was tested across HMN variants, parameter choices, and dynamical models.
Rare-region explanation: Rare regions with local λ(x) > λc retain activity for long periods, especially when large, although finite regions eventually become inactive.Their decay time follows τ ≃t0 exp[A(λ)ζ], while large regions are exponentially rare.
Rare-region explanation: Saddle-point evaluation of the rare-region convolution yields ρ(t) ∼t^-θ, with θ varying continuously with the disorder average λ̄.These generic power laws signal Griffiths phases, with HMN disorder encoded in the hierarchical contact pattern.

D. Diverging response in HMNs

Griffiths phases broaden the regime of strong network responsiveness beyond a single critical point. In synthetic and real brain networks, susceptibility, dynamic range, and avalanche statistics remain anomalously large across extended regions.

Response measures: Dynamic susceptibility Σ measures the response to a continuous localized stimulus through the difference between forced and spontaneous stationary activity densities.It is defined as Σ(λ) = N[ρf(λ) −ρs(λ)].
Finite-size response: In the Griffiths phase, system response increases with system size, unlike the decreasing response observed in the active phase.
Dynamic range: The broad response region is strongly asymmetric around the transition point, consistent with Griffiths-phase behavior rather than a symmetric cusp singularity.
Susceptibility: In the Griffiths phase, Σ becomes extremely large and grows as a λ-dependent power law of system size.
Dynamic range: The dynamic range Δ exhibits a broad region of huge values instead of a sharply peaked response at conventional criticality.
Functional implications: The critical-like region is broadened relative to regular disorder-free networks, extending large responses across a region rather than a singular point.
Real brain networks: Human connectome simulations produce truncated power-law avalanche distributions with continuously varying exponents across λ values.The fits use P(S) ∼S^-τe^-S/ξ and are selected using the Kolmogorov-Smirnov criterion.

E. Griffiths phases in real networks

Real brain networks with hierarchical organization exhibit broad critical-like regions rather than only a sharply tuned critical point. Simulations support Griffiths phases in both the human connectome and the C. elegans neural network, although finite network sizes limit observable scales.

C. elegans network: The detailed C. elegans neural network, consisting of N ≲300 neurons, shows similar results despite more severe finite-size effects.The reported results concern simulations of the dynamical models on the empirical network.
Real-network motivation: Hierarchical organization is shared by anatomical and functional brain networks, motivating tests of whether activity models on real networks exhibit Griffiths phases.The human connectome analyzed here is a highly coarse-grained, hierarchical mapping obtained using diffusion imaging.
Human connectome: The human connectome contains N = 998 brain areas and fiber tract densities between them, with hierarchical organization.
Finite-size effects: The human connectome’s N ≲1000 nodes necessarily cut off rare regions and associated power laws at small sizes and short times.The authors suggest that a model on the actual brain network, with about 10^12 neurons and 10^15 synapses, would exhibit a more robust GP over larger scales.
Human connectome: Human-connectome simulations show avalanche power laws with moderate finite-size effects across a broad range of λ-values.Truncated power laws with λ-dependent τ provide highly reliable fits under the Kolmogorov-Smirnov criterion, supporting a broad critical-like region.

F. Spectral fingerprints of Griffiths phases in HMNs

Spectral analysis links Griffiths phases in hierarchical-modular networks to localized eigenvectors and Lifshitz tails. These signatures identify rare, highly connected regions where activity can persist and explain the separation between linear instability and the critical point.

Spectral mechanism: For connected networks, linear stability gives λcΛmax = 1, but localized principal eigenvectors can produce activity confined to a few nodes rather than a true active state.In this case, the critical point shifts to a larger λ, while localized regions resemble Griffiths-phase rare regions.
Spectral mechanism: HMN spectra contain hundreds of eigenvalues below Λmax with localized eigenvectors, not only a localized principal eigenvector.The principal eigenvector is heavily peaked around a neighboring cluster of nodes.
Rare regions: Localized eigenvalue clusters correspond to above-average-connectivity rare regions where activity lingers for long times.
Rare regions: λc ≈0.41 exceeds 1/Λmax ≈0.33, and the interval between these values defines the Griffiths phase.
Lifshitz tails: Localized eigenvectors generate an exponential tail in the continuum spectrum and a Lifshitz tail in the cumulative Laplacian-eigenvalue distribution.The numerical Lifshitz-tail fit has exponent a ≈1.00.
Empirical network signature: The coexistence of localized eigenvectors and Lifshitz tails confirms Griffiths phases in networks with complex heterogeneous architectures.The human connectome displays corresponding localization of its principal eigenvector, with similar peak structures in weighted and unweighted representations.

II. DISCUSSION

The paper argues that hierarchical, modular, structurally disordered brain networks replace a finely tuned critical point with an extended Griffiths phase. In synthetic and empirical networks, this region produces critical-like dynamics and unusually large responses, potentially easing self-organization around criticality.

Structural basis: Hierarchical modular networks with finite topological dimension support Griffiths phases, whereas plain modular networks without a broad cluster-size distribution do not.The hierarchy supplies disorder across scales, allowing rare regions to become arbitrarily large.
Synthetic networks: Synthetic large-world networks exhibit anomalous slow relaxation, power-law avalanches with continuously varying exponents, and critical-like behavior across a broad active–inactive region.These signatures are accompanied by localized eigenvectors and Lifshitz tails, the spectral counterparts of rare regions.
Functional response: The dynamic susceptibility and dynamic range diverge with system size throughout the Griffiths phase, indicating an anomalously large response to stimuli.Unlike the active phase, the overall response increases for larger systems in the Griffiths phase.
Empirical networks: Griffiths phases also appear in the human connectome and C. elegans networks, although limited network sizes impose finite-size cutoffs on their power laws.The best fits are continuously varying power laws truncated by finite-size effects.
Mechanism: Structural disorder alone induces Griffiths phases in the modeled networks, without requiring additional neuronal or synaptic heterogeneity.Adding further heterogeneity is expected to enhance rare-region effects and Griffiths phases.
Implications: Because the response is large across a broad parameter region, Griffiths phases may facilitate mechanisms selecting for scale-free behavior without delicate parameter fine tuning.The authors consequently call for models of self-organization, adaptation, and evolution targeting the broad region between order and chaos.

III. MATERIALS AND METHODS

The study combines synthetic hierarchical networks, empirical brain networks, minimal spreading dynamics, and multiple response protocols to examine Griffiths phases. It links structural disorder and localized eigenvectors to extended critical-like behavior and enhanced responses.

Empirical networks: The empirical systems are the hierarchical-modular C. elegans neural network and the human connectome, with C. elegans reported as six times more hierarchical than comparable randomized networks.Human-connectome analyses likewise identify hierarchical organization.
Dynamical models: Two binary-state spreading models are simulated: Model A activates one inactive neighbor, whereas Model B independently tests all inactive nearest neighbors.Both models use activation and deactivation probabilities determined by λ and μ; real-network results use Model B on unweighted networks.
Dynamical protocols: The protocols measure activity decay, avalanche spreading, continuous stimulation, and Poissonian stimulation through density, survival, avalanche-size, and response measures.Avalanche size is defined as the number of activation events during an avalanche, while dynamic susceptibility compares forced and spontaneous steady-state densities.
Findings: For each λ, effective exponents approach asymptotic values, supporting persistence of generic power-law behavior in the thermodynamic limit.The authors distinguish this from finite-size corrections characteristic of Griffiths phases.
Analysis: Localized eigenvectors and rare regions are analyzed spectrally, while activity decay and avalanche distributions are assessed using asymptotic scaling and Kolmogorov-Smirnov fits.The analysis also evaluates dynamic susceptibility and dynamic range across spreading rates.
Findings: Across hierarchical networks and empirical brain networks, Griffiths phases produce extended anomalous responses and generic power-law activity decay over finite spreading-rate ranges.The response region broadens relative to an equivalent disorder-free lattice, while finite-size effects can alter apparent exponents without revealing exponential cutoffs.

PHkL

The figures characterize hierarchical organization, extended activity responses, finite-size scaling, and avalanche distributions in hierarchical and empirical brain networks. The connectome shows a less abrupt transition and a broader activity range than a disorder-free lattice.

Hierarchical structure: The human connectome adjacency matrix appears markedly hierarchical upon visual inspection.The matrix representation places black squares at non-zero entries.
Activity response: The connectome’s activity transition is significantly less steep than that of a same-size disorder-free square lattice.The comparison uses Model B simulations and arbitrary vertical-axis cuts for visual display.
Activity response: The connectome’s activity extends over a decade more than activity in the comparison lattice.This broader response accompanies the less-steep transition.
Finite-size scaling: For HMN-1 networks, activity-decay slopes approach asymptotic values across system sizes spanning 10^4–10^6 nodes.The plotted cases use λ = 2.25 and λ = 2.28 with N = 2^14, 2^17, and 2^20.
Avalanche distributions: At λ = 0.017, the truncated-power-law fit is best among the tested avalanche-distribution fits by the KS criterion and visual inspection.The comparison includes empirical data, truncated-power-law, power-law, and exponential fits.