Criticality without fine-tuning, spirals with conserved charge
The brain is the most controversial system in this series. For twenty years, neuroscientists have debated whether neural activity is critical — poised at the boundary between order and disorder. The debate has been fierce because the implications are large: if the brain is critical, then much of neural computation derives not from specific circuits but from the universal properties of the critical state itself.
In 2025, that debate shifted. A 140-dataset meta-analysis by Hengen and Shew in Neuron1 established near-consensus that the brain is near criticality. But the brain is not near a single critical point. It contains three independent critical phenomena — and a structural mechanism that makes criticality robust without requiring precise parameter tuning.
When one neuron fires, it can trigger its neighbors to fire, which trigger their neighbors, and so on. These cascading events are called neuronal avalanchesA cascade of neural activity where one neuron firing triggers its neighbors, which trigger their neighbors. Like a chain reaction. The size distribution reveals whether the system is at a critical point.5. At criticality, avalanche sizes follow a power lawA mathematical pattern where small events are common and large events are rare, with no characteristic scale. Doubling the size reduces frequency by a fixed ratio. The hallmark of scale-free, critical systems.: most are small (a few neurons), but occasionally a cascade sweeps across an entire cortical region. There is no characteristic size — the distribution is scale-freeA system with no characteristic size — events of all scales occur, from tiny to system-spanning. The ratio between scales is constant. Contrast with normal distributions, which cluster around a mean..
The 140-dataset meta-analysis consolidated evidence from EEG, MEG, two-photon imaging, and multi-electrode arrays across species (human, monkey, mouse, rat, zebrafish) and preparation types (in vivo, in vitro, awake, anesthetized). The conclusion: neural avalanche statistics are consistent with criticality across measurement modalities, species, and brain states.
More remarkable: how close the brain sits to criticality is substantially heritableUnder genetic control — the trait’s variation across people is partly explained by their genes, as measured by comparing identical (monozygotic) and fraternal (dizygotic) twins., and proximity to criticality correlates with cognitive functionPerformance on reasoning, memory, and problem-solving tasks. The cited twin study links how near the brain operates to its critical point with general cognitive ability. (Xin et al., PNAS 2025)4. Evolution has shaped the brain not for a particular circuit pattern, but for a particular distance from a phase transitionA sudden qualitative change in a system’s behavior at a threshold value — like water freezing at 0°C. In the brain, the transition is between ordered (synchronized) and disordered (random) neural activity..
The standard objection to brain criticality has always been the fine-tuning problemThe objection that a critical point occupies a single value in parameter space. Maintaining the brain exactly at that value seems implausibly precise — like balancing a pencil on its tip forever.: a critical point is a single point in parameter space. Why would billions of neurons be tuned to exactly that point? The slightest perturbation should push the system away from criticality.
The answer is Griffiths phasesExtended critical regions that appear in modular networks. Instead of a single critical point, modular architecture stretches criticality over a range of parameters. Named after Robert Griffiths (1969).2. In a modular-hierarchicalA network organized into distinct communities (modules) that are themselves grouped into larger communities, forming a hierarchy. The cerebral cortex has this structure: columns → areas → lobes. network — which is precisely the architecture of the cerebral cortex — the critical point is stretched into a critical phase. Over a finite range of parameters, the system shows power-law scaling, long-range correlations, and the other signatures of criticality. No fine-tuning required.
The mechanism is elegant: individual modules can be locally supercritical or subcritical, but the coupling between modules produces collective behavior that is critical over the entire range of module parameters. Moretti and Muñoz (Nature Communications) demonstrated this explicitly. The modular-hierarchical architecture of the cortex is not an accident of development — it is the structural feature that makes robust criticality possible.
Independently of the avalanche and Griffiths phenomena, the brain’s electrical activity contains spiral wavesRotating patterns of electrical activity that sweep across the cortex. Each spiral has a center (the topological defect) and a rotation direction that determines information flow. — rotating patterns of neural activity visible in EEG and MEG recordings. Xu and colleagues (Nature Human Behaviour, 2023)3 showed that these spirals are genuine topological defectsPoints where a continuous field has a singularity that cannot be smoothed away — like the eye of a hurricane. They carry conserved ‘charge’ and can only be created or destroyed in pairs. with conserved topological chargeA conserved integer (+1 or −1) assigned to each spiral center based on its rotation direction. Total charge is constant — spirals always appear and disappear in opposite-charge pairs. that annihilate in pairs, exactly like the defects in active matter and cardiac tissue.
The cortical spirals are not epiphenomena. They organize information flow: the rotation direction determines the direction of signal propagation, and spiral interactions gate the communication between cortical regions. The topological charge conservation means that spirals cannot simply appear or disappear — they must be created and destroyed in pairs, imposing a global constraint on the dynamics.
Drag the slider to change the inter-module coupling strength. Watch how modular architecture stretches a critical point into a critical phase.
The key insight we propose is that the same structural feature — modular-hierarchical cortical architecture — produces both the Griffiths phase and the organized topological defect dynamics.◆ our resultAn original, not-yet-peer-reviewed synthesis from this research program — the individual phenomena are published; the claim that one architecture is their common cause is ours. This is not a coincidence. It is a consequence of what modularity does to a dynamical system.
Modularity creates regions with different local dynamics. Some cortical modules may be slightly supercritical (tendency toward activation cascades), others slightly subcritical (tendency toward quiescence). The Griffiths phase arises because the supercritical modules create long-lived avalanches even when the global average is subcritical — the heterogeneity stretches the critical point into a phase.
Simultaneously, the boundaries between modules create the conditions for spiral wave formation and interaction. Spirals nucleate at module boundaries, rotate within modules, and interact when they reach the next boundary. The modular architecture organizes the topological defects the same way it organizes the avalanche dynamics.
This is a different type of multi-criticality from the previous tutorials. In the glass transition, two transitions couple at a bicritical point. In active matter, geometry selects between transition types. In the brain, a single architectural feature produces multiple critical phenomena simultaneously. The coupling is not between transitions — it is between the mechanisms that make each transition possible.
Hengen & Shew (Neuron, 2025) synthesized results from 140 datasets spanning:
Species: human, non-human primate, mouse, rat, cat, turtle, zebrafish, in vitro cortical cultures
Modalities: multi-electrode arrays, two-photon calcium imaging, EEG, MEG, ECoG, voltage-sensitive dye imaging
States: awake resting, task-engaged, anesthetized, sleeping
The meta-analysis evaluated evidence for three signatures of criticality: (1) power-law avalanche size distributions, (2) power-law avalanche duration distributions, (3) the crackling-noise scaling relation between the size and duration exponents (τ, α, and 1/σνz). All three were consistently observed across datasets, with the scaling relation providing the strongest evidence — it is a prediction specific to criticality that alternative mechanisms (e.g., oscillations, bistability) do not produce.
The strongest counterarguments — that subsampling artifacts can produce power laws from non-critical systems (Touboul & Destexhe) — were addressed through complete-system simulations and coarse-graining analyses. The scaling relation survives subsampling; spurious power laws from oscillatory dynamics do not satisfy it.
In a homogeneous network, the critical point is a single value of the control parameter (e.g., excitatory coupling strength). In a modular-hierarchical network, different modules have different effective coupling strengths. Moretti & Muñoz showed that this creates a Griffiths phase: a finite interval of the global control parameter where:
• Power-law distributions persist (with continuously varying exponents)
• SusceptibilityHow strongly a system responds to a small perturbation. At a critical point, susceptibility diverges — even a tiny push produces a large response. In a Griffiths phase, it stays elevated over a range. diverges (anomalously slow relaxation)
• Correlation lengths are long but finite
• The system responds to perturbations as if critical
The width of the Griffiths phase depends on the heterogeneity of the modular structure. More heterogeneous modules → wider Griffiths phase → more robust criticality. The cortex is extremely heterogeneous (area V1 differs from prefrontal cortex in cytoarchitecture, connectivity density, layer thickness, neurotransmitter balance), which predicts a wide Griffiths phase.
Key distinction from standard criticality: in a Griffiths phase, the exponents are non-universal. They depend continuously on the global control parameter. This explains why different studies measuring avalanche exponents get slightly different values — they may be sampling different points within the Griffiths phase, not measuring the same critical point with noise.
Xu et al. (Nature Human Behaviour, 2023) identified spiral wave patterns in human EEG/MEG data that satisfy the mathematical criteria for topological defects: (1) the phase field has a winding numberHow many times a field’s phase rotates around a point. A winding number of ±1 means the phase completes one full rotation — the mathematical definition of a topological defect. of ±1 around each spiral center, (2) spiral centers carry conserved topological charge (total charge is constant except when pairs are created or annihilated), (3) spirals of opposite charge annihilate on contact.
The spirals are large-scale structures spanning centimeters of cortex. They rotate at frequencies in the theta (4–8 Hz) and alpha (8–13 Hz) bands. The rotation direction determines the direction of information flow: clockwise spirals propagate signals inward (toward the center), counterclockwise spirals propagate signals outward. This creates a natural mechanism for gating information between cortical regions — two regions can communicate only when their spiral rotation directions are compatible.
The connection to active matter (Tutorial 02) is direct: both systems contain +1/2 and −1/2 topological defects that interact through the same mathematical framework. The difference is substrate (neurons vs. molecular rods) and scale (centimeters vs. micrometers). The question of whether the KT unbinding transition also applies to cortical spirals — whether there is a bound-to-unbound transition in the brain analogous to flutter-to-fibrillation in the heart — is open and testable.
Xin et al. (PNAS, 2025) analysed resting-state fMRI from a large twin cohort — hundreds of monozygotic and dizygotic twins plus unrelated subjects — using a model-based measure of each brain’s proximity to criticality. They reported that (1) proximity to criticality is substantially heritableUnder genetic control — how much of the variation across people is explained by their genes, estimated by comparing identical and fraternal twins.: genetic factors shape how close the brain operates to its critical point across individual regions, functional networks, and the whole brain, and (2) individual variation in proximity to criticality relates to cognitive function.
The heritability result is consistent with the Griffiths-phase mechanism: if criticality is a downstream consequence of modular-hierarchical architecture, then the genes that build that architecture — rather than any single parameter tuned to a knife-edge — are what selection would act on. Which specific genes are involved (ion-channel subunits, synaptic-plasticity mediators, or connectivity-shaping factors) is a natural follow-up question, but is not established by this twin-heritability study.◆ our readingThe interpretive link between the twin-heritability result and the Griffiths-phase architecture is this research program’s synthesis, not a claim of the cited paper.
Claim: The universality classA group of phase transitions that share the same critical exponents despite occurring in different physical systems. Determined by symmetry and dimensionality, not microscopic details. of the avalanche transition depends on the network topology. Modular-hierarchical networks produce Griffiths phases with non-universal exponents. Small-world networks without modularity produce standard mean-field criticality. Scale-free networks produce a different class entirely. The brain’s specific exponents are a consequence of its modular-hierarchical architecture, not of neural dynamics per se.
Status: Framework. Sun et al. (Nature Physics, 2025) independently showed that network topology governs criticality class in cytoskeletal networks — consistent with this prediction in a non-neural system.
Test: Simulate identical neural dynamics on three network topologies: modular-hierarchical (cortical), Erdös-Rényi (random), and Barabási-Albert (scale-free). If the avalanche exponents differ systematically between topologies, the network determines the universality class. If they are the same, the dynamics dominate and topology is secondary.
Claim: The onset of cortical modularity during development coincides with the onset of critical avalanche statistics. Before modular architecture is established, the developing brain should not show Griffiths-phase criticality. After modules form, criticality should appear without any change in the single-neuron dynamics.
Status: Planned. Requires longitudinal avalanche measurements during cortical development. Zebrafish and organoid models are the most tractable systems for this test.
Connected through Griffiths phase architecture. Both the brain and the tokamak are modular-hierarchical systems where the architecture produces robust criticality without fine-tuning. In the brain: cortical modules with heterogeneous dynamics. In the tokamak: nested transport barriers with heterogeneous radial profiles. The same mathematical mechanism — heterogeneity stretching a critical point into a critical phase — produces robust criticality in both, despite the substrates (neurons vs. plasma) having nothing else in common.
Three order parameters on nested node sets. Avalanche criticality operates on the full cortical network. The Griffiths phase arises from the modular structure — a coarse-grained view where nodes are cortical modules rather than individual neurons. Topological defects operate on the cortical surface (a 2D manifold embedded in 3D space).
The Griffiths phase is unusual from a detection perspective: the exponents vary continuously with the control parameter. This means a relational gradient operator measuring the avalanche field would see different gradient statistics at different points within the Griffiths phase — not the fixed universality of a standard critical point.
The question: Can a relational operator distinguish between a system at a true critical point (fixed exponents, one universality class) and a system in a Griffiths phase (continuously varying exponents, extended critical region)? If so, the operator would detect not just proximity to criticality, but the type of criticality — whether it arises from parameter tuning or from architectural robustness. This distinction has direct clinical relevance: a brain that is critical because of Griffiths phases is robust; a brain that is critical because of fine-tuning is fragile.
Hengen, K. B. & Shew, W. L. (2025). Neural criticality: A comprehensive meta-analysis. Neuron.
Moretti, P. & Muñoz, M. A. (2013). Griffiths phases and the stretching of criticality in brain networks. Nature Communications, 4, 2521.
Xu, Y. et al. (2023). Cortical spiral waves as topological defects. Nature Human Behaviour.
Xin, Y., Cui, Y., Yu, S. & Liu, N. (2025). Genetic contributions to brain criticality and its relationship with human cognitive functions. PNAS, 122(26), e2417010122. doi:10.1073/pnas.2417010122.
Sun, M. et al. (2025). Network topology governs cytoskeletal criticality. Nature Physics.
Beggs, J. M. & Plenz, D. (2003). Neuronal avalanches in neocortical circuits. J. Neurosci., 23, 11167–11177.
Touboul, J. & Destexhe, A. (2017). Power-law statistics and universal scaling in the absence of criticality. Phys. Rev. E, 95, 012413.