Neuronal-avalanche criticality: an annotated reference to fourteen papers

This is an annotated reference to fourteen papers that together define the modern toolkit for testing whether a system sits near a critical point through the statistics of its avalanches. The set spans the founding cortical-avalanche and crackling-noise work, the statistical-methodology literature, the critiques that establish what a power law does and does not prove, and the recent transfer of the same framework to self-assembled neuromorphic and memristive nanomaterials. Each entry follows a fixed structure — type, setup, core claims, quantitative findings, methodological details, connection to the wider framework, and important nuances (including notation conflicts) — and the entries are followed by a set of cross-cutting points worth keeping in mind when evaluating any criticality claim.

A note on sources. Quotations from the original papers are kept short and are paraphrased where a longer passage would otherwise be reproduced; the scientific content is unchanged. Figures from the original papers are not reproduced — where one appeared, a short caption and a link to the figure in the (open-access) source article are given in its place.

The papers, in order:

Beggs & Plenz (2003), J. Neurosci. 23(35), 11167 — “Neuronal Avalanches in Neocortical Circuits”
Sethna, Dahmen & Myers (2001), Nature 410, 242 — “Crackling Noise”
Friedman et al. (2012), Phys. Rev. Lett. 108, 208102
Touboul & Destexhe (2010), PLoS ONE 5(2), e8982
Touboul & Destexhe (2017), Phys. Rev. E 95, 012413
Clauset, Shalizi & Newman (2009), SIAM Review 51, 661
Marshall et al. (2016), Front. Physiol. 7, 250 — the NCC toolbox
Mariani et al. (2022), Sci. Rep. 12, 10770
Hochstetter et al. (2021), Nat. Commun. 12, 4008
Mallinson et al. (2019), Sci. Adv. 5, eaaw8438
Priesemann et al. (2014), Front. Syst. Neurosci. 8, 108
Muñoz (2018), Rev. Mod. Phys. 90, 031001
Fosque et al. (2021), Phys. Rev. Lett. 126, 098101
Dunham et al. (2021), J. Phys. Complex. 2, 042001

Notation

A single convention is used throughout these notes; each paper’s own symbols are mapped to it inside its entry.

τ — avalanche-size exponent, P(S) ∝ S^(−τ).
β — avalanche-duration exponent, P(T) ∝ T^(−β). Most of the papers call this α or τ_t. ⚠ It is not the same as Sethna’s β = 1/σνz − 1, which is the shape-collapse exponent; nor is it Mallinson’s β (an autocorrelation-decay exponent) or Priesemann’s β (a detrended-fluctuation exponent). Those collisions are flagged where they arise.
γ_B — the size-given-duration exponent, ⟨S⟩(T) ∝ T^(γ_B). This is the same quantity that Sethna, Friedman, Marshall and the NCC toolbox write as 1/σνz, and that Dunham and Fosque write as γ.
Crackling-noise relation: γ_B = (β − 1)/(τ − 1). It is necessary for criticality but not sufficient — see cross-cutting points 1–3.

Device-exponent classes at a glance

System (reference)	τ (size)	β (duration)	γ_B = ⟨S⟩(T)	Notes
Mean-field branching / directed percolation (theory)	3/2	2	2	Reference values; matched by Beggs–Plenz cortical cultures
Cortex, in vitro / in vivo (Friedman 2012)	≈ 1.6	≈ 1.7	≈ 1.3	Near but off mean-field; crackling δ ≈ 1.28
Nanowire network, Ag-PVP (Hochstetter 2021)	≈ 2.0 (1.95 sim / 2.05 exp)	≈ 2.25	≈ 1.3 sim / 1.2 exp	”Device class,” distinct from directed percolation
Nanoparticle network, Sn (Pike 2020)	2.0	2.3	1.3	Closest match to Hochstetter
Nanoparticle network, Sn (Mallinson 2019)	≈ 2.0	≈ 2.4–2.7	≈ 1.4 (pooled, direct) / ≈ 1.6 (single sample)	τ is shared cleanly; β and γ_B run higher and scatter
Nanowire network, Ag₂Se (Dunham 2021)	1.89 ± 0.02	2.12 ± 0.04	1.23 ± 0.04 (direct)	Crackling relation predicts 1.26 ± 0.05

Throughout the table, “sim” = network simulation, “exp” = experiment, “direct” = fitted from the ⟨S⟩(T) regression, and “relation” = computed from (β−1)/(τ−1). β is the duration exponent (written α or τ_t in several of these papers); γ_B is ⟨S⟩(T) (written 1/σνz or γ elsewhere). See Notation above.

0. Beggs & Plenz (2003), “Neuronal Avalanches in Neocortical Circuits,” Journal of Neuroscience 23(35), 11167–11177

Type: Foundational empirical paper. First identification of neuronal avalanches as a distinct mode of cortical network activity, with power-law size and lifetime distributions and a critical branching parameter σ ≈ 1. Every paper in this reference (Sections 1–10) either builds on this one directly or critiques the methodology established here.

Setup

Organotypic cultures: 7 mature cortical cultures (28 ± 3 DIV) from rat somatosensory cortex, grown on 60-channel multi-electrode arrays (8 × 8 minus corners, 200 μm inter-electrode distance, 30 μm electrode diameter). 70 hr total recording.
Acute slices: 9 slices from 6–8 week old rat primary motor and somatosensory cortex. Spontaneous activity induced pharmacologically by bath application of NMDA (6 μM) + the D1 agonist SKF-38393 (5 μM).
Recording: spontaneous LFPs sampled at 1 kHz, low-pass-filtered at 50 Hz, binned at Δt ∈ {1, 2, 4, 8, 16} ms. Threshold set per channel by receiver-operating-characteristic curve against a Gaussian noise baseline (mean threshold ≈ 2.86 ± 0.23 SD). LFPs identified as sharp negative peaks (population spikes) with 20 ms refractory period.
Rate: 58 000 ± 55 000 LFPs/hr per culture (range 10 000–240 000); average 58.9 ± 0.4 of 60 electrodes active per culture.

Core claims the paper establishes

Definition of “neuronal avalanche.” A spatiotemporal pattern on the multi-electrode array consisting of a sequence of consecutively active frames (frame width Δt) preceded and followed by an empty frame. Avalanche size S = number of electrode activations; avalanche lifetime/duration T = number of frames. This is the definition adopted by every subsequent paper in this reference and embedded in the NCC toolbox avprops.m.
Power-law avalanche size distribution with characteristic exponent ≈ −3/2. P(n) ∝ n^α with α = −1.50 ± 0.008 (number of active electrodes) and α = −1.49 ± 0.005 (summed LFP amplitude), R² = 0.98–0.99. Cutoff at n = 60 = the number of electrodes, attributed to finite-size effects.
Power-law avalanche lifetime distribution with characteristic exponent ≈ −2. Lifetime distributions are scale-invariant under the rescaling t’ = t/Δt across bin widths 1–16 ms, with initial slope near α_t ≈ −2 and an exponential cutoff.
Critical branching parameter σ ≈ 1. Direct measurement of the branching parameter σ (average number of descendant electrodes activated in the next time bin per active ancestor electrode, following de Carvalho & Prado 2000): σ = 1.04 ± 0.19 for avalanches starting from a single ancestor, σ = 0.90 ± 0.19 for multiple-ancestor avalanches (≈ 90 000 avalanches analysed). The critical value σ = 1 corresponds to the boundary between sub-critical (activity decays) and super-critical (runaway) regimes for a branching process (Harris 1989).
Theoretical interpretation: the measured exponents τ ≈ 1.5 and τ_t ≈ 2 are the mean-field critical-branching-process predictions (Zapperi, Lauritsen & Stanley 1995; Harris 1989). These are also the mean-field directed-percolation values. The paper interprets neuronal avalanches as evidence that cortical networks operate at or near the critical point of a mean-field branching process.
Information-transmission optimum at σ = 1. Feedforward neural-network simulations show that information transmission peaks at σ = 1.1 ± 0.30, asymptotically approaching σ = 1.04 ± 0.10 as the input layer size N grows (R² = 0.99). Per-input-unit information increases with the input-to-connections ratio N/C, suggesting the critical state is especially advantageous under the sparse connectivity characteristic of neocortex.

Quantitative findings

Organotypic cultures (n = 7, Δt = IEI_avg ≈ 4 ms, IED = 200 μm):

Size exponent α = −1.50 ± 0.008 (electrode count), −1.49 ± 0.005 (LFP sum)
Lifetime exponent ≈ −2 (initial slope, with exponential cutoff)
Branching parameter σ ≈ 1.04 ± 0.19 (single ancestor)
Inter-event interval IEI_avg varies 2.7–6 ms across cultures
Contiguity index 39.3 ± 8% — activity is not wave-like
Cross-correlations decline to baseline within 100–200 ms

Acute slices (n = 9, induced activity):

Size exponent α = −1.50 ± 0.08 (LFP), −1.58 ± 0.04 (electrodes)
Lifetime exponent ≈ −2 with exponential cutoff
Contiguity index 28 ± 9%
IEI_avg ≈ 3.73 ± 0.467 ms
Cross-correlations decline within 50–80 ms (faster than cultures — attributed to severed long-range cortical connections)
Avalanches more spatially compact than in cultures (exponential cutoff appears earlier)

Pharmacology (n = 3 cultures, picrotoxin 2 μM, GABA_A antagonist):

Pre-drug: α_pre = −1.45 ± 0.08 (consistent with critical)
Picrotoxin: α = −1.95 ± 0.02 — power law destroyed, distribution becomes bimodal (small events + large epileptic events)
24 hr wash: α_wash = −1.51 ± 0.03 — criticality recovers
ANOVA p < 0.05, Tukey test

Surrogate / robustness checks:

Threshold-independence: α unchanged for thresholds 3–10 SD (ANOVA: p = 0.66 for electrodes, p = 0.44 for LFPs).
Bin-width robustness: power law preserved at all Δt ∈ {1, 2, 4, 8, 16} ms; slope shifts smoothly with α(Δt) ∝ Δt^(−0.16 ± 0.01), R² = 0.99 ± 0.01.
Spatial rescaling: removing intermediate electrodes (effective IED = 400, 600 μm) preserves α ≈ −1.5 at the matched IEI_avg, demonstrating scale invariance.
Finite-size cutoff: cutting the array into halves and quarters shifts the cutoff to scale with the number of electrodes, consistent with finite-size scaling.
Jitter control: jittering LFP times by ±4 ms drops σ from ≈ 1.04 to ≈ 0.7 (ANOVA p < 0.05) — σ ≈ 1 is a property of the data’s temporal structure, not chance coincidence of marginal rates.
Refractory-period control: power law and exponent unchanged when refractory period is set to 2 ms (instead of 20 ms) at Δt = 4 ms.
Phase-space trajectory in (σ, α): as Δt varies from 1 to 16 ms, the (σ, α) trajectory crosses the critical point (σ = 1, α = −1.5) at Δt ≈ 4 ms, which matches the population-average IEI_avg ≈ 4.2 ms. This trajectory crossing is the paper’s strongest internal-consistency check.

Methodological details

Bin width Δt = IEI_avg, the population-average inter-event interval (mean interval between LFPs across all electrodes, not within a single electrode). This is the canonical Beggs & Plenz binning convention adopted by every subsequent paper in this reference (Marshall 2016 default; Hochstetter 2021; Mariani 2022; Mallinson 2019). The justification: at Δt > IEI_avg avalanches are spuriously concatenated; at Δt < IEI_avg they are spuriously fragmented; the matched value lets the data’s natural propagation time-scale set the binning.
Power-law fitting by linear regression in log–log space. No MLE. No KS goodness-of-fit. No Vuong likelihood-ratio test against alternatives. This is the methodology that Clauset, Shalizi & Newman (2009) — entry 5 — was written specifically to replace, and that Touboul & Destexhe (2010) — entry 3 — critiqued head-on. The Beggs–Plenz exponents are robust enough that the upgrade to MLE doesn’t change them substantially, but the statistical-significance machinery used to defend them is by modern standards inadequate.
Branching parameter σ estimated from the first two frames of each avalanche via Eq. (1) of the paper: σ = Σ d · p(d), where d is the number of descendants. For multiple-ancestor cases (Eq. 2), d is rounded(n_descendants / n_ancestors), with a correction factor for refractoriness (Eq. 3) accounting for the reduced number of electrodes available in the next bin. The σ ≈ 1 result is the paper’s most direct dynamical signature of criticality.
Avalanche-size > 60 events sometimes observed. Attributed to avalanches looping back to electrodes that have exited their refractory period — a single electrode can participate multiple times in one avalanche.
Information transmission modelled in a feedforward, not recurrent, network. Justified by the empirical observation that < 4.6% of electrodes are reactivated within an avalanche, and reactivations have ≥ 5 intervening empty frames. The simulations are not a model of the actual cortical circuit, only of avalanche propagation through the recorded layer of activity.

Connection to the rest of the framework

Pre-dates Sethna (entry 1) as a cortical reference. Beggs & Plenz arrive at the power-law analysis via the Bak–Tang–Wiesenfeld self-organised-criticality lineage (Bak et al. 1987; Paczuski et al. 1996) and the Zapperi–Lauritsen–Stanley 1995 / Harris 1989 critical-branching-process theory. They do not cite Sethna 2001 — the connection to the renormalisation-group / crackling-noise framework is made later, by Friedman et al. 2012 (entry 2).
The Friedman 2012 paper (entry 2) is the direct successor that adds the Sethna exponent-relation and shape-collapse framework to the Beggs–Plenz foundation. Friedman’s two near-critical samples are the cortical-culture analog of the cultures in Beggs–Plenz 2003.
The Touboul & Destexhe 2010 paper (entry 3) is the direct methodological critique. T&D analyse LFP data with the same Beggs–Plenz methodology (linear regression on log–log axes) and show it gives apparent power laws even on non-critical data. Their proposed remedy — Clauset’s KS + LR procedure — explicitly targets the statistical scaffolding of Beggs & Plenz 2003.
The NCC toolbox (Marshall 2016, entry 6) is built around the Beggs–Plenz avalanche definition. avprops.m implements the Beggs–Plenz frame-based avalanche extraction; rebin.m, asdf2, and the entire data-format design are downstream of the multi-electrode-array conventions established here. The toolbox’s improvements are in the statistics of the exponent fit, not in the definition of the object being fit.
Mariani 2022 (entry 7) cites Beggs & Plenz 2003 as ref 1. Hochstetter 2021 (entry 8) cites it as ref 32 and uses the Δt = ⟨IEI⟩ bin-width convention directly.

Important nuances

The exponents τ = 3/2 and τ_t = 2 are mean-field branching / mean-field directed-percolation values — these are the theoretical predictions Beggs & Plenz match, not empirical coincidences. Any subsequent finding that cortical exponents are near but not exactly these values (Friedman 2012: τ ≈ 1.6, α ≈ 1.7) is interpreted as deviations from mean-field due to non-trivial network structure. Hochstetter 2021’s finding that nanowire networks give τ ≈ 2, α ≈ 2.3 places them in a different class from Beggs–Plenz cortex.
The σ ≈ 1 finding is methodologically separate from the power-law finding and arguably stronger. A power-law slope of −1.5 in a histogram is a fit. σ ≈ 1 is a structural measurement of the dynamics: count descendants per ancestor and average. The fact that the two converge on the mean-field-branching prediction is the paper’s central argument. Subsequent critiques (T&D 2010, T&D 2017, Mariani 2022) target the power-law part of the argument; the σ-measurement is less commonly contested.
The pharmacology result connects criticality to E-I balance. Picrotoxin (disinhibition) pushes the network super-critical and destroys the power law. This is the first demonstration that the cortical critical state is controlled by inhibition, and it is the empirical seed of every subsequent “criticality is tunable by E-I balance” claim (Shew et al. 2009/2011, Poil et al. 2012, the Munoz colloquium review).
LFPs are interpreted as locally synchronised population spikes. This is the load-bearing interpretive step. If LFPs are not synchronised spikes — if they are instead, say, dendritic current dipoles or extrinsic-modulation artefacts — then the avalanche analysis is measuring something other than neural propagation. The Touboul–Destexhe 2010 critique (entry 3) targets this interpretive step. Subsequent papers that work with MUAs or spike-sorted units (Mariani et al. 2021 companion paper; Cramer et al. 2020) implicitly acknowledge that the LFP interpretation is contested and prefer higher-resolution data when available.
Notation convention: α negative. Beggs & Plenz write the size exponent as α ≈ −1.5, including the sign in the value. Downstream papers (Sethna’s τ, NCC’s τ, Friedman’s τ, Mariani’s τ) use the positive convention τ = −α = +1.5. The lifetime exponent is α_t ≈ −2 in Beggs–Plenz vs τ_t = +2 elsewhere. This sign convention is the most common cross-paper confusion when reading the early literature.
No truncated-power-law fit, no LR test against alternatives. By the standards of Clauset 2009 and Marshall 2016, the original Beggs–Plenz exponent estimates would not pass a modern goodness-of-fit test as stated. The exponents are robust to re-analysis with modern methods (verified by every subsequent re-analysis in this framework, e.g. the NCC paper’s own demonstrations), but the original linear-regression methodology is no longer field-standard.
The finite-size cutoff at n = 60 is the recording-system size, not a network property. Beggs & Plenz explicitly demonstrate (Fig. 4F) that the cutoff scales with the number of electrodes used in the analysis, predicting that an infinite array would give an unbounded power law. This is the canonical finite-size-scaling argument later formalised by Cardy 1996, Pruessner 2012, and applied as a positive diagnostic by Hochstetter 2021 (Supplementary Fig. 8).

1. Sethna, Dahmen & Myers (2001), “Crackling Noise,” Nature 410, 242–250

Type: Review article on the renormalization-group framework for systems that respond to slow driving through a broad distribution of discrete events (avalanches).

Core claims the paper establishes

Power-law avalanche-size distributions at critical points. At a continuous (second-order) phase transition, the avalanche-size distribution takes the form P(S) ∝ S^(−τ) and the correlation length diverges. Demonstrated in the paper’s worked example (random-field Ising model with hysteresis) at critical disorder R_c ≈ 2.16 J.
Universality. The asymptotic long-wavelength behavior is set by symmetry and dimensionality, not microscopic detail. Different microscopic systems flowing to the same RG fixed point share the same critical exponents and scaling functions. This is the central theoretical justification for using simple toy models to describe real experiments.
Critical exponents alone are not decisive. Sethna et al. argue (p. 248) that a power law on its own is almost never decisive: it becomes compelling only alongside independent evidence that the morphology is scale-invariant — for example, a successful scaling collapse. This is the paper’s headline methodological recommendation.
Universal scaling functions are more discriminating than exponents. The paper develops the scaling form for avalanche shape: ⟨V⟩(t, T) = T^β V(t/T) where β = 1/σνz − 1 (equation 3, p. 247). The full scaling function V is a universal prediction; collapses of differently-sized avalanche shapes onto a single curve are the recommended sharper test.

Notation

τ: avalanche-size distribution exponent at H_c.
σνz combinations: govern the relation ⟨S⟩(T) ∝ T^(1/σνz) (p. 248).
β = 1/σνz − 1: exponent in the shape-collapse form.
The paper does not introduce α as a duration-distribution exponent. The duration is treated via z (T ∝ L^z), and combinations of basic exponents handle the duration distribution.
The form γ = (α−1)/(τ−1) = 1/(σνz) is not literally written in the paper, although it follows from combining the paper’s exponent relations. This is a downstream form used by Friedman 2012, the NCC literature, and the Touboul–Destexhe critique.

Worked example: disorder-induced criticality

The paper’s specific model is the random-field Ising model with hysteresis: each cubic-grid domain S_i has neighbors S_j coupled with J, plus a quenched random field h_i drawn from N(0, R²). The disorder R is tuned to R_c ≈ 2.16 J for critical behavior.
This is “plain old criticality” — tuned to a critical point — explicitly distinguished from self-organized criticality (Bak-Tang-Wiesenfeld lineage, Fig. 5b and surrounding text on pp. 245–246).
The ε-expansion (ε = 6 − d) is mentioned but not derived; numerical estimates of exponents in d = 3 are given (Fig. 8).

Important nuances to note

The “three-γ Δ_triple” packaging used in downstream criticality pipelines (γ_direct from ⟨S⟩(T) regression, γ_predicted from (α−1)/(τ−1), γ_collapse from shape collapse) is not Sethna’s own formulation. Sethna recommends scaling collapse alongside exponent estimates; the explicit three-way γ-consistency check is a downstream operationalization (Friedman 2012, Marshall 2016, Hochstetter 2021).
The paper’s universality class is the disorder-induced nonequilibrium RFIM.

2. Friedman et al. (2012), “Universal Critical Dynamics in High Resolution Neuronal Avalanche Data,” Phys. Rev. Lett. 108, 208102

Type: Empirical paper applying the Sethna framework to spiking activity from organotypic cortical cultures.

Setup

400 μm slices of rat cortical tissue, cultured to maturity, placed on 512-electrode array (60 μm spacing).
Time series from 100–340 individual neurons; recordings up to 8 hours.
Avalanches defined from spike rasters as consecutive 5-ms bins with activity, bracketed by silent bins.
Triggering matrix p_ij extracted from data via transfer entropy.
Compared to DeVille et al. spiking-network model run with the empirically-determined p_ij.

What the paper claims to show

The paper claims four signatures of criticality in cultured cortical networks (abstract):

Approximate power-law distributions of avalanche sizes and durations
Subcritical and supercritical samples in the dataset
Scaling relations between exponents
Quantitative universal scaling function for the mean avalanche shape (shape collapse) — described as “among the most striking predictions”

Quantitative findings

Two near-critical samples, with averaged exponents reported in text: τ = 1.6 ± 0.2, α = 1.7 ± 0.2, 1/σνz = 1.3 ± 0.05.
Sample 8 specifically (the cleanest critical sample, shown in Figure 2 top row): dashed-line slopes give τ = 1.7, α = 1.9, 1/σνz = 1.3. The figure caption notes that these values satisfy the exponent relation (α−1)/(τ−1) = 1/σνz, as expected for a system near criticality.
Exponent-relation check, two readings:
- Using globally averaged exponents (τ=1.6, α=1.7): (α−1)/(τ−1) = 0.7/0.6 ≈ 1.17, vs measured 1.3 → ~10% disagreement.
- Using Sample-8 specific exponents (τ=1.7, α=1.9): (α−1)/(τ−1) = 0.9/0.7 ≈ 1.286, vs measured 1.3 → ~1.1% disagreement.
- The paper does not quantify the agreement level — only says values “are consistent with this relation” and “satisfy the exponent relation.”
Mean-field comparison (critical): The paper explicitly states (p. 5): “If p_ij is a constant p for all pairs of neurons, 1/σνz = 2.0.” The measured value 1.3 differs from this mean-field prediction by ~35%. The paper attributes this deviation to non-trivial network structure.

Number of γ estimates

The paper effectively involves three uses of 1/σνz, but only two free estimates:

1/σνz from the regression of ⟨S⟩(T) vs T (equation 3) → 1.3
1/σνz from the exponent relation (α−1)/(τ−1) (equation 4) → 1.17 averaged, or 1.29 for sample 8
Shape collapse (equation 5) is performed using 1/σνz = 1.3 as input — i.e., it’s a consistency check that the collapse works with that value, not an independent fit yielding a separate γ_collapse.

The explicit “three γ estimates” packaging is from the NCC toolbox (Marshall 2016), where avshapecollapse.m searches for the best-collapse exponent as a free parameter.

Other findings

8 of 10 samples are subcritical or supercritical, identified via histograms and shape-collapse failure (Fig. 4).
Drugs and developmental stage can move samples along the critical manifold.
Dissociated cultures (separate ~40-neuron preparations): 7 of 10 showed approximate shape collapse but with different critical exponents than organotypic samples.
Mean-field theory gives parabolic shape; experimental shapes are parabolic (in contrast to other systems like Barkhausen noise which show asymmetric shapes).

3. Touboul & Destexhe (2010), “Can power-law scaling and neuronal avalanches arise from stochastic dynamics?” PLoS ONE 5(2), e8982

Type: Critique of LFP-based criticality claims, with a positive methodological recommendation.

Setup

Negative LFP peaks (nLFPs) from awake cat parietal cortex (8-electrode linear array, area 5–7).
Avalanches defined by binning nLFP rasters into Δt = 4–16 ms bins, clusters separated by silent bins.
Avalanche “size” defined as either summed amplitude or count of peaks.
Compared to: positive LFP peaks (control with no neural correlate), event-shuffled surrogates, analytical shot-noise and Ornstein-Uhlenbeck stochastic processes, and the Levina et al. SOC neural network (positive control).

Key empirical findings

Threshold-dependence of apparent power-laws. High detection threshold gives apparent power-law statistics; low threshold gives clearly exponential statistics (Fig. 3 and Tables 1–2).
Figure — threshold-dependence of the apparent power law: a high detection threshold yields power-law-like statistics, a low threshold gives clearly exponential statistics. Not reproduced here; see Fig. 3 in Touboul & Destexhe (2010), PLoS ONE (open access).
Positive LFP peaks (no neural correlation, Fig. 4B WTA) show the same apparent power-law structure as nLFPs.
Event-shuffled surrogates also show the same apparent power-law structure (Fig. 5).
Analytical shot-noise and Ornstein-Uhlenbeck processes (Section “Peak Size Distributions from Stochastic Processes”) generate power-law-looking peak amplitude distributions, with closed-form derivations (equations 7–14).

Key methodological message — POSITIVE

Linear regression on log-log axes cannot distinguish artifacts from genuine power-laws. Demonstrated extensively.
The Clauset 2009 procedure (KS goodness-of-fit + Vuong log-likelihood ratio) CAN distinguish them. Demonstrated by:
- On cat LFP data (Tables 1, 3): KS p-values can be high for power-law fits, but LLR is strongly negative with significance p-val = 0.0 → rejected in favor of exponential.
- On Levina et al. SOC network (Table 5, positive control): power-law p = 0.85, exponential p = 0.00, LLR = +1645 with p-val = 0.0 → clearly identified as power-law.
Discussion conclusion: power-law scaling — especially when read off a log–log plot — does not by itself prove self-organised criticality, and should be backed up by more sophisticated statistical analysis. The paper advocates using statistical procedures, not abandoning them.

Surrogate construction in this paper

Only event-time shuffling of nLFP peaks (preserving amplitude distribution, randomizing peak times). Plus analytical comparison to shot-noise and OU processes computed in closed form. The paper does NOT introduce a phase-randomization surrogate (that comes from Theiler et al. 1992, Physica D 58, 77).

Important nuance

There is a narrow technical sense in which KS goodness-of-fit alone can be ambiguous — Table 2 (avalanche size = number of peaks) shows both exponential and power-law KS fits passing at conventional thresholds. What disambiguates them is the LR test added on top. The paper’s overall message is to use the combined KS + LR procedure, not to throw out goodness-of-fit testing.

The other figures referenced above — Fig. 4B (positive LFP peaks: the same apparent power law with no neural correlate) and Fig. 5 (event-shuffled surrogates) — are in the same open-access article: Touboul & Destexhe (2010), PLoS ONE.

4. Touboul & Destexhe (2017), “Power-law statistics and universal scaling in the absence of criticality,” Phys. Rev. E 95, 012413

Type: Theoretical critique with a positive discriminator proposal.

Setup

Brunel’s sparsely-connected spiking neural network model in its synchronous-irregular (SI) state. Critical slowing-down explicitly absent (Appendix D, Fig. 8) — the system is confirmed to be away from any phase transition.
“Boltzmann molecular chaos” surrogate: N independent Poisson processes sharing a common time-varying firing rate. Two rate constructions: positive part of Ornstein-Uhlenbeck process (main text, Fig. 2) and positive part of reflected Brownian motion (Appendix E, Fig. 9).
Biophysical LFP model: Vogels-Abbott network with Coulomb’s-law LFP construction (Section V, Fig. 5).

Main result

Power-law statistics with critical exponents (τ ≈ 1.5, α ≈ 2) and shape collapse emerge robustly from:

The Brunel SI state (away from criticality)
The Boltzmann-chaos surrogate (no critical point exists)
Analytical derivation (Section III.4) in the slow-rate, large-n limit

Quantitative numbers:

Brunel SI (Fig. 1): τ = 1.42, α = 2.11, measured γ = 1.50
Independent-Poisson + OU rate (Fig. 2): τ = 1.47, α = 1.9, measured γ = 1.4
Analytical Boltzmann-chaos: τ = 3/2, α = 2, measured γ = 3/2

POSITIVE methodological proposal (often missed)

The paper does NOT only criticize — it proposes a positive discriminator. Section III.4 concludes that although a power-law relationship still holds between the mean-shape amplitude A_τ and the duration τ, the measured scaling exponent does not equal the value of 2 that Sethna’s crackling-noise relation predicts from the size (3/2) and duration (2) exponents.

From the Discussion (Section VI): the authors invoke Sethna’s point that a hallmark of criticality, beyond the power laws themselves, is the particular relationship among the exponents; the power laws they generate in the absence of criticality fail that relationship, and they propose using it as a way to tell apart power laws due to criticality from those due to Boltzmann-style molecular chaos.

Quantitative violations of the crackling-noise exponent relation γ = (α−1)/(τ−1):

Brunel SI: (2.11−1)/(1.42−1) = 2.64 predicted, vs measured 1.50 → ~76% mismatch
Poisson-OU: (1.9−1)/(1.47−1) = 1.91 predicted, vs measured 1.4 → ~36% mismatch
Analytical: (2−1)/(3/2−1) = 2 predicted, vs measured 3/2 → ~33% mismatch

For comparison, the Friedman 2012 critical cortical-culture data satisfies the exponent relation to within ~10% (averaged) or ~1% (Sample 8).

Other results

Entropy/information capacity (Section IV, Fig. 4): Maximized in both SI (power-law) AND AI (no avalanches) regimes. Therefore the “criticality → maximal information capacity” link is undermined.
LFP statistics (Section V, Fig. 5): SI, AI, and stochastic-surrogate Vogels-Abbott networks all give similar power-law LFP avalanche statistics with the same exponent ~−1.5. LFP avalanche analysis cannot distinguish synchronous-irregular from asynchronous-irregular network activity.

Surrogate construction in this paper

NOT phase-randomization. The paper builds a population of N independent Poisson processes sharing a common stochastic firing rate. This is closer to a population-level matched-Poisson with stochastic shared rate, NOT a phase-randomization surrogate (Theiler 1992).

5. Clauset, Shalizi & Newman (2009), “Power-law distributions in empirical data,” SIAM Review 51, 661–703

Type: Methodological review and methods paper for fitting and testing power-law distributions.

The Clauset procedure (Box 1, p. 663)

Three-step recipe:

Estimate parameters x_min and α via MLE (Section 3).
Goodness-of-fit test for the power-law hypothesis using KS-based bootstrap p-values (Section 4). If p > 0.1, power law is plausible; otherwise rejected.
Likelihood-ratio test comparing power law against alternative distributions (Section 5).

Section-by-section content

Section 3 (MLE):

Continuous MLE (equation 3.1): α̂ = 1 + n / Σ ln(x_i/x_min) — equivalent to the Hill estimator.
Discrete MLE (equation 3.4): solve ζ’(α̂, x_min)/ζ(α̂, x_min) = −(1/n)Σ ln(x_i) numerically.
Discrete approximation (equation 3.7): α̂ ≈ 1 + n / Σ ln(x_i/(x_min−1/2)) — accurate to ~1% for x_min ≥ 6.
Linear regression in log space is shown to be biased (Table 2, Figure 2). For data with true α = 2.5: LS+PDF gives 1.5, LS+CDF gives 2.37, discrete MLE gives 2.49 (correct).
Recommended minimum sample size: n ≳ 50 for reliable estimates.

Section 3.3 (x_min estimation):

Minimize KS distance D = max |S(x) − P(x)| over x ≥ x_min between empirical CDF S(x) and fitted power-law CDF P(x).
BIC-based alternative (Handcock & Jones) tends to underestimate x_min.
Anderson-Darling alternative is overly conservative.

Section 4 (Goodness-of-fit):

Procedure: fit power law to data; compute empirical KS statistic; generate many semiparametric bootstrap synthetic data sets matching the fit above x_min and resampling the empirical distribution below x_min; for each, refit and compute its own KS statistic; p-value = fraction of synthetic KS statistics exceeding the empirical one.
The bootstrap is semiparametric, not purely parametric — important detail.
Recommended rule of thumb: power law ruled out if p ≤ 0.1 (explicitly described as “relatively conservative” — alternative thresholds like 0.05 are noted as common).
A large p-value does NOT confirm the power law; it only fails to rule it out.
Number of synthetic data sets: ~¼ε^(−2) for accuracy ε, so ~2500 for 2-digit precision.

Section 5 (Likelihood Ratio Test) — note: Section 5, NOT Section VI:

Vuong’s normalized log-likelihood ratio: R/(√n·σ), where σ is the standard deviation of per-observation log-likelihood differences (equations C.3–C.4).
Two-sided p-value: p = erfc(|R|/√(2nσ)) (equation C.6) — the test integrates both tails.
p < 0.1 means the sign of R is statistically reliable (rule of thumb, not derived standard).
For nested distributions (e.g., pure power-law vs power-law-with-cutoff), R follows a chi-squared distribution rather than Gaussian (Section C.1).

Alternative distributions compared (Tables 4, 5)

Continuous data (Table 4): 4 alternatives — exponential, log-normal, stretched exponential (Weibull), power law with exponential cutoff.
Discrete data (Table 5): 5 alternatives — adds Poisson.

Support of distributions

All power-law distributions in the paper are defined on [x_min, ∞), NOT [x_min, x_max].
The “power law with cutoff” (Table 1) has the form x^(−α) e^(−λx) — exponentially tempered tail, not a hard upper bound x_max.
If a pipeline uses fits restricted to a finite [x_min, x_max], that is a refinement of the Clauset procedure, not part of it. The doubly-truncated discrete extension is the central contribution of Marshall et al. 2016 (entry 6).

Empirical findings from 24 real-world datasets (Section 6)

Only ONE of the 24 datasets (“word frequency in Moby Dick”) was judged unambiguously power-law with all alternatives ruled out.
Most datasets that pass the power-law goodness-of-fit test cannot reject log-normal or stretched-exponential alternatives.
Log-normal is the alternative consistently hardest to distinguish from power-law. Section 6 reports that log-normal and power-law behaviour are extremely hard to tell apart: over realistic ranges of x the two distributions sit so close together that no test is likely to separate them without an extremely large dataset.

6. Marshall, Timme, Bennett, Ripp, Lautzenhiser & Beggs (2016), “Analysis of Power Laws, Shape Collapses, and Neural Complexity: New Techniques and MATLAB Support via the NCC Toolbox,” Front. Physiol. 7, 250

Type: Methods paper and MATLAB software release. Three methodological contributions plus the NCC (Neural Criticality and Complexity) toolbox.

Core claims the paper establishes

Automated MLE fitting of doubly truncated, discrete power-law distributions (Section 3.2). The fit is performed on a fixed support [x_min, x_max] with a normalisation constant A(α, x_min, x_max) (Eq. 10) that depends on truncation, so both the likelihood and the goodness-of-fit test are computed self-consistently for the truncated PMF — not the open-tailed Clauset form. This is the headline statistical contribution and is the central refinement of the Clauset 2009 procedure.
Automated avalanche shape collapse (Section 4.2) — the first published automated implementation. Quality of the collapse is quantified as mean variance / span² of the interpolated, rescaled profiles; the scaling exponent γ is found by lattice search minimising that error; a quadratic polynomial is fitted to the collapsed shape and its mean absolute curvature reported as a flatness/sharpness measure.
MATLAB toolbox that bundles all of the above (Section 2.4 and the function-by-function listings at the end of each Methods section).

How the NCC toolbox operationalises the three exponent estimates

The paper does not phrase it as “three γ estimates” — but the three quantities that downstream pipelines treat as the consistency-check triple are all produced by named NCC functions, and the paper does explicitly compare two of them and report agreement:

γ from the size–duration regression (1/σνz from ⟨S⟩(T) ∝ T^(1/σνz)): Eq. 3 and Figure 5. Implemented by sizegivdurwls.m — weighted least squares of log⟨S⟩ on log T, weights = counts per duration, fit range inherited from the duration MLE fit’s [x_min, x_max]. On the cortical-branching demo data the paper reports 1/σνz = 1.503.
γ from the scaling relation (α − 1)/(τ − 1) = 1/σνz: Implicit in the structure but never explicitly computed by a single toolbox function. The avalanche size and duration exponents come out of avpropvals.m (which wraps plmle.m and plparams.m); the (α−1)/(τ−1) combination is the user’s job to form.
γ from the shape collapse: Eqs. 13–15. The avalanche shape is s(t,T) ∝ T^γ F(t/T) (Eq. 13) and γ = 1/σνz − 1 (Eq. 15), so avshapecollapse.m finds γ by lattice-search minimisation of the collapse error and adds 1 before reporting 1/σνz. On the cortical-branching demo data the paper reports 1/σνz = 1.498, and notes “This represents a difference of 0.3%” relative to the ⟨S⟩(T) estimate (1.503). That 0.3% statement is the closest the paper itself gets to a quantitative γ-consistency claim, and it concerns only two of the three estimates, not three.

The toolbox provides the machinery for all three γ estimates, but it does not itself perform the three-way comparison. The “three-γ Δ_triple” packaging is downstream of Marshall, not in it.

How the NCC toolbox implements the Clauset 2009 procedure — and where it deviates

The MLE procedure is structurally the Clauset 2009 procedure, with one substantive refinement and several practical refinements:

Likelihood and exponent search (Eq. 12). The log-likelihood for the truncated PMF is maximised by a three-stage lattice search on α ∈ [1, 5], terminating at precision 10⁻³. Implemented in plmle.m. Structurally the discrete MLE of Clauset 2009 Eq. 3.4 generalised to a finite upper cutoff.
Refinement: doubly truncated support [x_min, x_max]. Section 3.1 paragraph 4 frames the entire fitting contribution as fixing “the vast majority of real potentially power-law data are doubly truncated.” Burroughs & Tebbens (2001) and Yu et al. (2014) are cited for the observation that truncated power laws produce bent CDFs and CCDFs — so CDF-based fitting (Clauset’s approach when displayed on CCDF) wrongly rejects good fits on truncated data.
Figure 4 is the central demonstration of this departure. Same continuous power-law data, artificially truncated at 10⁴: the Clauset method gives p = 0 (rejected); the NCC method gives p = 0.978 (accepted). Both methods recover τ ≈ 1.5 in the exponent; the difference is in the p-value, because the CDF of truncated data is intrinsically bent and the goodness-of-fit test must account for that bend.
Figure — the same power-law data (τ ≈ 1.5) artificially truncated at 10⁴: Clauset’s goodness-of-fit test rejects it (p = 0) while the NCC truncated-support test accepts it (p = 0.978), and both recover the same exponent. Not reproduced here; see Fig. 4 in Marshall et al. (2016), Front. Physiol. (open access).
Goodness-of-fit (Section 3.2, second half). pvcalc.m generates N_PLM = 500 synthetic data sets from the fitted truncated power law over the fit range, computes the KS statistic of each against its own MLE refit, and reports p = fraction of synthetic KS values exceeding the empirical one. The acceptance threshold is p ≥ p_thresh = 0.2, not the 0.1 of Clauset 2009. The synthetic ensemble is generated entirely from the fitted truncated power law, not Clauset’s semiparametric mix.
Fit-range search (plparams.m). If pvcalc rejects the full fit range, the function searches progressively smaller windows, ranked by log(x_max)/log(x_min), and returns the largest window that passes p ≥ p_thresh. This is the toolbox’s automation of the x_min selection step — more aggressive than Clauset, because it also searches x_max.
The toolbox does not perform Vuong’s likelihood-ratio test against named alternatives. This is the largest single gap relative to the Clauset 2009 procedure. Clauset 2009 Section 5 (LR vs exponential, log-normal, stretched exponential, power-law-with-cutoff, plus Poisson for discrete) is not implemented in the NCC toolbox. The paper acknowledges this in Section 3.3 as future work, noting log-normal and exponentially-modified power-law have two parameters and require more careful treatment.
The randomizeasdf2.m null-model generator is the only built-in alternative-comparison facility. It produces jittered, swapped, Poisson-randomised, or wrapped surrogates of the asdf2 spike data, which the user can feed back through avprops and plplottool to compare against the real-data fits visually. It is not an LR test against named parametric distributions.

Empirical/demonstration findings

Cortical-branching demo data (p_trans = 0.26, 100 neurons on torus, 2794 avalanches). Used throughout to illustrate every component of the pipeline. The size-given-duration regression returns 1/σνz = 1.503, the shape collapse returns 1/σνz = 1.498 — the paper’s only quantitative within-paper γ-comparison.
Distribution-model tests (Section 3.2, Figure 3). 100 samples × 4 distributions (power-law, truncated power-law, exponential, log-normal). On true PL: recovered exponent very close to the input τ = 2.5, fit range typically the whole support. On TPL: fit range close to the true [10, 75], slight overestimation of x_max, slight underestimation of τ. On exponential and log-normal data: short tail segments are wrongly accepted as power-law in a noticeable fraction of cases. Section 3.3 explicitly flags this as the toolbox’s main failure mode and recommends LR comparison against null/alternative models — the step the toolbox does not itself automate.
Speed comparison vs Clauset’s plfit. On 10 sets of 10⁴ discretised PL data points (τ = 2), plmle takes ~0.1 s/set vs plfit’s 33.5 s/set. For continuous data, 0.005 s vs 0.013 s.

Important nuances to note

The “three γ” packaging is not in Marshall. The paper computes two estimates and reports a 0.3% agreement on the demo data. The full three-way mutual-compatibility test (γ_predicted from (α−1)/(τ−1), γ_direct from sizegivdurwls, γ_collapse from avshapecollapse) is downstream operationalisation — by Mallinson 2019, Hochstetter 2021, and subsequent device-criticality pipelines.
The p_thresh = 0.2 threshold is more permissive than Clauset’s 0.1. This refinement makes the toolbox more likely to accept a power-law fit than Clauset’s protocol would.
The toolbox does not perform LR tests against named alternatives. If the NCC toolbox is described as “implementing the Clauset procedure,” this should be qualified: it implements the MLE + KS Monte-Carlo half of the procedure, generalised to truncated support; the Vuong LR step is not present.
Notation conflict on γ. Marshall calls the shape-collapse exponent γ in Eq. 13 (s ∝ T^γ F(t/T)) and gives the relation γ = 1/σνz − 1 (Eq. 15), so Marshall’s γ in the shape-collapse formula is β in Sethna 2001’s notation. The exponent that the rest of the avalanche literature calls γ (= 1/σνz) is γ_Marshall + 1 inside avshapecollapse.m. Cross-paper comparison requires keeping track of this offset.
The paper itself flags the over-fitting failure mode targeted by the Touboul–Destexhe critique. Section 3.3 is more cautious than the abstract or the automated machinery suggests. The remedy recommended in the discussion is precisely the LR step that the toolbox does not implement.
The shape-collapse quality-of-collapse metric is not normalised across datasets. Section 4.3 acknowledges that the authors could not devise a method to decide when a given dataset genuinely exhibits shape collapse, nor a quantitative metric for the quality of a collapse. The toolbox returns a γ and an error number, but the error is not comparable across datasets. Shaukat & Thivierge (2016) is cited as an attempt to fix this.
Cuts to the avalanche set. The toolbox enforces a minimum duration of 4 and a minimum of 20 occurrences per duration for the shape-collapse analysis, and the same cuts are applied to the size/duration MLE fits (Section 3.2 paragraph 8) so that the power-law fits use the same portion of the data as the shape-collapse analysis. Any deviation from these defaults (e.g., lowering occ_min_sc to retain statistics on sparse runs) is a pipeline choice that should be documented.

7. Mariani, Nicoletti, Bisio, Maschietto, Vassanelli & Suweis (2022), “Disentangling the critical signatures of neural activity,” Scientific Reports 12, 10770

Type: Empirical + theoretical paper that proposes a positive discriminator for criticality (scale-free spatial correlations) and shows how to disentangle the contributions of extrinsic stochastic modulation from intrinsic interaction networks.

Setup

Empirical: LFPs from rat primary somatosensory (barrel) cortex via a 256-channel multi-electrode array organised as a 64 × 4 matrix with 32 μm inter-electrode pitch. The probe spans all six cortical layers vertically. 20 trials × 7.22 s of basal activity in 4 rats. CMOS-based EOSFET arrays; 7.4 μm electrode diameter; signal sampled at 976.56 Hz and band-pass filtered 2–300 Hz. LFP events (both negative and positive peaks) detected at a 3-SD threshold per channel.
Theoretical: Two models. (1) A paradigmatic multivariate Ornstein–Uhlenbeck (mOU) process with a shared stochastic noise modulation D(t) and an interaction matrix A, tractable in closed form. (2) A coupled stochastic Wilson–Cowan model where the external modulation h is itself the firing rate of an unobserved WC population in a balanced E–I state — more biophysically realistic, simulated only.

Core claims the paper establishes

Power-law avalanches and the crackling-noise relation can be reproduced by extrinsic stochastic modulation alone, without any interactions between units. In the mOU model with W_ij = 0 and a thresholded noise process D(t), the avalanche size and duration distributions are power-law and satisfy δ = (τ_t − 1)/(τ − 1) to within the Monte-Carlo precision of the fit. This is a direct refinement of the Touboul–Destexhe 2017 critique: T&D 2017 showed Boltzmann-chaos populations produce power-law avalanches but proposed the crackling-noise relation as a discriminator; Mariani et al. show that under a properly shaped extrinsic modulation, even the crackling-noise relation can be reproduced by non-critical populations.
Scale-free spatial correlations cannot be reproduced by extrinsic modulation alone. In the same mOU model without interactions, the correlation length ξ(L) is trivially flat: ξ = 1 for any system size L. Switching on the interaction matrix A (inferred from data by solving the Lyapunov equation) recovers the empirically observed linear scaling ξ ∝ L. The correlation length is therefore an interaction-network signature, not an extrinsic-noise signature. This is the paper’s positive methodological proposal.
The two contributions are formally disentangled in the mutual information. In the limit of slow modulation (γ_D ≫ γ_i), the mutual information receives an interaction-dependent shift that is independent of D* plus a modulation-dependent contribution that vanishes only as D* → ∞. The two contributions are additive and independent in this model.

Quantitative findings

LFP data, averaged across four rats:

τ (size) = 1.75 ± 0.10
τ_t (duration) = 2.10 ± 0.30
δ_pred = (τ_t − 1)/(τ − 1) = 1.47 ± 0.18
δ_fit (from ⟨S⟩(T)) = 1.46 ± 0.14
Correlation-length scaling: ξ(L) linear in L, slope ≈ 0.33 ± 0.01, no plateau within 5–55 channel rows

MUA data (referenced from companion paper Mariani et al. 2021, Front. Syst. Neurosci. 15): δ ≈ 1.28 — matches the Fontenele 2019 / Buendia 2021 “near-universal” cortical exponent. The LFP δ ≈ 1.47 differs from MUA δ ≈ 1.28; the authors attribute this to LFP finite-size effects, with finite-size scaling done in the companion paper.

mOU model, low-D* regime:

τ^ext = 1.60 ± 0.01, τ_t^ext = 1.77 ± 0.01
δ^ext_fit = 1.21 ± 0.01, δ^ext_pred = 1.28 ± 0.02
Interacting model gives nearly identical τ ≈ 1.55, τ_t ≈ 1.74, δ ≈ 1.28

Wilson–Cowan model with balanced unobserved input:

τ = 1.72, τ_t = 1.94
δ_fit = 1.24 ± 0.02, δ_pred = 1.31 ± 0.04

Methodological details

Power-law fitting uses a corrected MLE method from Gerlach & Altmann 2019, PRL 122, 168301 — not vanilla Clauset 2009. The correction addresses correlated samples: avalanche sizes are not independent draws, so Gerlach–Altmann sub-sample the data at a decorrelation lag τ* before performing the KS bootstrap. This is a refinement of the Clauset procedure that addresses a problem the Clauset paper itself flags but does not solve. (Reference 42 in Mariani.)
x_min selected by KS minimisation; x_max fixed at the empirical maximum. Closer to the Clauset procedure than to the Marshall procedure (which also searches x_max). Standard 1000-surrogate KS bootstrap; power-law accepted if surrogate-KS-greater-than-data-KS fraction exceeds 0.1.
Correlation length defined by the zero-crossing of the spatial correlation function of fluctuations (Eq. 14), averaged over all channel pairs at a given separation. Sub-sampling is done by selecting L × 4 windows from the 55 × 4 array, with the mean activity recomputed inside each window — the Cavagna et al. 2010 / Martin et al. 2021 “box scaling” procedure for inferring whether the correlation length diverges with system size.
Effective interaction matrix A is inferred from the data by solving a Lyapunov equation σA + Aσ = Q where σ is the empirical correlation matrix and Q is a diagonal matrix set by the noise modulation. This is the inverse problem from observed correlations to inferred interactions — Gilson et al. 2016 lineage.

Connection to the existing framework

Sethna 2001 → cited (ref 21) for the crackling-noise relation and the universality framework.
Friedman 2012 → cited (ref 22) for shape collapse and the empirical near-critical exponents.
Touboul–Destexhe 2010 and 2017 → cited (refs 16 and 17) and engaged with directly. The mOU model is explicitly designed as a more careful version of the T&D 2017 Boltzmann-chaos surrogate.
Clauset 2009 → cited (ref 66) but the fitting procedure used is the Gerlach–Altmann 2019 correlated-data refinement.
Marshall 2016 → not cited. The paper is in the Suweis/Munoz lineage rather than the Beggs/NCC lineage; the fitting machinery is independent.

Important nuances

The “near-universal δ ≈ 1.28” is recovered as a non-critical model artefact. Both the mOU model with extrinsic modulation and the Wilson–Cowan model with balanced unobserved input give δ ≈ 1.28, matching the value reported in cortical experiments (Fontenele 2019, Buendia 2021). If δ = 1.28 is invoked as evidence of criticality, this paper is the explicit counter-construction: the same value emerges from non-critical extrinsic-modulation nulls. This sharpens what was already a soft point (the universality class to which δ = 1.28 belongs is contested) by adding a non-critical generative process to the list of mechanisms producing that exponent.
The empirical LFP δ ≈ 1.47 disagrees with MUA δ ≈ 1.28 in the same animals. The paper attributes the LFP value to finite-size effects in the LFP signal and treats the MUA value as the “true” cortical scaling exponent. Citing δ ≈ 1.28 as “the cortical scaling exponent” without specifying the recording type is ambiguous.
The crackling-noise relation derivation (Eq. 7) is the cleanest “this relation does not require criticality” statement in the literature. If p(s) ∼ s^(−τ) and p(T) ∼ T^(−τ_t) and s ∼ T^δ with negligible fluctuations, then p(s) |ds/dT| dT = p(T) dT gives (T^δ)^(−τ) · δ T^(δ−1) = T^(−τ_t), from which δ = (τ_t − 1)/(τ − 1). No criticality assumption enters this derivation. In critical systems the relation follows from scaling laws (Sethna’s universality machinery); in non-critical power-law-producing systems satisfying s ∼ T^δ, it follows from elementary change-of-variables. The relation is necessary for criticality, but not sufficient.
The model is paradigmatic, not biophysically tight. The authors are upfront that, while not biophysically realistic, the model is simple enough to treat analytically and to interpret physically, yet complex enough to display non-trivial behaviours. The Wilson–Cowan extension is offered as the biophysically more realistic case where the same qualitative conclusions hold.
Sub-sampling and box scaling as a proxy for system size. The 55 × 4 array is sub-sampled into smaller windows and ξ is recomputed in each. This is the Cavagna et al. 2010 procedure for finite-size scaling, recently shown by Martin et al. 2021 (ref 45) to be equivalent in critical models to varying the system size. This is the actionable method for any pipeline that has spatially extended data — it lets you do finite-size scaling without varying the recording.

8. Hochstetter, Zhu, Loeffler, Diaz-Alvarez, Nakayama & Kuncic (2021), “Avalanches and edge-of-chaos learning in neuromorphic nanowire networks,” Nature Communications 12, 4008

Type: Empirical + simulation paper applying the Sethna / Friedman / NCC criticality framework to a non-biological neuromorphic system (memristive nanowire networks). Distinguishes two operationally distinct kinds of criticality — avalanche criticality (DC drive, V* = 1) and edge-of-chaos criticality (AC drive, λ ≈ 0) — and shows that they do not coexist in the same dynamical regime.

Setup

System: self-assembled Ag-PVP nanowire networks with memristive Ag∣PVP∣Ag cross-point junctions. Junctions modelled as voltage-controlled memristors with threshold-driven filament growth (V_set, V_reset). Both simulation and experiment.
Simulations: main figures use a 100-nanowire, 261-junction network at V* = V/V_th = 1; avalanche statistics from an ensemble of 1000 independently generated 2250-nanowire, 6800–7100-junction networks (150 × 150 μm², density 0.10 nw/μm²).
Experiments: 500 × 500 μm² physical NWNs at same density, electrically driven at constant DC bias just above the network’s switching threshold; current sampled at 30 kHz; avalanche statistics combined across three repeats per network.
Avalanche definition: standard NCC/Beggs convention — a maximal run of contiguous time-frames (Δt) containing events, bounded by empty frames. Frame width Δt set to the average inter-event interval ⟨IEI⟩, following Beggs & Plenz 2003 and Mallinson 2019.

Core claims the paper establishes

First-order (discontinuous) phase transition at the avalanche-critical voltage. The steady-state network conductance G*_∞ jumps by ~3 orders of magnitude as V* crosses 1, with multistability and hysteresis. The transition is universal in V* = V/(nV_set), where n is the shortest source–drain path length. Avalanches with power-law statistics appear near a first-order transition, which is unusual — power-law avalanches are conventionally associated with second-order critical points. The authors note this is also seen in disordered hysteretic systems (Sethna et al. 1993, ref 56) and in spiking-network models (Scarpetta et al. 2018, ref 57), and that disorder plus hysteresis are the likely key ingredients.
Avalanche criticality demonstrated with the full Sethna/Friedman three-exponent battery. Power-law P(S), P(T), and ⟨S⟩(T), with the crackling-noise relation (α−1)/(τ−1) = 1/σνz satisfied within uncertainties from three independent estimates (size–duration regression, exponent-relation combination, shape collapse). This is one of the cleanest worked operationalisations of the three-γ check in the literature.
Edge-of-chaos criticality as a distinct dynamical regime. Under AC bipolar drive, the maximal Lyapunov exponent λ varies smoothly through zero as a function of drive amplitude A and frequency f. Ordered (λ < 0), edge-of-chaos (λ ≈ 0) and chaotic (λ > 0) regimes are mapped out. Information-processing performance on a non-linear waveform-transformation task is optimised at λ ≈ 0 for computationally complex targets (double-frequency, π/2-shifted), supporting the edge-of-chaos hypothesis (Langton 1990, Bertschinger & Natschläger 2004).
The two criticalities do not coexist. Avalanche distributions taken at AC drive parameters with λ ≈ 0 are not power-law — KS-test fits fail unless the fitting range is shrunk to x_max/x_min ≲ 2. The paper attributes this to fast AC drive breaking the time-scale separation between drive and network response that avalanche analysis requires. Citing Kanders, Lorimer & Stoop 2017 (ref 69): “avalanche and edge-of-chaos criticality do not necessarily co-occur in neural networks.”

Quantitative findings

Avalanche exponents at V* = 1:

Simulation (N = 2250 NWNs, ensemble of 1000 networks):
- τ = 1.95 ± 0.05, KSD = 0.006, p = 0.54
- α = 2.25 ± 0.05, KSD = 0.004, p = 0.54
- 1/σνz from ⟨S⟩(T) regression = 1.3 ± 0.05
- 1/σνz from (α−1)/(τ−1) = 1.3 ± 0.1
- 1/σνz from shape collapse ≈ 1.3 (Supplementary Fig. 9)
- Three independent γ estimates agree within stated uncertainties.
Experiment (500 × 500 μm² physical NWN):
- τ = 2.05 ± 0.10, KSD = 0.032, p = 0.80
- α = 2.25 ± 0.10, KSD = 0.037, p = 0.59
- 1/σνz from ⟨S⟩(T) regression = 1.2 ± 0.05
- 1/σνz from (α−1)/(τ−1) = 1.2 ± 0.15
- Crackling-noise relation satisfied within uncertainties.

Voltage dependence (simulation):

V* < 1: small-scale avalanches with exponential cutoffs.
V* ≈ 1: clean power-law over the available range.
V* > 1: bi-modal distribution — power-law tail plus a characteristic-size bump corresponding to anomalously large pathway-spanning avalanches. Identified as supercritical.

Universality-class observations: the measured exponents (τ ≈ 2, α ≈ 2.3, 1/σνz ≈ 1.3) match those of percolating tin-nanoparticle networks (Pike et al. 2020, ref 62: τ = 2.0, α = 2.3, 1/σνz = 1.3) but are not the mean-field directed-percolation values (τ = 3/2, α = 2, 1/σνz = 2) found in branching-process / cortical-culture data (Friedman 2012; Cramer 2020). The authors suggest NWNs and NPNs may belong to a universality class distinct from DP — a separate device-physics class associated with the discontinuous-transition + disorder + hysteresis combination, with self-organised bistability (di Santo et al. 2016, ref 59) and quasi-criticality (Fosque et al. 2021, ref 60) as candidate frameworks.

Methodological details

Power-law fitting uses the NCC toolbox (Marshall et al. 2016, explicitly cited as ref 75) together with the Clauset 2009 KS goodness-of-fit (ref 76). This is one of the two foundational device-criticality papers (alongside Mallinson 2019) that established the Marshall–Clauset pipeline as the field-standard machinery for neuromorphic systems.
Doubly-truncated power-law MLE. Both x_min and x_max are estimated; the scaling exponent is determined to precision 10⁻². x_min and x_max are determined to the nearest integer.
Acceptance threshold p > 0.5 — substantially stricter than either Marshall’s p ≥ 0.2 or Clauset’s p ≥ 0.1. This is the strictest threshold used among the eight papers in this reference and gives the Hochstetter exponents an unusually high level of confidence.
500 KS-bootstrap synthetic data sets per fit (consistent with Marshall’s default N_PLM = 500).
Fit-range selection: x_min and x_max chosen to maximise log(x_max/x_min) subject to the p > 0.5 acceptance, with explicit verification that exponents do not change within uncertainties as the range is varied (Supplementary Fig. 17). This is best-practice robustness testing of the fit.
⟨S⟩(T) fitted by linear regression on log–log axes — standard NCC sizegivdurwls.m pattern (weighted least squares on count-weighted bins).
Binning: linear for simulation (high statistics), logarithmic for experiment (sparser tail). Time-frame Δt = ⟨IEI⟩ following the rate-normalising convention of Beggs & Plenz 2003 and Priesemann et al. 2014.
Event detection uses a threshold on |dG/dt|/G (~10⁻³ s⁻¹ for junction-level simulation events, ~1 s⁻¹ for network-level experimental events). Robustness check: changing the threshold leads to negligible change in avalanche statistics.

Connection to the existing framework

Sethna 2001 → cited (ref 46) for the crackling-noise relation framework. The paper explicitly invokes Sethna’s logic: power laws plus the exponent relation are stronger evidence than power laws alone.
Friedman 2012 → cited (ref 33) for the cortical-culture worked example.
Touboul–Destexhe 2017 → cited (ref 47) as the source for the discriminator argument. From the avalanche-analysis section: the paper states that the agreement of the two independent estimates of 1/σνz tests avalanches for consistency with criticality more rigorously than power laws alone, since thresholded stochastic processes can produce power-law size and lifetime distributions yet fail the exponent relation. This is the clearest engagement with the T&D 2017 positive proposal in the device-criticality literature.
Marshall 2016 → cited (ref 75) as the source of the fitting machinery. Hochstetter is, alongside Mallinson 2019, the canonical device-physics application of the NCC toolbox.
Clauset 2009 → cited (ref 76) for the KS-bootstrap goodness-of-fit.
Mallinson 2019 (ref 34) and Pike 2020 (ref 62) — the nanoparticle-network lineage — cited for comparison; the avalanche exponents in NWNs and NPNs are reported as close, suggesting a common universality class for self-assembled memristive devices.
Mariani 2022 → not cited (Hochstetter is 2021, predates Mariani 2022). The Mariani extrinsic-modulation critique therefore postdates this paper; an updated reading would need to ask whether the NWN crackling-noise satisfaction is also reproducible by an extrinsic-modulation null. The Hochstetter answer would presumably draw on the existence of an underlying discontinuous transition with measurable hysteresis and finite-size scaling (Supplementary Fig. 8) as evidence that the NWN avalanches are not a mere extrinsic-modulation artifact.

Important nuances

Power-law avalanches at a first-order transition. Standard avalanche-criticality theory (Sethna; Friedman) is built on second-order transitions. The authors handle this by invoking the disorder + hysteresis ingredient set from Sethna et al. 1993 (RFIM with hysteresis) and Scarpetta et al. 2018 (spiking networks near a first-order transition). The candidate frameworks discussed are self-organised criticality (Bak–Tang–Wiesenfeld, second-order), self-organised bistability (di Santo et al. 2016, first-order), and quasi-criticality (Fosque et al. 2021). The paper does not commit to a specific framework for NWNs.
Three-γ check operationalised in full. This is the cleanest published instance of the (γ_predicted, γ_direct, γ_collapse) triple-comparison the NCC toolbox enables. The paper does not perform a formal pairwise compatibility test (no Δ_triple statistic), but it does report all three values with uncertainties and verify mutual agreement.
Acceptance threshold p > 0.5 is unusually strict. Worth noting if Hochstetter is cited as the field-standard threshold — many other applications of the NCC toolbox use Marshall’s default p ≥ 0.2 or Clauset’s p ≥ 0.1. The p > 0.5 choice means roughly half of the resampled KS distances must exceed the data KS distance, which is a much stronger consistency requirement than either upstream method demands.
Avalanche and edge-of-chaos criticality are distinct regimes and do not coexist. This is an important point for any presentation that conflates the two — citing Kanders, Lorimer & Stoop 2017 (ref 69) as the prior literature on the distinction. In NWNs specifically, avalanche criticality is a DC-drive phenomenon at V* = 1, while edge-of-chaos is an AC-drive phenomenon at appropriately tuned (A, f); the avalanche statistics measured at AC edge-of-chaos parameters are not power-law.
Universality class is not directed percolation. NWN exponents (τ ≈ 2, α ≈ 2.3, 1/σνz ≈ 1.3) differ from the mean-field directed-percolation prediction (τ = 3/2, α = 2, 1/σνz = 2) that holds in critical branching processes and cortical-culture data. The closest match is the nanoparticle-network result (Pike 2020). If a presentation claims “directed-percolation universality” for a device system, this paper is the relevant counter-data point in the same class of systems.
Finite-size scaling explicitly demonstrated (Supplementary Fig. 8a–c): the power-law break increases with network size while the power-law slope is unchanged. This is the standard finite-size-scaling signature of true critical behaviour rather than an apparent power law from another mechanism, and is one of the few signatures in the eight papers that is not reproducible by simple extrinsic-modulation nulls.

9. Mallinson, Shirai, Acharya, Bose, Galli & Brown (2019), “Avalanches and Criticality in Self-Organized Nanoscale Networks,” Science Advances 5, eaaw8438

Type: Empirical device-physics paper. The first systematic demonstration of avalanche criticality in self-organized percolating nanoparticle networks (Sn nanoparticle films with atomic-switch dynamics), obtained by importing the Beggs–Plenz avalanche definition and the Sethna/Friedman/NCC three-exponent criticality battery wholesale into a non-biological electronic system. Together with Hochstetter et al. (2021) this is one of the two foundational device-criticality references in this reference, and it is the earlier of the two — Hochstetter cites it.

Setup

Devices and fabrication. Seven-nanometre Sn nanoparticles deposited between gold electrodes (electrode spacing 100 μm) on a silicon-nitride surface. Deposition is terminated at the onset of conduction, which coincides with the percolation threshold; the critical surface coverage is p_c ≈ 68%, and the device conductance near threshold follows ~(p − p_c)^1.3. Simple two-terminal contact geometry.
Stimulus. Constant DC voltage applied to one contact, opposite edge held at ground. DC is chosen over pulsed/ramped stimulus because it facilitates long-time observation of ongoing switch reconfigurations and avoids power-law cutoffs from short records.
Devices presented. Four devices under DC stimulus (samples I–IV) are presented in the main figures; the authors state the conclusions are consistent with DC, pulsed and ramped stimulus on a further 10 devices.
Two measurement chains / two time scales. A picoammeter gives the slow chain (0.1 s sampling interval); a fast digital oscilloscope gives the fast chain (200 μs sampling interval, i.e. ~1000× faster). Sample I and III use the slow chain; sample II uses the fast chain; sample IV is measured on both. Voltages span 4–6 V (sample I), repeated 6 V DC (sample II), and 6–10 V (samples III, IV). One representative record is 2400 s long.
Observable. Crucially, the measured quantity is a single global time series — the total two-terminal network conductance G(t) — not a spatially resolved multi-electrode array. Events are discrete changes ΔG in this one signal. There is no spatial channel structure, so no branching parameter and no spatial-correlation length are measurable (see nuances).

Core claims the paper establishes

Percolating nanoparticle networks fabricated at the phase transition exhibit power-law avalanche statistics and pass the rigorous criticality battery. Avalanche size S and duration T are power-law distributed (Eqs. 1–2), the mean size grows as a power of duration (Eq. 3), and the three independent estimates of 1/σνz — from ⟨S⟩(T) regression, from the crackling relation (Eq. 4), and from avalanche shape collapse — agree within uncertainty. This three-way agreement is presented as the decisive evidence for criticality.
The signals are qualitatively and quantitatively similar to cortical neuronal avalanches. The authors frame the discrete conductance-switching cascades as the device analog of LFP/spike avalanches, analysed with the same neuroscience machinery (Beggs–Plenz lineage).
Long-range temporal correlations underlie the avalanches. Inter-event-interval (IEI) distributions are power-law; the autocorrelation function (ACF) decays as a power law over several decades; shuffling the IEI sequence destroys both the slow ACF decay and the power-law avalanche distributions (which become exponential), establishing that correlations and avalanches are intimately linked.
Spatial self-similarity of the switching events. The PDF of conductance-change magnitudes P(ΔG) is heavy-tailed over ~3 orders of magnitude, attributed to switches lying on differently-sized branched pathways through the percolating structure.
Self-tuned criticality. Across devices, voltages and repeated measurements, the three 1/σνz estimates remain in agreement even while τ and α themselves drift due to internal reconfiguration of the device — read by the authors as evidence for self-tuned criticality.
Criticality requires the percolation threshold. Devices fabricated below or above p_c show no avalanches: low-coverage devices suffer irreversible conductance drops toward open-circuit, high-coverage devices show no switching at low V and melt at high V.

Quantitative findings

The two worked example samples (Fig. 2–3): sample I = slow chain, sample II = fast chain.

Avalanche size exponent τ ≈ 2 (P(S) ∝ S^(−τ)): S^(−2.01) (sample I, Fig. 3A), S^(−1.96) (sample II, Fig. 3F). Power-law over at least three orders of magnitude in S.
Avalanche duration exponent α ≈ 2.7 (P(T) ∝ T^(−α)): T^(−2.66) (sample I, Fig. 3B), T^(−2.69) (sample II, Fig. 3G). Power-law over about two orders of magnitude in T.
Size-given-duration exponent 1/σνz ≈ 1.6 (⟨S⟩(T) ∝ T^(1/σνz)): T^(1.55) (sample I, Fig. 3C), T^(1.64) (sample II, Fig. 3H). Shape collapse (Fig. 3 D/E, I/J) yields a consistent 1/σνz ≈ 1.6 with parabolic scaling functions.
Pooled “demonstration of criticality” means across all devices/voltages (Fig. 4): 1/σνz = 1.46 ± 0.05 (crackling relation), 1.40 ± 0.03 (⟨S⟩(T)), 1.40 ± 0.04 (shape collapse), uncertainties 1 SD. A single-factor ANOVA finds no significant difference among the three methods, P = 0.47. (Note these pooled means are lower than the two single-sample Fig. 3 values of ~1.6; see nuance 4.)
Conductance-change PDF P(ΔG): ΔG^(−2.59) (sample I), ΔG^(−2.36) (sample II) — a separate exponent from τ; this is the magnitude-of-single-event distribution, not the avalanche-size distribution.
IEI distribution P(t): t^(−1.39) (sample I), t^(−1.30) (sample II), power-law over ~2 orders of magnitude with both a lower and an upper cutoff.
ACF decay exponent β ≈ 0.2 (A(t) ∝ t^(−β)): experimental decay t^(−0.19) (sample I) / t^(−0.23) (sample II), versus much steeper shuffled-surrogate decays t^(−0.66) / t^(−0.64). Via β = 2 − 2H this gives a Hurst exponent H ≈ 0.9.
Stated exponent uncertainties. The authors quote ±0.1 on α and τ, explicitly noting that the formal ML uncertainties are smaller but do not reflect the scatter in the data.

Methodological details

Avalanche definition is the Beggs–Plenz frame convention. Events are conductance changes exceeding a threshold; time bins are set to the mean IEI; an avalanche is a maximal run of consecutive non-empty bins bracketed by empty bins; duration is the number of such bins. Threshold choice (fig. S3) and bin size (fig. S4) are reported not to affect the analysis significantly. This is identical to the convention used by Hochstetter 2021 and inherited from Beggs & Plenz 2003.
Power-law fitting via MLE in the NCC toolbox. The maximum-likelihood approach of Clauset 2009 as implemented in the Marshall/NCC MATLAB toolbox is used for all quantities that form a PDF (IEIs, S, T). Both x_min and x_max are searched (doubly-truncated fit), matching the NCC refinement rather than the open-tailed Clauset form.
Goodness-of-fit by KS bootstrap, acceptance threshold p > 0.2. 500 random power laws are generated from the ML estimators (the NCC default N_PLM = 500), cutoffs iterated, and the fraction of surrogate KS distances at least as large as the data’s is computed. The null of power-law form is not rejected when P > 0.2 — i.e. Mallinson uses the permissive Marshall default (p ≥ 0.2), not Clauset’s 0.1 and emphatically not Hochstetter’s strict p > 0.5. Conversely, shuffled data are rejected as power-law and found exponential.
Model comparison is power-law vs exponential only, via BIC — not a Vuong LR battery. The Bayesian information criterion (Vrieze 2012) selects between power-law and exponential, finding power-law preferred in all cases (fig. S5). There is no likelihood-ratio test against log-normal, stretched-exponential or power-law-with-cutoff — the load-bearing alternatives of Clauset 2009 Section 5 are not tested here.
⟨S⟩(T) and A(t) fitted by linear regression on log–log axes, because the ML machinery applies only to PDFs; this follows the Sethna/Friedman/NCC pattern.
Surrogate = IEI-sequence shuffling. The single surrogate is random shuffling of the IEI sequence, used both to inflate the ACF slope and to collapse the avalanche distributions to exponential. This destroys temporal correlations while preserving the marginal event rate; it is closest to the event-time-shuffling surrogate of Touboul–Destexhe 2010, not a phase-randomization (Theiler 1992) nor a Mariani-style extrinsic-modulation null.
No branching-parameter and no finite-size-scaling demonstration. Because the observable is a single global signal, σ (descendants per ancestor) cannot be measured; and unlike Hochstetter (Supp. Fig. 8) there is no systematic scaling of the cutoff with device size. Long records are used to push back the temporal cutoff, but finite-size scaling is not a positive diagnostic here.

Connection to the existing framework

Beggs & Plenz 2003 (entry 0) → cited (ref 24) and used directly for the avalanche definition and the Δt = ⟨IEI⟩ binning convention; the whole analysis pipeline is described as “substantially the same as those developed in the neuroscience community” (refs 24–27).
Sethna 2001 (entry 1) → cited (ref 23) for the crackling-noise framework, the key exponent 1/σνz, the crackling relation (their Eq. 4), and the fractal-dimension interpretation of the ⟨S⟩(T) slope. The paper explicitly adopts Sethna’s logic that exponents alone are insufficient and that multiple independent exponent estimates must agree.
Friedman 2012 (entry 2) → cited (ref 25) for shape collapse and the parabolic scaling function; Mallinson’s collapsed shapes are stated to be consistent with the parabolic neuronal-avalanche shapes of Friedman.
Touboul & Destexhe 2017 (entry 4) → cited (ref 39) and used as the criticality discriminator: the agreement of the three 1/σνz estimates is invoked specifically to rule out non-critical power-law generators (“random walks”). This is the same T&D-2017 exponent-relation test that Hochstetter later adopts, and Mallinson is the earlier device paper to apply it.
Touboul & Destexhe 2010 (entry 3) → not cited. Only the 2017 paper appears (ref 39); the 2010 LFP critique is absent.
Clauset 2009 (entry 5) → cited (ref 46) for the MLE approach. The KS-bootstrap goodness-of-fit half is used; the LR-against-alternatives half is replaced by a simpler BIC power-law-vs-exponential comparison (see methodological caveat above).
Marshall/NCC 2016 (entry 6) → cited (ref 45) as the toolbox that supplies the fitting and shape-collapse machinery. Mallinson is, alongside Hochstetter, the canonical device-physics application of the NCC toolbox, and is the reference that uses the toolbox’s default p ≥ 0.2 threshold without tightening it.
Mariani 2022 (entry 7) → not cited; predates it (Mallinson is 2019). The extrinsic-modulation critique and the ξ ∝ L spatial discriminator therefore post-date this paper, and — because Mallinson has only a single global observable — the spatial discriminator could not have been applied here even in principle.
Hochstetter 2021 (entry 8) → not cited by Mallinson; Mallinson predates Hochstetter and is cited by it. The two are the paired foundational device-criticality references. Hochstetter places NWNs in a device class with τ ≈ 2, α ≈ 2.3, 1/σνz ≈ 1.3 and explicitly groups the Sn-nanoparticle networks of this lineage with it; Mallinson is the nanoparticle-network progenitor of that comparison (Pike et al. 2020 being the direct follow-on).

Important nuances

Notation is mostly field-standard, with one genuine conflict on β. Mallinson uses τ for the size exponent (Eq. 1), α for the duration exponent (Eq. 2 — the τ_t convention, not a duration “α” in the Beggs negative-sign sense), and 1/σνz for the size-given-duration exponent (Eq. 3) — all consistent with Sethna/Friedman/NCC. But β here denotes the ACF power-law decay exponent (A(t) ∝ t^(−β), β ≈ 0.2), which is not Sethna’s shape-collapse exponent β = 1/σνz − 1. Anyone cross-reading Mallinson’s β against Sethna’s or Marshall’s β will mismatch them; Mallinson’s β is tied to the Hurst exponent via β = 2 − 2H, not to the shape collapse.
The crackling exponent is written as 1/σνz, not as γ or δ. Mallinson keeps Sethna’s 1/σνz notation throughout and calls Eq. 4 the “crackling relationship.” This is the same object that Mariani 2022 and Touboul–Destexhe 2017 call δ. When tabulating across the reference, Mallinson’s 1/σνz ≈ 1.4–1.6 is directly comparable to Mariani’s δ and to the NCC 1/σνz.
Single global observable — several standard discriminators are simply unavailable. Unlike Beggs–Plenz (60 electrodes, branching parameter σ) and Mariani (256-channel array, ξ ∝ L), Mallinson measures one two-terminal conductance trace. No σ, no spatial correlation length, no finite-size scaling of a spatial cutoff. The criticality case rests entirely on the temporal three-γ agreement plus the shuffling control. This is a narrower evidential base than the spatially-resolved papers, and it is worth stating plainly when Mallinson is cited as “demonstrating criticality.”
Two different exponent sets are reported and should not be conflated. The Fig. 3 single-sample values are τ ≈ 2.0, α ≈ 2.67, 1/σνz ≈ 1.6; the Fig. 4 pooled means are 1/σνz ≈ 1.40–1.46 (implying, with τ ≈ 2, a pooled α nearer 2.4–2.5). Quoting “1/σνz ≈ 1.6” (single sample) and “1/σνz = 1.40 ± 0.03” (pooled) as the same number is an error; the pooled means are the statistically defensible figures and are the ones to use for cross-paper comparison.
Universality-class placement: τ ≈ 2 is the robust shared device feature; α and 1/σνz are looser. Mallinson’s τ ≈ 2 matches the device-physics class (Hochstetter NWNs and Pike NPNs) and is not the mean-field directed-percolation / cortical value τ = 3/2. However, Mallinson’s α ≈ 2.4–2.7 and 1/σνz ≈ 1.4–1.6 run somewhat higher than Hochstetter/Pike’s α ≈ 2.3 and 1/σνz ≈ 1.3. Treating “the device class” as a single tight exponent triple overstates the agreement: τ ≈ 2 is shared cleanly, but the duration and size-given-duration exponents scatter across the NPN/NWN papers.
Avalanches here are associated with a continuous (percolation) transition, in contrast to Hochstetter’s first-order transition. The network is fabricated at the percolation threshold p_c ≈ 68%, a continuous/second-order structural critical point, and the avalanches are atomic-switch cascades on that fixed percolating backbone. Hochstetter 2021 instead finds power-law avalanches near a first-order (discontinuous) conductance transition. Both are “device criticality,” but the underlying transition order differs — a point to flag if the two are cited as the same phenomenon.
“Self-organized” refers chiefly to fabrication, and the critical point is partly tuned, not purely self-organized. The network self-assembles, but it is fabricated at the percolation threshold by terminating deposition at the onset of conduction; the authors report that attempts to tune the operating point post-hoc by varying surface coverage were unsuccessful. The “self-tuned criticality” claim rests on the stability of the three-γ agreement under internal reconfiguration, not on a demonstrated dynamical self-organization to the critical point in the Bak–Tang–Wiesenfeld sense.
Model comparison is incomplete relative to the Clauset standard. Power-law is tested only against exponential (via BIC). Log-normal — the alternative Clauset 2009 identifies as hardest to exclude — is never tested, nor are stretched-exponential or power-law-with-cutoff. The KS p > 0.2 acceptance plus a power-law-beats-exponential BIC result is weaker evidence than the full MLE + KS + Vuong-LR battery, and weaker than Hochstetter’s p > 0.5.
The shuffling control rules out renewal/uncorrelated nulls but not an extrinsic-modulation null. Destroying the IEI ordering turns the distributions exponential, which excludes a memoryless/renewal generator and shows the avalanches depend on genuine temporal correlations. It does not exclude a Mariani-2022-style shared slow stochastic drive, which would also produce temporally correlated, power-law, crackling-relation-satisfying events. By Mariani’s later logic the three-γ agreement is necessary-not-sufficient, and the discriminator that would settle it (ξ ∝ L) is unavailable in this single-channel data.
The ΔG, IEI and ACF power laws are correlation evidence, not the criticality test. The criticality demonstration is built strictly on P(S), P(T), ⟨S⟩(T), and shape collapse. The ΔG/IEI/ACF power laws (and H ≈ 0.9) support long-range correlation but are not part of the exponent-relation battery; they should not be quoted as additional “critical exponents.”

10. Priesemann, Wibral, Valderrama, Pröpper, Le Van Quyen, Geisel, Triesch, Nikolić & Munk (2014), “Spike Avalanches in Vivo Suggest a Driven, Slightly Subcritical Brain State,” Frontiers in Systems Neuroscience 8, 108

Type: Empirical + modeling critique of the self-organized-criticality (SOC) hypothesis for the brain. Using highly parallel in vivo spike recordings (rats, monkeys, cats) and human intracranial LFP, it argues that mammalian cortex is not SOC but operates in a driven, slightly subcritical regime without a separation of time scales. It introduces two subsampling-robust avalanche measures and is the foundational paper of the “subsampling / subcritical / reverberating-regime” lineage (Priesemann 2009, 2013; later Wilting & Priesemann 2018). It is the principal dissenting voice among the entries here.

Setup

In vivo spikes (17 experiments, three species). Awake rats — hippocampus CA1, CRCNS/Mizuseki–Buzsáki data, five datasets with {37, 77, 32, 58, 58} single units plus {4, 8, 8, 8, 8} multi-units (single + multi combined, since unit identity is irrelevant to the avalanche definition). Anesthetized cat — visual cortex area 18, CRCNS/Blanche–Swindale “pvc3,” 50 sorted single units, spontaneous (stimulus-free) activity. Awake monkeys — three macaques, lateral prefrontal cortex, up to 16 single-ended microelectrodes (ø 80 μm) or tetrodes (ø 96 μm) on a 0.5/1.0 mm grid, visual short-term-memory task, 11 sessions, ~12,000 trials.
Human LFP. Five refractory-epilepsy patients, intracranial depth electrodes, 44–63 contacts each (44, 48, 45, 50, 61 used after artifact rejection), digitized at 400 Hz, low-pass-filtered at 40 Hz. Events extracted as the area under positive LFP deflection lobes between zero crossings, thresholded so each site has a fixed event rate r = 1/4 Hz; 1 ⟨IEI⟩ ≈ 80 ms.
Models. (1) A Bak–Tang–Wiesenfeld SOC model: 2500 non-leaky integrate-and-fire neurons on a 50 × 50 nearest-neighbour grid, coupling strength α, driven at rate h; for h → 0 and α = 1 it is the canonical BTW/SOC model. (2) A stochastic branching model: 2500 neurons, k = 4, random topology, activation probability p = α/k, open boundaries (p_diss = 0.001). Both are critical at α = 1 and subcritical for α < 1. Subsampling implemented by recording N = 100 of the 2500 neurons (randomly or in fixed 4×4 / 8×8 geometries at distance 1 or 5).

Core claims the paper establishes

In vivo spike avalanches are not power-law — and not exponential either. In 16 of 17 spike experiments the size distribution f(s) is best fit by a lognormal; power-law and Poisson-exponential are both excluded. This is the empirical core: the marker most often assumed to be power-law (spiking) is not.
The brain is driven and slightly subcritical, not SOC. A battery of measures (⟨s⟩, f(s=1,bs), σ*, DFA β) all point to a coupling strength α ≈ 0.98–0.99 — near, but below, the critical α = 1. Phrased dynamically: on average one spike triggers a little less than one spike in the next step.
Three modifications reconcile critical models with in vivo data: (1) subsampling the model as the array subsamples the brain, (2) raising the drive h to eliminate the separation of time scales (STS), and (3) tuning slightly subcritical (α < 1). Only all three together reproduce the data, across both models.
Separation of time scales (STS) is the conceptual pivot. SOC requires slow drive and long pauses so avalanches are temporally separated and well-defined. In vivo activity has no such pauses, so avalanches form a mélange — they meet, merge, intermingle and split — and “avalanches” must be imposed by temporal binning. Binning-dependent avalanches are therefore not the cascade-from-a-single-input avalanches of SOC theory.
There is no unique neural critical exponent τ. Because neural f(s) changes with bin size, τ is not fixed: Beggs–Plenz’s τ runs from 2 to 1.2 as the bin goes 1 → 16 ms (τ = 1.5 only at 4 ms); the human LFP τ here runs from 3 to 1 as the bin goes 2.5 ms → 2.5 s (τ = 1.5 only at ~80 ms ≈ 1 ⟨IEI⟩). So neural avalanches cannot be assigned a universality class by τ.
Two new, subsampling-robust measures. The mean avalanche size ⟨s⟩(bs) and the frequency of size-one avalanches f(s = 1, bs), both as functions of bin size, discriminate driven-SOC (power-law) from Poisson (exponential); f(s=1,bs) is notably robust to the subsampling strategy and is proposed as the better criticality probe under subsampling.
Apparent universality across species/areas/states is a generic driven-subcritical property, not a critical universality class. f(s) and σ*(bs) are strikingly similar across rat/cat/monkey, hippocampus/visual/prefrontal, and anesthetized/awake, despite a ~50× range in population rate.

Quantitative findings

Lognormal fit (spikes): μ = 0.89 ± 0.25, variance σ² = 1.2 ± 0.1 at bin size 1 ⟨IEI⟩; the distribution maximum sits at s = 0.87 ± 0.38, i.e. f(s) is monotonically decreasing. Stretched-exponential and power-law-with-cutoff fits are ~1% worse in likelihood.
Model size exponents: fully sampled BTW/SOC gives f(s) ∝ s^(−τ) with τ ≈ 1 and a finite-size cutoff at s ≈ 1000; the critical branching model gives τ = 1.5.
Population rate R ranged 37 Hz to ~1560 Hz across the 17 experiments (~50×); the driven model used h set so subsampled R ≈ 320 Hz (N = 100, single-unit r = R/N = 3.2 Hz; the α-sweep figures fix r = 5 Hz).
Branching parameter σ:* in vivo maximum only ≈ 1.4 (vs ≈ 3 at bs ≈ 100 ms for the α = 1 model); σ* → 1 for large bins regardless of state; best in vivo match at driven α ≈ 0.98.
DFA exponent β: in vivo β ranged 0.55–0.9 (human LFP mean β = 0.6), matching the driven subcritical model with 0.98 ≤ α < 0.999. Reference points: α = 1 → β ≈ 1 (1/f noise); α = 0 (Poisson) → β ≈ 0.5.
Input-spike (drive) fraction at α = 0.99: only ~1 spike in ~3600 is an external input spike — i.e. the subcritical regime that fits the data is barely driven.
Human LFP f(s) follows an approximate power law (τ = 1.5 at bs ≈ 80 ms), but the slope changes with bin size (τ: 3 → 1 over bs 2.5 ms → 2.5 s) and every other measure (⟨s⟩, f(s=1,bs), σ*, β) indicates subcritical — power-law f(s) coexisting with subcritical diagnostics.

Methodological details

Avalanche definition is the Beggs–Plenz temporal-binning convention: an avalanche is the events in a run of consecutive non-empty bins bracketed by empty bins; size s = total spikes. f(s) (and τ) change with bin size — treated here as a feature to be modelled, not a nuisance.
Normalized bin size. Bins are expressed in units of the average inter-event interval, 1 ⟨IEI⟩ = 1/R, to remove the dependence on population rate and to compare across experiments and against subsampled models. (This is the same ⟨IEI⟩ convention later adopted by the device papers; Hochstetter cites this paper for it.)
Branching parameter σ* estimated as the mean over bins of (events in bin t_i)/(events in bin t_{i−1}), restricted to bins whose predecessor is non-empty (Beggs–Plenz / Priesemann 2009). The paper then demonstrates σ is unreliable* — it depends on bin size, subsampling geometry, and STS; σ* ≈ 1 is not unique to criticality, and σ* > 1 does not imply supercriticality under subsampling.
DFA applied to the summed population activity, window widths 2⁴–2¹¹ ms.
Distribution fitting for the lognormal/alternatives uses the MLE method of Clauset, Young & Gleditsch 2007 (the severe-terrorist-events paper) via Priesemann 2013 — not the canonical Clauset–Shalizi–Newman 2009 SIAM Review procedure, and with no KS-bootstrap goodness-of-fit or Vuong LR battery reported here.
What is deliberately not done: no avalanche-duration distribution, no 1/σνz, no crackling-noise exponent relation, no avalanche shape collapse. The analysis rests entirely on the size distribution f(s), the branching parameter σ*, DFA β, and the two new bin-size-dependence measures. The novel measures’ power-law scaling is empirical, not analytically derived.

Connection to the existing framework

Beggs & Plenz 2003 (entry 0) → cited throughout; the paper uses the Beggs–Plenz binning definition and σ* estimator but is a direct critique of the SOC inference built on them. It specifically undermines the “σ* ≈ 1 ⇒ critical” diagnostic that is one of Beggs–Plenz’s load-bearing arguments, showing σ* is bin-size-, subsampling-, and STS-dependent.
Sethna 2001 (entry 1) → cited only in passing (Methods, for the renormalization/scaling-law context). The paper does not use the crackling-noise exponent battery.
Friedman 2012 (entry 2) → cited as one of the SOC-supporting in-vitro studies (and for the existence of sub-/supercritical samples). Their conclusions partly oppose: Friedman finds near-critical cortical cultures (with shape collapse and the exponent relation); Priesemann finds in vivo spiking is subcritical and does no shape-collapse analysis.
Touboul & Destexhe 2010 (entry 3) → not cited, despite predating this paper. The two are parallel critiques of the “power-law ⇒ criticality” inference from different angles — T&D show stochastic processes mimic power laws; Priesemann shows in vivo spikes are not even power-law and the brain is subcritical.
Touboul & Destexhe 2017 (entry 4) → postdates (2017 > 2014); not cited. Its exponent-relation discriminator is a different strategy from Priesemann’s subsampling-robust measures.
Clauset, Shalizi & Newman 2009 (entry 5) → not cited; the fitting here uses the related Clauset 2007 MLE method. Note the convergence with the 2009 entry’s central warning: lognormal is the hardest alternative to exclude — and here lognormal actually beats power law for spikes (16/17), a concrete realisation of that warning.
Marshall/NCC 2016 (entry 6) → postdates; not cited; the NCC toolbox is not used.
Mariani 2022 (entry 7) → postdates; not cited. (Conceptually adjacent: both are “the obvious critical signature can be reproduced/explained without true criticality,” but via different mechanisms — subsampling+drive vs extrinsic modulation.)
Hochstetter 2021 (entry 8) and Mallinson 2019 (entry 9) → postdate this paper and are not cited by it; but Priesemann 2014 is cited by Hochstetter for the Δt = ⟨IEI⟩ rate-normalising binning convention. So this paper is upstream of the device lineage on the binning methodology, even though it reaches the opposite conclusion (subcritical) about the biological system the devices are emulating.

Important nuances

Notation collisions — the single most important thing to track. Three of the four Greek symbols here mean something different from the rest of this reference:
- α = synaptic/connection strength (the model control parameter, equal to the true branching parameter in the models). The headline “α ≈ 0.98–0.99” is a coupling strength, not the avalanche-duration exponent that α denotes in Sethna/Friedman/Mallinson/Hochstetter.
- σ / σ* = branching parameter (Beggs sense), not the σ inside Sethna’s 1/σνz.
- β = DFA scaling exponent (Hurst-family). The paper itself notes β “is often denoted as α in the literature.” It is not Sethna’s shape exponent β = 1/σνz − 1, and not a duration exponent.
- τ = avalanche-size exponent — this one is standard. When tabulating across the framework, every α, σ, and β from this paper must be relabelled before comparison.
The paper denies a unique neural τ — a framework-level claim. Because τ varies with bin size (Beggs 2→1.2; LFP 3→1; Hahn finds 1.8 at 1 ⟨IEI⟩ in cats), this paper argues neural avalanches cannot be assigned a universality class via τ. Any citation that pins “neural τ = 1.5 = mean-field DP” should be read against this objection.
It runs only half the battery. Size distribution + branching parameter + DFA + two new measures — but no durations, no 1/σνz, no crackling relation, no shape collapse. Do not cite it for duration exponents, the exponent relation, or shape collapse; cite it for the SOC critique, subsampling effects, the STS argument, the ⟨IEI⟩ binning rationale, and the f(s=1,bs)/⟨s⟩(bs) measures.
“Slightly subcritical” is near-critical, with a deliberate safety margin. α ≈ 0.98–0.99 is just below 1; the paper frames the regime as retaining most computational benefits of criticality while avoiding the runaway (supercritical) activity linked to epilepsy. It is a refinement of the criticality picture, not a wholesale rejection.
Lognormal beats power law for spikes (16/17). This is the concrete instance of Clauset 2009’s “log-normal is the load-bearing alternative”: when the proper alternatives are fit, the canonical neural marker is better described as lognormal than power-law. Relevant whenever a power-law claim rests on size-distribution fitting alone.
σ* is shown to be an unreliable criticality diagnostic. It depends on bin size, subsampling geometry, and STS; σ* ≈ 1 is not unique to criticality and σ* > 1 ⇏ supercritical under subsampling. This is a direct methodological caveat on the Beggs σ ≈ 1 signature (entry 0) — worth pairing with that entry’s claim that the σ measurement is its strongest evidence.
The STS argument applies to any driven avalanche analysis, including devices. SOC presupposes a separation of time scales; a continuously driven system (no long pauses) yields a mélange, and binning-imposed “avalanches” are not SOC avalanches. The device papers (Mallinson, Hochstetter) use DC drive over long records — whether they enjoy an effective STS is exactly the question Priesemann’s framework raises. The clean device power laws and the messy in-vivo lognormals may differ precisely because the drive/STS conditions differ, not only the substrate.
Subsampling is unavoidable and distorting. Measuring a fraction of units splits single avalanches and biases f(s), σ*, and DFA; the normalized ⟨IEI⟩ bin and the f(s=1,bs) measure are the paper’s mitigations. Any in-vivo or device comparison that fixes an absolute bin size (e.g. 1 ms) rather than ⟨IEI⟩ inherits the subsampling distortion this paper warns against.
f(s=1,bs) is this paper’s positive discriminator — the analogue of T&D 2017’s exponent relation or Mariani’s ξ ∝ L. Each critique paper proposes its own criticality probe; this one is subsampling-robust but, as the authors note, lacks an analytical derivation, so it functions as a model-vs-data comparison tool rather than a closed-form criterion.
Power-law f(s) coexisting with subcritical diagnostics (human LFP). The LFP looks power-law (τ = 1.5 at ~80 ms) yet every other measure says subcritical. This is a clean within-paper demonstration of the framework-wide refrain that a single power-law fit is insufficient — here made for the subcritical side rather than the stochastic-mimicry side.

11. Muñoz (2018), “Colloquium: Criticality and dynamical scaling in living systems,” Reviews of Modern Physics 90, 031001

Type: Broad review (RMP Colloquium) of the criticality hypothesis in living systems, written from a dynamical / non-equilibrium standpoint. It is not an avalanche-analysis methods paper; it is the reference that fixes the vocabulary — what “critical”, “self-organised critical”, “quasi-critical”, “generically scale-invariant” and “statistically critical” each mean, and how they differ. Its principal value is (i) the theoretical origin of the mean-field τ = 3/2, α = 2 exponents that everything else is compared against, (ii) the precise statement of when self-organisation does and does not reach a true critical point, and (iii) a clean catalogue of the avalanche-analysis caveats (thresholding, binning, subsampling). The entry below extracts only the parts load-bearing for an externally driven avalanche analysis.

Core claims the paper establishes

Criticality = a continuous (second-order) phase transition, reached by tuning a control parameter to a precise value. Born in equilibrium (Ising/ferromagnet at T_c; liquid–gas critical opalescence) and extended to non-equilibrium dynamical transitions, which is the viewpoint the paper adopts for living systems. The defining signatures are diverging correlation length, diverging susceptibility/response, critical slowing down, and scale-invariant (power-law) fluctuations. Section II.
Power laws are necessary but not sufficient. The paper enumerates non-critical mechanisms that produce power laws — random walks (return-time and excursion-”avalanche” statistics), multiplicative processes, preferential attachment, optimisation — and explicitly endorses Clauset et al. 2009 statistical rigour. This is the same “necessary-not-sufficient” position as Sethna 2001 and Touboul–Destexhe.
The contact process / branching-process mean field is the theoretical anchor for cortical avalanche exponents. Worked through in detail (Section II.3): a mean-field absorbing-state transition with bifurcation at λ_c = 1 separating a quiescent/absorbing phase (λ < 1) from an active phase (λ > 1). At criticality there is critical slowing down (activity decays as ρ(t) ∼ t⁻¹) and a diverging susceptibility (Ξ ∝ δ⁻¹, δ = |λ − 1|). “Spreading experiments” from a single seed give avalanches whose mean-field exponents are duration F(T) ∼ T⁻ᵅ with α = 2 and size P(S) ∼ S⁻ᵗ with τ = 3/2, coinciding with the unbiased (Galton–Watson) branching process. These are exactly the values Beggs & Plenz report (τ ≈ 3/2, α ≈ 2) and the values the device-criticality exponents are usually contrasted with.
Finite-size scaling is the diagnostic that distinguishes real criticality from an apparent power law. At criticality the cut-offs scale with system size: P(S, N) ∼ S⁻ᵗ G(S/N), so plotting P(S, N)·Sᵗ against the rescaled variable S/N collapses curves of different N onto one. Because perfect power laws/divergences exist only in the infinite-size limit, finite systems are diagnosed by a peak in susceptibility or correlation length used as a proxy for “approximate” criticality. This is the theoretical justification for the Mariani 2022 ξ ∝ L test and for the finite-size criterion later stated by Dunham (entry 13, criterion b).
Self-organisation to criticality (SOC), and its important fine print. Section II.4. Bak–Tang–Wiesenfeld SOC (sandpile) reaches the critical point without external tuning, via a feedback loop with separated timescales: slow drive in the quiescent phase, fast dissipation/redistribution above a local threshold. Crucially, this feedback self-organises the system to the true critical point only if (i) the separation of timescales is infinite and (ii) the dynamics is conservative (Bonachela & Muñoz 2009; Vespignani–Dickman–Muñoz–Zapperi 1998/2000; Zapperi et al. 1995). Otherwise — finite timescale separation and/or non-conservative dynamics, which is the realistic case — the system is only self-organised to the neighbourhood of the critical point, hovering with excursions to either side: “self-organised quasi-criticality” (SOqC) (Bonachela & Muñoz 2009; Dickman et al. 2000). A network variant (“adaptive criticality”, rewiring-based) is also described. This is the single most useful conceptual result in the paper for framing a driven device.
Generic scale invariance — power laws over extended regions, not at a point. Section II.7 / Appendix A. In systems with broken continuous symmetry (low dimensions), quenched disorder (→ Griffiths phases), or neutral dynamics, scale-invariant behaviour can appear across a finite region of parameter space without any fine-tuning to a critical point. This is an alternative explanation for empirical power laws and an important caveat: e.g. the Millman et al. up/down-state model produces scale-free avalanches in its whole active phase via neutral dynamics, not criticality (Martinello et al. 2017).
Functional advantages of criticality (Section III): maximal dynamic range (Kinouchi–Copelli 2006), long correlations, maximal dynamical repertoire / metastable-state count, and optimal computation. The “edge of chaos” is identified explicitly with the critical point, and reservoir computing is cited as the AI instantiation that performs best near criticality — the conceptual bridge to the neuromorphic-device literature (Hochstetter, Dunham).

Quantitative findings (the citable anchors)

Mean-field branching / contact-process exponents: τ = 3/2 (size), α = 2 (duration). The theoretical reference values for “directed-percolation / branching mean field.”
1-D random walk (non-critical example): P(T) ∼ T⁻³ᐟ², P(S) ∼ S⁻⁴ᐟ³. Illustrates power laws from a non-critical mechanism.
Beggs & Plenz cortical avalanches: τ ≈ 3/2, α ≈ 2, with size-dependent cut-offs obeying finite-size scaling.

Notation (several collisions worth flagging)

τ = avalanche size exponent (P(S) ∼ S⁻ᵗ). Matches τ here. ✓
α = avalanche duration exponent (F(T) ∼ T⁻ᵅ). This is the field-standard “α = duration” convention shared with Friedman, Hochstetter, Mallinson — and it is the duration exponent β in the convention used here. When importing a Muñoz statement, his α is that duration exponent β.
β in Muñoz is the effective inverse temperature of the Mora–Bialek maximum-entropy / “statistical-criticality” construction (β_c ≈ 1). It is not a duration exponent and has nothing to do with the duration exponent β used here.
δ in Muñoz is the distance to criticality, δ = |λ − 1|. It is not the ⟨S⟩(T) scaling exponent that Mariani and Touboul–Destexhe call δ. Two different δ’s across the reference.
σ_S = the size-vs-system-size exponent (⟨S⟩ ∝ L^σ_S, with σ_S = d_S(2 − τ_S) when τ_S < 2). Unrelated to the σ in Sethna’s “1/σνz”, and unrelated to a branching ratio.
ξ = correlation length; λ = control parameter (activity-creation rate) of the contact process.
Muñoz does not assign a dedicated symbol to the direct ⟨S⟩(T) exponent (here γ_B); he routes the size–system-size relation through σ_S instead.

Connection to the existing framework

Beggs & Plenz 2003 (entry 0) → Muñoz supplies the theory (contact process / branching mean field) behind the τ ≈ 3/2, α ≈ 2 values Beggs reports, and behind the bin = mean inter-event-interval convention (he lists it as the standard binning that recovers the mean-field exponents).
Sethna 2001 (entry 1) → same “power law + collapse, not power law alone” stance; Muñoz adds the non-equilibrium (absorbing-state) version of the universality argument.
Touboul–Destexhe 2010/2017 (entries 3–4) → Muñoz lists their non-critical interpretation of avalanches among the standing caveats (Section IV.A.2(v)); he treats the question as open.
Clauset 2009 (entry 5) → cited as the statistical-rigour standard for power-law claims.
Mariani 2022 (entry 7) → Muñoz’s finite-size-scaling argument (cut-offs/ξ diverging with size, with collapse) is the general principle of which Mariani’s ξ ∝ L test is the spatial-correlation implementation.
Hochstetter 2021 / Mallinson 2019 (entries 8–9) → Muñoz’s edge-of-chaos = critical-point identification and his discussion of reservoir computing near criticality are the conceptual frame these device papers operate in; his SOqC and the “first-order/discontinuous transition with disorder + hysteresis” possibility connect to Hochstetter’s self-organised-bistability discussion.
Priesemann 2014 (entry 10) → Muñoz cites Priesemann directly for the subsampling → driven, slightly subcritical reading, and lists subsampling as caveat (iii).
Relevance to driven-device studies: Muñoz is the reference to cite for (a) the SOC-vs-criticality distinction — and specifically the SOqC fine print, which is what makes an externally driven device “near-/quasi-critical” rather than self-organised; (b) the theoretical provenance of the τ = 3/2, α = 2 mean-field values that device exponents are compared with; and (c) the finite-size-scaling rationale that underpins both the Mariani spatial test and the point that a single global observable G(t) cannot, on its own, close the spatial leg.

Important nuances

SOC reaches a true critical point only under idealised conditions. Infinite timescale separation and conservative dynamics. Real, finite, dissipative, externally driven systems sit near the point, not on it — Muñoz’s “self-organised quasi-criticality.” For an externally driven device this is the precise statement behind “tuned/near-critical, not self-organised critical”: even systems that do self-organise generically land in SOqC, and a bias-driven device does not even self-organise — it is tuned by the applied drive.
“Self-organised” in the title is a structural claim in the device literature, a dynamical one here. Muñoz’s SOC/SOqC is about the dynamics tuning itself to (near) criticality. In Mallinson/Hochstetter titles “self-organised” usually refers to the structural self-assembly of the network. Keep the two senses separate.
Power laws have many non-critical origins; generic scale invariance is the subtle one. Quenched disorder (Griffiths phases) and neutral dynamics can produce scale-free avalanches over an extended parameter region without a critical point. For a disordered, hysteretic device this is a live alternative hypothesis, not a remote one.
The mean-field exponents are a reference, not a target. τ = 3/2, α = 2, and the implied ⟨S⟩(T) slope of 2, hold for branching/DP mean field. Real cortex (Friedman) and devices (Hochstetter/Mallinson/Ag₂Se) sit elsewhere; matching the mean-field values is neither expected nor required, and a device γ_B ≈ 1.3 sits firmly in the device range, not the mean-field one.
Muñoz is candid that there is “no smoking gun.” The Discussion flags the epistemological risk that models only generate feature-rich complexity at criticality, so fitting “best at the critical point” can be near-tautological. His recommended antidotes — finite-size analyses and explicit identification of the two phases the transition separates — are exactly the two things hardest to do on a single global conductance trace, and are worth citing as the reason the present claim is framed as compatible with near-critical scaling.

12. Fosque, Williams-García, Beggs & Ortiz (2021), “Evidence for Quasicritical Brain Dynamics,” Physical Review Letters 126, 098101

Type: Primary research Letter (theory + data). The reference for quasicriticality as a concrete, predictive theory: it is the paper that explains why avalanche exponents vary with stimulus/drive while still obeying a scaling relation, and it is the correct primary citation behind the “quasi-criticality” mentioned by Dunham (entry 13). Directly relevant because it describes precisely the situation in an externally driven device — τ and β moving with drive at an approximately preserved scaling exponent.

Setup

The paradox addressed. Cortex looks near-critical, yet no single universality class (single exponent set) is ever recovered: estimated exponents differ across species, across individuals, over time, and with stimulus — while still approximately satisfying the dynamical scaling relation. A single fixed (τ_S, τ_T) universality class seems contradicted by the data.
The resolution proposed. The theory of quasicriticality (Williams-García, Moore, Beggs & Ortiz 2014): a healthy cortex adapts to operate near a Widom line — the locus of maximal dynamical susceptibility — and is truly critical only in the limit of vanishing external drive (stimulus, noise, dissipation). The framework rests on absorbing-state phase transitions, principally directed percolation.
What is tested. Predictions are checked against the Cortical Branching Model (CBM) and against rodent cortical-culture data.

Core claims the paper establishes

Four hypotheses, distinguished by where the (τ̃_S, τ̃_T) pairs sit (Fig. 1). (a) Critical: a single point on the dynamical scaling line. (b) Critical, many universality classes: several points, each on the line. (c) Non-critical: points scattered, generally off the line. (d) Quasicritical: multiple effective pairs, all adhering to the scaling line and possibly moving along it over time. Panel (d) is the cleanest single picture of what “quasi-criticality” means.
As external drive increases, the effective exponents and the branching ratio decrease in magnitude, while the scaling relation is approximately preserved. Moving along the Widom line away from the critical point, dynamical-susceptibility and information-transmission peaks remain finite (no singularity) and shrink; the measured size and duration exponents and the branching ratio all decline in a testable, coordinated way. A system ideally belongs to a unique class (τ_S, τ_T) reached only as drive → 0, but the measured effective exponents τ̃_q wander along the dynamical scaling line as a function of stimulus/noise.
Confirmed in model and data. CBM simulations reproduce the predicted exponent trends; three rodent datasets reproduce the prediction that susceptibility χ and branching ratio σ grow together with the effective exponents.

Quantitative findings

Crackling/scaling relation (derived, Section “Critical vs quasicritical vs noncritical”): from finite-size scaling P(q; L) = q⁻^{τ_q} Ψ_q(q/L^{d_q}) one gets ⟨S⟩ ∝ T^γ with γ = (τ_T − 1)/(τ_S − 1), and the size–size-system relation ⟨S⟩ ∝ L^{σ_S} with σ_S = d_S(2 − τ_S) for τ_S < 2 (so ⟨S⟩ diverges with system size).
CBM (Table I), effective exponents vs spontaneous-activation probability p_s and refractory period τ_r (exponents decrease as drive p_s rises):
- τ_r = 1, p_s = 10⁻³: τ̃_S = 1.57, τ̃_T = 1.82, (τ̃_T−1)/(τ̃_S−1) = 1.44, γ = 1.51
- τ_r = 1, p_s = 10⁻⁴: τ̃_S = 1.64, τ̃_T = 1.99, ratio = 1.55, γ = 1.57
- τ_r = 1, p_s = 10⁻⁵: τ̃_S = 1.69, τ̃_T = 2.14, ratio = 1.65, γ = 1.66
- Increasing τ_r (to 5) shifts κ_w up and increases the effective exponents.
Model/data scaling-line slope ≈ 1.45(4) (differs from the Fontenele et al. 2019 slope; they stress that the slope is tunable but the trends in susceptibility and branching ratio are the real, falsifiable predictions).
Empirical data (Table II): 3 datasets (mouse cortical cultures L = 310 and L = 180 binned at 1 ms; rat culture L = 107 binned at 5 ms):
- Set 1: χ = 0.0096, σ = 0.7101, τ̃_S = 1.57(2), τ̃_T = 1.73(4), ratio = 1.29(3), γ = 1.33(1)
- Set 2: χ = 0.019, σ = 0.7322, τ̃_S = 1.58(3), τ̃_T = 1.77(5), ratio = 1.33(5), γ = 1.33(2)
- Set 3: χ = 0.0475, σ = 0.7433, τ̃_S = 1.65(6), τ̃_T = 1.98(5), ratio = 1.50(6), γ = 1.48(7)
- As predicted, χ and σ (branching ratio) grow together with the effective exponents.

Methodological details

CBM: non-equilibrium stochastic cellular automaton; strongly connected (irreducible-adjacency) graphs, L = 128, fixed in-degree k_in = 3, exponential connection-strength distribution; analytic mean-field exponents coincide with (mean-field) directed percolation.
Locating the Widom line: for given (p_s, τ_r), vary the branching parameter κ until the dynamical susceptibility χ = L[⟨ρ₁²⟩ − ⟨ρ₁⟩²] (variance of the active-node density ρ₁) is maximal, at κ_w; effective exponents are measured at that point.
Local time fluctuation (LTF) = (1/⟨ρ₁⟩)·√(χ/L), used to bin empirical data; effective exponents are computed from bins with intermediate LTF, which correspond to maxima of χ and intermediate branching ratio σ.
Branching ratio σ estimated as descendants/ancestors (more advanced estimators give the same trends with smaller spread).
Both the scaling-relation prediction (τ̃_T − 1)/(τ̃_S − 1) and the directly fitted ⟨S⟩(T) slope γ are reported per dataset and checked for agreement — the same predicted-vs-direct logic as the NCC three-γ comparison.
Dataset-selection criterion: require a “bump” in the avalanche distributions when in the supercritical regime (large bins) and no bump when subcritical (small bins).

Connection to the existing framework

Sethna 2001 / Friedman 2012 (entries 1–2) → cited (refs 12–13) for crackling noise and the scaling relation; quasicriticality keeps the relation but reinterprets the spread of exponents around it.
Priesemann 2014 (entry 10) → cited (ref 6) as the driven, slightly-subcritical reading; quasicriticality is the constructive, predictive version of “driven near-critical.”
Muñoz 2018 (entry 11) → same absorbing-state / directed-percolation foundation; the Widom-line surrogate-for-criticality idea is exactly Muñoz’s footnote that driven systems are never truly critical and the Widom line stands in for the point of maximal susceptibility (Williams-García et al. 2014).
Touboul–Destexhe (entries 3–4) → addresses the same “apparent criticality under drive” worry, but turns it into a positive theory with falsifiable trends rather than a null.
Dunham 2021 (entry 13) → Dunham’s “quasi-criticality” is this theory; Dunham is the device-level invocation, Fosque is the primary source.
Relevance to driven-device studies: the rigorous, citable basis for interpreting drive-dependent exponents. Size and duration exponents that shift with applied voltage (and across nominally similar runs) at an approximately preserved γ_B are the quasicritical picture — effective exponents wandering along the scaling line as drive changes, with the system near (not at) a critical point. It reframes such variability from a caveat into a recognised phenomenon. Note the direction prediction: more drive → smaller effective exponents; this is a concrete, testable expectation for any applied-voltage series.

Important nuances

The effective exponents are never exact critical exponents when drive is non-zero. For p_s ≠ 0 the system is not scale-invariant; a finite-size extrapolation to L → ∞ is possible in principle (Supplemental Material) but the measured τ̃_q will not equal the ideal class exponents. They move along the line; they do not sit on the fixed point. This is the honest counterpart to claiming criticality from a driven dataset.
The slope is tunable; the trends are the prediction. Model parameters can be set to match almost any scaling-line slope, so a matching slope is not the evidence. The falsifiable content is that susceptibility and branching ratio decrease as drive increases while the points stay near the line — that is what should be tested, not the slope value.
Quasicriticality vs. a threshold artefact are different things. The drive-dependent and genuine run-to-run physical variation of (τ, β) is what quasicriticality describes. The threshold-placement sensitivity you document (same run, different θ → different τ, β) is a separate, methodological effect and should not be folded into the quasicritical interpretation. Cite Fosque for the former, keep the threshold caveat for the latter.

Notation (collisions to watch)

τ_S = size exponent (= τ here ✓); τ_T = duration exponent (= β here); γ = ⟨S⟩(T) exponent (= γ_B here ✓). Effective/measured versions carry tildes (τ̃_S, τ̃_T).
σ in Fosque is the branching ratio (a proxy for the control parameter κ) — not the Sethna “1/σνz” exponent and not a scaling exponent at all. A standalone σ here means branching ratio.
σ_S (subscripted) is the size-vs-system-size exponent — distinct again from the bare σ.
τ_r is the refractory period (a model parameter, integer time-steps), not an exponent — a τ that is not a τ.
χ = dynamical susceptibility (order-parameter variance); κ = branching parameter, κ_w its value on the Widom line; p_s = spontaneous-activation probability (the drive knob).

13. Dunham, Lilak, Hochstetter, Loeffler, Zhu, Chase, Stieg, Kuncic & Gimzewski (2021), “Nanoscale neuromorphic networks and criticality: a perspective,” Journal of Physics: Complexity 2, 042001

Type: Perspective / topical review (with original Ag₂Se experimental data). Argues that abiotic neuromorphic nanowire networks are a tunable platform for studying criticality, free of the confounds of biological tissue. It is the most directly comparable published device study for cluster-assembled memristive films: another group running essentially the same DC-bias → threshold → crackling-relation analysis on a similar memristive system, with exponents in the same range, and an explicit invocation of quasi-criticality.

Setup

Systems reviewed: self-assembled atomic-switch / nanowire networks (Ag₂S, Ag-PVP, and the new Ag₂Se) with metal–insulator–metal memristive junctions; ∼10⁸ junction “synapses” per cm²; switching via filament formation/rupture (ionic migration + redox); used for reservoir computing, with best task performance near edge-of-chaos / critical states (Hochstetter 2021; Cramer 2020).
New experiment: an Ag₂Se nanowire network drop-cast onto a microelectrode array (MEA), characterised by (i) I–V cycling and (ii) a long DC-bias time series, then analysed for avalanche criticality.

Core claims the paper establishes

Power laws are necessary but not sufficient; the paper states five criteria for criticality (Section 1.2): (a) power-law relations among order parameter, control parameter and system size; (b) finite-size scaling of correlation length and susceptibility (cut-offs diverge with system size); (c) mathematical (scaling) relations between exponents; (d) shape/data collapse; (e) tunability into sub-/super-critical regimes. This is a compact, citable checklist.
Ag₂Se nanowire networks exhibit avalanche criticality (alongside pinched hysteresis, a power-law PSD, and chaotic-attractor dynamics). The avalanche size, duration and ⟨S⟩(T) exponents satisfy the crackling-noise relation within uncertainties, which the authors read as confirming avalanche criticality.
Across materials the exponents differ but all roughly obey the crackling relation, which the paper interprets as either different universality classes or quasi-criticality.

Quantitative findings

Ag₂Se avalanches (from a 5 V DC-bias, 10 h current trace; events where |ΔI/I| > 5%): τ_S = 1.89 ± 0.02, τ_T = 2.12 ± 0.04, γ = 1.23 ± 0.04, vs the crackling prediction (τ_T − 1)/(τ_S − 1) = 1.26 ± 0.05 (consistent → “confirming avalanche criticality”).
PSD exponent β = 1.7 (Fig. 4B), explicitly noted to differ from γ = 1.23; the difference is attributed to network inhomogeneity (in homogeneous mean-field systems, e.g. RFIM, there is a predicted γ–β relation).
Cross-material comparison (Fig. 5 discussion): Ag-PVP (Hochstetter 2021): τ_S = 2.1, τ_T = 2.3, γ = 1.2; Sn nanoparticles (Mallinson 2019): τ_S = 2.0, τ_T = 2.6, γ = 1.6. All ∼obey the crackling relation. Tension noted: Pike et al. 2020 reported Sn networks inconsistent with the relation.
Quasi-criticality evidence: longer Ag₂Se runs (up to 72 h) showed avalanches with varying exponents in different epochs (stated as “data not shown”).

Methodological details

I–V cycling: 1 Hz triangular input, ±1 V. Pinched bipolar hysteresis loops (memristive memory), occasional hard switching, and chaotic-attractor dynamics — trajectories that never coincide but stay confined to a localised phase-space region (possible edge-of-chaos, to be tested via Lyapunov exponents).
Avalanche analysis: DC bias → relative threshold on the current change (|ΔI/I| > 5%) → ML power-law fits to P(S) and P(T) → ⟨S⟩(T) fit → crackling-relation check. No explicit shape collapse and no finite-size test are reported for Ag₂Se.

Connection to the existing framework

Sethna 2001 / Friedman 2012 (entries 1–2) → the crackling-relation and ⟨S⟩(T) machinery is the same; Dunham applies it to a device.
Hochstetter 2021 / Mallinson 2019 (entries 8–9) → the two prior device studies Dunham compares against; Ag₂Se is presented as a third member of the same device-exponent class.
Fosque 2021 (entry 12) → Dunham’s “quasi-criticality” is Fosque’s theory; Dunham is the device-level invocation, Fosque the primary source. (Cite Fosque as primary, Dunham as the device application — Dunham’s own 72 h epoch evidence is “data not shown”.)
Mariani 2022 (entry 7) / Muñoz 2018 (entry 11) → Dunham’s criterion (b), finite-size scaling of correlation length, is the same principle as Mariani’s ξ ∝ L and Muñoz’s cut-off collapse; it backs the “necessary-not-sufficient” framing and the acknowledgement that a single global G(t) cannot close the spatial leg.
Relevance to driven-device studies: the closest published comparator for cluster-assembled memristive films. Typical device exponents in this class sit in the band spanned by Ag₂Se (1.89/2.12), Ag-PVP (2.1/2.3) and Sn (2.0/2.6), with ⟨S⟩(T) slopes (γ_B) of roughly 1.2–1.4 — bracketing Ag₂Se’s 1.23, Ag-PVP’s 1.2 and Friedman’s ≈1.3. Dunham is also a methodological precedent for the DC-bias + threshold + crackling approach, with one choice worth noting: it uses a relative |ΔI/I| > 5% threshold, as opposed to an absolute noise-edge threshold on the conductance change |ΔG|.

Important nuances

Notation collision on β (critical). Dunham’s τ_S = τ here (size); Dunham’s τ_T = β here (duration); Dunham’s γ = γ_B here (the ⟨S⟩(T) exponent). But Dunham’s β is the PSD (power-spectrum) exponent (= 1.7), not a duration exponent — do not conflate it with the duration exponent β. Three symbols, two of them colliding with the convention used here in different ways.
Dunham’s quoted γ for other networks are partly crackling-relation values, not direct fits. For Sn, Dunham’s “γ = 1.6” is the relation value (τ_T − 1)/(τ_S − 1) = (2.6 − 1)/(2.0 − 1), whereas Mallinson’s directly fitted ⟨S⟩(T) exponent (entry 9) is ≈ 1.4. Compare like with like — direct-to-direct — which actually reinforces the point that the directly fitted γ_B is the quantity to trust.
It is a perspective, and the headline avalanche result rests on one DC trace. A single 10 h Ag₂Se run, no shape collapse, no finite-size test, and the quasi-criticality (72 h) evidence is “data not shown.” Useful and directly comparable, but it is corroborating context, not a heavyweight criticality demonstration; weight it accordingly when citing.
Internal tension on the Sn class. Mallinson (Sn obeys crackling, γ = 1.6) vs. Pike (Sn inconsistent with crackling) — Dunham reports both without resolving it, a reminder that “device systems obey the relation” is not yet a settled statement.

Cross-cutting points worth remembering when evaluating any criticality claim

Power-law fit alone is insufficient. (Sethna 2001, Touboul & Destexhe 2010, Touboul & Destexhe 2017.)
Scaling collapse alone is insufficient. (Touboul & Destexhe 2017 — explicit demonstration that Boltzmann molecular chaos produces both power-laws AND shape collapse without criticality.)
The crackling-noise exponent relation γ = (α−1)/(τ−1) is a necessary condition but is no longer a clean discriminator. T&D 2017 proposed it as a discriminator and showed Boltzmann-chaos fails it; Mariani 2022 showed that a more carefully constructed extrinsic-modulation null (thresholded slow noise) does satisfy it, including at the seemingly-universal δ ≈ 1.28 found in cortex. The relation is therefore necessary for criticality and useful for ruling out the crudest nulls, but not sufficient.
Scale-free spatial correlations (ξ scaling linearly with system size, no plateau) are the strongest currently available positive discriminator. (Mariani 2022.) They cannot be reproduced by extrinsic-modulation nulls; they require an interaction matrix with the right spectral structure. Where spatial data exist, finite-size scaling of ξ via the Cavagna/Martin box-scaling procedure should be done.
The Clauset 2009 procedure (MLE + KS bootstrap + Vuong LR against named alternatives) is the standard statistical scaffolding for any power-law claim. Linear regression in log space is biased and should not be used. The NCC toolbox (Marshall 2016) implements the MLE + KS half on doubly-truncated support; the Vuong LR step against named alternatives is not implemented and must be added separately to make the claim complete. In practice the toolbox’s p_thresh = 0.2 default is sometimes tightened — Hochstetter et al. 2021 require p > 0.5 — and sometimes left at the default (Mallinson 2019).
Log-normal is the load-bearing alternative. It is closest to a truncated power-law over realistic dynamic ranges, and the LR test against it is typically what fails to be decisive. (Clauset 2009 Section 6.)
Mean-field (all-to-all) network exponents differ from real cortical exponents. Friedman 2012 measured 1/σνz = 1.3 vs MF prediction of 2.0; attributed to network structure. Claims of “mean-field directed-percolation consistency” should be checked against actual numbers — for typical cortical data the agreement at the γ level is at ~35% (i.e. inconsistent), even when τ and α individually are close to MF values.
Self-organized criticality is a different mechanism from disorder-induced (“plain old”) criticality. Sethna 2001’s main worked example is the latter; conflating the two when citing Sethna is a common error.
For discrete data, Poisson must be included as a 5th alternative in the LR test. (Clauset 2009 Table 5.) Most neural avalanche data are discrete.
Surrogate attribution matters. Phase-randomization preserving autocorrelation is Theiler et al. 1992, Physica D 58, 77 — not Touboul & Destexhe. Event-time shuffling is T&D 2010. Independent stochastic processes with shared rate is T&D 2017. Thresholded slow-noise OU modulation of independent units is Mariani 2022.
The threshold p < 0.1 is a “relatively conservative” rule of thumb (Clauset 2009), not a derived statistical standard. Convention varies by field. The NCC toolbox uses the more permissive p ≥ 0.2.
The “near-universal” δ ≈ 1.28 has at least three candidate explanations in the literature — a synchronous-asynchronous transition (Buendia 2021), subsampling of a directed-percolation process (Carvalho 2020), and extrinsic stochastic modulation (Mariani 2022). Citing it as “the cortical critical exponent” without engaging with these alternatives is incomplete. Note also that the device-criticality literature finds a different exponent set entirely (τ ≈ 2, α ≈ 2.3, 1/σνz ≈ 1.3) — Hochstetter et al. 2021 (nanowire networks) and Pike et al. 2020 (nanoparticle networks) — which is the natural class for self-assembled memristive films and is not the cortical “δ ≈ 1.28” / DP class. Any comparison between device-system data and a cortical baseline should specify which lineage is being invoked.
Notation discipline. τ for size exponent, α (or τ_t) for duration exponent are field-standard. The size–duration scaling exponent has multiple conventions: 1/σνz (Sethna, Friedman, Marshall, NCC), γ (in some derived treatments), δ (Mariani, T&D 2017). The shape-collapse exponent is β = 1/σνz − 1 (Sethna) or γ_Marshall (NCC Eq. 13). When cross-citing, verify which convention each paper uses.
Pipelines that fit on a finite support [x_min, x_max] are using an extension of Clauset 2009, not Clauset 2009 itself. The doubly-truncated discrete MLE was introduced by Marshall 2016 (the NCC toolbox); when citing Clauset for a truncated-power-law fit, the truncation extension should be flagged as a refinement and attributed to Marshall (or Deluca & Corral 2013 for the continuous case).
Self-organised criticality, criticality, and quasi-criticality are three distinct things. Criticality is the critical state of a continuous phase transition, generally reached by tuning a control parameter (Muñoz; Sethna). Self-organised criticality reaches that state without tuning, via slow-drive/threshold/dissipation feedback — but only lands on the true critical point under infinite timescale separation and conservative dynamics; otherwise it is self-organised quasi-criticality, hovering near the point (Bak–Tang–Wiesenfeld 1987; Bonachela & Muñoz 2009; Muñoz 2018). Quasi-criticality (Fosque et al. 2021) is the externally driven case: the system sits near a Widom line of maximal susceptibility, with effective exponents that vary with drive while approximately obeying the scaling relation. The three share identical statistical signatures, so power laws do not distinguish them — only the mechanism does. For an externally driven device, the correct label is tuned / near-critical (quasi-critical), not self-organised critical.
Driven systems are never exactly critical; the Widom line is the surrogate. A system under non-zero external drive is, strictly, not scale-invariant (Muñoz, footnote on the Widom line; Williams-García et al. 2014; Fosque et al. 2021). The defensible claim for driven data is therefore “near-/quasi-critical,” which is consistent with Priesemann’s “driven, slightly subcritical” reading (entry 10). Quasi-criticality additionally predicts a direction: increasing drive → smaller effective size/duration exponents and smaller branching ratio, at roughly preserved scaling-relation slope — a falsifiable trend to test against a voltage series.
The device-exponent class now has three members, but mind which γ is quoted. Ag₂Se (τ_S = 1.89, τ_T = 2.12, γ = 1.23; Dunham 2021) joins Ag-PVP (2.1/2.3, γ = 1.2; Hochstetter 2021) and Sn (2.0/2.6, γ ≈ 1.4–1.6; Mallinson 2019 / Pike 2020) — the natural class for self-assembled memristive films, distinct from the cortical “δ ≈ 1.28 / directed-percolation” baseline (extends main point 12). Caution: across these papers the quoted ⟨S⟩(T) exponent is sometimes a direct fit and sometimes the crackling-relation value; comparisons should be direct-to-direct — the direct ⟨S⟩(T) fit is in any case the most threshold-stable of the three quantities.