--- title: "Why a power law cannot prove the brain is critical" description: "Fit a power law properly, then watch uncoupled models pass the test. With the truth known, only perturbing the network separates critical cascades from mimics." canonical_url: "https://sandboxphysics.com/stories/neural-statistical-physics/when-a-power-law-misleads/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/neural-statistical-physics/when-a-power-law-misleads/" last_modified: "2026-10-31" --- [Statistical physics of neurons](https://sandboxphysics.com/stories/neural-statistical-physics/) L4 · Chapter 7 # Why a power law cannot prove the brain is critical Sandbox Physics · 2026-10-31 The last chapter found that a branching process at its critical point gives avalanche sizes falling with exponent three halves. Experiments often argue the other way: the recorded avalanches look like a power law, so the brain must be critical. This chapter checks that argument. First it fits power laws properly; then, in simulations where the truth is known, it asks which tests can be fooled and by what; finally it looks at a preprint that offers another reason why fitted models so often sit near a critical point. ## Before you begin Branching processes, cutting avalanches and multistep regression from chapter 6; the basic ideas of maximum-likelihood estimation and the Kolmogorov–Smirnov distance. - Estimate a power-law exponent and its statistical error by discrete maximum likelihood, choose the lower cutoff by the Kolmogorov–Smirnov distance, and read the goodness-of-fit p-value correctly. - Name two mechanisms that give a power law, or a branching estimate near one, without critical cascades. - Read a false-positive table: which model fools which test. - Explain why a fitted maximum-entropy model sitting near a critical point is not evidence of criticality either. ## Fit the power law properly first A straight line on log–log axes is not a test. Clauset, Shalizi and Newman give a three-step recipe. For each candidate lower cutoff $s_{\min}$, estimate the exponent by maximum likelihood, and keep the cutoff that minimises the Kolmogorov–Smirnov distance $D$ between the data and the fitted law. Then draw many synthetic data sets from that law and count the fraction whose distance exceeds the measured one; that fraction is the goodness-of-fit $p$. A value $p\ge0.1$ means the power law is not rejected, not that it is proved. For discrete data the exponent has an approximate formula, given below, which they report accurate to about 1% once $s_{\min}$ is above about 6. We tested the Lab’s fit on synthetic data of known exponent, 40 sets of 3000 values each: a true exponent of 1.5 gives $1.4969\pm0.0020$, 2 gives $1.988\pm0.006$ and 2.5 gives $2.434\pm0.015$. The last is 4.5 standard errors low, because the median selected cutoff is only 3, below the range where the approximation holds. The Lab’s $p$ takes one shortcut: the synthetic sets keep the cutoff chosen for the measured data, whereas the original method chooses a new cutoff for every synthetic set. Without that optimisation the synthetic distances tend to be larger, so the Lab’s $p$ is more lenient than the published one. $$ \hat\alpha\simeq1+n\Bigl[\sum_{i=1}^{n}\ln\frac{s_i}{s_{\min}-\frac12}\Bigr]^{-1},\qquad \sigma_{\hat\alpha}=\frac{\hat\alpha-1}{\sqrt n},\qquad D=\max_{s\ge s_{\min}}\bigl|S(s)-P(s)\bigr| $$ Left: the approximate maximum-likelihood exponent for discrete data, using the $n$ events with $s_i\ge s_{\min}$. Middle: its statistical error. Right: the Kolmogorov–Smirnov distance between the cumulative distribution of the data $S$ and the fitted law $P$. Source: A. Clauset · J. Touboul and A. Destexhe ## Power laws without critical cascades Touboul and Destexhe found that thresholding a random signal, or shuffling a recording into a surrogate, can produce scaling much like neuronal avalanches; a straight line on a log–log plot does not survive testing. Priesemann and Shriki go further: a Poisson process with a constant rate gives no power law, but once the input rate switches among a few levels, avalanche sizes come close to a power law with exponent near 3/2 over a limited range. Morrell, Sederberg and Nemenman let independently firing neurons share a few slowly varying inputs and reproduced the scaling under coarse-graining that had been read as a sign of criticality. The Lab has two such mimics, with no coupling between units at all. The one with a shared slow input, at the default array and a bin of 4 steps, gives $\hat\alpha=2.13$ and $p=0.87$ over 2.7 decades, and multistep regression gives 0.997. The one with a drifting input gives a multistep estimate of 1.000. Multistep regression is fooled for a simple reason. It only measures how long activity stays correlated in time, and a slowly varying input keeps activity correlated for a long time without any unit triggering another. Priesemann and Shriki conclude that passive analysis cannot rule out an external drive that happens to mimic critical activity; small, controlled perturbations are needed. Source: J. Touboul and A. Destexhe · V. Priesemann and O. Shriki · M. C. Morrell · J. Wilting and V. Priesemann ## A false-positive table To see what fools each test we took four models with a known truth: a coupled network near the critical point, a subcritical coupled network, and two uncoupled mimics, one with a drifting input and one with a shared slow input. Each model was analysed with 3 electrode arrays (covering 60.8%, 10% and 2% of the units), 3 bin widths (1, 4 and 8 steps) and 10 random seeds: 90 analyses per model. A plausible power law ($p\ge0.1$ over at least 1.5 decades): 0.178 for the near-critical network, 0.378 for the shared slow input, 0.022 for the drifting input and 0 for the subcritical network. The shared-slow-input mimic passes more than twice as often as the network that really is near critical. That network fails in 82% of analyses: all 60 with the 10% and 2% arrays fail, and with the largest array 3, 8 and 5 of 10 pass at bins of 1, 4 and 8 steps. So failing a power-law test does not show that a network is not critical either. The naive verdict, a plausible power law with exponent within 0.2 of 1.5, passes only the near-critical network, at 0.178; the mimics’ plausible fits all have exponents between 1.75 and 2.74. A multistep estimate of at least 0.95 passes both mimics every time, the near-critical network 0.433 of the time and the subcritical one never. The last test is an intervention: pause the external input, kick one recorded unit, and ask whether more than 2 recorded spikes follow on average. The near-critical network passes all 90 times, neither mimic ever does, and the subcritical network passes 0.667 of the time. A kick shows whether there is coupling; how close to critical the network is needs the number of spikes one kick triggers in the whole network. Source: A. Clauset · V. Priesemann and O. Shriki · J. Wilting and V. Priesemann ## Why fitted models so often sit near a critical point Another line of evidence comes from maximum-entropy models: fit an Ising-type model using only each neuron’s mean activity and the pairwise correlations, and the parameters often land near a phase transition. A 2026 preprint by Carcamo and Lynn offers an explanation. In the Curie–Weiss model, which keeps only the mean activity $m$ and a correlation strength $\chi$, a small patch of parameters near the critical point maps onto a large region of statistics, and that region grows with the number of neurons. Inferring parameters from data therefore pushes models toward the critical line without any fine-tuning. We checked their Eq. (3) on the model only; we did not reanalyse any recording. Take $m=-0.8$ and $\chi=2$, already above the mean-field upper limit of 0.712. Solving the inverse problem exactly gives $\lambda$ of 1.478, 1.374 and 1.371 for $N=20$, 100 and 1000, all above the critical value 1, and $h$ of $-0.110$, $-0.033$ and $-0.0048$, approaching 0 as $N$ grows, that is, approaching the critical line. The double-well approximation of Eq. (3) gets $\lambda$ with an error falling from 1.8% to 0.29%. Page 11 of the preprint says the inferred interactions exceed the critical strength but writes $\lambda\ge0$ in parentheses. By Eq. (3), $\lambda=\tanh^{-1}(m_0)/m_0$ is at least 1 for every $0