--- title: "Neural Avalanches and Criticality: When a Power Law Misleads Interactive Lab | Sandbox Physics" description: "Does a power-law distribution of avalanche sizes show that a network is critical? No." canonical_url: "https://sandboxphysics.com/labs/neural-avalanche-criticality/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/neural-avalanche-criticality/" last_modified: "2026-10-31" --- FR42 · Neural statistical physics # Neural Avalanches and Criticality: When a Power Law Misleads **Does a power-law distribution of avalanche sizes show that a network is critical?** No. A near-critical network does give avalanche sizes close to a power law with exponent three halves, but uncoupled units that share a slow input also pass a power-law test and give a branching estimate near one. Only a causal test, such as kicking one unit and counting what follows, separates real cascades from the mimic. Record cascades in a simulated network. Uncoupled units that only share a slow input also give a plausible power law and a branching estimate near one. > Interactive simulation: open this page in a web browser to change the parameters and run the model. Model assumptions A sheet of 1000 binary units: each spike activates each of its ten wired neighbours in the next step with probability one tenth of the branching parameter, a unit cannot fire twice in a row, and random external input starts new cascades. The magenta array records a block of units. Avalanches are runs of non-empty time bins in the recording. The power-law fit is the discrete maximum-likelihood method of Clauset, Shalizi and Newman (2009); the branching estimate is the multistep regression of Wilting and Priesemann (2018). The mimic presets switch the coupling off and give the units a drifting or shared slow input instead. The kick test fires one recorded unit in a silent network and counts what follows; only a simulation can do that cleanly. Everything is simulated. Recorded avalanche sizes and the power-law fit Regression slopes across lags and the multistep fit Causal test: recorded spikes after kicking one unit of a silent network More numbers and how they are computed - **Lower cutoff and range of the fit**: $0$ - **Naive test: plausible power law near three halves**: $0$ - **Conventional one-step slope**: $0$ - **Multistep estimate on every unit (model truth)**: $0$ - **Network spikes per kick (model truth)**: $0$ - **Units under the array**: $0$ - **Analysed record**: $0$ - **Time bin**: $0$ The size chart, the fit and the slopes use only the recorded, binned spikes. The kick test and the whole-network estimate need the model itself and are marked as model truth. The exported file holds the binned recording, the avalanche sizes, the fit and the kick response. Physics tutorial ## A power law is not proof of criticality **Background** Recordings of brain tissue show bursts of activity, called avalanches, whose sizes spread over orders of magnitude. A branching process poised at its critical point predicts a power law with exponent three halves, so a fitted power law is often read as a sign of criticality. **Why it matters** The inference runs backwards: from statistics of a partial recording to the dynamics that made them. A simulation where the truth is known is the cleanest place to test which analyses can be fooled, and by what. Start with the essentials - **Focus question**: Does a power-law distribution of avalanche sizes show that a network is critical? - **One-sentence intuition**: A coupled network near its critical point gives sizes close to $s^{-3/2}$. Units with no coupling at all, sharing only a slow input, also give a plausible power law and a branching estimate near one. Kicking a single unit separates them: the coupled network answers with about a hundred recorded spikes, the mimic with just the kicked one. ### Core mathematical model #### Critical branching process $$ P(s)\simeq\sqrt{\tfrac{2}{\pi}}\;s^{-3/2}\quad(m=1,\ \text{binary branching, odd }s) $$ For binary branching at the critical point, Zapperi, Lauritsen and Stanley (1995) derive this large-size tail. Below the critical point an exponential cutoff appears. #### Discrete maximum-likelihood exponent $$ \hat\alpha\simeq1+n\Big[\sum_{i=1}^{n}\ln\frac{s_i}{s_{\min}-\tfrac12}\Big]^{-1} $$ Only avalanches with $s_i\ge s_{\min}$ enter. A goodness-of-fit $p$ above 0.1 means the power law is not rejected; it does not mean it is proved. #### Multistep regression $$ r_k=\frac{\operatorname{Cov}(a_{t+k},a_t)}{\operatorname{Var}a_t}=b\,m^k $$ Subsampling scales every slope by the same factor $b$, so fitting the decay with $k$ recovers $m$. Any slow input that correlates activity across time also gives $m\approx1$. ### Common difficulties #### A good fit is not a mechanism **Typical misconception** If the power-law test passes, the network must be critical. **Better mental model** The shared slow input preset has no coupling, yet its fit gives $p\approx0.9$ over more than two decades. #### The exponent depends on how you look **Typical misconception** The avalanche exponent is a property of the network alone. **Better mental model** Recording 20 of 1000 units from the same near-critical network moves the fitted exponent from about 1.46 to about 3.1, and the one-step slope falls to about 0.49. ### Run the experiment 1. 01 #### Record near-critical cascades Leave the first preset running and watch the size chart fill. **What to observe:** The fitted exponent is about 1.46 and the multistep estimate about 0.985. One kicked unit leads to about 97 recorded spikes, itself included. 2. 02 #### Switch the coupling off Choose Shared slow input, no coupling. **What to observe:** The fit is still plausible, now with $\hat\alpha\approx2.1$, and the multistep estimate rises to about 0.997. The kick now produces one spike: the kicked unit itself. 3. 03 #### Record from a corner Choose A few electrodes see little, then drag the array or its corner. **What to observe:** The network is the same near-critical one, yet the exponent moves to about 3.1. Enlarging the array brings it back toward 1.5. Physical conclusion Avalanche statistics are a property of the network, the input, the array and the analysis together. A power law can support criticality but cannot establish it; a causal perturbation is what tells coupling from a shared drive. ## Related experiments and articles Statistical physics of neurons · Reading path ### From a charged membrane to avalanches in a network of neurons Articles on this path that use this experiment, followed by the other experiments they suggest. 1. [Article **Neural avalanches from branching spikes, seen through a partial recording**](https://sandboxphysics.com/stories/neural-statistical-physics/from-neurons-to-a-recording/) 2. [Article **Why a power law cannot prove the brain is critical**](https://sandboxphysics.com/stories/neural-statistical-physics/when-a-power-law-misleads/) Fields & matter · Starter path ## From a field to an electron microscope Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface. 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