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.

Power-law fit to recorded avalanche sizes $\hat\alpha=1.462\pm0.010,\ p=0.21$

Branching estimate from the recording $\hat m_{\mathrm{MR}}=0.98479\pm0.00074$

Recorded spikes after kicking one unit $\bar n_{\mathrm{kick}}=97.4$

Model truth: coupling and kick cascade ratio $\sigma=1.05,\ m_{\mathrm{kick}}=0.994$

Couple the network, then test the recording

External input

Coupling: units one spike tries to fire

1.05

\sigma

External input, events per step

-2

\log_{10}h

Time bin of the analysis, steps

1

\Delta t

Drag the magenta array across the sheet, or its corner to resize it. The analysis only sees what the array records.

Recording, seed and playback

Detector miss probability

0

p_{\mathrm{miss}}

False detections per unit per step

0

r_{\mathrm{false}}

Steps in the analysed record

300000

T_{\mathrm{rec}}

Random seed

42

k

Model steps shown per second

25

\dot n

Model, method and sources

$P(s)\propto s^{-\alpha}\ (s\ge s_{\min}),\quad \hat\alpha\simeq1+n\Big[\sum_i\ln\frac{s_i}{s_{\min}-\tfrac12}\Big]^{-1},\qquad r_k=b\,m^k$

The power-law exponent uses the discrete approximation of Clauset, Shalizi and Newman (2009). The lower cutoff $s_{\min}$ minimises the Kolmogorov–Smirnov distance, and the goodness-of-fit $p$ comes from 100 synthetic sets with that cutoff fixed; a fit counts as plausible when $p>0.1$ . The multistep estimator fits $r_k=b\,m^k$ to the regression slopes between bins $k$ apart, which removes the bias that subsampling puts on the one-step slope. The model truth comes from kicking one unit of a silent network 200 times.

Clauset, Shalizi & Newman (2009), arXiv:0706.1062v2 · Wilting & Priesemann (2018), arXiv:1608.07035v3 · Priesemann & Shriki (2018), PLoS Comput. Biol. 14, e1006081 · Morrell, Sederberg & Nemenman (2021), arXiv:2008.04435v1

Read when a power law misleads

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

BackgroundRecordings 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 mattersThe 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 misconceptionIf the power-law test passes, the network must be critical.

Better mental modelThe 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 misconceptionThe avalanche exponent is a property of the network alone.

Better mental modelRecording 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

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.
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.
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.

Neural Avalanches and Criticality: When a Power Law Misleads

A power law is not proof of criticality

Core mathematical model

Critical branching process

Discrete maximum-likelihood exponent

Multistep regression

Common difficulties

A good fit is not a mechanism

The exponent depends on how you look

Run the experiment

Record near-critical cascades

Switch the coupling off

Record from a corner

Related experiments and articles

From a charged membrane to avalanches in a network of neurons