Criticality & Neural Avalanches

Drop grains of sand onto a pile one at a time. Most grains just settle, but every so often one triggers a small slide — and once in a long while a single grain sets off an avalanche that reshapes the whole slope. You can't predict which grain does what, yet the pile keeps poising itself right at the edge of sliding. Many neuroscientists think active brain tissue sits at a similar edge: one spike usually fizzles, sometimes sparks a small cascade of firing, and rarely ignites a wave across the network. Those cascades of firing are called neural avalanches.

When one neuron fires, it nudges the neurons it connects to. On average, each firing neuron makes $\sigma$ (sigma) others fire — this number is the branching ratio. If $\sigma<1$, activity fades out (the network is "subcritical" and quiet); if $\sigma>1$, activity runs away (seizure-like); right at $\sigma=1$ the network is "critical," balanced on a knife-edge. Count how many neurons fire in one cascade and call it the size $s$. At the critical point there is no typical avalanche size — tiny and huge cascades both occur, and the chance of a cascade of size $s$ falls off as a power law, $P(s)\propto s^{-3/2}$. Put in numbers: a cascade of 4 neurons is about $2^{3/2}\approx 2.8$ times rarer than one of 2 neurons, yet it is never forbidden.

Written precisely, the size and duration distributions are $P(s)\sim s^{-\tau}$ with $\tau\approx 3/2$ and $P(T)\sim T^{-\alpha}$ with $\alpha\approx 2$, and the two exponents lock together through the scaling relation $\langle s\,|\,T\rangle\sim T^{(\alpha-1)/(\tau-1)}=T^{2}$. The branching ratio is the ratio of mean descendant spikes to ancestor spikes, and $\sigma=1$ marks a genuine phase transition between order and disorder — the same kind of critical point physicists study in magnets and percolation. Sitting there is not an accident: Kinouchi & Copelli (2006) proved that a network's dynamic range and its information transmission both peak exactly at $\sigma=1$, which is why a brain might tune itself to that edge. On this page the controls stand in for those knobs — the parameter sliders set excitability / branching ratio, Input $I$ sets the external drive, and T sim sets how long you watch.

Push the branching parameter below 1: cascades should stay small and short-lived, with activity dying out. Nudge it up toward 1 and avalanches of every size appear — the straight-line power-law signature on a log-log histogram. Push past 1 and watch activity blow up into runaway, seizure-like bursts. Then hold the setting at criticality and raise Input $I$: the response stays graded across a huge range of drives, which is the "maximum dynamic range" that makes the critical point so useful.

Beggs & Plenz (2003): MEA recordings, bin spikes in Δt ≈ interelectrode interval. Avalanche: contiguous non-empty bins, bounded by empty bins. P(size) ~ s^{-3/2} and P(duration) ~ T^{-2} are exact signatures of critical branching process. Observed in cortical slices, awake animals, fMRI.

Each spike generates σ descendants. σ=1 (critical): P(s) ~ s^{-3/2} (mean-field critical branching theory, Otter–Harris). σ<1: exponentially truncated (subcritical). σ>1: exponential growth → seizure. Duration-size scaling: ⟨s|T⟩ ~ T^{(α-1)/(τ-1)} = T² (confirmed experimentally).

At criticality: (1) maximum dynamic range (equal sensitivity to all input amplitudes); (2) peak information transmission (I(input;output) maximised); (3) maximum susceptibility (fastest response to input). Debate: power laws arise from many mechanisms beyond criticality (super-position artifact, etc.). Kinouchi & Copelli (2006) theoretically proved dynamic range maximisation at σ=1.