This is the 2D lattice counterpart to the 3D cubic-lattice avalanche model. Roughly 1,900 model neurons sit on a flat square grid with 4 nearest neighbours (k = 4, von Neumann neighbourhood) instead of a cube with 6. When a neuron fires, it independently activates each quiescent neighbour with probability:
p = σ / k (k = 4 here, k = 6 on the cube)
so the expected number of "child" firings
per firing neuron is still exactly σ.
What's genuinely different from the 3D version: instead of an orbiting camera over a sphere lattice, the whole grid is rendered flat and paired, live, with a log-log histogram of completed avalanche sizes — the actual statistical signature (a power law, P(s) ~ s-τ) that defines the critical-brain hypothesis, but which the 3D view never plots directly. Watch the histogram bars stretch into a straight diagonal line in log-log space as σ approaches 1, and curl over into a short, steep drop-off away from criticality.
- σ < 1 (subcritical) — cascades die out almost immediately; small, exponentially-bounded avalanches, a steep histogram.
- σ = 1 (critical) — cascades die or grow with roughly equal likelihood at every step; the size histogram flattens toward a straight line in log-log space, with no characteristic scale — exactly what Beggs & Plenz measured in cortical slice recordings.
- σ > 1 (supercritical) — cascades tend to grow without bound, like runaway seizure-like activity, and start to saturate the whole grid.
A small external drive rate spontaneously fires a random quiescent neuron whenever the grid falls silent. Each neuron also has a strict refractory period after firing during which it cannot re-fire — enforced here with a dedicated "firing" state distinct from the refractory countdown, so a neuron never mistakenly re-triggers itself the instant its refractory period ends; it can only be re-activated by an actual neighbour.
The critical-brain hypothesis proposes that healthy cortex self-organises near σ ≈ 1 because this maximises dynamic range and information transmission. This 2D pair lets you watch that transition both spatially (on the grid) and statistically (in the live size distribution) at the same time.