knife edge
From threshold to scale invariance
Many systems share one structural fact: there is a control parameter with a critical value. Below it, perturbations decay and the system has a typical scale of response. Above it, perturbations amplify and the system commits to a new regime. At the threshold, scale disappears: correlation length diverges, response becomes unbounded, and the same equations describe magnets, branching cascades, neuronal avalanches, percolation networks, and pre-seizure cortex.
Subcritical: the medium absorbs the message.
Critical: the medium passes the question to itself.
Supercritical: the medium becomes the fuel.
Power laws as a fingerprint
Off criticality, avalanche sizes follow distributions with a characteristic scale, exponential tails, Gaussian shoulders. At criticality the cutoff disappears and you get a power law:
For mean-field branching processes the exponent is . The avalanches tab fits this from the live distribution; nudge gain through 1 and watch the empirical slope drift toward the theoretical line.
Sheaves illuminate criticality
A cellular sheaf assigns local stalks to overlapping patches, with restriction maps that demand agreement on overlaps. The sheaf Laplacian satisfies
so its kernel is the space of global sections: distributed states that are locally consistent everywhere. Near criticality, the wave field develops scale-free fluctuations across patch boundaries, restriction maps weaken, and the spectral gap collapses, many almost-global sections, but no rigid global one. This is the structural counterpart to the long-range correlations you see in the lattice.
Why this matters for consciousness
Cortical recordings show avalanche distributions with branching ratio σ ≈ 1 and an exponent close to -3/2 (Beggs & Plenz, 2003). Anaesthetic agents that abolish consciousness, propofol, xenon - shift dynamics toward the subcritical exponential regime; ketamine, which preserves dream-like experience, leaves dynamics near criticality (Maschke et al., 2024). This does not show that criticality is consciousness, but it suggests it is an enabling regime: the place where local activity can both differentiate and integrate, and where local-to-global compatibility (sheaf gluing) becomes flexible rather than rigid.
What is solid, what is speculative
The wave equation, branching-process exponent, and bifurcation analysis are textbook physics. The Beggs/Plenz avalanche evidence and the Maschke anaesthesia results are empirical. The cellular sheaf Laplacian construction is mathematically standard (Hansen & Ghrist, 2019). The interpretive claim, that the spectral gap of a sheaf over neural activity tracks consciousness, is a speculative bridge, well-motivated but unproven. The 96-site lattice is a cartoon, not a brain.
Model changelog
- 96-site periodic 1D lattice with linear growth, local Laplacian coupling, mean-field long-range coupling, and cubic saturation
- Five regime presets: subcritical, critical, supercritical, avalanche edge (Beggs/Plenz-style), over-synchronised
- Live avalanche detection with log-binned histogram and OLS τ-exponent fit
- Cellular sheaf Laplacian over 16 patches with data-dependent restriction maps; Jacobi eigendecomposition exposes kernel dimension and spectral gap
- Analytical (gain, damping) phase diagram with critical-curve overlay and current-point marker
- Sensitivity tornado on λ_max across all parameters
- Calibration table linking presets to known empirical regimes (cortical avalanches, propofol, ketamine, percolation, seizures, BTW sandpile)