ODE Bifurcations vs Criticality

"Criticality" in brain models means at least three different things, and confusing them has real consequences for what you can claim about how the brain computes.

The paper starts from a simple observation: the most popular whole-brain model (the Hopf model) fits each brain region with a Stuart–Landau oscillator and calls the system "near criticality" when the oscillator's control parameter sits close to zero — the point where spontaneous oscillations switch on. Meanwhile, a separate tradition fits Ising spin models to fMRI data and locates the brain relative to a genuine thermodynamic phase transition. Both communities say "criticality." The paper asks whether those two uses of the word refer to the same thing. The answer is: usually not, and the gap matters.

The key concept is the order parameter — a single macroscopic number that is zero in a disordered phase and nonzero in an ordered one (magnetization in a magnet, synchrony in an oscillator population). Landau showed long ago that a continuous phase transition is a pitchfork bifurcation of the order-parameter equation, so bifurcations and critical points are not automatically different animals. But two conditions must hold for the identification to be legitimate. First, the bifurcating variable must actually be an order parameter of a large many-body system — not just a phenomenological coordinate fitted to data. The standard Hopf whole-brain model fails this test: its parameters are tuned to match fMRI, not derived from neurons. Second, even a genuine order-parameter bifurcation only captures mean-field criticality — it throws away the spatial fluctuations that produce diverging correlation lengths, scale-free spatial structure, and non-classical exponents. Those are the hallmarks of Wilson–Fisher criticality, and they are what Landau theory famously misses.

There is one class of models that partially bridges the gap: the Montbrió–Pazó–Roxin (MPR) neural mass equations, which are an exact reduction of a large spiking network in the thermodynamic limit. Here the dynamical variables are provably related to the Kuramoto order parameter of the underlying population, so an MPR bifurcation genuinely signals a collective transition in the neurons — not just a property of a fitted curve. But even MPR only delivers mean-field criticality, because it assumes all-to-all coupling and has no spatial geometry. It cannot develop a diverging spatial correlation length, which is precisely the property the paper argues matters most for computation.

Why does that spatial property matter? The paper's central functional argument is that a brain implementing hierarchical, compositional models of the world — the kind an "algorithmic agent" needs — requires information to flow freely across spatial and temporal scales simultaneously. Scale-free correlations at a genuine critical point provide exactly that: no characteristic scale acts as a bottleneck. A fitted Hopf oscillator near its bifurcation is locally noise-sensitive but has no spatial extent; it cannot supply cross-scale coupling. The Ising framework, by contrast, directly quantifies the spatial correlation length, and the finding that LSD pushes the brain's Ising temperature further from criticality is interpretable as a partial collapse of that cross-scale hierarchy — consistent with the perceptual fragmentation reported in psychedelic states.

The practical upshot is a vocabulary recommendation and a hierarchy of claims: "Hopf-proximal" for fitted ODE models, "nonequilibrium macroscopic transition" for MPR bifurcations, and "critical point" reserved for the statistical-mechanics sense with diverging correlations. The distinction is not pedantic — it separates phenomenological curve-fitting from a mechanistic claim about the computational substrate of biological intelligence.