Dynamical criticality (overview)

The hypothesis

The criticality hypothesis (a.k.a. “computation at the edge of chaos”): systems poised between an ordered regime (perturbations die out — robust but rigid) and a disordered/chaotic regime (perturbations spread unboundedly — flexible but unreliable) attain the highest computational capability and the best robustness↔flexibility trade-off. Proposed by Kauffman for living systems and by Packard/Langton/Crutchfield for computation. This is a review of the evidence and the open problems. The focus is dynamical criticality (order/chaos regimes separated by critical surfaces in parameter space), distinct from — though related to — self-organized criticality (SOC, spontaneous tuning to a critical state, e.g. the sandpile).

What the critical point buys (statistical-physics preliminaries)

At a second-order transition: universality (order parameters follow power laws with critical exponents shared across whole classes of systems); a correlation length that diverges, $\Gamma\sim(T-T_c)^{-\xi}$ , so distant elements become maximally coupled; power-law cluster/avalanche distributions and $1/f$ power spectra; a scale-free response to perturbations; and critical slowing down.

Computation at the edge of chaos

Cellular automata (Langton’s $\lambda$ ). As the transition-function parameter $\lambda$ moves toward $\approx 0.5$ , CA cross from order to chaos with second-order signatures — critical slowing down, power-law transients, maximal mutual information between cells — coinciding with Wolfram’s complex Class 4 (some of which are universal computers).
Crutchfield $\epsilon$ -machines. The minimal statistical model reconstructed from a data stream diverges in size at the transition (e.g. logistic map at the onset of chaos) — intrinsic computational complexity peaks there.
Random Boolean networks (RBNs). The order/chaos transition sits on the critical line $2p(1-p)K = 1$ (bias $p$ , in-degree $K$ ). Information measures peak on it: perturbation-size entropy (Rämö), pairwise mutual information between nodes (Ribeiro), basin-entropy scaling with system size (Krawitz & Shmulevich), and set-based complexity (Galas). Lizier’s decomposition is the sharp result: information storage peaks approaching the edge, information transfer peaks just inside chaos, and the critical line optimally balances the two. Fisher information is also maximal at criticality.
Excitable / threshold networks. Kinouchi & Copelli’s excitable net has a transition in active-node density vs. average branching, with critical networks showing maximal variance in extinction times and power-law avalanche sizes; Bertschinger & Natschläger’s threshold-gate (and spiking) networks compute best on time series at the edge of chaos. Even combinatorial-optimization local search peaks in performance at a phase transition (Macready).

Critical living systems

The standard method: build an ensemble of network models, compare their statistics to real biological data across ordered/critical/chaotic parameters, and read criticality off the best fit.

Cells. Gene-expression avalanches after knock-out match RBNs best at (slightly sub-)critical parameters — indexed by the Derrida parameter $\zeta$ ( $\zeta<1$ ordered, $\zeta>1$ chaotic, $\zeta=1$ critical); Serra/Villani find S. cerevisiae matches at $\zeta$ slightly below 1. Similar best-fit-at-critical results via Lempel–Ziv complexity (HeLa), compression distance (macrophages), and across four kingdoms (Balleza).
Brain. Neuronal avalanches follow power laws only at normal activity, breaking under over-/under-activation — evidence the cortex is poised near criticality.
Also flocks (Mora & Bialek) and morphogenesis; Bailly & Longo’s “extended critical situations” — life inhabits a critical region, not a point.
Two (non-contradictory) rationales: (i) evolvability — critical systems best balance mutational robustness against phenotypic innovation; (ii) fitness — they best balance information storage / modification / transfer, and behavioural repertoire against reliability.

Does evolution reach criticality? — the crux

Mixed. Artificial evolution drives RBNs to the edge of chaos (Torres-Sosa), and RBNs evolved for combinatorial tasks converge to critical (Goudarzi); Hidalgo et al. find criticality is a stable evolutionary solution only in variable / complex environments, not low-complexity ones. But Benedettini/Roli show not all tasks lead to criticality — the fitness landscape and task can dominate the dynamical regime. Task-dependence is the central unresolved point.

Open questions

Definitional slipperiness. Criticality is defined via average perturbation response, but the ensemble one averages over (all initial conditions vs. reachable states vs. states on an attractor vs. structure ensembles) changes the definition and can mislead — formalised as a hierarchy of sensitivities.
When does evolution favour criticality? Needs a fitness function and a proper categorisation of tasks by their fitness-landscape features (correlation length, rate/amount of change); the guess is time-varying, complicated tasks.
Environment & openness, and the evolution/adaptation/learning relation, are usually overlooked.
Artificial design. Enforcing criticality could be a general, task-agnostic design criterion for learning systems — an alternative to ad-hoc fitness functions.
Common internal organisation? Whether critical systems share a structural signature (e.g. via the dynamical cluster index / relevant subsets) is still open.

References

Roli, Villani, Filisetti & Serra (2016), arXiv:1512.05259 · Langton (1990), Physica D 42, 12 · Crutchfield & Young (1990), Complexity, Entropy & Physics of Information · Kauffman (1993), The Origins of Order · Kinouchi & Copelli (2006), Nat. Phys. 2, 348 · Lizier, Prokopenko & Zomaya (2008), ALIFE XI · Serra, Villani, Graudenzi & Kauffman (2007), J. Theor. Biol. 246, 449 · Hidalgo et al. (2014), PNAS 111, 10095 · Mora & Bialek (2011), J. Stat. Phys. 144, 268 · Beggs & Timme (2012), Front. Physiol. 3, 163.