Quasi-Criticality in Complex Systems

• Quasi-criticality is a theoretical framework describing systems operating in a finite region of parameter space that shows high susceptibility and scale-invariant fluctuations.
• It manifests through mechanisms like finite drive, system heterogeneity, and adaptive dynamics, broadening the transition from strict criticality.
• This regime confers practical benefits such as enhanced dynamic range, robust information processing, and improved adaptability in physical, biological, and neural systems.

The quasi-criticality hypothesis postulates that many complex systems—particularly in physics, biology, and neuroscience—operate not at a singular critical point but within a finite window of parameter space exhibiting critical-like features. In this regime, systems demonstrate large susceptibilities, scale-free fluctuations, and optimal information processing characteristics associated with criticality, though strictly infinite correlation lengths and diverging susceptibilities do not occur. Quasi-criticality thus emerges both as a consequence of intrinsic system heterogeneity, adaptation, or external drive and as a robust attractor in dynamically evolving or spatially extended systems where perfect fine-tuning or conservation is unachievable.

1. Fundamental Definitions and Theoretical Framework

A critical point is a singular locus in parameter space where a system undergoes a continuous (second-order) phase transition: correlation lengths diverge, order parameters change nonanalytically, and susceptibilities (e.g., variance of the order parameter) diverge. Hallmarks include scale invariance, power-law distributions, and maximal sensitivity to perturbations.

Quasi-criticality generalizes this notion to situations in which strong but finite susceptibility, power-law scaling over several decades, or maximal information transfer persists within a finite ("broad" or "smeared") region of parameter space. The concept admits several realizations:

• Finite drive or dissipation (e.g., external input in neural networks, spontaneous activity in percolation models): displaces the stationary state from the critical point but leads to a "Widom line"—a locus of maximal response in parameter space (Tian et al., 2023).
• System heterogeneity (e.g., site temperature variance in the Ising model, in-degree variability in Boolean networks): broadens the critical transition into a parameter band exhibiting critical-like fluctuations (Sánchez-Puig et al., 2022).
• Adaptive or evolutionary dynamics: homeostatic feedback or mutual adaptation continuously reorganizes control parameters toward critical or near-critical regions, but non-conservative dynamics and stochasticity maintain the system in a hovering, fluctuating quasi-critical state (Kinouchi et al., 2020, Menesse et al., 2021, Hidalgo et al., 2013).

2. Quasi-Criticality in Biological and Neural Systems

In biological collectives and neural circuits, quasi-criticality provides a mechanistic explanation for the ubiquity of critical-like statistics and optimal processing without requiring unrealistic global tuning:

• Neural avalanches: Empirical and theoretical studies confirm that neuronal networks demonstrate avalanche-size and duration distributions approximating power-laws with mean-field exponents over a finite range, truncated by finite-size or cutoff effects (Tian et al., 2023, Fosque et al., 2020).
• Self-organized quasi-criticality (SOqC): Models with adaptive synaptic weights, neuronal gains, or firing thresholds—updated by antagonistic drive-depression homeostasis—hover the network around the critical point, but with stochastic sawtooth fluctuations due to the lack of strict conservation or infinite time scale separation (Kinouchi et al., 2020, Menesse et al., 2021).
• Evidence from experiment and simulation: Large-scale multi-electrode recordings reveal that the cortex generally operates near the Widom line—maximal susceptibility under finite input—with drift in critical exponents and cutoff but preservation of scaling relations, as expected from quasi-critical dynamics (Fosque et al., 2020).

These behaviors confer major functional advantages: high dynamic range, improved control, and maximal information capacity (Finlinson et al., 2019, Hidalgo et al., 2013). For instance, control of the collective firing rate is easiest in the quasi-critical regime, resolving the conflict between computational flexibility and excessive noise at exact criticality (Finlinson et al., 2019).

3. Quasi-Criticality in Condensed Matter and Disordered Systems

Quasi-criticality also arises generically in diverse condensed-matter models near quantum or thermal phase transitions. Key examples include:

• Disordered magnets and low-dimensional quantum spin systems: Systems such as Ca₂Y₂Cu₅O₁₀, governed by J₁–J₂ Heisenberg chains, exhibit strong sensitivity of high-energy local excitations (e.g., Zhang–Rice singlet intensities) and thermodynamics to proximity to the critical frustration ratio α_c, even when the low-energy collective modes remain nearly unrenormalized (Kuzian et al., 2012).
• Extended critical regions in quantum materials: Organic conductors (TMTSF)₂PF₆ under pressure, iron-based superconductors, and heavy-fermion compounds show linear-in-T resistivity and logarithmic specific heat over broad regions near quantum critical points, accounted for by two-fluid models with "hot" and "cold" Fermi surface regions (Meier et al., 2012, Varma, 2015). In these cases, the quasi-critical/extended critical regime is defined by finite hot-spot fractions and crossovers persisting over a wide control-parameter window.
• Quantum XY models and topological excitations: The onset of criticality in 2D dissipative quantum XY models is often controlled by the proliferation of topological defects (vortices, warps). The resulting critical region is characterized by separable correlation functions, effective dynamical exponent z=∞, ω/T scaling, and anomalous transport (e.g., T-linear resistivity, C/T ~ ln T) over a finite range—strongly supporting the universality of quasi-criticality at two-dimensional quantum critical points (Varma, 2015).

4. Quasi-Criticality in Discrete, Excitable, and Epidemic Systems

• Percolation and absorbing-state transitions: In percolation and branching-process models with spontaneous activation (e.g., background noise or seed input), criticality is destroyed in the strict sense, but quasi-criticality remains robust. Dynamic susceptibility peaks at a shifted pseudo-critical point (Widom line), while spatial and temporal correlations show power laws at distinct, nearby thresholds (Jasna et al., 29 Aug 2025). This operationally decouples maximal sensitivity and maximal correlation, reflecting distinct functional optima.
• Stochastic epidemic models: Heavy-tailed, stochastically varying infectivity generates critical-like plateaus with reproduction number R(t) ≈ 1 over long intervals—suppressed exponential growth or decay—without external tuning (Ariel et al., 2021). This mechanism extends naturally to other spreading and adaptive systems exhibiting long-lived, self-organized quasi-steady states near instability boundaries.

5. Mechanisms: Heterogeneity, Adaptation, and Environmental Complexity

Several concrete mechanisms generically broaden the critical window and drive systems into quasi-criticality:

Mechanism	Effect on Criticality	Example Systems
Heterogeneity	Broadens sharp critical transitions into finite critical regions	Ising models, Boolean networks (Sánchez-Puig et al., 2022)
Adaptive homeostasis	Dynamically steers control parameters toward criticality, but with finite oscillation amplitude	Neural networks, gene-regulatory systems (Kinouchi et al., 2020, Menesse et al., 2021)
Environmental complexity	Evolution selects for maximal Fisher information (susceptibility) in model representations, favoring critical regions	Adaptive/evolutionary population models (Hidalgo et al., 2013)

Heterogeneity in control parameters, connectivity, or adaptation timescales smears out phase transitions, making the quasi-critical regime wide and robust to perturbation or environmental drift (Sánchez-Puig et al., 2022). Homeostatic feedback in neural and biological systems further reinforces hovering near the critical boundary (Kinouchi et al., 2020).

6. Mathematical Formalism: Scaling, Susceptibility, and Widom Lines

Quasi-criticality is formalized via the scaling of fluctuations, susceptibilities, and correlation lengths as functions of the distance to the critical surface:

• Avalanche statistics and scaling: In mean-field (directed-percolation-like) neural or spreading systems near criticality, the avalanche size distribution follows

$P(S) \sim S^{-\tau} f(S/S_c)$

where the scaling cutoff $S_c$ diverges as a (negative) power of the distance $\Delta$ to the critical point: $S_c \sim \Delta^{-1/\sigma}$ (Tian et al., 2023, Fosque et al., 2020).

• Susceptibility maxima and the Widom line: In externally driven systems, the locus along which susceptibility is maximized defines the Widom line. Along this line, scaling exponents and finite-size cutoffs drift with control parameters, but the system maintains near-maximal responsiveness and critical-like scaling laws (Fosque et al., 2020, Jasna et al., 29 Aug 2025).
• Nontrivial scaling relations: In networks and statistical models, empirical exponents for, e.g., avalanche sizes and durations, may drift but remain near-satisfying exact scaling relations (e.g., $\gamma = (\tau_T - 1)/(\tau_S - 1)$), testifying to quasi-criticality (Fosque et al., 2020).

Such scaling behaviors persist under changes in system size, input rate, or heterogeneity, confirming the robustness of quasi-criticality as an operational regime.

7. Implications, Functional Advantages, and Limitations

Quasi-criticality confers several advantages in both biological and artificial systems:

• Enhanced dynamic range: Operation near the Widom line provides broad response without the need for perfect fine-tuning.
• Robust computational function: Quasi-criticality balances fluctuation-driven sensitivity with control, facilitating large information transmission and flexible adaptation (Finlinson et al., 2019, Fosque et al., 2020).
• Evolvability and heterogeneity: Natural selection may favor quasi-critical regimes as they accommodate complexity, heterogeneity, and metaplasticity, enabling adaptation to changing environments or system components (Hidalgo et al., 2013, Sánchez-Puig et al., 2022).

However, susceptibility remains finite, power-law scaling is truncated at a critical scale, and exact universality classes may become blurred due to drift in exponents, cutoff effects, and environmental or architectural complexity. There is ongoing investigation into extending field-theoretic frameworks for quasi-criticality across coupled dynamical phases, incorporating network topology, and identifying experimental markers of quasi-critical operating points in complex systems (Tian et al., 2023, Fosque et al., 2020).

Quasi-criticality thus unifies a wide spectrum of phenomena in physics, biology, and complex adaptive systems, providing a rigorous foundation for the observed preponderance of critical-like behavior under generic, real-world conditions where idealized, fine-tuned criticality cannot be sustained.