# Introduction to Self-Organised Criticality

> A primer on self-organised criticality (SOC): how systems of many interacting parts reach, on their own, a state where disturbances occur on all scales.

- URL: https://www.poil.dk/research-background/self-organized-criticality/
- Author: Simon-Shlomo Poil
- First published: 2007-08-20
- Last updated: 2026-11-15
- Section: Research Background (https://www.poil.dk/research-background/)

*This essay is adapted from my MSc thesis (University of Copenhagen & VU University Amsterdam, 2007) and was first published on this site in two parts (2009 and 2012). The first sections give a short introduction; the later sections keep the more formal treatment from the thesis. The foundational concepts have remained stable, though the application of SOC to specific systems — particularly the brain — has matured considerably since then, with sharper debates about when criticality claims are warranted and improved tools for testing them. For the neuroscience application, see [The Brain as a Critical System](https://www.poil.dk/research-background/brain-criticality/); for the propagation mechanism behind avalanches, see [Branching Processes](https://www.poil.dk/research-background/branching-processes/).*

*If citing this material: S.-S. Poil, Temporal correlations and criticality in models of neuronal networks, MSc Thesis, University of Copenhagen & VU University Amsterdam, 2007.*

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## What is self-organised criticality?

Spatial structures with a scale-invariant, fractal (self-similar) structure, and temporal dynamics with so-called 1/f noise, characterise many systems in nature (Buchanan, 2000). In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld introduced **self-organised criticality** (SOC) as a unifying explanation of why these systems exhibit power-law correlations in space and time (Bak et al., 1987, 1988). The main idea was that scale-free structures in space and time are a consequence of each other, and that they emerge only because of internal interactions — without any external fine-tuning (Bak, 1997).

To understand SOC, it is first necessary to understand what *critical* means in physics. In statistical physics, a *critical point* is a special configuration of a system at the boundary between two phases — between order and disorder. At this point, the system exhibits **scale-free** behaviour: fluctuations and correlations occur on all length and time scales, with no single characteristic scale dominating. The mathematical signature of such a state is often a **power-law** distribution.

What Bak and colleagues proposed was that a wide class of systems can reach such a state spontaneously. This is what distinguishes self-organised criticality from ordinary criticality: the system is not tuned to criticality from the outside, but tends to organise itself toward that regime.

## The sandpile

The first model of SOC was a sandpile model (Bak et al., 1987). Consider a pile of sand that is slowly driven by the addition of grains. Gravity and the "stickiness" of the grains ensure that avalanches of different sizes spread over the surface of the pile, just like snow avalanches.

During the initial build-up, the response — the avalanche sizes — is proportional to the input. In the critical state, the slope of the pile fluctuates around a critical angle: a slope that is too steep produces larger avalanches, whereas a more moderate slope allows build-up. This is the basis of self-organising to criticality in the sandpile: the slope is attracted to the critical angle.

In this critical state, **avalanches of all sizes occur**, from the smallest to the largest, limited only by the size of the pile. The response of the pile is thus not proportional to the input, but unpredictable. Adding a single grain can cause a disturbance on any scale up to the size of the pile, and the structural impact of any disturbance is remembered by the pile until many other disturbances erase it.

The core of criticality is the scale-free size distribution of disturbances, which shows that the correlation length has approached the size of the system. Criticality is dynamic in nature, sitting on the edge between disorder and structure. The core of *self-organised* criticality is that criticality is the natural state of the system: no externally imposed tuning is needed to keep it there (Bak, 1997; Christensen and Moloney, 2005).

SOC also introduced the idea that all events are intrinsic to the system. No "freak" events exist, because all events, large and small, arise from the interaction among the elements of the system. This is a useful way of thinking about forest fires, earthquakes and fluctuations in stock markets.

## What is scale invariance?

A general SOC medium is in a critical state if an energy-dependent observable — for example the size of avalanches in a sandpile, or the size of fires in a forest-fire model — has a scale-invariant probability distribution, within the limits of finite-size effects (Bak, 1997; Christensen and Moloney, 2005). Criticality is defined from energy-dependent observables because, in an SOC system, the critical state should depend only on the existence of a flow of energy through the system; otherwise the system is considered fine-tuned (Bak, 1997; Flyvbjerg, 1996).

A scale-invariant probability distribution is most often a power law,

$$P(s) \propto s^{-\beta},$$

where $s$ is the observable and $\beta$ is the power-law (scaling) exponent. A power law is scale-invariant because no scale is special: no feature exists at one scale that makes that scale stand out. Consider two sizes $s_1$ and $s_2$. If we multiply both by a constant $k$, their relative probability does not change:

$$\frac{P(k s_2)}{P(k s_1)} = \frac{(k s_2)^{-\beta}}{(k s_1)^{-\beta}} = \frac{s_2^{-\beta}}{s_1^{-\beta}}.$$

Whatever scale we look at, we find patterns with the same quantitative relations — as in a Koch curve or the coastline of Norway, where only finite-size effects limit the conclusion (Bak, 1997; Mandelbrot, 1983). A scale-free time signal — 1/f noise — shows a similar pattern: doubling the frequency from $f$ to $2f$ halves the power spectral density, and doubling again to $4f$ halves it again. We can zoom in on the signal and find fluctuations on all time scales with the same pattern (Linkenkaer-Hansen, 2002).

## A working definition: energy flow and the critical medium

The definition of self-organised criticality is flexible: SOC is defined for specific systems rather than by one all-inclusive definition. The conditions for a sandpile may not be comparable to those for a neuronal network. This may not be satisfying from a physical point of view, but SOC is an abstract concept that explains many different complex systems that self-organise to a critical state. Its core is that structure emerges from the intrinsic interactions among the elements of the system, without external pressure. The following common characteristics define a system in a self-organised critical state.

**Self-organising through energy flow.** The system should self-organise into a critical state through a driving force. It is not allowed to be fine-tuned, but should be attracted to the critical state by the in-flux and intermittent dissipation of energy. Intermittent dissipation, often seen as avalanches, is essential because it allows order to build up in the system. The driving force is defined by three rules:

- The driving rate is slow compared to the relaxation events in the system.
- The intensity and spatial position of the drive are independent of the system's state.
- Correlations in the intensity or spatial position that are not an intrinsic part of the system are allowed, provided they are not needed for the critical state.

With this definition we allow a time series of driving forces, under the condition that the system would not change its critical state had another time series been chosen.

![Two sandpile diagrams. In A, energy flows steadily in and out; in B, the SOC sandpile releases energy in intermittent bursts.](intermittent-dissipation.png "Intermittent dissipation of energy. (A) When energy flows through a system steadily, no order builds up. (B) When energy is dissipated intermittently, as in avalanches, order builds up between events. This is what makes the scale-free spatio-temporal patterns of SOC systems possible. From the MSc thesis, 2007.")

**The critical medium.** The energy in an SOC system interacts with a medium — a sandpile, a neuronal network or a forest. This interaction changes the state of the medium so that it is in the critical state. The medium must be able to store energy, in a local memory or in thresholds, and must have many meta-stable states; in the critical state it moves between them. The ability to store energy is what makes the intermittent dissipation possible.

## Characteristics of SOC systems

SOC systems are usually slowly driven, steady-state, non-equilibrium systems. That leaves a huge set of possible systems, because most systems in nature are non-equilibrium and often in a steady state. Equilibrium systems are the textbook systems, such as the Ising model, whose thermodynamic properties can be calculated from the free energy. Many processes in nature relax slowly to equilibrium — if ever — and non-equilibrium systems often react unpredictably, which makes them hard to fine-tune to criticality. *Steady state* means that the in-flux and the average out-flux of energy are equal over long time scales; *non-equilibrium* means that the energy flow at the boundaries drives the system away from equilibrium, so that its response to a perturbation is not always linear (Christensen and Moloney, 2005).

SOC systems often have the following characteristics:

- Local interactions among elements cause the dynamics.
- Activity propagates in avalanches, i.e. energy is dissipated intermittently.
- Insensitivity to initial conditions.
- High susceptibility to perturbations.
- Power-law scaling in temporal dynamics and spatial structure: no characteristic scales exist, apart from finite-size effects.
- Many meta-stable states.
- Intrinsic memory or thresholds in the medium.
- The system is "old": it has evolved for a long time relative to its size.
- Robustness to local failures or random perturbations.

Because of the dynamics of the critical state, self-organised systems are highly susceptible to perturbations. Their scaling statistics, however, are robust to random perturbations and local failures, because the critical state is an attractor — which is also why they are insensitive to initial conditions.

## Self-organisation in nature

The critical state of the Ising model is not SOC: it appears only when the system is fine-tuned, with the physicist setting the temperature. In nature it is hard to see how systems could be fine-tuned. One could argue that biological systems have been "fine-tuned" by evolution and by the environment they live in, so it is essential to distinguish *reorganisation* driven by external forces from *self-organisation* that depends only on interactions within the system.

![Two drawings of a figure and a wheel: in A the figure pushes the wheel from outside; in B the figure runs inside the wheel and drives it from within.](self-organisation-vs-forcing.png "Reorganisation versus self-organisation. (A) Order imposed by an external force that pushes the system in a particular direction. (B) Order that arises only from the interactions among the system's own elements. From the MSc thesis, 2007.")

Biological systems organise through internal interactions among their elements, but over long time scales they have evolved through interaction with their environment, which changes the local rules that govern those interactions. In the brain, the local rules are the mechanisms that support interactions among neurons. They do not change on short time scales (seconds, minutes), but depend on external influences over longer ones (days, generations). That influence acts like a biased driving force and is not itself SOC: had it been different, the mechanisms would have evolved differently.

Even so, organisation towards criticality and scale invariance seems ubiquitous in nature (Buchanan, 2000). Chialvo (2006) made the argument for sensory systems: in a sub-critical world, where everything is steady, sensors with a large dynamic range would be useless; in a super-critical world, where everything changes too fast, they would be useless too. Real environments lie in between — with extreme events and long-tailed probabilities — and that is where a wide dynamic range pays off. On long time scales, then, it is the advantage of the critical state that "fine-tunes" biological organisation towards criticality; on short time scales, it is internal interaction — self-organised criticality — that attracts the system to the critical state (see also Johnson, 2001).

## Separation of time scales

A system that builds internal structure must export entropy to its surroundings (Haynie, 2001). To do so, an SOC system needs a separation between the time scales of energy in-flux and out-flux: energy should come in slowly and be released quickly.

An analogy may help. Consider a collection of articles. Every day new articles arrive and others are deleted — the in-flux and out-flux. Suppose each article is either good or bad, and that only bad articles are deleted. The way to keep the collection in good order is a slow rate of incoming articles and a fast rate of deleting bad ones. With a fast in-flux and a slow out-flux, the collection would on average hold a huge number of both good and bad articles, and no order would emerge.

In an SOC system, the flow of energy depends on local thresholds in the medium. The thresholds store energy and ensure that it is dissipated intermittently rather than continuously. The driving must be slow compared with the typical duration of an avalanche, so that the system can relax through its local interactions without being disturbed by the drive.

## Examples beyond sandpiles

SOC has been used to explain rice piles (Frette et al., 1996; Aegerter et al., 2003), earthquakes (Bak and Tang, 1989; Bak et al., 2002; Turcotte and Malamud, 2004), forest fires (Malamud et al., 1998), solar flares (Charbonneau et al., 2001; Paczuski and Hughes, 2004), evolution (Bak and Sneppen, 1993; Sneppen et al., 1995), financial markets (Lux and Marchesi, 1999; Bartolozzi et al., 2005) and neuronal networks (Linkenkaer-Hansen et al., 2001; Chialvo, 2004; de Arcangelis et al., 2006; Poil et al., 2008).

**Earthquakes.** The Gutenberg–Richter law describes a scale-free relationship between earthquake magnitude and frequency, which has often been interpreted through a critical-systems lens (Bak and Tang, 1989; Sornette, 2004).

**Forest fires.** The Drossel–Schwabl forest-fire model is a classic SOC model in which trees grow slowly and fires, once ignited, spread through connected regions. The resulting fire-size distribution can show scale-free behaviour (Drossel and Schwabl, 1992).

**Neuronal networks.** The brain, with its billions of threshold-based neurons supporting excitation, inhibition and plasticity, has many of the structural ingredients required for SOC. This application is developed in the companion essay, [The Brain as a Critical System](https://www.poil.dk/research-background/brain-criticality/).

**Evolution.** Bak and Sneppen (1993) proposed a model of co-evolution in which extinction events follow scale-free dynamics, suggesting that large and small extinctions may arise from the same underlying mechanism.

## Boundaries and debates

SOC is not a universal explanation for every heavy-tailed or scale-free pattern. The claim that a particular system exhibits self-organised criticality requires more than the observation of a power-law distribution. Similar statistical patterns can arise from multiple generative mechanisms, and careful empirical testing is needed to distinguish genuine SOC from other forms of complex behaviour (Christensen and Moloney, 2005; Clauset et al., 2009).

This matters especially in neuroscience, where the criticality hypothesis remains active but contested. Arguments have been made for sub-critical and near-critical regimes, and methodological concerns have been raised about inference from finite data and about the interpretation of power-law fits. The most defensible use of SOC is therefore as a framework that generates testable questions — about event-size distributions, temporal correlations, sensitivity to perturbation and system organisation — rather than as a label to be applied too quickly.

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