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Introduction to self-organized criticality (SOC) – Part 1 – Simon-Shlomo Poil
(This is an excerpt from my master thesis: Please cite as; S.-S. Poil, Temporal correlations and criticality in models of neuronal networks, M.sc. thesis, University of Copenhagen & VU University Amsterdam, 2007 and/or link to http://www.poil.dk/s/self-organized-criticality-soc/55 )

Spatial structures with a scale invariant, fractal (self-similar) structure, and temporal dynamics with so-called 1/f -noise characterized many systems in nature [Buchanan, 2000]. In 1987 Bak, Tang, and Wiesenfeld introduced self-organized 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 these only emerge because of internal interactions. Later studies used self-organized criticality to explain several systems; e.g., rice piles [Aegerter et al., 2003; Frette et al., 1996], earthquakes [Bak et al., 2002; Lippiello et al., 2005; Turcotte and Malamud, 2004], forest fires [Malamud et al., 1998; Turcotte and Malamud, 2004], solar flares [Charbonneau et al., 2001; Paczuski and Hughes, 2004], evolution [Bak and Sneppen, 1993; Sneppen et al., 1995], commodity markets [Bartolozzi et al., 2005; Lux and Marchesi, 1999], Open-Source Software Evolution [Wu, 2006], and recently to neuronal networks [Chialvo, 2004; de Arcangelis et al., 2006; Linkenkaer-Hansen et al., 2001; Poil et al., 2008; Montez et al., 2009]. The book with the ambitious title ‘How nature works’ describes the amazing story of SOC [Bak, 1997].

The sand-pile model

The first model of SOC was a sand-pile model [Bak et al., 1987]. We consider a pile of sand that is slowly driven by the addition of grains. Gravity and the ‘stickiness’ of the grains will ensure that avalanches of different sizes will spread on the surface of the pile (in topplings of grains), just like snow avalanches. During the initial build-up of the sand pile the response, i.e., the avalanche sizes will be proportional to the input. In the critical state, the slope of the pile will fluctuate around a critical angle; a too steep slope will facilitate larger avalanches, whereas a more moderate slope will cause a build-up. This is the basics of self-organizing to criticality in the sand pile model – the slope is attracted to the critical angle. In this critical state, avalanches of all sizes will occur; from the smallest to the largest that are only limited by the size of the pile. The response of the pile is, thus, not proportional to the input but unpredictable. The probability distribution of avalanche sizes will be scale-free, i.e., the size distribution is independent on what scale we look. This is a simple mathematical property of power-laws. An addition of a single grain will have the potential of causing a huge disturbance on a scale dependent on the size of the pile. The structurally impact of any disturbance will be remembered by the pile until many other disturbances erase it.

The core of criticality; is the scale-free size distribution of disturbances (avalanches) that show the correlation length has approach the size of the system. Criticality is explicit dynamic in nature, sitting on the edge between disorder and structure. The core of self-organized criticality is that criticality is the ‘natural’ state of the system, i.e., no externally imposed tuning is needed to keep the system in this state [Bak, 1997; Christensen and Moloney, 2005].

SOC introduced the idea that all events are intrinsic to the system – no ‘freak’ events exist, because all events can be described by the interaction among the elements in the system. This is a useful understanding for problems in society like forest fires, earthquakes, or fluctuations in stock markets.

A rigorous definition of self-organized criticality?

The definition of self-organized criticality is flexible, this means that SOC is defined for specific systems, and not as an all-inclusive definition. The conditions for a sand pile might not be comparable to, e.g., the conditions for a neuronal network. We cannot force the world in to a square box, and make a rigorous definition of SOC. This might not be satisfactory from the physical point of view, but SOC is an abstract concept to explain many different complex systems that ’self-organize’ to a ’critical state’. The core of SOC is that structure emerges from the intrinsic interaction among the elements in the system without external pressure. The following common characteristics define a system to be in a self-organized critical state:

> Self-organizing through energy flow

The system should self-organize into a critical state through a driving force, which means the system is not allowed to be fine-tuned, but should be attracted to the critical state by the in-flux and intermittent dissipation of energy. The intermittent dissipation of energy, which is often seen in avalanches, is an essential property, because it allows the build-up of entropy in the system.

The following rules define the driving force.

• The driving rate is slow compared to the relaxation events in the system.

• The intensity and spatial position of the drive is independent of the systems state.

• Correlations in the intensity or spatial position, that are not an intrinsic part of the system, are allowed if they are not needed for the critical state.

Figure 1: Illustration of intermittent dissipation of energy in SOC systems. (A) In a non-SOC system energy flows through the system, but because the energy flow is high, no order is created in the system. (B) In a system with intermittent dissipation of energy, which is often seen in avalanche like activity, a build-up of order appears (build-up of entropy). Intermittent dissipation of energy is, therefore, important for the scale-free spatio-temporal patterns seen in SOC systems.
Figure 1: Illustration of intermittent dissipation of energy in SOC systems. (A) In a non-SOC system energy flows through the system, but because the energy flow is high, no order is created in the system. (B) In a system with intermittent dissipation of energy, which is often seen in avalanche like activity, a build-up of order appears (build-up of entropy). Intermittent dissipation of energy is, therefore, important for the scale-free spatio-temporal patterns seen in SOC systems.

With this definition, we allow a time-series of driving forces, under the condition the system does not change its critical state in the case we had chosen another time-series.

> The critical medium

The energy in the SOC system interacts with a medium, e.g., the sand pile, a neuronal network, or a forest. This interaction causes a change in the state of the medium, such that the medium is in the critical state. The medium is defined to be able to store energy (i.e., in a local memory or in thresholds), and have many meta-stable states. In the critical state, the medium changes between meta-stable states. The ability to store energy in the medium is crucial for the intermittent dissipation of energy.

See part 2 of “Self-organized criticality” here.

References

C. M. Aegerter, R. Günther, and R. J. Wijngaarden. Avalanche dynamics, surface roughening, and self-organized criticality: Experiments on a three-dimensional pile of rice. Phys. Rev. E, 67, 2003.

P. Bak. How nature works. Oxford University Press, 1997.

P. Bak and K. Sneppen. Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett., 71:4083, 1993.

P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality: An explanation of 1/f noise. Phys. Rev.Lett., 59(4):381 – 384, July 1987.

P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality. Phys. Rev. A, 38:364–374, 1988.

P. Bak, K. Christensen, L. Danon, and T. Scanlon. Unified scaling law for earthquakes. Phys. Rev. Lett., 88(17):178501, Apr 2002.

M. Bartolozzi, D. Leinweber, and A. Thomas. Self-organized criticality and stock market dynamics: an empirical study. Physica A, 350:451–465, 2005.

M. Buchanan. Ubiquity. Weidenfeld and Nicolson, The Orion Publishing Group Ltd., 2000.

P. Charbonneau, S. W. McIntosh, H.-L. Lie, and T. J. Bogdan. Avalanche models for solar flares. Solar Physics, 203:321–353, 2001.

D. R. Chialvo. Critical brain networks. Physica A, 340:756 – 765, 2004.

K. Christensen and N. R. Moloney. Complexity and Criticality. Imperial College Press, 2005.

L. de Arcangelis, C. Perrone-Capano, and H. J. Herrmann. Self-organized criticality model for brain plasticity. Phys. Rev. Lett., 96, 2006.

V. Frette, K. Christensen, A. Malthe-Sørenssen, J. Feder, T. Jøssang, and P. Meakin. Avalanche
dynamics in a pile of rice. Nature, 379:49–52, 1996.

V. Frette, K. Christensen, A. Malthe-Sørenssen, J. Feder, T. Jøssang, and P. Meakin. Avalanche dynamics in a pile of rice. Nature, 379:49–52, 1996.

K. Linkenkaer-Hansen, V. V. Nikouline, J. M. Palva, and R. J. Ilmoniemi. Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations. J. Neurosci., 21(4):1370–1377, 2001.

E. Lippiello, L. de Arcangelis, and C. Godano. Memory in self-organized criticality. Europhys. Lett., 72:678–684, 2005.

T. Lux and M. Marchesi. Scaling and criticality in a stochastic multi-agent model of a financial market. Nature, 397:498 – 500, 1999.

Malamud, Morein, and Turcotte. Forest fires: An example of self-organized critical behavior. Science,281(5384):1840–1842, Sep 1998.

T. Montez, S.-S. Poil, B. F., Jones, I. Manshanden, J. P. A. Verbunt, B. W. van Dijk, A. B. Brussaard, A. van Ooyen, C. J. Stam, P. Scheltens, K. Linkenkaer-Hansen. Altered temporal correlations in parietal alpha and prefrontal theta oscillations in early-stage Alzheimer disease. Proc Natl Acad Sci U S A 106: 5. 1614-1619 Feb 2009

K. Sneppen, P. Bak, H. Flyvbjerg, and M. H. Jensen. Evolution as a self-organized critical phenomenon. PNAS, 92:5209–5213, 1995.

D. L. Turcotte and B. D. Malamud. Landslides, forest fires, and earthquakes: examples of self-organized critical behavior. Physica A, 340:580–589, 2004.

M. Paczuski and D. Hughes. A heavenly example of scale-free networks and self-organized criticality. Physica A, 342:158–163, 2004.

S.-S. Poil, Arjen van Ooyen, Klaus Linkenkaer-Hansen. Avalanche dynamics of human brain oscillations: relation to critical branching processes and temporal correlations. Hum Brain Mapp 29: 7. 770-777 Jul 2008

J. Wu. Open Source Software Evolution and Its Dynamics. PhD thesis, University of Waterloo, 2006.

 


Introduction to self-organized criticality (SOC) – Part 1