# Self-organized criticality

An image of the 2d Bak-Tang-Wiesenfeld sandpile, the original model of self-organized criticality.

Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to a precise value, because the system, effectively, tunes itself as it evolves towards criticality.

The concept was put forward by Per Bak, Chao Tang and Kurt Wiesenfeld ("BTW") in a paper[1] published in 1987 in Physical Review Letters, and is considered to be one of the mechanisms by which complexity[2] arises in nature. Its concepts have been applied across fields as diverse as geophysics,[3] physical cosmology, evolutionary biology and ecology, bio-inspired computing and optimization (mathematics), economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology[4][5][6] and others.

SOC is typically observed in slowly driven non-equilibrium systems with many degrees of freedom and strongly nonlinear dynamics. Many individual examples have been identified since BTW's original paper, but to date there is no known set of general characteristics that guarantee a system will display SOC.

## Overview

Self-organized criticality is one of a number of important discoveries made in statistical physics and related fields over the latter half of the 20th century, discoveries which relate particularly to the study of complexity in nature. For example, the study of cellular automata, from the early discoveries of Stanislaw Ulam and John von Neumann through to John Conway's Game of Life and the extensive work of Stephen Wolfram, made it clear that complexity could be generated as an emergent feature of extended systems with simple local interactions. Over a similar period of time, Benoît Mandelbrot's large body of work on fractals showed that much complexity in nature could be described by certain ubiquitous mathematical laws, while the extensive study of phase transitions carried out in the 1960s and 1970s showed how scale invariant phenomena such as fractals and power laws emerged at the critical point between phases.

The term self-organized criticality was firstly introduced by Bak, Tang and Wiesenfeld's 1987 paper, which clearly linked together those factors: a simple cellular automaton was shown to produce several characteristic features observed in natural complexity (fractal geometry, pink (1/f) noise and power laws) in a way that could be linked to critical-point phenomena. Crucially, however, the paper emphasized that the complexity observed emerged in a robust manner that did not depend on finely tuned details of the system: variable parameters in the model could be changed widely without affecting the emergence of critical behavior: hence, self-organized criticality. Thus, the key result of BTW's paper was its discovery of a mechanism by which the emergence of complexity from simple local interactions could be spontaneous—and therefore plausible as a source of natural complexity—rather than something that was only possible in artificial situations in which control parameters are tuned to precise critical values. The publication of this research sparked considerable interest from both theoreticians and experimentalists, producing some of the most cited papers in the scientific literature.

Due to BTW's metaphorical visualization of their model as a "sandpile" on which new sand grains were being slowly sprinkled to cause "avalanches", much of the initial experimental work tended to focus on examining real avalanches in granular matter, the most famous and extensive such study probably being the Oslo ricepile experiment[7][citation needed]. Other experiments include those carried out on magnetic-domain patterns, the Barkhausen effect and vortices in superconductors.

Early theoretical work included the development of a variety of alternative SOC-generating dynamics distinct from the BTW model, attempts to prove model properties analytically (including calculating the critical exponents[8][9]), and examination of the conditions necessary for SOC to emerge. One of the important issues for the latter investigation was whether conservation of energy was required in the local dynamical exchanges of models: the answer in general is no, but with (minor) reservations, as some exchange dynamics (such as those of BTW) do require local conservation at least on average. In the long term, key theoretical issues yet to be resolved include the calculation of the possible universality classes of SOC behavior and the question of whether it is possible to derive a general rule for determining if an arbitrary algorithm displays SOC.

Alongside these largely lab-based approaches, many other investigations have centered around large-scale natural or social systems that are known (or suspected) to display scale-invariant behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including: earthquakes (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the Gutenberg–Richter law describing the statistical distribution of earthquake size, and the Omori law describing the frequency of aftershocks[10][3]); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires; landslides; epidemics; neuronal avalanches in the cortex;[5][11] 1/f noise in the amplitude of electrophysiological signals;[4] and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould). These "applied" investigations of SOC have included both modelling (either developing new models or adapting existing ones to the specifics of a given natural system) and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.

In addition, SOC has been applied to computational algorithms. Recently, it has been found that the avalanches from an SOC process, like the BTW model, make effective patterns in a random search for optimal solutions on graphs.[12] An example of such an optimization problem is graph coloring. The SOC process apparently helps the optimization from getting stuck in a local optimum without the use of any annealing scheme, as suggested by previous work on extremal optimization.

The recent excitement generated by scale-free networks has raised some interesting new questions for SOC-related research: a number of different SOC models have been shown to generate such networks as an emergent phenomenon, as opposed to the simpler models proposed by network researchers where the network tends to be assumed to exist independently of any physical space or dynamics. While many single phenomena have been shown to exhibit scale-free properties over narrow ranges, a phenomenon offering by far a larger amount of data is solvent-accessible surface areas in globular proteins.[13] These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity.[14]

Despite the considerable interest and research output generated from the SOC hypothesis, there remains no general agreement with regards to its mechanisms in abstract mathematical form. Bak Tang and Wiesenfeld based their hypothesis on the behavior of their sandpile model.[1] However, it has been argued that this model would actually generate 1/f2 noise rather than 1/f noise.[15] This claim was based on untested scaling assumptions, and a more rigorous analysis showed that sandpile models generally produce 1/fa spectra, with a<2.[16] Other simulation models were proposed later that could produce true 1/f noise,[17] and experimental sandpile models were observed to yield 1/f noise.[18] In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon information theory,[19] mean field theory,[20] the convergence of random variables,[21] and cluster formation.[22] A continuous model of self-organised criticality is proposed by using tropical geometry.[23]

## Examples of self-organized critical dynamics

In chronological order of development:

## References

1. ^ a b Bak, P., Tang, C. and Wiesenfeld, K. (1987). "Self-organized criticality: an explanation of 1/f noise". Physical Review Letters. 59 (4): 381–384. Bibcode:1987PhRvL..59..381B. doi:10.1103/PhysRevLett.59.381. PMID 10035754.CS1 maint: multiple names: authors list (link) Papercore summary: http://papercore.org/Bak1987.
2. ^ Bak, P., and Paczuski, M. (1995). "Complexity, contingency, and criticality". Proc Natl Acad Sci U S A. 92 (15): 6689–6696. Bibcode:1995PNAS...92.6689B. doi:10.1073/pnas.92.15.6689. PMC 41396. PMID 11607561.CS1 maint: multiple names: authors list (link)
3. ^ a b c Smalley, R. F., Jr.; Turcotte, D. L.; Solla, S. A. (1985). "A renormalization group approach to the stick-slip behavior of faults". Journal of Geophysical Research. 90 (B2): 1894. Bibcode:1985JGR....90.1894S. doi:10.1029/JB090iB02p01894.CS1 maint: multiple names: authors list (link)
4. ^ a b K. Linkenkaer-Hansen; V. V. Nikouline; J. M. Palva & R. J. Ilmoniemi. (2001). "Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations". J. Neurosci. 21 (4): 1370–1377. doi:10.1523/JNEUROSCI.21-04-01370.2001. PMC 6762238. PMID 11160408.
5. ^ a b J. M. Beggs & D. Plenz (2006). "Neuronal Avalanches in Neocortical Circuits". J. Neurosci. 23 (35): 11167–77. doi:10.1523/JNEUROSCI.23-35-11167.2003. PMC 6741045. PMID 14657176.
6. ^ Chialvo, D. R. (2004). "Critical brain networks". Physica A. 340 (4): 756–765. arXiv:cond-mat/0402538. Bibcode:2004PhyA..340..756R. doi:10.1016/j.physa.2004.05.064.
7. ^ Malthe-Sørenssen, Anders. "The Rice Pile Project". Retrieved 18 August 2020.
8. ^ Tang, C. and Bak, P. (1988). "Critical exponents and scaling relations for self-organized critical phenomena". Physical Review Letters. 60 (23): 2347–2350. Bibcode:1988PhRvL..60.2347T. doi:10.1103/PhysRevLett.60.2347. PMID 10038328.CS1 maint: multiple names: authors list (link)
9. ^ Tang, C. and Bak, P. (1988). "Mean field theory of self-organized critical phenomena". Journal of Statistical Physics (Submitted manuscript). 51 (5–6): 797–802. Bibcode:1988JSP....51..797T. doi:10.1007/BF01014884.CS1 maint: multiple names: authors list (link)
10. ^ a b Turcotte, D. L.; Smalley, R. F., Jr.; Solla, S. A. (1985). "Collapse of loaded fractal trees". Nature. 313 (6004): 671–672. Bibcode:1985Natur.313..671T. doi:10.1038/313671a0.CS1 maint: multiple names: authors list (link)
11. ^ Poil, SS; Hardstone, R; Mansvelder, HD; Linkenkaer-Hansen, K (Jul 2012). "Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks". Journal of Neuroscience. 32 (29): 9817–23. doi:10.1523/JNEUROSCI.5990-11.2012. PMC 3553543. PMID 22815496.
12. ^ Hoffmann, H. and Payton, D. W. (2018). "Optimization by Self-Organized Criticality". Scientific Reports. 8 (1): 2358. Bibcode:2018NatSR...8.2358H. doi:10.1038/s41598-018-20275-7. PMC 5799203. PMID 29402956.CS1 maint: multiple names: authors list (link)
13. ^ Moret, M. A. and Zebende, G. (2007). "Amino acid hydrophobicity and accessible surface area". Phys. Rev. E. 75 (1): 011920. Bibcode:2007PhRvE..75a1920M. doi:10.1103/PhysRevE.75.011920. PMID 17358197.CS1 maint: multiple names: authors list (link)
14. ^ Phillips, J. C. (2014). "Fractals and self-organized criticality in proteins". Physica A. 415: 440–448. Bibcode:2014PhyA..415..440P. doi:10.1016/j.physa.2014.08.034.
15. ^ Jensen, H. J., Christensen, K. and Fogedby, H. C. (1989). "1/f noise, distribution of lifetimes, and a pile of sand". Phys. Rev. B. 40 (10): 7425–7427. Bibcode:1989PhRvB..40.7425J. doi:10.1103/physrevb.40.7425. PMID 9991162.CS1 maint: multiple names: authors list (link)
16. ^ Laurson, Lasse; Alava, Mikko J.; Zapperi, Stefano (15 September 2005). "Letter: Power spectra of self-organized critical sand piles". Journal of Statistical Mechanics: Theory and Experiment. 0511. L001.
17. ^ Maslov, S., Tang, C. and Zhang, Y. - C. (1999). "1/f noise in Bak-Tang-Wiesenfeld models on narrow stripes". Phys. Rev. Lett. 83 (12): 2449–2452. arXiv:cond-mat/9902074. Bibcode:1999PhRvL..83.2449M. doi:10.1103/physrevlett.83.2449.CS1 maint: multiple names: authors list (link)
18. ^ Frette, V., Christinasen, K., Malthe-Sørenssen, A., Feder, J, Jøssang, T and Meaken, P (1996). "Avalanche dynamics in a pile of rice". Nature. 379 (6560): 49–52. Bibcode:1996Natur.379...49F. doi:10.1038/379049a0.CS1 maint: multiple names: authors list (link)
19. ^ Dewar, R. (2003). "Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states". J. Phys. A: Math. Gen. 36 (3): 631–641. arXiv:cond-mat/0005382. Bibcode:2003JPhA...36..631D. doi:10.1088/0305-4470/36/3/303.
20. ^ Vespignani, A., and Zapperi, S. (1998). "How self-organized criticality works: a unified mean-field picture". Phys. Rev. E. 57 (6): 6345–6362. arXiv:cond-mat/9709192. Bibcode:1998PhRvE..57.6345V. doi:10.1103/physreve.57.6345. hdl:2047/d20002173.CS1 maint: multiple names: authors list (link)
21. ^ Kendal, WS (2015). "Self-organized criticality attributed to a central limit-like convergence effect". Physica A. 421: 141–150. Bibcode:2015PhyA..421..141K. doi:10.1016/j.physa.2014.11.035.
22. ^ Hoffmann, H. (2018). "Impact of Network Topology on Self-Organized Criticality". Phys. Rev. E. 97 (2): 022313. Bibcode:2018PhRvE..97b2313H. doi:10.1103/PhysRevE.97.022313. PMID 29548239.
23. ^ Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (2018-08-15). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences. 115 (35): E8135–E8142. arXiv:1806.09153. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.