Self-organized criticality

In physics, self-organized criticality (SOC) is a property of (classes of) dynamical systems which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to precise values.

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 enthusiastically applied across fields as diverse as geophysics, physical cosmology, evolutionary biology and ecology, bio-inspired computing and optimization (mathematics), economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology^{[3][4][5][6][7][8][9][10]} and others.

SOC is typically observed in slowly-driven non-equilibrium systems with extended degrees of freedom and a high level of nonlinearity. 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.

Bak, Tang and Wiesenfeld's 1987 paper linked together these factors: a simple cellular automaton was shown to produce several characteristic features observed in natural complexity (fractal geometry, 1/f noise and power laws) in a way that could be linked to critical-point phenomena. Crucially, however, the paper demonstrated 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 behaviour (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 the lab (or lab computer) where it was possible to tune control parameters to precise values. The publication of this research sparked considerable interest from both theoreticians and experimentalists, and important papers on the subject are among 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. 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), and examination of the necessary conditions 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 behaviour 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 centred around large-scale natural or social systems that are known (or suspected) to display scale-invariant behaviour. 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 behaviour such as the Gutenberg-Richter law describing the statistical distribution of earthquake sizes, and the Omori law describing the frequency of aftershocks); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires; landslides; epidemics; 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). Worryingly, given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of wars. These "applied" investigations of SOC have included both attempts at 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.

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.

Realistic applications of SOC rely on exponents derived from assumptions that are often oversimplified. A promising exception to this limitation is the discovery by Marcelo Moret and Gilney Zebende that the globular compaction of protein chains by hydrophobic interactions is describable in non-Euclidean terms, with fractal critical exponents inferred from the evolution of solvent-accessible surface areas in chain segments of increasing length^[11]. Protein amino acid chains are conventionally represented by strings of letters, and their similarities quantified by counting letter identities in protein "words" (segments). Replacing the (alpha) letters with (alphanumeric) exponents enables non-Euclidean (contextual) analysis of universal water-mediated protein interactions^[12]. Membrane viral fusion proteins are ideal targets for SOC analysis, as they exhibit up to ~60% (hundreds of) mutations in public data bases.

Examples of self-organized critical dynamics

In chronological order of development:

• Bak-Tang-Wiesenfeld sandpile
• Forest-fire models
• Olami-Feder-Christensen model
• Bak-Sneppen model

See also

• 1/f noise
• Complex systems
• Fractals
• Power laws
• Scale invariance
• Self-organization
• Critical exponents
• Ilya Prigogine, a systems scientist who helped formalize dissipative system behavior in general terms.
• Red Queen effect

References

1. ^ 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. 
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. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=41396. 
3. ^ K. Linkenkaer-Hansen, V. V. Nikouline, J. M. Palva, and R. J. Ilmoniemi. (2001). "Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations". J. Neurosci. 21 (4): 1370–1377. PMID 11160408. 
4. ^ J. M. Beggs and D. Plenz (2006). "Neuronal Avalanches in Neocortical Circuits". J. Neurosci 23. 
5. ^ Chialvo, D. R. (2004). "Critical brain networks". Physica A 340 (4): 756–765. arXiv:cond-mat/0402538. Bibcode 2004PhyA..340..756C. doi:10.1016/j.physa.2004.05.064. 
6. ^ Stefan Boettcher (1999). "Extremal optimization of graph partitioning at the percolation threshold". J. Phys. A: Math. Gen 32 (28): 5201–5211. arXiv:cond-mat/9901353. Bibcode 1999JPhA...32.5201B. doi:10.1088/0305-4470/32/28/302. 
7. ^ D. Fraiman, P. Balenzuela, J. Foss and D. R. Chialvo (2004). "Ising-like dynamics in large scale brain functional networks". Physical Review E 79 (6): 061922. Bibcode 2009PhRvE..79f1922F. doi:10.1103/PhysRevE.79.061922. 
8. ^ L. de Arcangelis, C. Perrone-Capano, and H. J. Herrmann (2006). "Self-organized criticality model for brain plasticity". Phys. Rev. Lett. 96. 
9. ^ Poil, Ss; Van, Ooyen, A; Linkenkaer-Hansen, K (Jul 2008). "Avalanche dynamics of human brain oscillations: relation to critical branching processes and temporal correlations". Human brain mapping 29 (7): 770–7. doi:10.1002/hbm.20590. PMID 18454457. 
10. ^ Manfred G. Kitzbichler, Marie L. Smith, Søren R. Christensen, Ed Bullmore1 (2009). Behrens, Tim. ed. "Broadband Criticality of Human Brain Network Synchronization". PLoS Comput Biol 5 (3): e1000314. doi:10.1371/journal.pcbi.1000314. PMC 2647739. PMID 19300473. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2647739. 
11. ^ Moret, M. A. and Zebende, G. F. (2007). "Amino acid hydrophobicity and accessible surface area". Physical Review E 75 (1): 011920. Bibcode 2007PhRvE..75..011920B. doi:10.1103/PhysRevE.75.011920. 
12. ^ Phillips, J. C. (2009). "Scaling and self-organized criticality in proteins: Lysozyme c". Physical Review E 80 (5): 051916. Bibcode 051916B 2009PhRvE..80.. 051916B. doi:10.1103/PhysRevE.80.051916. 

Further reading

• Adami, C. (1995). "Self-organized criticality in living systems". Physics Letters A 203 (1): 29–32. Bibcode 1995PhLA..203...29A. doi:10.1016/0375-9601(95)00372-A. 

• Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality. New York: Copernicus. ISBN 0-387-94791-4. 

• Bak, P. and Paczuski, M. (1995). "Complexity, contingency, and criticality". Proceedings of the National Academy of Sciences of the USA 92 (15): 6689–6696. Bibcode 1995PNAS...92.6689B. doi:10.1073/pnas.92.15.6689. PMC 41396. PMID 11607561. http://pnas.org/cgi/content/abstract/92/15/6689. 

• Bak, P. and Sneppen, K. (1993). "Punctuated equilibrium and criticality in a simple model of evolution". Physical Review Letters 71 (24): 4083–4086. Bibcode 1993PhRvL..71.4083B. doi:10.1103/PhysRevLett.71.4083. PMID 10055149. 

• 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. 

• Bak, P., Tang, C. and Wiesenfeld, K. (1988). "Self-organized criticality". Physical Review A 38 (1): 364–374. Bibcode 1988PhRvA..38..364B. doi:10.1103/PhysRevA.38.364. 

• Buchanan, M. (2000). Ubiquity. London: Weidenfeld & Nicolson. ISBN 0-7538-1297-5. 

• Jensen, H. J. (1998). Self-Organized Criticality. Cambridge: Cambridge University Press. ISBN 0-521-48371-9. 

• Turcotte, D. L.; Smalley, R. F., Jr.; Solla, S. A. (1985). "Collapse of loaded fractal trees". Nature 313 (6004): 671. Bibcode 1985Natur.313..671T. doi:10.1038/313671a0. http://www.nature.com/nature/journal/v313/n6004/abs/313671a0.html. 

• 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. 

• Katz, J. I. (1986). "A model of propagating brittle failure in heterogeneous media". Journal of Geophysical Research 91 (B10): 10412. Bibcode 1986JGR....9110412K. doi:10.1029/JB091iB10p10412. 

• Kron, T./Grund, T. (2009). "Society as a Selforganized Critical System". Cybernetics and Human Knowing 16: 65–82. 

• Paczuski, M. (2005). "Networks as renormalized models for emergent behavior in physical systems". ArXiv.org. The Science and Culture Series - Physics: physics/0502028. arXiv:physics/0502028. Bibcode 2005cmn..conf..363P. doi:10.1142/9789812701558_0042. ISBN 9789812565259. 

• Turcotte, D. L. (1997). Fractals and Chaos in Geology and Geophysics. Cambridge: Cambridge University Press. ISBN 0-521-56733-5. 

• Turcotte, D. L. (1999). "Self-organized criticality". Reports on Progress in Physics 62 (10): 1377–1429. Bibcode 1999RPPh...62.1377T. doi:10.1088/0034-4885/62/10/201. 
• A. N. Sekar Iyengar (2007). "Realization of {SOC} behavior in a dc glow discharge plasma". Physics Letters A 360: 717–721. arXiv:physics/0611069. Bibcode 2007PhLA..360..717N. doi:10.1016/j.physleta.2006.09.005. 

• Self-organized criticality on arxiv.org