In his contribution about self-organization Haken asks the important question whether there is an underlying principle in self-organization. He let the question open. However, I think, there is a common principle: self-enhancement balanced by an antagonistic reaction. It has been proposed by Gierer and Meinhardt in 1972 (see Gierer-Meinhardt model) and worked out further subsequently. Depending on time constants and spread of the components, either pattern in space or patterns in time can spontaneously emerge. For biological systems we have now firm evidence for this mechanism (Meinhardt 2008). In the following I will pick up some other examples, mostly touched by Haken, and show that these are based on this principle.
Pattern formation in the non-animated world
Star-formation is possible since a local increase in mass density by a fluctuation attracts more material, such that the local density increases further. A crucial antagonistic effect results from the light emission of an incipient star after ignition. The light pressure repels the cosmic dust.
Clouds and Bénard phenomenon: Warmed-up air or liquid expands, becomes lighter and moves upwards. There, the surrounding matter is even colder and this accelerates the upstream. By the Bernoulli-effect adjacent regions join the upstream, are sucked in. So, upstream is a self-enhancing process. However, an upstream is necessarily connected with a downstream at other positions and upstream and downstream are locally exclusive. If the upstream/downstream mechanism is essentially symmetric, stripes are the preferred pattern since for each upstream exist a downstream nearby. Stripes in insect segmentation are based on the same principle: the long range inhibition is replaced by a long-range activation of a second self-enhancing process that is locally exclusive to the first.
Sand dunes arise from a sand deposition behind a small wind shelter; this increases the wind shelter and thereby accelerates further sand deposition - a clear self-enhancing reaction. However, sand deposited at one position cannot collect somewhere else. The depletion of the sand in the air is the long-ranging antagonistic reaction. Thus, the homogeneous sand distribution is an instable situation.
Erosion proceeds faster at some injury. More water collects in the incipient valleys, accelerating the erosion there. Meandering, also a patterning process, makes the incipient valley wider. Like in the sand dune example, water collected in a river does no longer contribute to the erosion elsewhere. Next to a large river no second river will emerge.
Examples from social interactions: If it is easier to earn more money if one has some, the distribution of wealth will become necessarily inhomogeneous. Politicians who want to be elected try to enhance their own position (self-amplification) and to inhibit their competitors. The same is true for the competition of related shops in a city by which only few eventually survive.
Localization of gas discharges as observed by Purwin and his group and mentioned by Haken are also based on this mechanism. A gas discharge leads to more ions. Under the influence of the voltage-difference these ions are accelerated, producing more ions, which lead to the further increase of the local current, etc. The resistance in the current supply and global voltage breakdown restricts this self-enhancing process. The formation and localization of a lightning has the same base.
I am not a specialist for lasers. However, I expect that the same principle is there at work too. Some frequencies are more prevalent and cause 'light amplification by stimulated emission of radiation' - a self-enhancing effect which goes on the expense of less-privileged frequencies. At the end, one frequency survives. Is the enslavement principle equivalent to the long-range inhibition (in this case, in the frequency space)?
Patterns in time
Whether the resulting patterns are stable or oscillating depends on the relation of the time constants. If the antagonistic reaction has a longer time constant than the self-enhancing reaction, oscillations or burst-like activations will occur. An example is the time course of an infection: The infection with few viruses could be sufficient to trigger a sickness since the viruses replicate themselves (self-enhancement). The antagonistic reaction, mediated by the immune system, is much slower. It takes a day to become sick, but a week to become healthy again. (What appears on a first inspection as a disadvantage is in fact a good strategy. If the immune system would be much faster, an equilibrium between virus production and virus removal would be established. After a single infection we would fight for the rest of our life against the virus. However, due to the burst mode, we become sick for a short while but the virus is subsequently completely eliminated).
This example is also convenient to illustrate the requirement for wave formation: the self-enhancement has to spread moderately while the antagonistic reaction must be local (i.e., the condition is completely contrary to the condition for making stable patterns in space). This is clearly satisfied in the case of infections: the virus can spread, while the immune reaction is local. Therefore, infections can occur in epidemic waves. The same is true for a forest fire. Burning is a self-enhancing process since more heat releases more burnable gases, which lead to higher temperatures, triggering the burning of adjacent trees. etc. The antagonistic reaction results because the fire is no longer sustained by the ashes, i.e., the antagonistic effect is local.
An example from the economy: Obviously it is impossible to bring the economy into a stable state. If the economy is in an upward phase, the tendency of the industry for further investments raises - a self-enhancing effect. However, after some times, the storages are full, most have a new car, etc, and the downwards phase follows. Self-enhancement and delayed antagonistic reactions are also typical for the fluctuations of the stock marked.
In conclusion, the most interesting question for me would be whether there are major self-organizing processes that are NOT based on the interplay of a self-enhancing and an antagonistic reactions. An answer to this question would bring us closer to the question raised by Haken whether a universal principle for self-organization exist. There is certainly another mechanism that allows the formation of stable patterns: self-assembly by direct molecular recognition. It is involved, for instance, in the assembly of virus envelops. For oscillating patterns self-enhancement may no longer be required if the antagonistic reaction occurs with a delay. These are the only exceptions I know.
Meinhardt, H. (2008). Models of biological pattern formation: from elementary steps to the organization of embryonic axes. Curr. Top. Dev. Biol. 81, 1-63
Hermann Haken excellent paper investigates the self-organizing phenomena from the perspective of syneregetics.
Synergetics was founded by Hermann Haken. The goal has been to find general principles governing self-organization of elements independently of their nature. A variety of disciplines such as physics (lasers, fluids, plasmas), meteorology, chemistry (pattern formation by chemical reactions), biology (morphogenesis, brain, evolution theory, motor coordination), computer sciences (synergetic computer), sociology (e.g. regional migration), psychology and psychiatry were approached. Haken's synergetics grew up from his research in laser physics. Synergetics extended the concept of phase transition (which is a jump-like change in some variables) between so-called nonequilibrium structures.
As is well-known, synergetics developed together with two other scientific schools. The theory of dissipative structures labeled with the name of Ilya Prigogine grew out from the thermodynamic theory of open systems, and intended to describe the formation of (temporal, spatial and spatiotemporal) patterns first in physico-chemical, later, more ambitiously as well in biological and social systems. Somewhat earlier Rene Thom established catastrophe theory. One of his big goals was to explain the mathematical basis of morphogenesis of biological organisms. Though the schools did not often refer to each others' works, there is a big overlap in the phenomena they studied. The transitions among different dynamical states are the common themes. While the theory of dissipative structures and of synergetics used both deterministic and stochastic models and emphasized the role of fluctuations in switching systems from one state to another, catastrophe theory was purely deterministic. A stochastic version of catastrophe theory was elaborated by Cobb, L 1978. Behavioral Sci. 23(360-373)1998
The modern use of the concept of self-organization grew out from cybernetics (Ashby). Cybernetics emphasized the importance of feedback loops, and behind many (if not all) self-organizing phenomena there is a balance between positive and negative feedback. Roughly speaking negative feedback reduces the error or deviation from a goal state, therefore has stabilizing effects. Positive feedback increases the deviation from an initial state, so it has a destabilizing effects.
Self-organization implies the emergence of temporal, spatial and spatiotemporal patterns. Temporal patterns are identified with periodic and chaotic behavior, (temporally stationary) spatial patterns are blobs, stripes and other ordered structures, while waves are spatiotemporal patterns.
Examples(based on Érdi,P: Complexity Explained, Springer 2007)
In chemical kinetics self-organized patterns occur due to interaction of autocatalytic (i.e. positive feedback) compensated by reaction step blocking the unbounded growth. The well-known BZ reactions are the paradigm. Turing structures are spatial structures, spiral waves (Winfree) are spatiotemporal patterns.
Somitogenesis, a segmentation process, which produces a periodic pattern along the head-tail axis are somewhat analogues to Turing structures, while spiral waves occur also in cardiac muscles.
Population dynamics, ecological systems: connectivity, stability, diversity, resilience, resistance The fundamental question is how does the stability of an ecological system change if there is a change in the connectivity of the network of interacting populations, and/or in the strength of the interactions. Model studies show that weak connections enhance stability, and they ...may be the glue that binds natural communities together. (McCann et al Nature 395(794)1998)
Epidemics. Epidemics is characterized by the rapid increase of the size of infected population due to consequence of the interaction between infected and susceptible individuals. Infected can converted to a removed pool spontaneously or by external control.
War dynamics. R.W. Lanchester derived and analyzed a model of classical warfare. An famous extended version of the model given by GF Gause (a biologist in Moscow) in 1934, as as a model of the struggle for existence. The model is able to show the principle of competitive exclusion (one side wins, and the other becomes extinct), and coexistence, which can be interpreted as permanent war.
Segregation dynamics. Thomas Schelling celebrated model demonstrates how local rules (micromotives in Schelling terminology) imply globally ordered social structures (macrobehavior). Simulations results suggested that slight preference to live among their owns imply global segregation, and the formation of ghettos.
Opinion dynamics. Interaction of people in a group imply changes in their opinions about different issues may lead to to consensus, fragmentation and polarization.
Business cycles. In a very influential model Nicolas Kaldor gave a mechanism for the generation of temporal oscillatory dynamics in income and capital by assuming a nonlinear dependence of investment and saving on income.
Self-organization in the nervous system. According to embryological, anatomical and physiological studies the wiring of neural networks is the result of the interplay of purely deterministic (genetically regulated) and random (or highly complex) mechanisms. Fluctuations may operate as ``organizing forces in accordance with the theory of noise-induced transitions or stochastic resonance. Self-organizing developmental mechanisms (considered as pattern formation by learning) are responsible for the formation and plastic behavior of ordered neural structures. Evolvability, the basis of selforganization poses constraints on brain dynamics. Stable internal representation of the external world indicate the presence of attractors. Here, an attractor means one of the states of the system where the system settles after starting from a given initial condition. Self-organization needs these attractors to have a sufficient instability to be able to alter in order to adapt to the environment.
The stable dynamic operation of the brain is based on the balance of excitatory and inhibitory interactions. The impairment of the inhibitory synaptic transmission implies the onset of epileptic seizures. Epileptic activity occurs in a population of neurons when the membrane potentials of the neurons are "abnormally" synchronized. If inhibition falls below a critical level, the degree of synchrony exceeds the threshold of normal patterns, and the the system dynamics switches to epileptic pattern.