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    • br Self organized pattern formation at the edge of chaos

    2018-10-29


    Self-organized pattern formation at the ‘edge of chaos’ Self-organization is a key concept in theoretical physics that ICG001 refers to a state far from thermodynamic equilibrium that arises when microscopic fluctuations are too great for a system to contain (Ball, 2001; Chaisson, 2004; Gollub and Langer, 1999; Nicolis and Prigogine, 1977). Self-organization appears particularly relevant at sub-cellular levels (Denton et al., 2003; Lehn, 2002; Misteli, 2001a, b; Whitesides and Grzybowski, 2002), and is closely related to ‘edge of chaos’ dynamics (Bonabeau, 1997; Goodwin, 1994; Kauffman, 1993, 1995; Lewin, 2002), a concept best illustrated with an iron magnet. Note, however, that whereas a magnet held at its Curie temperature is an equilibrium system, self-organized systems are typically not (reviewed by Halley and Winkler, 2008a). Systems that are very ordered, such as a magnet at room temperature or equilibrium systems more generally, cannot undergo rapid changes in collective behaviour because perturbations do not readily propagate. At the opposite extreme, chaotic systems, such as a magnet above its Curie point, have too little order to change in a coherent way and do not appear as having recognisable pattern. Systems with dynamics between order and chaos are said to be at the edge of chaos and often display striking spatiotemporal pattern formation, a classic example of which is the pattern of magnetic domains within an iron magnet held at its Curie temperature (Ball, 2001; Binney et al., 1992; Haken, 1983; Ward, 2001). Here, the effects of perturbation of a single spin may propagate through the entire system. Whether or not any particular disturbance does indeed become system spanning is a different question however, as all spin perturbations have similar potential but exist within the one limited system. It has been argued that the intrinsic physical dynamics of a system at the edge of chaos generate the raw ingredients necessary for computation, where computation is defined loosely as an ability to store, manipulate and transmit information (i.e., order) (Langton, 1990, 1991). Systems at the edge of chaos retain a ‘history’ or ‘memory’ of previous behaviour but are also sensitive to fluctuations because they may propagate. Such systems balance stability or robustness of collective behaviour with flexibility (Andrecut et al., 2009; Ball, 2001; Binney et al., 1992; Bonabeau, 1997; Edelstein-Keshet, 1994; Haken, 1983; Hiett, 1999; Kauffman, 1991, 1995; Langton, 1990, 1991; Nykter et al., 2008; Ward, 2001; Watmough and Edelstein-Keshet, 1995; Wilson, 1998; Wolfram, 1984a, b). Although some systems benefit from edge of chaos dynamics because they facilitate rapid shifts between distinct collective behaviours, such dynamics can be neutral or detrimental to other systems (Cole, 2002; Parrish et al., 2002). Edge of chaos dynamics is therefore expected to be targeted by natural selection within systems that need to compute good solutions to complex problems (Halley and Winkler, 2008b), such as embryos that develop via regulative multilineage specification.
    Branching process theory and critical-like self-organization Branching process theory was introduced to explain the propagation of family names in British peerage (Harris, 1963; Watson and Galton, 1874) and can be considered ICG001 as a class of stochastic processes that model the reproduction of a population of similar units (Adami and Chu, 2002). The branching ratio, m, describes how the process propagates through time and can be simply regarded, in the context of a family tree for example, as the number of offspring a couple has. By definition, a process is critical if it has an average branching ratio, m, equal to 1. If 1, the branching process is subcritical and invariably decays but if m>1, it is supercritical and has the potential to propagate indefinitely. Note that a supercritical branching process does not always expand in terms of the number of components it engages, but it must be considered as propagating through time.