Dynamical systems approaches Determinism and indeterminism

Langevin's equation leads to a distinction that lies at the heart of the sort of determinism espoused by dynamical systems approaches. This is between order parameters (synergetics), behavioural variables (catastrophe theory) or collective variables (the state vector q) on the one hand and control parameters (the p term) on the other (see Hopkins 2001, for further details).

An order parameter is a single macroscopic entity that captures the behaviour of a complex system in terms of low dimensional descriptors (e.g., relative phase between moving limbs). It operates to create the collective state (or ordering) of a system and as such is a topographical variable and not a control entity. The latter is the function of a control parameter and which is analogous to an inus condition. In short, it functions to move systems between its collective states.

There are four important features about a control parameter. Firstly, when it is linearly scaled beyond some critical value, it may induce stochastic behaviour in the order parameter followed by a discontinuous change to another qualitatively different state. Secondly, it is completely aspecific with regard to the change it induces. Thus, a control parameter does not contain prescriptions as to how a system should change unlike its symbolic counterparts such as genetic programmes or schemata. Thirdly, different mechanisms can serve as control parameters at different ages (e.g., hormonal at one age, neural at another and cognitive at yet another). Fourthly, some form of experimental manipulation is required to identify such age-specific control parameters with any confidence. The last two points are brought into relief by a study of testosterone effects in seagulls (Groothuis & Meeuwissen 1992). When administered to young chicks, it induced precociously complete agonistic displays. With adult birds, high testosterone levels were not obligatory to trigger this display as it also occurred when this hormone was at a low level. All told, these findings identify testosterone as an age-specific control parameter in that when it is scaled-up beyond some critical value a later developmental transition was simulated.

The non-specific catalysing effect of control parameters is only applicable to non-linear systems that operate in far-from-equilibrium conditions. A hallmark of such open systems is that they respond by self-organizing when faced by external perturbations in these conditions. Self-organization, which complies with the laws of co-existence, refers to a process by which new structures and functions spontaneously emerge without specification from the external environment. Its meaning can be illustrated with reference to formal logic, the starting point for which is logical calculus, and to the humble snowflake.

A logical calculus involves, among other things, rules for the formation of complex expressions. In a sequence of such formation rules, the extremum (or last rule) states that no expression is valid unless its existence follows from the preceding rules. The type of logic required for understanding the meaning of 'follows from' in the extremum rule led to new types of logic known as formal systems. Such systems are depicted as being self-contained in that they make no appeal to an external logical system. In a similar way, self-organizing systems exhibit an internal logic arising from the system's own dynamics and not through an external, independently existing mode of organization. For example, the lattice structure of a snowflake arises from quantum exchange interactions in which exchange particles or quanta of thermal energy are manifested by self-propagating excitatory loops (Nittman & Stanley 1986). These loops create and maintain a cooperative effect responsible for the macroscopic order of the system. Thus, given certain boundary conditions based on a field of temperature sensitive gradients, thermodynamic (or extremum) processes are triggered resulting in the non-linear transformation of structureless water droplets into the lattice structure ofsnowflakes. Self-organization then is a temporal and spatial process in which an essentially structureless system becomes organized by non-linear, partly random mechanisms involving the self-assembly of interacting constituents - none of which contain prescriptions for the new state of organization.

So far, we have emphasised how dynamical systems approaches cater for determinism (viz., that control parameters deterministically, but unspecifically, induce changes in the order parameter of self-organizing systems). What then is to be gained by a deterministic system having some degree of indeterminism (i.e., stochastic properties)? One potential benefit is that it amplifies the emergence of new forms generated by dynamical, but deterministic, regimes associated with periodic, quasi-periodic and chaotic attractors. In doing so, it enables new behaviours to be assembled creatively so that novel and specific task demands can be addressed throughout development. Another is that stochastic fluctuations superimposed on deterministic processes facilitate exploration of the local environment and thereby maximize the extraction of relevant sources of information. The ability to capitalise on such fluctuations in this way may constitute a source of individual differences in development. This last point raises the possibility that deviant development could be a consequence of either a lack of deterministic constraints on stochastic processes (e.g., as might be the case in hyperactive children with attentional deficits) or of a rigidly determined system (e.g., in those children with cerebral palsy). Further speculations along these lines can be found in Hopkins (2002).

Dynamical systems approaches such as synergetics that allow for stochastic behaviour in an order parameter are especially germane to the study of developmental transitions (i.e., a change from one attractor state to another in a well-defined developing system during a restricted period of development). Defined in terms of non-equilibrium phase shifts, they constitute the simplest manifestation of self-organization in high dimensional complex systems.

Such shifts in an order parameter are referred to as bifurcations. As a bifurcation point is approached, an order parameter begins to manifest stochastic behaviour or what are termed critical fluctuations or anomalous variance (Gilmore 1981). In addition, its number of degrees of freedom is reduced and its behaviour can be captured in low dimensional terms.

According to the theory of noise-induced transitions (Horsthemke & Lefever 1984), such random fluctuations in the behaviour of an order parameter provide the material upon which deterministic processes can act to create new spatiotemporal patterns defined in terms of one or other attractor state. Thus, in complex systems determinism and stochasticity are complementary. Having said that, the problem remains of being able to distinguish stochastic-ity whose dimension is infinite from deterministic chaos that has a broad-based spectral component like noise as standard waveform analyses such as the Fast Fourier Transform cannot reliably make this distinction. Fortunately, solutions are available to overcome this problem (e.g., see Pincus 1991).

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