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A class of differential equations, which captures the logical structure of discrete time logical switching networks composed of many elements, displays deterministic chaos if each element has many inputs. Statistical features of the dynamics are approximated by using a mean field Langevin-type equation with a random telegraph signal as a stochastic forcing function, and also by considering a random walk on an N-dimensional hypercube. [S0031-9007(97)03599-0]

During the evolution of many systems found in nature, both the system composition and the interactions between components will vary. Equating the dimension with the number of different components, a system which adds or deletes components belongs to a class of dynamical systems with a finite dimensional phase space of variable dimension. We present two models of biochemical systems with a variable phase space, a model of autocatalytic reaction networks in the prebiotic soup and a model of the idiotypic network of the immune system. Each model contains characteristic meta-dynamical rules for constructing equations of motion from component properties. The simulation of each model occurs on two levels. On one level, the equations of motion are integrated to determine the state of each component. On a second level, algorithms which approximate physical processes in the real system are employed to change the equations of motion. Models with meta-dynamical rules possess several advantages for the study of evolving systems. First, there are no explicit fitness functions to determine how the components of the model rank in terms of survivability. The success of any component is a function of its relationship to the rest of the system. A second advantage is that since the phase space representation of the system is always finite but continually changing, we can explore a potentially infinite phase space which would otherwise be inaccessible with finite computer resources. Third, the enlarged capacity of systems with meta-dynamics for variation allows us to conduct true evolution experiments. The modeling methods presented here can be applied to many real biological systems. In the two studies we present, we are investigating two apparent properties of adaptive networks. With the simulation of the prebiotic soup, we are most interested in how a chemical reaction network might emerge from an initial state of relative disorder. With the study of the immune system, we study the self-regulation of the network including its ability to distinguish between species which are part of the network and those which are not.

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