Random simulation and confiners: Their application to neural networks

Random simulation of complex dynamical systems is generally used in order to obtain information about their asymptotic behaviour (i.e., when time or size of the system tends towards infinity). A fortunate and welcome circumstance in most of the systems studied by physicists, biologists, and economists is the existence of an invariant measure in the state space allowing determination of the frequency with which observation of asymptotic states is possible. Regions found between contour lines of the surface density of this invariant measure are called confiners. An example of such confiners is given for a formal neural network capable of learning. Finally, an application of this methodology is proposed in studying dependency of the network's invariant measure with regard to: 1) the mode of neurone updating (parallel or sequential), and 2) boundary conditions of the network (searching for phase transitions).