A Genetic System Based on Simulated Crossover: Stability Analysis and Relationships with Neural Nets

We consider a marginal distribution genetic model based on crossover of sequences of genes and provide relations between the associated infinite population genetic system and the neural networks. A lower bound on population size is exhibited stating that the behaviour of the finite population system, in case of sufficiently large sizes, can be approximated by the behaviour of the corresponding infinite population system. Assumptions on fitness and individual chromosomes are provided implying that the behaviour of the finite population genetic system remains consistent with the behaviour of the associated infinite population genetic system for suitably long trajectories. The attractors (with binary components) of the infinite population genetic system are characterized as equilibrium points of a discrete (neural network) system that can be considered as a variant of a Hopfield's network; it is shown that the fitness is a Lyapunov function for the variant of the discrete Hopfield's net. Our main result can be summarized by stating that the relation between marginal distribution genetic systems and neural nets is much more general than that already shown elsewhere for other simpler models.