Applications of the exponential ordering in the study of almost-periodic delayed Hopfield neural networks

This paper studies almost-periodic neural networks of Hopfield type described by delayed differential equations. The authors introduce an exponential ordering to analyze the long term behavior of the solutions. They prove some theorems of global convergence and deduce the stabilization role of the fast inhibitory self-connections. The proof, which in the case of the neural networks considered in this paper requires uniform stability, uses arguments of comparison of differential equations and methods of the non-autonomous monotone theory of dynamical systems. When the connections between different neurons are excitatory, improved conditions of convergence are obtained and the stabilization role of strongly positive inputs is also shown. The applicability of the results is illustrated with several numerical experiments based on two different families of neural networks.

► We study neural networks of Hopfield type described by non-autonomous FDE.
► The corresponding semiflow is monotone for an exponential ordering.
► We prove convergence to trajectories with the dynamics of the coefficients.
► We improve the conditions of convergence when the connections are excitatory.
► In the almost-periodic case all the solutions are asymptotically almost-periodic.