Editorial

Dynamical behaviors of coupled neural networks with reaction-diffusion terms: analysis, control and applications

Recently, the dynamical behaviors of neural networks (NNs) have been extensively investigated by researchers. The main reason for that is their wide applications in optimization, associative memory, signal processing, pattern classification, image processing, and so on. Therefore, a great many important results on analysis and control of dynamical behaviors have been established for various NNs. More recently, coupled neural networks (CNNs) have received considerable attention. Because CNNs can exhibit some interesting phenomena and can be utilized in engineering fields such as harmonic oscillation generation, chaos generators design, secure communication, and so on. CNNs is a special class of complex networks, in which a node denotes an NN. Although research on CNNs has attracted so much attention, little of that has been devoted to the dynamical behaviors of CNNs with reaction–diffusion terms. Strictly speaking, the diffusion phenomena cannot be avoided in NNs when electrons are moving in asymmetric electromagnetic fields, thus we must consider the diffusion effects in CNNs.

The aim of this special issue is to introduce novel research work on dynamical behaviors of CNNs with reaction–diffusion terms. With the help of Reviewers and the Editor-in-Chief, 18 papers were selected for publication after two or three rounds of rigorous reviews. We divide these accepted papers into three parts.

In the first part, we have 9 papers on dynamical behaviors analysis of CNNs with reaction–diffusion terms. The first paper, “Passivity and pinning passivity of complex dynamical networks with spatial diffusion coupling”, presents a coupled delayed reaction–diffusion neural networks with spatial diffusion coupling, and respectively analyzes passivity and pinning passivity of such network model. The second paper, “Zero singularities of codimension two in a delayed predator–prey diffusion system”, proves that the unique positive equilibrium of the diffusion system is a Bogdanov–Takens singularity under certain conditions. The papers “Finite-time state observer for delayed reaction–diffusion genetic regulatory networks”, “State estimation for recurrent neural networks with unknown delays: a robust analysis approach”, “State estimation and input-to-state stability of impulsive stochastic BAM neural networks with mixed delays” and “Non-fragile mixed ${\mathcal{H}}_{\infty }$ and passive asynchronous state estimation for Markov jump neural networks with randomly occurring uncertainties and sensor nonlinearity” respectively discuss state estimation problem for delayed reaction–diffusion genetic regulatory networks, recurrent neural networks, impulsive stochastic BAM neural networks and uncertain discrete-time Markov jump neural networks. The seventh paper, “Synchronization of multi-group coupled systems on networks with reaction–diffusion terms based on the graph-theoretic approach”, obtains two kinds of sufficient criteria to guarantee the synchronization of multi-group coupled systems on networks with reaction–diffusion terms on the basis of graph-theoretic method and vertex-Lyapunov functions set. The eighth paper, “Global attractivity of memristor-based fractional-order neural networks”, considers the global attractivity of memristor-based fractional-order neural networks. The ninth paper, “Circuit implementation of digitally programmable transconductance amplifier in analog simulation of reaction–diffusion neural model”, simulates a reaction–diffusion neural network by utilizing circuit.

In the second part, we have 3 papers on dynamical behavior control of CNNs with reaction–diffusion terms. The first paper, “Synchronization for coupled reaction–diffusion neural networks with and without multiple time-varying delays via pinning-control”, considers two types of coupled reaction–diffusion neural networks with and without time-varying delays, and studies synchronization of these network models by utilizing pinning control technique. The second paper, “Pinning synchronization of spatial diffusion coupled reaction–diffusion neural networks with and without multiple time-varying delays”, discusses pinning synchronization of two coupled reaction–diffusion neural networks with spatial diffusion coupling. The third paper, “Pinning sampled-data synchronization of coupled inertial neural networks with reaction–diffusion terms and time-varying delays”, investigates the problem of synchronization of coupled reaction–diffusion neural networks with added inertia and time-varying delays using a pinning sampled-data controller.

In the third part, we have 6 papers on applications of CNNs with reaction–diffusion terms. The first paper, “Secure communication based on the synchronous control of hysteretic chaotic neuron”, proposes a secure communication strategy based on the chaotic masking method. The second paper, “Modeling and control with neural networks for a magnetic levitation system”, designs a neural network controller for the trajectory tracking of the magnetic levitation system. The third paper, “Uniform stable radial basis function neural network for the prediction in two mechatronic processes”, presents a method to obtain a stable algorithm for the learning of a radial basis function neural network, and this method is applied for the prediction of the warehouse process and the brain behavior. The fourth paper, “Adaptive robust speed control based on recurrent Elman neural network for sensorless PMSM servo drives”, develops an adaptive robust control scheme to achieve high-performance speed tracking despite of the existence of system uncertainties for the sensorless permanent magnet synchronous motor servo drive. The papers “A novel memristive Hopfield neural network with application in associative memory” and “Memristive pulse coupled neural network with applications in medical image processing” respectively construct appropriate memristive neural networks for associative memory and medical image processing.

This special issue can serve as a stepping stone to study the CNNs with reaction–diffusion terms, and there are still some interesting and challenging problems deserving further investigation.