Introduction

Artificial neural network research began in the 1940s with the M-P neuron network model, which firstly proposed by biologist McCulloch and mathematician Pitts, kicking off neural network research [95]. In 1949, Hebb studied adaptive laws in neural systems and proposed Hebb’s rule for improving the strength of connections between neurons, breaking new ground in the study of neural networks. In 1982, Hopfield proposed a recurrent neural network model, which could perform optimal computation and associative memory. In 1986, Rumelhart et al. gave an error back propagation algorithm (i.e. the BP algorithm) for training weights of multilayer perceptrons. Both the forward neural network model and the recurrent neural network model are based on the nonlinear characteristics, adaptability and excitation inhibition of biological neurons, and the main purpose is to design high-performance controllers based on the bionic principle [152]. A controller that can be implemented online must be relatively simple in structure and have online adaptive and learning capabilities. The neural controllers are the preferred method to meet this requirement, especially recurrent neural networks. They not only have the universal approximation property of the forward neural network, but also the dynamical properties and associative memory capabilities that the forward neural network does not have. In early neural network research, neural networks were discussed as single individual to solve relatively homogeneous task, which was related to the task requirement and the scientific problem at the time. The correction of connection weights between neurons is a key concern, which has led to research on the design of neural network learning algorithms. With the advent of the network era, the complex neural networks (i.e. coupled neural networks) show potential advantages in large-scale pattern recognition and image processing [3, 17, 30, 42, 94, 131, 132, 151, 203]. As a special kind of complex networks, the complex neural networks have a number of characteristics that a single neural network does not have. As a result, the dynamic properties of complex neural networks (such as synchronization properties and chaos properties) are widely studied. The research of complex neural networks focuses on the topological structure or connection strength between nodes, rather than focusing on the connection relationships between microscopic neurons. Each node in complex neural networks is composed of neural networks with appropriate dimensions, rather than a single neuron. In this way, the complex neural networks are endowed with some complex networks characteristics, and they have the following features compared with the earlier neural networks.