Phase transitions widely exist in nature and occur when some control parameters are changed. In neural systems, their macroscopic states are represented by the activity states of neuron populations, and phase transitions between different activity states are closely related to corresponding functions in the brain. In particular, phase transitions to some rhythmic synchronous firing states play significant roles on diverse brain functions and disfunctions, such as encoding rhythmical external stimuli, epileptic seizure, etc. However, in previous studies, phase transitions in neuronal networks are almost driven by network parameters （e.g., external stimuli）, and there has been no investigation about the transitions between typical activity states of neuronal networks in a self-organized way by applying plastic connection weights. In this paper, we discuss phase transitions in electrically coupled and lattice-based small-world neuronal networks （LBSW networks） under spike-timing-dependent plasticity （STDP）. By applying STDP on all electrical synapses, various known and novel phase transitions could emerge in LBSW networks, particularly, the phenomenon of self-organized phase transitions （SOPTs）： repeated transitions between synchronous and asynchronous firing states. We further explore the mechanics generating SOPTs on the basis of synaptic weight dynamics.
All dynamic complex networks have two important aspects, pattern dynamics and network topology. Discovering different types of pattern dynamics and exploring how these dynamics depend or/network topologies are tasks of both great theoretical importance and broad practical significance. In this paper we study the oscillatory behaviors of excitable complex networks （ECNs） and find some interesting dynamic behaviors of ECNs in oscillatory probability, the multiplicity of oscillatory attractors, period distribution, and different types of oscillatory patterns （e.g., periodic, quasiperiodic, and chaotic）. In these aspects, we further explore strikingly sharp differences among network dynamics induced by different topologies （random or scale-free topologies） and different interaction structures （symmetric or asymmetric couplings）. The mechanisms behind these differences are explained physically.
This review describes the investigations of oscillatory complex networks consisting of excitable nodes, focusing on the target wave patterns or say the target wave attractors. A method of dominant phase advanced driving （DPAD） is introduced to reveal the dynamic structures in the networks supporting osciUations, such as the oscillation sources and the main excitation propagation paths from the sources to the whole networks. The target center nodes and their drivers are regarded as the key nodes which can completely determine the corresponding target wave patterns. Therefore, the center （say node A） and its driver （say node B） of a target wave can be used as a label, （A, B）, of the given target pattern. The label can give a clue to conveniently retrieve, suppress, and control the target waves. Statistical investigations, both theoretically from the label analysis and numerically from direct simulations of network dynamics, show that there exist huge numbers of target wave attractors in excitable complex networks if the system size is large, and all these attractors can be labeled and easily controlled based on the information given by the labels. The possible applications of the physical ideas and the mathematical methods about multiplicity and labelability of attractors to memory problems of neural networks are briefly discussed.
