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Pattern estimation and selection in media can give important clues to understand the collective response to external stimulus by detecting the observable variables. Both reaction-diffusion systems (RDs) and neuronal networks can be treated as multi-agent systems from molecular level, intrinsic cooperation, competition. An external stimulus or attack can cause collapse of spatial order and distribution, while appropriate noise can enhance the consensus in the spatiotemporal systems. Pattern formation and synchronization stability can bridge isolated oscillators and the network by coupling these nodes with appropriate connection types. As a result, the dynamical behaviors can be detected and discussed by developing different spatial patterns and realizing network synchronization. Indeed, the collective response of network and multi-agent system depends on the local kinetics of nodes and cells. It is better to know the standard bifurcation analysis and stability control schemes before dealing with network problems. In this review, dynamics discussion and synchronization control on low-dimensional systems, pattern formation and synchronization stability on network, wave stability in RDs and neuronal network are summarized. Finally, possible guidance is presented when some physical effects such as polarization field and electromagnetic induction are considered. [ABSTRACT FROM AUTHOR]

Appropriate noise can enhance the regularity in electrical activity of neuron, and stochastic resonance emerges when periodic forcing is imposed on the neuron and/or neuronal network. In this paper, periodic forcing is imposed on some neurons of a two-dimensional square lattice with nearest-neighbor connection (the boundary of one side in the network of neurons), which the local kinetics of node is described by Hindmarsh-Rose neuron, while noise is imposed on another side of the two-dimensional lattice. Indeed, spiral waves can be developed in the side of the network driven by appropriate noise intensity while the another side is occupied by plane wave. A statistical factor of synchronization is defined to discern the occurrence of regularity induced by local noise and local pacing. The emitted plane wave from one side of network can be suppressed by noise and disordered state emerges with stronger noise being used on all nodes of the network, furthermore, spiral wave is induced in the network under appropriate noise intensity. The network shows spatial regularity when spiral wave is developed and it is interesting to find spiral wave can coexist with plane wave in the network. Stochastic resonance-like behavior could be observed in the two-dimensional lattice composed of neurons even if noise and periodic forcing are imposed on different areas of the network. The occurrence of spatial regularity under coherence could rely on the competition between noise driving-induced segments and wave fronts induced by periodic forcing. [ABSTRACT FROM AUTHOR]

Regular spatial patterns could be observed in spatiotemporal systems far from equilibrium states. Artificial networks with different topologies are often designed to reproduce the collective behaviors of nodes (or neurons) which the local kinetics of node is described by kinds of oscillator models. It is believed that the self-organization of network much depends on the bifurcation parameters and topology connection type. Indeed, the boundary effect is every important on the pattern formation of network. In this paper, a regular network of Hindmarsh–Rose neurons is designed in a two-dimensional square array with nearest-neighbor connection type. The neurons on the boundary are excited with random stimulus. It is found that spiral waves, even a pair of spiral waves could be developed in the network under appropriate coupling intensity. Otherwise, the spatial distribution of network shows irregular states. A statistical variable is defined to detect the collective behavior by using mean field theory. It is confirmed that regular pattern could be developed when the synchronization degree is low. The potential mechanism could be that random perturbation on the boundary could induce coherence resonance-like behavior thus spiral wave could be developed in the network. [ABSTRACT FROM AUTHOR]