Chimera states and cluster solutions in Hindmarsh-Rose neural networks with state resetting process

Yi Yang; Changcheng Xiang; Xiangguang Dai; Xianxiu Zhang; Liyuan Qi; Bingli Zhu; Tao Dong

doi:10.1007/s11571-021-09691-0

Chimera states and cluster solutions in Hindmarsh-Rose neural networks with state resetting process

Cogn Neurodyn. 2022 Feb;16(1):215-228. doi: 10.1007/s11571-021-09691-0. Epub 2021 Jun 30.

Authors

Yi Yang^{1

2}, Changcheng Xiang³, Xiangguang Dai^{1

2}, Xianxiu Zhang^{1

2}, Liyuan Qi^{1

2}, Bingli Zhu^{1

2}, Tao Dong⁴

Affiliations

¹ College of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing, 404100 China.
² Key Laboratory of Intelligent Information Processing and Control of Chongqing Municipal Institutions of Higher Education, Chongqing Three Gorges University, Chongqing, 404100 China.
³ School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000 China.
⁴ College of Electronics and Information Engineering, Southwest University, Chongqing, 400715 China.

Abstract

The neuronal state resetting model is a hybrid system, which combines neuronal system with state resetting process. As the membrane potential reaches a certain threshold, the membrane potential and recovery current are reset. Through the resetting process, the neuronal system can produce abundant new firing patterns. By integrating with the state resetting process, the neuronal system can generate irregular limit cycles (limit cycles with impulsive breakpoints), resulting in repetitive spiking or bursting with firing peaks which can not exceed a presetting threshold. Although some studies have discussed the state resetting process in neurons, it has not been addressed in neural networks so far. In this paper, we consider chimera states and cluster solutions in Hindmarsh-Rose neural networks with state resetting process. The network structures are based on regular ring structures and the connections among neurons are assumed to be bidirectional. Chimera and cluster states are two types of phenomena related to synchronization. For neural networks, the chimera state is a self-organization phenomenon in which some neuronal nodes are synchronous while the others are asynchronous. Cluster synchronization divides the system into several subgroups based on their synchronization characteristics, with neuronal nodes in each subgroup being synchronous. By improving previous chimera measures, we detect the spike inspire time instead of the state variable and calculate the time between two adjacent spikes. We then discuss the incoherence, chimera state, and coherence of the constructed neural networks using phase diagrams, time series diagrams, and probability density histograms. Besides, we further contrast the cluster solutions of the system under local and global coupling, respectively. The subordinate state resetting process enriches the firing mode of the proposed Hindmarsh-Rose neural networks.

Keywords: Chimera state; Cluster solution; Hindmarsh-Rose; Neural networks; State reset process.