Abstract
Time delay and coupling strength are important factors that affect the synchronization of neural networks. In this study, a modular neural network containing subnetworks of different scales was constructed using the Hodgkin–Huxley (HH) neural model; i.e., a small-scale random network was unidirectionally connected to a large-scale small-world network through chemical synapses. Time delays were found to induce multiple synchronization transitions in the network. An increase in coupling strength also promoted synchronization of the network when the time delay was an integer multiple of the firing period of a single neuron. Considering that time delays at different locations in a modular network may have different effects, we explored the influence of time delays within each subnetwork and between two subnetworks on the synchronization of modular networks. We found that when the subnetworks were well synchronized internally, an increase in the time delay within both subnetworks induced multiple synchronization transitions of their own. In addition, the synchronization state of the small-scale network affected the synchronization of the large-scale network. It was surprising to find that an increase in the time delay between the two subnetworks caused the synchronization factor of the modular network to vary periodically, but it had essentially no effect on the synchronization within the receiving subnetwork. By analyzing the phase difference between the two subnetworks, we found that the mechanism of the periodic variation of the synchronization factor of the modular network was the periodic variation of the phase difference. Finally, the generality of the results was demonstrated by investigating modular networks at different scales.
摘要
时间延迟和耦合强度是影响神经网络同步的重要因素. 本文利用霍奇金—赫胥黎(HH)神经元模型构建一个包含不同尺度子网络的模块化神经网络, 即小尺度随机网络通过化学突触与大尺度小世界网络单向连接. 研究发现, 时间延迟在网络中诱发了多个同步转换. 当时间延迟是单个神经元放电周期的整数倍时, 耦合强度增加也促进网络同步化. 考虑到模块化网络中不同位置的时间延迟可能具有不同作用, 我们探讨子网络之间以及子网络内部的时间延迟对模块化网络同步的影响. 我们发现, 当子网络内同步良好时, 两个子网络内部时间延迟增加会诱发其自身出现多个同步转换. 此外, 小尺度网络的同步状态会影响大尺度网络的同步. 进一步发现, 两个子网络之间的时间延迟诱导模块化网络的同步转换, 但对接收信号的子网络内的同步基本无影响. 通过分析两个子网络之间的相位差, 我们发现模块化网络出现同步转换的机制是相位差的周期性变化. 最后, 通过对不同尺度模块化网络的研究, 证明了本文结果的泛化性.
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Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Weifang HUANG and Lijian YANG designed the research and processed the data. Weifang HUANG drafted the paper. Xuan ZHAN and Ziying FU helped organize the paper. Lijian YANG and Ya JIA revised and finalized the paper.
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Weifang HUANG, Lijian YANG, Xuan ZHAN, Ziying FU, and Ya JIA declare that they have no conflict of interest.
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Project supported by the National Natural Science Foundation of China (No. 12175080) and the Fundamental Research Funds for the Central Universities, China (No. CCNU22JC009)
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1 Results and discussions
Fig. S1 Spatio-temporal firing raster plots of neuronal membrane potentials of the modular neural network at different coupling strengths g
Fig. S2 Spatio-temporal firing raster plots of neuronal membrane potentials of the modular neural network at different time delays τ2
Fig. S3 Distribution of synchronization factors of modular neural networks over different network parameters with increasing time delay
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Huang, W., Yang, L., Zhan, X. et al. Synchronization transition of a modular neural network containing subnetworks of different scales. Front Inform Technol Electron Eng 24, 1458–1470 (2023). https://doi.org/10.1631/FITEE.2300008
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DOI: https://doi.org/10.1631/FITEE.2300008