Synchronization transition of a modular neural network containing subnetworks of different scales

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.