Synchronization in scale-free neural networks with heterogeneous time delay

The functional network of the human brain exhibits scale-free topology, and there are inevitable time delays in information transmission between neurons. This study explores the relationship between synchronization transitions and heterogeneous time delays in scale-free neuronal networks, as well as the influence of coupling strength on the synchronization process under the premise of introducing heterogeneous time delays. Inspired by small-world network construction methods, we designed a scale-free neural network model with heterogeneous time delays based on the Rulkov neuron model, referred to as the Heterogeneous Scale-Free Neural Network (HSFNN). In this paper, we propose a time delay determination mechanism (TDDM). Subsequently, we conducted numerical simulations to analyze the effects of various parameters related to time delays on the synchronization of HSFNNs. The results were analyzed using spatiotemporal state diagrams, recursion diagrams, and node state diagrams. The findings indicate that HSFNNs exhibit various dynamical phenomena, such as asynchronous, synchronous, and alternating states of synchronization and asynchrony. Furthermore, we found that altering coupling strength under the premise of introducing heterogeneous time delays also affects the synchronization states of HSFNNs.