Synchronization transitions induced by partial time delay in a excitatory–inhibitory coupled neuronal network

Abstract

Information transmission delays are an inherent factor of neuronal systems as a consequence of the finite propagation speeds and time lapses occurring by both dendritic and synaptic processes. In real neuronal systems, some delay between two neurons is too small and can be ignored, which results in partial time delay. In this paper, we focus on investigating influences of partial time delay on synchronization transitions in a excitatory–inhibitory (E–I) coupled neuronal networks. Here, we suppose time delay between two neurons equals to $\tau$ with probability $p_{\mathrm{delay}}$ and investigate effect of partial time delay on synchronization transitions of the neuronal networks by controlling $\tau$ and $p_{\mathrm{delay}}$ under three cases. In these three cases, excitatory synapses are always considered to delayed with probability $p_{\mathrm{delay}}$, while inhibitory synapses are considered to be without delays (case I), delayed with probability $p_{\mathrm{delay}}$ (case II), and always delayed (case III), respectively. It is revealed that, in the first two cases, partial time delay has little influences on synchronization of the neuronal network for small $p_{\mathrm{delay}}$, while it could induce synchronization transitions at $\tau$ around integer multiples of the period of individual neuron T when $p_{\mathrm{delay}}$ is large enough, while in the case III, partial time delay could induce synchronization transitions at $\tau$ being around odd integer multiples of T / 2 for small $p_{\mathrm{delay}}$ and at $\tau$ being around integer multiples of T for large $p_{\mathrm{delay}}$. Most interesting observation is that partial time delay could induce frequent synchronization transitions at $\tau$ being around integer multiples of T / 2 for intermediate $p_{\mathrm{delay}}$. Moreover, effect of rewiring probability on synchronization transitions induced by partial time delay has been discussed. It is found that synchronization transitions induced by partial time delay are robust to rewiring probability for large $p_{\mathrm{delay}}$ under the three cases.