Synchronization and stable phase-locking in a network of neurons with memory

We consider a network of three identical neurons whose dynamics is governed by the Hopfieldʹs model with delay to account for the finite switching speed of amplifiers (neurons). We show that in a certain region of the space of (α, β), where α and β are the normalized parameters measuring, respectively, the synaptic strength of self-connection and neighbourhood-interaction, each solution of the network is convergent to the set of synchronous states in the phase space, and this synchronization is independent of the size of the delay. We also obtain a surface, as the graph of a continuous function of τ = τ(α, β) (the normalized delay) in some region of (α, β), where Hopf bifurcation of periodic solutions takes place. We describe a continuous curve on such a surface where the system undergoes mode-interaction and we describe the change of patterns from stable synchronous periodic solutions to the coexistence of two stable phase-locked oscillations and several unstable mirror-reflecting waves and standing waves.