Synchronization of map-based neurons with memory and synaptic delay

Synchronization of two synaptically coupled neurons with memory and synaptic delay is studied using the Rulkov map, one of the simplest neuron models which displaysspecific features inherent to bursting dynamics. We demonstrate a transition from lag to anticipated synchronization as the relationship between the memory duration and the synaptic delay time changes. The neuron maps synchronize either with anticipation, if the memory is longer than the synaptic delay time, or with lag otherwise. The mean anticipation time is equal to the difference between the memory and synaptic delay independently of the coupling strength. Frequency entrainment and phase-locking phenomena as well as a transition from regular spikes to chaos are demonstrated with respect to the coupling strength.