Synchronization of a computationally efficient neuron model with memory and synaptic delay

Experiments have recently shown synchronization in pairs of biological neurons. Several models have been studied, from biological plausible ones like the Hodgkin-Huxley model to formal models (Rulkov). We use the modified Kinouchi-Tragtenberg (KTz) model to study the synchronization of two coupled identical neurons with memory and synaptic delay. The KTz is a logistic neuron model with the membrane potential described by the discrete time recursion relation $x(t+1) = f(u(t))$, where $f$ is a sigmoidal function, $u(t )= [x(t)-K y(t) + z(t) + I(t)] / T$, I(t) is an external current and $z(t)$ is a slow current. We study the case where $f(u) = u / [1 +|u|]$. This is a computationally efficient neuron model with many dynamical behaviors similar to biological neurons: excitable fixed point, fast and slow regular spiking, bursts and spikes with plateau etc. Using a master-slave configuration we demonstrate that depending on the relationship between memory and synaptic delay times the neurons synchronize either with anticipation or with lag. We use the similarity function $S^2(\phi)$ to characterize anticipation and lag. The difference between memory and synaptic delay times corresponds to the mean delay time. We vary the coupling strength between two neurons in order to show that it controls phase-locking and frequency entrainment of the system.