Anticipated synchronization in neuronal circuits unveiled by a phase response curve analysis

Anticipated synchronization (AS) is a counterintuitive behavior that has been observed in several systems.
When AS occurs in a sender-receiver configuration, the latter can predict the future dynamics of the former for
certain parameter values. In particular, in neuroscience AS was proposed to explain the apparent discrepancy
between information flow and time lag in the cortical activity recorded in monkeys. Despite its success, a clear
understanding of the mechanisms yielding AS in neuronal circuits is still missing. Here we use the well-known
phase-response-curve (PRC) approach to study the prototypical sender-receiver-interneuron neuronal motif. Our
aim is to better understand how the transitions between delayed to anticipated synchronization and anticipated
synchronization to phase-drift regimes occur. We construct a map based on the PRC method to predict the
phase-locking regimes and their stability.We find that a PRC function of two variables, accounting simultaneously
for the inputs from sender and interneuron into the receiver, is essential to reproduce the numerical results obtained
using a Hodgkin-Huxley model for the neurons. On the contrary, the typical approximation that considers a sum
of two independent single-variable PRCs fails for intermediate to high values of the inhibitory coupling strength
of the interneuron. In particular, it loses the delayed-synchronization to anticipated-synchronization transition.