Mode locking and Arnold tongues in integrate-and-fire neural oscillators

An analysis of mode-locked solutions that may arise in periodically forced integrate-and-fire (IF) neural oscillators is introduced based upon a firing map formulation of the dynamics. A $q:p$ mode-locked solution is identified with a spike train in which p firing events occur in a period $q\Delta ,$ where $\Delta$ is the forcing period. A linear stability analysis of the map of firing times around such solutions allows the determination of the Arnold tongue structure for regions in parameter space where stable solutions exist. The analysis is verified against direct numerical simulations for both a sinusoidally forced IF system and one in which a periodic sequence of spikes is used to induce a biologically realistic synaptic input current. This approach is extended to the case of two synaptically coupled IF oscillators, showing that mode-locked states can exist for some self-consistently determined common period of repetitive firing. Numerical simulations show that such solutions have a bursting structure where regions of spiking activity are interspersed with quiescent periods before repeating. The influence of the synaptic current upon the Arnold tongue structure is explored in the regime of weak coupling.