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 firin g 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, whe re Delta is the forcing period. A linear stability analysis of the map of f iring times around such solutions allows the determination of the Arnold to ngue structure for regions in parameter space where-stable solutions exist. The analysis is verified against direct numerical simulations for both a s inusoidally forced IF system and one in which a periodic sequence of spikes is used to induce a biologically realistic synaptic input current. This ap proach is extended to the case of two synaptically coupled IF oscillators, showing that mode-locked states can exist for some self-consistently determ ined common period of repetitive firing. Numerical simulations show that su ch solutions have a bursting structure where regions of spiking activity ar e interspersed with quiescent periods before repeating. The influence of th e synaptic current upon the Arnold tongue structure is explored in the regi me of weak coupling. [S1063-651X(99)09207-7].