Bifurcation, bursting, and spike frequency adaptation

Many neural systems display adaptive properties that occur on time scales that are slower than the time scales associated with repetitive firing of action potentials or bursting oscillations. Spike frequency adaptation is the name given to processes that reduce the frequency of rhythmic tonic firing of action potentials, sometimes leading to the termination of spiking and the cell becoming quiescent. This article examines these processes mathematically, within the context of singularly perturbed dynamical systems. We place emphasis on the lengths of successive interspike intervals during adaptation. Two different bifurcation mechanisms in singularly perturbed systems that correspond to the termination of firing are distinguished by the rate at which interspike intervals slow near the termination of firing. We compare theoretical predictions to measurement of spike frequency adaptation in a model of the LP cell of the lobster stomatogastric ganglion.