Phase resetting reduces theta–gamma rhythmic interaction to a one-dimensional map

Gamma and theta oscillations of the hippocampus are known to interact, but the mechanisms underlying such interaction are not well understood. We focus on a previously published computational model of hippocampal activity that shows the gamma rhythms nesting in the theta rhythms, and investigate the dynamical mechanisms underlying that interaction. There are three types of neurons in the model: pyramidal cells, fast-spiking interneurons, and “oriens lacunosum-moelculare” (O-LM cells); the latter is an inhibitory cell whose inhibition has a longer time scale, and which has currents associated with intrinsic theta-rhythm behavior. We identify two main modes of interaction among the slow and the fast rhythms in the model, modulated by the strength of the excitatory synapse on the O-LM cells. Using resets of phases after each pyramidal cell and O-LM spike, we extend the use of the phase transition map (PTM) to encode the stability type of spiking patterns in networks where different frequencies interact. The tailored application of the PTM to the model network measures how the interaction between the shape of the phase response curves and the length of the gamma period determines the number of gamma spikes in theta cycles, and provides an explicit formula for the length of theta intervals in nesting regimes. Using the PTM, we also explain the covariance of the gamma and theta rhythms as drive is changed over some intervals.