Excitation dominated or inhibition dominated: Different mechanisms behind rhythmic interaction in a hippocampal model

Biophysically based computational models of brain rhythms offer insights into physiological mechanisms, and naturally set up mathematical questions about what dynamical mechanisms underlie oscillatory behaviors.;The analysis in this thesis is motivated by work of Tort et al., which gave a computational model of the nesting of gamma oscillations (30-90Hz) in theta rhythms (3-8Hz) in the hippocampus, the brain area considered the gate of memory. This work showed that a specific kind of cell (O-LM cell) that tends to spike at theta frequency, and which sends extended inhibitory signals to its targets, can recruit and organize the nesting of gamma oscillations in theta rhythms. In the model, gamma spiking is provided by the interaction of two other kinds of cells, excitatory and inhibitory, that each spike in the gamma range, with the decay time of fast inhibition determining the gamma period.;Computational studies show that there are 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 theta-spiking O-LM cells. In the excitation dominated (ED) regime the excitatory cells make the O-LM cells fire, while in the inhibition dominated (ID) regime the O-LM cells usually fire by inhibitory rebound. These regimes have different coordination properties: for example, the phases of the gamma spikes in the theta rhythm are more variable in the ID regimes.;For the mathematical analysis, the system is reduced to a 2 cell network, and the rhythmic interaction is read as phases of the gamma and theta oscillations that vary on a torus and reset each other with spiking. This leads to the definition of a new map that we call the Phase Transition Map, which encodes the stability type of spiking patterns in networks where different frequencies interact and represents stable patterns as torus knots. This theory extends the previous Spike Time Map Theory to settings where multiple inputs per cycle affect the timing of a cell. The map predicts the different coordination of theta and gamma oscillations in ED and ID regimes.