Roger Traub and collaborators study very fast (>80 Hz) oscillations (VFOs) seen experimentally in the hippocampus. High amplitude VFOs often arise immediately prior to epileptic seizures; this is one of several motivations for studying VFOs. Traub and his colleagues found that VFOs can be obtained even when chemical synapses are blocked and somata and dendrites are removed, but are abolished by gap junction blockers. They therefore hypothesized that the mechanism underlying VFOs involves a network of neurons coupled by axo-axonal gap junctions (electrical synapses), the "axonal plexus". Their simulations confirmed that VFOs can indeed arise in networks of electrically coupled axons. Traub also proposed a radically simplified model to clarify the mechanism, one in which neurons are replaced by the cells of a cellular automaton (CA). Tim Lewis and John Rinzel analyzed Traub's CA, showing how VFOs arise in such a model when there is random external stimulation. They note that these VFOs are stimulus-driven (they stop when external stimulation stops) and not re-entrant VFOs, which persist without stimulation. Lewis and Rinzel then modeled the axon of Traub's full model and saw stimulus-driven VFOs. They hypothesized that stimulus-driven VFOs occur in the axonal plexus.

We further investigate the behavior the axonal plexus. We do this by taking the axon of the Traub's full model, like Lewis and Rinzel, and we apply a fixed somatic voltage V_S to the end of the axon. We then vary V_S and the gap junction conductance g_gj and find three different parameter regimes for network behavior. We find that for high V_S and g_gj, we see stimulus-driven VFOs as described by Lewis and Rinzel. For low V_S and g_gj we see noisy, non-oscillatory activity. Interestingly, there is a large intermediate parameter regime where we see re-entrant VFOs. Furthermore, we find that when cells with the maximal number of connections (4-connected cells) (1) can always fire, we see stimulus-driven VFOs, (2) rarely fire, we see noise, (3) can fire but have a longer refractory period than other cells, we see re-entrant activity. Analysis on cellular automata confirms that assigning 4-connected cells a longer refractory period produces re-entrant VFOs.