Statement Erin Munro November 30 , 2007

A gap junction is a perforation connecting the cytoplasms of two neurons so that ions flow freely from one to the other. Thus when one neuron fires, it may induce its electrically coupled neighbors to fire. There is increasing evidence that gap junctions among pyramidal cell axons play a role in oscillations. The gamma oscillations (30-80 Hz) seen in hippocampal experiments in [8, 9] and models in [7, 8, 11] cease when gap junctions are blocked. Blocking gap junctions also annihilates the very fast oscillations (> 80 Hz, VFOs) seen in thalamocortical experiments in [1, 2] and in the model of [7]. The fact that gap junctions connect pyramidal cell axons is strongly indicated in both experiment and model in [9]. Moreover, gap junctions and VFOs may play a role in seizure initiation [2, 6, 10]. In [12], Traub et al. presented a model of an axonal plexus — a network of axons connected by gap junctions. They simulated a network of CA3 pyramidal cells connected only by gap junctions on their axons. When cells were stimulated randomly via a Poisson process, the network would periodically burst with VFOs occurring during the burst. This was a first step to understanding how such a network of gap junctionally connected axons would behave. In order to understand the behavior, Traub et al. suggested a cellular automaton where cells had three possible states: excitable, firing, and refractory. Cells start in the excitable state until they are induced to fire. After cells fire, they are refractory for a fixed number of time steps during which they are prevented from firing again. Then they return to the excitable state. Cells fire either by external random stimulation via a Poisson process or when one of their neighbors fires. Lewis and Rinzel studied this cellular automaton in [3]. They found that once a cell was stimulated randomly, an expanding wave would emanate from it. Because the waves were topologically closed (the only way to travel from outside the wave to inside via the network was through the wave), waves would expand and merge into each other forming larger waves until they reached the edge of the network. Each new wave must be initiated by a random stimulation. Lewis and Rinzel showed that such waves can produce a VFO, which we refer to as noise-driven oscillations. VFOs can also occur due to re-entrant activity. In re-entrant activity, waves re-enter themselves, exciting cells behind the wave front and topologically forming a spiral wave. The rules for the cellular automaton do not allow re-entrant activity to occur spontaneously. However, if initial conditions are set so that there is a broken wave, then a spiral wave can form around the broken end of the wave because it is not topologically closed. No outside stimulation is needed for re-entrant activity to persist. We refer to oscillations caused by re-entrant activity as re-entrant oscillations. In [4], Lewis and Rinzel showed that topologically closed waves are a possible mechanism for VFOs in an axonal plexus, by modelling only the axon from the CA3 pyramidal cell used in [12]. They hypothesized that re-entrant activity is unlikely, as they had to apply heavy stimulation in order to induce it.