Structure and dynamics of neural network oscillators.

Techniques are given to represent oscillating neural networks by asynchronous logical switching networks, and to analyze the oscillating networks using a directed graph called a state transition diagram. Consideration is restricted to network oscillators containing no rhythm determining pacemaker neurons, and no neurons exhibiting self-limiting properties such as post-inhibitory rebound or accumulating refractoriness. In the state transition diagrams, stable oscillations are associated with a particular geometric configuration called a cyclic attractor (the heavy cycle in Fig. 2). We show that given the network connectivity it is possible to predict autonomous dynamic behaviour, as well as behaviour following hyperpolarizing or depolarizing inputs to neurons of the network. Conversely, given information about patterns of firing activity during cycles and transients of neural networks, the network connectivity can be predicted. The theoretical techniques can be used to generate a census of network structures capable of generating stable oscillations. Several representative network oscillators are discussed in the context of previous theoretical and experimental studies of the structure of neural network oscillators. Although the number of theoretically possile network oscillators capable of generating sustained oscillations is very large, the techniques which are given should be useful in the design of experiments capable of distinguishing between equally plausible hypotheses.