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We describe a modification to our recent model of the action potential which introduces two additional equilibrium points. By using stability analysis we show that one of these equilibrium points is a saddle point from which there are two separatrices which divide the phase plane into two regions. In one region all phase paths approach a limit cycle and in the other all phase paths approach a stable equilibrium point. A consequence of this is that a short depolarizing current pulse will change an initially silent model neuron into one that fires repetitively. Addition of a third equation limits this firing to either an isolated burst or a depolarizing afterpotential. When steady depolarizing current was applied to this model it resulted in periodic bursting. The equations, which were initially developed to explain isolated triggered bursts, therefore provide one of the simplest models of the more general phenomenon of oscillatory burst discharge.

1. Certain nonlinear properties of molluscan neurones suggest that network activity could be described in relation to nonlinear relaxation oscillators. 2. A specific example is fitted by the Van der Pol equation with an exponentially decaying damping coefficient, and changes in the associated energy and sensitivity are considered. 3. More precise details of the interaction pattern determined by analysis of a single cycle of burst activity, and it is shown that there are two states of activity, depending on the balance of presumed inhibitory components. 4. Previous results are discussed in relation to an overall decaying oscillation. 5. Changes in subcomponents of a cycle are described for decaying sequences and the constancy of synaptic interaction is demonstrated. The results are briefly discussed in relation to catastrophe theory and learning.