Spontaneous secondary spiking in excitable cells

Kepler & Marder (1993, Biol. Cybern. 68, 209-214) proposed a model describi ng the electrical activity of a crab neuron in which a train of directly in duced action potentials is sometimes followed by one or more spontaneous ac tion potentials, referred to as spontaneous secondary spikes. We reduce the ir five-dimensional model to three dimensions in two different ways in orde r to gain insight into the mechanism underlying the spontaneous spikes. We then treat a slowly varying current as a parameter in order to give a quali tative explanation of the phenomenon using phase-plane and bifurcation anal ysis. We demonstrate that a three-dimensional model, consisting of a two-di mensional excitable system plus a slow inward current, is sufficient to pro duce the behaviour observed in the original model. The exact dynamics of th e excitable system are not important, but the relative time constant and am plitude of the slow inward current are crucial. Using the numerical bifurca tion analysis package AUTO (Doedel & Kernevez, 1986, AUTO: Software for Con tinuation ann Bifurcation Problems in Ordinary Differential Equations. Cali fornia Institute of Technology), we compute bifurcation diagrams using the maximum amplitude of the slow inward current as the bifurcation parameter. The full and reduced models have a stable resting potential. for all values of the bifurcation parameter. At a critical value of the bifurcation param eter, a stable tonic firing mode arises via a saddle-node of periodics bifu rcation. Whether or not the models can exhibit transient or continuous spon taneous spiking depends on their position in parameter space relative to th is saddle-node of periodics. (C) 2000 Academic Press.