Phase plane description of endogenous neuronal oscillators in Aplysia

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Abstract
Phase plane techniques are used to describe graphically the limit cycle behavior of identified endogenous neuronal oscillators in the isolated abdominal ganglion of Aplysia. Intracellularly recorded membrane potential from a bursting neuron and its first derivative with respect to time are used as coordinates (state variables) in phase space. The derivative is either measured electronically or calculated digitally. Each trajectory in phase space represents the entire output of the bursting neuron, i.e., both the rapid action potentials and slow pacemaker potentials. Phase plane portraits are presented for the free run limit cycle before and after a change in a system parameter (applied transmembrane current) and also for phase resetting produced by direct synaptic inhibition from an identified interneuron. The complex topology of the trajectory suggests that the bursting oscillator is a higher order system. Therefore, the second time derivative is used as another state variable. This type of phase plot can help to relate biophysical and mathematical analyses.