Reduction of a model for an Onchidium pacemaker neuron.

The eight-variable model for the giant neuron localized in the esophageal ganglia of the marine pulmonate mollusk Onchidium verruculatum is reduced to four- and-three-dimensional systems by regrouping variables with similar time scales. These reduced models replicate the complex behavior including beating, periodic bursting and aperiodic bursting displayed by the original full model when the parameter Iext representing the intensity of the constant DC current stimulation is varied across a wide range. The complex behavior of the full model arises from the interaction of fast and slow dynamics, and depends on the time scale Cs of the slow dynamics. The four-variable reduced model is constructed independently from the parameter Cs so that it reproduces the two-dimensional bifurcation structure of the full model for the two parameters Iext and Cs. The three-variable reduced model is derived for a specific value of Cs. The parameters of this model are tuned so that its one-parameter bifurcation diagram for Iext closely matches that of the full model. Correspondence between bifurcation structures ensures that both reduced models reproduce the various discharge patterns of the full model. Similarity between the full and reduced models is also confirmed by comparing mean firing frequencies and membrane potential waveforms in various regimes. The reduction exposes the factors essential for reproducing the dynamics of the full model; indeed, it shows that the eight variables representing the membrane potential and seven gating variables of six ionic currents in the full model account, in fact, for three basic processes responsible for excitability, post-discharge refractoriness and slow membrane modulation.