Analysis of models for crustacean stretch receptors

The mechanisms underlying the diverse responses to step current stimuli of models [Edman et al. (1987)J Physiol (Lond) 384: 649-669] of lobster slowly adapting stretch receptor organs (SAO) and fast-adapting stretch receptor organs (FAO) are analyzed. In response to a step current, the models displa y three distinct types of firing reflecting the level of adaptation to the stimulation. Low-amplitude currents evoke transient firing containing one t o several action potentials before the system stabilizes to a resting state . Conversely, high-amplitude stimulations induce a high frequency transient burst that can last several seconds before the model returns to its quiesc ent state. In the SAO model, the transition between the two regimes is char acterized by a sustained pacemaker firing at an intermediate stimulation am plitude. The FAO model does not exhibit such a maintained firing; rather, t he duration of the transient firing increases at first with the stimulus in tensity, goes through a maximum and then decreases at larger intensities. B oth models comprise seven variables representing the membrane potential, th e sodium fast activation, fast inactivation, slow inactivation, the potassi um fast activation, slow inactivation gating variables, and the intra cellu lar sodium concentration. To elucidate the mechanisms of the firing adaptat ions, the seven-variable model for the lobster stretch receptor neuron is f irst reduced to a three-dimensional system by regrouping variables with sim ilar time scales. More precisely, we substituted the membrane potential V f or the sodium fast activation equivalent potential V-m, the potassium fast inactivation V-n for the sodium fast inactivation V-h, and the sodium slow inactivation V-l for the potassium slow inactivation V-r. Comparison of the responses of the reduced models to those of the original models revealed t hat the main behaviors of the system were preserved in the reduction proces s. We classified the different types of responses of the reduced SAO and FA O models to constant current stimulation. We analyzed the transient and sta tionary responses of the reduced models by constructing bifurcation diagram s representing the qualitatively distinct dynamics of the models and the tr ansitions between them. These revealed that (1) the transient firings prior to reaching the stationary state can be accounted for by the sodium slow i nactivation evolving more slowly than the other two variables, so that the changes during the transient firings reflect the bifurcations that the two- dimensional system undergoes when the sodium slow inactivation, considered as a parameter, is varied; and (2) the stationary behaviors of the models a re captured by the standard bifurcations of a two-dimensional system formed by the membrane potential and the potassium fast inactivation. We found th at each type of firing and the transitions between them is due to the inter play between essentially three variables: two fast ones accounting for the action potential generation and the post-discharge refractoriness, and a th ird slow one representing the adaptation.