Threshold for repetitive activity for a slow stimulus ramp: a memory effect and its dependence on fluctuations

We have obtained new insights into the behavior of a class of excitable systems when a stimulus, or parameter, is slowly tuned through a threshold value. Such systems do not accommodate no matter how slowly a stimulus ramp is applied, and the stimulus value at onset of repetitive activity shows a curious, nonmonotonic dependence on ramp speed. (Jakobsson, E. and R. Guttman. Biophys. J. 1980. 31:293-298.) demonstrated this for squid axon and for the Hodgkin-Huxley (HH) model. Furthermore, they showed theoretically that for moderately slow ramps the threshold increases as the ramp speed decreases, but for much slower ramp speeds threshold decreases as the ramp speed decreases. This latter feature was found surprising and it was suggested that the HH model, and squid axon in low calcium, exhibits reverse accommodation. We have found that reverse accommodation reflects the influence of persistent random fluctuations, and is a feature of all such excitable systems. We have derived an analytic condition which yields an approximation for threshold in the case of a slow ramp when the effect of fluctuations are negligible. This condition predicts, and numerical calculations confirm, that the onset of oscillations occurs beyond the critical stimulus value which is predicted by treating the stimulus intensity as a static parameter, i.e., the dynamic aspect of a ramp leads to a delay in the onset. The condition further demonstrates a memory effect, i.e., firing threshold is dependent on the initial state of the system. For very slow ramps then, fluctuations diminish both the delay and memory effects. We characterize the class of excitable systems for which these behaviors are expected, and we illustrate the phenomena for the HH model and for a model of cAMP-receptor dynamics in Dictyostelium discoideum.