Asymptotic analysis of noise sensitivity in a neuronal burster

A combination of asymptotic approaches provides a new analysis of the effect of small noise on the bursting cycle of a neuronal burster of elliptic type (type III). The analysis is applied to a stochastic model of an excitable spine, with an activity-dependent stem conductance, that exhibits conditional burst dynamics. First, we give an asymptotic approximation to the probability density for the state of the system. This density is used to compute several quantities which describe the influence of the noise on the transition from the silent to the active phase. Second, we also use a multiscale method to provide a reduced system for analysing the effect of noise on the transition out of the active phase. The combination of these two approaches results in a new framework for a quantitative description of how noise shortens the burst cycle, which measures the significant influence of small noise. For the stochastic spine model, this study suggests that small amplitude noise can significantly influence the activity-dependent morphological plasticity of dendritic spines. The techniques used in this paper combine probabilistic and asymptotic methods, and have been generalized for other noisy nonlinear systems.