Identifying bifurcations underlying a neuronal bursting of mixed-mode oscillations with two slow variables in inner hair cell

Neuronal bursting patterns are nonlinear behaviors related to various brain functions and diseases. Bursting modulated by two slow variables manifest more complex dynamics than that with one slow variable, due to involved with more bifurcations of fast subsystem. The traditional dissection method or singular perturbation method often fails to analyze the bursting with two slow variables, for instance, a complex mixed-mode oscillation (MMO) bursting observed in the immature inner hair cell related to the development of auditory function. The MMO bursting exhibits alternation between a single spike with large amplitude and a pseudo-plateau burst behaving as fast burst with small oscillations. In the present paper, the MMO bursting is successfully analyzed using a novel dissection process. With two slow variables c and h regarded as bifurcation parameters of the fast subsystem, various dynamical behaviors and bifurcation curves are obtained. By identifying real intersections between the bursting trajectory and bifurcation curves in the (c, h, V) space instead of (c, h) plane, the pseudo-plateau burst is built relationships to the depolarization block and subcritical Hopf bifurcation of the fast subsystem, and the spike is associated with coexisting firing and limit point bifurcation of cycles. Interestingly, no quiescent state is identified to be related to the resting state or saddle-node bifurcation on an invariant cycle (SNIC), due to that c is not slow enough. As c becomes sufficiently slow, the bursting pattern is associated with the resting state and the SNIC. In addition, the bursting trajectory exhibits a narrow shape around an oblique line in the (c, h) plane, and the bifurcations along the line presents a simple and effective candidate to characterize the bursting. The results present effective process of dissection method and deep understanding of the MMO bursting, which are helpful for modulating the development of auditory functions related to the inner hair cell.