Strange Nonchaotic Bursting in A Quasiperiodially-Forced Hindmarsh-Rose Neuron

We study the transition from a silent state to a bursting state by varying the dc stimulus in the Hindmarsh-Rose neuron under quasiperiodic stimulation. For this quasiperiodically forced case, a new type of strange nonchaotic (SN) bursting state is found to occur between the silent state and the chaotic bursting state. This is in contrast to the periodically forced case where the silent state transforms directly to a chaotic bursting state. Using a rational approximation to the quasiperiodic forcing, the mechanism for the appearance of such an SN bursting state is investigated. Thus, a smooth torus (corresponding to a silent state) is found to transform to an SN bursting attractor through a phase-dependent subcritical period-doubling bifurcation. These SN bursting states, together with chaotic bursting states, are characterized in terms of the interburst interval, the bursting length, and the number of spikes in each burst. Both bursting states are found to be aperiodic complex ones. Consequently, aperiodic complex burstings may result from two dynamically different states with strange geometry (one is chaotic and the other one is nonchaotic). Thus, in addition to chaotic burstings, SN burstings may become a dynamical origin for complex physiological rhythms which are ubiquitous in organisms.