Dynamical phases of the Hindmarsh-Rose neuronal model: studies of the transition from bursting to spiking chaos

The dynamical phases of the Hindmarsh-Rose neuronal model are analyzed in detail by varying theexternal current I. For increasing current values, the model exhibits a peculiar cascade of nonchaoticand chaotic period-adding bifurcations leading the system from the silent regime to a chaoticstate dominated by bursting events. At higher I-values, this phase is substituted by a regime ofcontinuous chaotic spiking and finally via an inverse period doubling cascade the system returns tosilence. The analysis is focused on the transition between the two chaotic phases displayed by themodel: one dominated by spiking dynamics and the other by bursts. At the transition an abruptshrinking of the attractor size associated with a sharp peak in the maximal Lyapunov exponent isobservable. However, the transition appears to be continuous and smoothed out over a finite currentinterval, where bursts and spikes coexist. The beginning of the transition from the bursting side issignaled from a structural modification in the interspike interval return map. This change in the mapshape is associated with the disappearance of the family of solutions responsible for the onset of thebursting chaos. The successive passage from bursting to spiking chaos is associated with a progressivepruning of unstable long-lasting bursts.