Robustness and analysis of mixed mode (bursting) oscillations in three-timescale neuronal systems

Natural systems that exhibit patterns involving multiple timescales are present in many aspects of our daily lives. The human heartbeat, for example, speeds up in response to stress or exercise and slows down during rest or relaxation. Similarly, the forest ecosystem experiences quick daily changes, such as plants absorbing sunlight and water, and slow seasonal changes, like animals hibernating in winter and awakening in spring. In this work, we focus on investigating mixed mode (bursting) oscillations, which are complex rhythmic behaviors commonly observed in multiple-timescale systems. We aim to uncover the properties and mechanisms underlying these complex dynamics with three distinct timescales. Additionally, we explore how changes in parameters, like adjusting the volume on a stereo, can affect the timescales and features of these rhythms. Our work leads to a better understanding of how different mechanisms that remain separated in the two-timescale context can interact to produce more robust mixed mode (bursting) oscillations in three-timescale systems, as well as their roles in flexible phase-locking of neural oscillators.