Robustness and analysis of mixed mode (bursting) oscillations in three-timescale neuronal systems
Creators: name: Phan, Ngoc Anh
Publisher: name: University of Iowa
Year: 2025
Subjects: subject: Bifurcation theory
subject: Geometric singular perturbation theory
subject: Mixed mode bursting oscillations
subject: Mixed mode oscillations
subject: Multiple-timescale systems
subject: Neurosciences
subject: FOS: Clinical medicine

description: Mixed mode (bursting) oscillations (MM(B)Os) are complex oscillatory behaviors of systems with multiple timescales. These patterns are characterized by small-amplitude oscillations (SAOs) interspersed with large-amplitude excursions or multiple consecutive fast spikes within bursts. Despite the recent advancements in the analysis of MMOs in three-timescale systems, our understanding of their properties and mechanisms remains constrained and primarily focused on two-timescale systems. Moreover, while MMBOs have been observed in numerous three-timescale systems, no studies have yet examined MMBOs arising from three timescales. Motivated by this gap, we aim to explore MMOs and MMBOs in three-timescale settings. To this end, we consider two three-timescale neuronal systems: a coupled Morris-Lecar system and a cortical theta oscillator model.It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov-Hopf bifurcation (DHB) of the fast subsystem. By considering coupled Morris-Lecar neurons, we examine the mechanisms of MMOs involving three timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how the features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work provides crucial insights into conditions under which two separate mechanisms in the two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, especially when there are timescale changes.
description: By considering a cortical theta oscillator model, we explore the mechanisms and properties of both MMOs and MMBOs within a three-timescale system. Using bifurcation analysis and geometric singular perturbation theory, we elucidate how transitions among different solution patterns, including MMOs, MMBOs, and bursting without SAOs, can occur as the timescales are perturbed. Our analysis uncovers four different mechanisms underlying the SAOs in the theta model: the DHB, the interplay between canard and DHB near a CDH singularity, the saddle-focus equilibrium, and a combination of DHB and saddle-focus equilibrium. Notably, we demonstrate that SAOs governed by saddle-focus equilibrium exhibit the greatest robustness to variations in timescales. However, these oscillations display significantly less flexible phase-locking in the theta oscillator model compared with oscillations arising from other mechanisms such as CDH.
description: In summary, we focus on uncovering the properties and mechanisms of MMOs and MMBOs involving three timescales by studying two different three-timescale neuronal systems. This work is aimed to advance the understanding of MMOs and MMBOs in the three-timescale setting from the perspective of dynamical systems. Furthermore, the insights gained from this work will be useful in guiding future modeling work and analyzing experimental data related to MMOs and MMBOs.