Stuart-landau oscillators with pairwise and higher order interactions and application in PGES

Abstract

This thesis focuses on pairwise and higher-order modeling of coupled Stuart-Landau oscillators, with an emphasis on the suppression of oscillations and its application to a neuropathological condition known as Postictal Generalized EEG Suppression (PGES). To investigate the suppression of oscillations in various nonlinear model systems, coupled Stuart-Landau oscillators prove to be particularly suitable for understanding the origin and implications of such behavior. Quenching of oscillations in these systems is primarily achieved through three mechanisms, introducing parameter mismatch, communication delays, and conjugate coupling. A quenched or “death” state of an oscillator can be broadly classified into two categories, amplitude death (AD) and oscillation death (OD), based on the spatial configuration and symmetry of the associated fixed points. The AD state corresponds to all oscillators converging to the same fixed point, which is typically an unstable fixed point in the uncoupled system. In this case, coupling stabilizes the AD state via a Hopf bifurcation while preserving parity symmetry. In contrast, in the OD state, oscillators settle at different fixed points that emerge due to coupling-induced and parity symmetry-breaking bifurcations. First, we propose a coupling setup that yields suppression in coupled Stuart-Landau oscillators in the form of AD and OD. We derive the necessary and sufficient condition for attaining the AD state for this setup. Moreover, we develop a generalized theoretical framework to obtain the analytical condition for AD state for similar coupling forms. Furthermore, we attempt to identify applications of dynamical models that demonstrate the suppression of oscillations in a neuropathological condition known as PGES. PGES is defined by marked suppression in brain activity just after a tonic-clonic seizure.