[en] Bursting dynamics appear across a wide range of physical and biological systems, particularly in neuroscience, where they play a central role in modeling epilepsy. Epilepsy is characterized by abnormal, excessive or synchronous neuronal activity in the brain, marked by abrupt transitions from quiescent periods to seizures. Although EEG recordings help identify seizure patterns and affected brain regions, they often cannot predict precisely the onset, termination, or propagation of seizures. This poses a challenge for designing surgical interventions for the one-third of epilepsy patients who do not respond to medication. To deepen our understanding, phenomenological models based on nonlinear dynamics and bifurcation theory have been developed to classify seizure mechanisms by capturing fast-slow oscillations between ictal and interictal states. Control-based continuation (CBC) techniques provide a powerful framework for experimentally exploring complex bifurcation diagrams of nonlinear systems. In this work, we present modifications to a recent extension of CBC, termed arclength controlbased
continuation (ACBC), for identifying fixed points and limit cycles in autonomous systems featuring fast-slow oscillations. While ACBC has been successfully applied in nonlinear vibration testing, where synchronization between the forcing and response frequencies simplifies signal analysis, autonomous systems present a unique challenge. Their self-sustained limit cycles have frequencies governed by internal dynamics which are not known a priori. To address this, we leverage angleencoding to estimate the system’s instantaneous phase and demonstrate its application within the context of seizure modeling.
