Exploring Synchronization Mechanisms via Bifurcation Analysis – A Unified Approach Across Neuronal, Ecological and Physical Realms

Synchronization mechanisms, while inherently complex, are central to a wide range of dynamical systems, from neuronal networks to physical systems like superconductive junctions. The aim of this contribution is to introduce a unified approach using the continuation program, MatCont, to explore these phenomena through the lens of bifurcation theory, specifically employing Arnold tongues and limit points of cycle manifolds on tori as analytical tools. Our findings suggest that this approach may explain the synchronization scenarios in various fields. Firstly, we focus on networks of neurons connected by gap-junctions, which can be modeled as neurons excited by external alternating currents or by interconnected neurons. Whether addressing a single neuron or a complex network, our approach provides a comprehensive understanding of the possible synchronization scenarios. This methodology is also applied to shed light on bistability observed in in-phase and anti-phase synchronization patterns in neuronal networks. Our research proposes an explanation that these patterns could be linked to very high-frequency EEG signals observed near epileptic foci. While the definitive connection between these bistable synchronization patterns and very high-frequency oscillations is yet to be established, our methodology offers a promising direction for investigation, potentially contributing to a deeper understanding of pathological brain activity. Further demonstrating the applicability of our approach, we present its successful implementation in deciphering Shapiro steps in superconductive Josephson junctions and seasonal synchronization in population models. These applications underscore the power of our methodology not only in neuroscience but also in the broader context of complex dynamical systems. Through the exposition of this MatCont-based method for bifurcation analysis, we aim to inspire further utilization and development of this approach, catalyzing advancements in modeling and understanding synchronization mechanisms across a diverse range of systems.