Dynamical Systems Analysis of Diffusively Coupled FitzHugh–Nagumo Neural Networks: Synchronization, Stability, and Bifurcation on Weighted Graphs

Abstract

Understanding how individual neuron dynamics translate into collective network behavior remains a fundamental challenge in computational neuroscience. Analytical understanding of network dynamics is therefore often limited by assumptions regarding behavior or network topology, with many studies relying largely on numerical simulation. Abnormal neuron synchronization patterns are characteristic of several neurological disorders, such as epileptic seizures. Unlike existing analyses that rely on specific network architectures, we develop a topology-independent analytical framework for studying diffusively coupled FitzHugh–Nagumo networks on general weighted undirected graphs. Primarily using dynamical systems theory, we derive conditions for bifurcations—which characterize transitions between qualitatively distinct neuronal behaviors—of individual neurons with respect to model parameters and external current. We then study synchronous equilibria of the coupled n-neuron system, analytically deriving the parameter regions in which these equilibria are stable. We classify local single-parameter bifurcations of these equilibria in terms of eigenvalues of the graph Laplacian matrix, establishing a relationship between network structure and synchronous behavior. Using these Laplacian eigenvalue conditions, we derive upper bounds on maximum weighted degree for equilibrium stability and bifurcation regimes. Finally, we develop a generalized framework for extending stability and bifurcation analysis beyond fully synchronized states to more complex collective behaviors, including partial synchronization and non-synchronized states. These results provide a mathematically rigorous framework for understanding how connectivity patterns shape collective neural behavior and may inform future studies of abnormal synchronization associated with neurological disorders. This work aligns with the United Nations Sustainable Development Goal #3: Good Health and Well-Being.