Unstable dimension variability as a key dynamical factor shaping neural network synchronization mechanisms

Neuronal network models play a crucial role in advancing our understanding of real coupled neurons and the complex dynamics that emerge from their interactions. These models make it possible to explore how connectivity patterns, coupling strengths, and intrinsic neuronal properties shape collective behaviors such as (phase) synchronization. Synchronized state stability is a key determinant of the system’s long-term behavior, depending on both the intrinsic properties of the individual units and the coupling strength. In general, phase synchronization is a robust regime, but it can become fragile near critical parameter values and when it is affected by Unstable Dimension Variability (UDV) in at least one of its attractors. Here we study how UDV may be involved in the transition to a phase synchronized state of a Hindmarsh-Rose neuron-model network. We show that the level of UDV and how Lyapunov exponents oscillates around zero modifie key characteristics of the network, transforming the transition from a non-synchronized state to a phase-synchronized state originally involving bistability into an intermittent transition. We further demonstrate that, for larger coupling values, the same UDV causes the phase-synchronized state to lose global stability, leading to intermittent escapes of the trajectory from the synchronized manifold. This effect becomes stronger as the coupling increases, implying that strongly coupled networks may exhibit more phase-unsynchronized states than networks with weaker coupling.