Dual role of Poisson impulses in Hindmarsh-Rose networks: Disorder, pattern formation, and synchronization

We investigate how stochastic Poisson impulsive forcing influences the spatiotemporal dynamics of a two-dimensional network of Hindmarsh-Rose neurons. Unlike continuous noise, impulsive forcing introduces discrete, state-dependent perturbations, making the system response highly sensitive to both the statistics and the spatial structure of the input. In most of the parameter space, stochastic impulses destabilize the initial coherent spiral-wave regime and lead to unstable spiral-wave activity. At the same time, they can also produce constructive effects. Both spatially uniform and spatially non-uniform forcing induce stable target waves that are absent in the autonomous system, suggesting a common mechanism of structure formation driven by discrete excitation events. In addition, spatially uniform forcing suppresses spatial heterogeneity and leads to coherent, nearly synchronous oscillations, demonstrating that even random impulses can promote global synchronization. By contrast, spatially uncorrelated forcing enhances front fragmentation and incoherence, especially at higher amplitudes. To identify and compare the resulting regimes in a highly multistable parameter space, we use a feature-based clustering approach based on statistical descriptors of wave morphology and evolution across multiple time scales. This results in a clear separation of physically distinct regimes. Overall, stochastic Poisson impulses serve not only as a source of disorder but also as an effective control mechanism for destabilizing, reorganizing, and synchronizing collective wave dynamics in networks of coupled neuronal oscillators.