Noise-induced bursting near a crisis bifurcation in a map-based neuron model.

Bursting oscillations, characterized by alternating patterns of rapid oscillations and quiescence, is a prototypical form of complex dynamics that can emerge in both in low-dimensional systems and networks of coupled dynamical systems. Such bursting patterns are frequently observed in a wide range of systems, including neuronal models, electronic circuits, and laser models. However, the mechanisms governing the conditions under which bursting is initiated remain challenging from the nonlinear dynamics viewpoint. In this work, we investigate the emergence of bursting in the Chialvo model, a map-based neuron model, focusing on a critical parameter region near a crisis bifurcation where the deterministic system rests at a stable equilibrium. In the deterministic setting, we demonstrate that (i) chaotic bursting arises suddenly due to the bifurcation structure, (ii) interburst intervals follow a power-law distribution, and (iii) long chaotic transients may precede convergence to the stable state. Under stochastic influences, we uncover a range of nontrivial phenomena, including noise-induced bursting, coherence resonance, and transitions from regular to chaotic activity. By employing stochastic sensitivity analysis and confidence ellipses, we predict noise-induced transition thresholds and confirm them via numerical simulations. These findings provide new insights into how stochastic fluctuations interact with underlying bifurcation structures to generate rich and complex bursting patterns in low-dimensional systems, shedding light on mechanisms that may also be relevant for neuronal networks.