Transition and basin stability in a delayed FitzHugh–Nagumo neural model under non-Gaussian colored noise

Non-Gaussian colored noise and time delay are pivotal factors regulating the dynamical behaviors of neuronal systems, yet their combined influences on the state transition and regional stability of the FitzHugh–Nagumo (FHN) neural model remain insufficiently explored. To address the non-Markovian nature of the original system, we first convert it into an equivalent Markovian system by integrating the Unified Colored-Noise Approximation (UCNA) and small delay approximation theories. We then employ Mean First Exit Time (MFET) and First Escape Probability (FEP) to quantify the excited-to-resting state transition and introduce the Stochastic Basin of Attraction (SBA) to evaluate the stability of the excited region. Key results indicate that (i) The non-Gaussian colored noise intensity [Formula: see text], time delay [Formula: see text], and deviation parameter [Formula: see text] act as critical control parameters inducing the excited-to-resting transition; (ii) Increasing [Formula: see text], [Formula: see text], or [Formula: see text] could lead to a reduction in MFET and an increase in FEP, thereby enhancing the system’s propensity for escaping from the excited region to the resting one; (iii) Stronger stochastic perturbations and longer time delays result in a shrinkage of the SBA size, which indicates a weakening of the stability of the excited region. This study provides new insights into the underlying regulatory mechanisms of noise and delay in neuronal dynamics, thus facilitating a deeper understanding of the complexity and dynamics of neuronal activity.